2019मจDouble Field Theory ʹΔήʔδରশͱLieѥରʹΑΔ Vaismanѥͷߏ

ཬେେӃ ཧڀݚՊ

ࢠՊઐ߈ ߨཧࢠ

ང

ཁ

Double Field Theory (DFT)ɼTରΛനͳରশͱॏཧͰΓɼ

ͷഎܠʹഒԽزԿ (doubled geometry)ͷଘߟࡏΒΕΔɽɼDFTͷήʔδ

ରশ VaismanΑͼ Courant ѥ (algebroid) ʹΑهड़ΕΔɽຊमจͰ

ɼLieͷ Drinfelrsquod doubleΛग़ͱɼVaismanѥͷ Lieѥͷ

ΒಘΒΕΔͱΛɽΒʹɼഒԽزԿͷݱԽͱύϥΤϧϛʔτଟମΛಋೖ

ɼͷଟମʹ VaismanѥΛߏஙΔɽ

3

ୈ 1ষ Ίʹ 5

ୈ 2ষ Double Field Theory 9

21 DFT ԿزΕΔݱʹ 9

22 DFT ͷ༻ͱ C ހ 11

23 ཧత߆ଋ 12

ୈ 3ষ ѥ (Algebroid)ͷಋೖ 15

31 Lieʢʣͱ Drinfelrsquod double 15

32 Lie ѥ 18

33 Courant ѥ 20

34 Double ʹΑΔ Vaisman ѥͷߏ 23

ୈ 4ষ Vaisman algebroidݱΕΔزԿ 31

41 ύϥෳૉߏ 31

42 ύϥΤϧϛʔτଟମ 32

43 ύϥυϧϘʔίϗϞϩδʔ 35

44 DFT ʹΔ Vaisman ѥ 37

45 Derivaiton ͷΕ 39

46 ҰൠزԿͱͷ 41

ୈ 5ষ ͱΊ 43

A ϊʔτܭ 46

A1 C ͷހ Jacobiator ͷಋग़ 46

A2 C Βހ Courant ͷހ reduction 48

A3 T (e1 e2 e3) ͷ (336) ͷಋग़ 49

A4 T (e1 e2 e3) ͷ (337) ͷಋग़ 51

A5 Axiom C1 ͷ 52

A6 (A45) ͷಋग़ 54

A7 (A46) ͷಋग़ 55

A8 Axiom C2 ͷ 56

A9 Axiom C3 ͷ 58

4

A10 Axiom C4 ͷ 59

A11 Axiom C5 ͷ 62

A12 NK = NP +NP ͰΔͱͷ 63

ݙจߟ 65

5

ୈ 1ষ

Ίʹ

ΛѻΊͷཧͱ༺ޓ૬ڧཧɼϋυϩϯΛΔͱΛతʹɼݭ 1960

ʹੜΕɽݭཧͰɼͷݯతͳ୯ҐͱͷΘΓʹ 1 ݭͷݩ (string) Λ༻

ཧΛߏஙΔɽݭͷৼಈϞʔυͷҧʹΑɼҟͳΔछͷϋυϩϯΛҰهड़Ͱ

ΔɼͱͷͰɽɼ1970 ʹͳΔͱࢠ৭ (quantum chromodynamics

QCD) Εɼڧ૬ޓ༻Λهड़Δཧͱਖ਼ͳͷ QCD ͰΔͱΕɽ

ɼดݭͷىঢ়ଶॏʹΔͱࢦఠΕɼݭཧɼࠓࢠॏཧ

ͷͱͼͷΛΈΔͱͱͳɽॏΛهड़ΔཧͱݭཧΛղऍΔʹ

ΓɼҎԼͷΑͳΑಛΔͱΘɽ

bull ॏࢠ (graviton) ཧͷεϖΫτϧʹݱΕΔɽ

bull ݭཧͰɼͷݩ 10 ఆΔɽʹݩ

bull ͷுݭཧʹΔਓҝతͳύϥϝʔλʔݭ T ͷΈͰΔɽ

ͷఆΔҰɼզʑΔݩͷ 4 ͷॲཧݩͰΔɽɼ༨ݩ

ΛߟͳͳΒͳɽ༨ݩͷରॲͷͻͱɼΤωϧΪʔͰݕग़ΕͳΑ

ʹɼΛίϯύΫτԽΔͱͰΔɽίϯύΫτԽͷ೦ɼKaluza-Klein (KK) ཧ [12]

ͰߟҊΕɽݭཧͷಛɼ૬ޓ༻ͰͳͷுҬͰىΔͱͰΔɽ

ͷࢠ (Quantum Field Theory QFT) ͰɼॏͷಈܭΛߦͱىɽ

ΕɼQFT Ͱͷখ୯ҐʹମΛͳࢠΛ༻ΔΊʹੜΔͰɼ

ΕݴॏཧͰΔɽݶۃཧͷΤωϧΪʔݭཧͰຊతʹճආͰΔɽݭ

ɼݭཧطଘͷཧͱॏཧΛแΔ [3]ɽ

ݟΘΔɽΕΒͷʹذͷͷͷʹଟɼݟཧզʑʹΒݭ

ͷݩͱͳͷ ʮରʯͰΔɽͳରɼAdSCFT ରԠͰΖ [4]ɽΕ

ɼʮD ͷυɾδολʔݩ (Anti-de Sitter AdS) ͰఆΕॏཧɼDܠഎ minus 1

ཧܗڞͷݩ (Conformal Field Theory CFT) ͱରͰΔʯͱ༧ͰΔɽͰ

ʮରͰΔʯͱɼʮ2 ͷҟͳΔཧಉཧΛදͱʯΛҙຯΔɽͷ༧ͷ༻ͳ

ɼҰͷཧͷڧͱɼରͳยͷཧͰऑͱʹΓɼݭཧͷ

ඇಈతͳཧղΛΊΔͱͱͳɽݭཧʹݱΕΔରɼଞʹ S ର [5]ɼU ର

[6]ɼຊमจͷதͱͳΔ T ର [7] ΔɽU ରɼT ରͱ S ରΛ

6

ΘΑͳରশͰΔɽݭཧɼՁͳཧ 5 ΓɼΕΒͷରΛ௨

ҠΓͱΒΕΔɽಛʹ T ରɼݭͷʹདྷΔɼݭཧʹಛͳର

ͰΔɽT ରݱΕΔ؆୯ͳɼ1 ݩ S1 ʹίϯύΫτԽͰΔɽ

ݭ 1 ͷମͰΔͱΒɼίϯύΫτԽΕݩ S1 ͷΤרɽͷרʹ

ωϧΪʔ S1 ͷܘΛ R ͱɼݭͷுΛઃఆΕܭͰΔɽͷՌݭͷεϖΫ

τϧʹد༩ɼݭͷεϖΫτϧ M2 ͶҎԼͷΑʹॻͱͰΔɽ

M2 prop n

R

2+

mR

αprime

$2

nm isin Z (11)

ҰݭͷӡಈʹɼೋݭͷுʹདྷΔΤωϧΪʔΛදɽɼαprime ݭͷு

ΛΊΔύϥϝʔλͰΔɽͷݭͷεϖΫτϧܘ RhArr αprimeR ͷೖΕସͷԼͰɼݭͷӡಈ

n ͱר m ͷΛΔɽͷೖΕସͷԼͰཧෆมͱͳΔͱΛ T ର

ͱɽT ରʹΑΓɼݭཧͰרͱݱΕɼݭࢠͱҟͳΔͰ

ʹཧݭड़ΔͱಡΈऔΕΔɽɼهΛ T ରӅΕରশͰΔɽT

ରΛനͳରশͱΔཧߟΒΕͳΖʁ

Double Field Theory (DFT) ɼT ରΛཧͷରশͱΔॏཧͰΔ [8]ɽ

DFT ͷಛɼݭͷӡಈʹ Fourier ʹרͷݭඪͱͳڞ Fourier ඪΛՁͳڞ

ʹѻͱͰɼඪΛ double ʹுɼΛ O(DD) ΜͱʹΔɽΕʹܗมͳڞ

ʹΑΓɼT ରΛനͳରশͱݱΔɽDFT ுΕͰఆΕΔҰͰɼ

ݭཧͰͷݩ 10 ͱΔɽDFTݩ ཧతʹਖ਼Λɼཧ

ͱͳΑʹΔΊʹɼԿΒͷ߆ଋΛ༩ͷݩͷΛΘΔඞ

ཁΔɽͷΛཧ section ͱɽͷʹΑΓɼுͷதΒཧతʹҙຯ

Δ෦ΛࢦఆΔɽDFT Ͱɼ2D ͷഒԽݩ (doubled) ͷதΒɼD ͷ෦ݩ

ΛબͿͱʹͳΔɽDFT Ͱ section ΛຬͻͱͷɼഒԽʹఆΕ

ΒΏΔύϥϝʔλΒɼרඪͷґଘΛऔΓআͱͰΔɽ

ͷΑͳɼରʹجཧҰݟඒݟɼΕͰѻͳߴΤωϧΪʔε

έʔϧͰͷͷࢠΛهड़ΔΑʹࢥΔɽɼݭཧͷΤωϧΪʔεέʔϧݱ

ݕՄͳϨϕϧΛͱΖʹΔΊɼཧਖ਼ͲΛݕͰݧͷज़Ͱࡏ

ΔͱɽͰɼগͳͱߏஙཧʹͳͱΛɼͷΛआΓ

ΊɽΛѻɼͷΛௐΔͱɼزԿͰΔɽ

Ұൠ૬ରཧ Einstein ʹΑΕͷɼ1915 ͷͱͰΔ [10]ɽͷཧʹ

ΑΔॏཁͳओுɼʮॏͱͷΈͰΔʯͱͱͰΔɽɼҰൠ૬

ରཧΛఆԽΔʹɼۂ () Λهड़ΔزԿඞཁෆՄͰɽͷ

Ίɼ() ϦʔϚϯزԿॏΛهड़ΔπʔϧͱɼҰൠ૬ରཧͷͱͱʹ

ΕΔΑʹͳɽҰൠ૬ରཧͱϦʔϚϯزԿͷΈΘɼॏཧΛߏஙɼ

ΔΖɽݴͱͰΔͱΛثͳڧԿزʹΛཧղΔߏ

७ਮͱͷزԿɼݱలΛଓΔɽදతͳɼ2004 ʹ Hitchin Βʹ

ΑఏҊΕɼҰൠԽزԿ (generalized geometry) ͰΔ [11]ɽҰൠԽزԿͰɼଟ

ମͷଋ (tangent bundle) Λɼଋͱ༨ଋ (cotangent bundle) ͷʹுΔɽҰൠԽزԿ

ɼΔछͷॏཧΛهड़ΔπʔϧͱΔɽҰൠ૬ରཧͷରশΛ

7

ΊுͱɼॏཧଘࡏΔɽҰൠԽزԿɼݭཧΒಘΒΕΔॏཧ

ͷɼಛʹ NS-NS sector ʹରͷزԿతͳඳΛ༩Δɽ

ΕΒͷલʹΑߟΔͱɼDFT ॏཧͷ T ରෆมͳఆԽΛతͱߏங

ΕॏཧͰΔΒɼͷഎܠʹΔزԿతͳඳʹΑఆԽΕΔΖɽDFT

ͷഎܠʹΔزԿͱΕΔͷɼഒԽزԿ (doubled geometry) [12] ͰΔɽഒ

ԽزԿड़ͷϦʔϚϯزԿҰൠԽزԿͱҟͳΔͷɼతʹਖ਼ʹߏஙΕ

ͳͰΔɽͷҙຯͰɼഒԽزԿݱஈ֊Ͱ DFT ݴͷ૯শͱߏΕΔతͳݱʹ

Αɽͱ ഒԽزԿΛඋΔͱͰɼDFT ͷཧߏΛཧղͰΔͱظΕ

ΔɽΕɼT ରΛ௨ɼݟݭΔߴΤωϧΪʔͷΈΛඥղେͳख

ΓʹͳΔΖɽ

ຊमจɼ2020 1 ʹग़൛จ [13] ͷ༰ΛϕʔεʹɼDFT ͷزԿతͳඳʹ

ணҎԼͷΑʹߏͷͰΔɽ

ୈ 2ষ ͷষͰɼDFT ͱɼͷഎܠʹΔ ഒԽزԿΛಋೖΔɽॳΊʹɼDFT Ͱ༻Δ

ഒԽඪΛఏΔɽʹɼҰൠԽ Lie ඍΛఆɼDFT ͷήʔδରশΛهड़Δ

C ଋͱԿʹड़߆ڧͷষͰͷओͱͳΔޙΛ༩Δɽɼހ

Δɽ

ୈ 3ষ લͷষͰड़ɼC ΒΔɽCʹɼతͳߏఆΔنހ

ߏఆΔنހ Vaisman ѥ (algebroid) ͱݺΕΔɽLie (algebra)

Λग़ͱɼߏΛঃʑʹҰൠԽͱͰ Vaisman ѥͷఆΛ༩Δɽ

ɼͷͰΑΒΕΔ Lie ͷ Drinfelrsquod double ͱΛݩʹɼ

ͷ Lie ѥΒ Vaisman ѥΛߏஙͰΔͱΛΔɽ

ୈ 4ষ ୈ 3 ষͷՌΛɼDFT ԽύϥΤϧϛʔτݱԿͷతͳزΔɽഒԽݩʹ

(para-Hermitian) ଟମʹΑͳΕΔͱࢦఠΔ [1415]ɽͷՌʹج

ɼύϥΤϧϛʔτଟମʹɼલͷষͰఆɼdouble ʹΑಘΒΕΔ Vaisman

ѥΛߏஙΔɽɼܭͷՌΒɼ߆ڧଋͷతͳݯىʹ

ΔɽՃɼഒԽزԿͱҰൠԽزԿͷʹΔɽ

ୈ 5ষ ҎͷΛͱΊΔɽɼޙࠓͷలʹड़Δɽ

ຊจதʹݱΕʹΔܭͷࡉɼAppendix ʹճɽಛʹɼୈ 3 ষͰߦ

Vaisman ѥͷߏʹඞཁͳܭશࡌهɽஅΓͳݶΓɼຊจதͷهҎԼͷҙ

ຯͰ༻Δɽ༻ޠͷఆඞཁʹԠຊจதͰ༩Δɽ

8

M ଟମ

M ഒԽ (ύϥΤϧϛʔτଟମ)

g Lie algebra

E (L L) Lie ѥ ( Lagrangian ෦ Dirac (ߏ

Elowast E ͷରɼରଋ

ρbull bullrarr TM Anchor map

C Courant ѥ

V Vaisman ѥ

[middot middot] TM ͷ Lie ހ

[middot middot]ELL E L L ͷ Lie ހ

[middot middot]S Schouten(-Nijenhuis) ހ

[middot middot]c C ͷ Courant ހ

[middot middot]V V ͷ Vaisman ހ

[middot middot]C C ހ

⟨middot middot⟩

(middot middot) ઢܗܕ

9

ୈ 2ষ

Double Field Theory

ͷষͰɼDouble Field Theory ͷجతͳ෦ʹ؆୯ʹड़Δɽಛʹɼ3 ষҎ

ͰඞཁͱͳΔɼDFT ʹΔήʔδʹݟʹࡉɽɼDFT ΕΔഒԽݱʹ

ɼDFTʹԿͷཁૉΛ༩Δɽز ʹΔ༻ͱήʔδมΛ༩ɼΕΛఆΔހͱ

ɼC ଋΛಋೖΔɽ߆ڧͷดΔΑʹɼΛಋೖΔɽήʔδมހ

21 DFTʹݱΕΔزԿ

DFT ɼtype II ॏཧͷ NS-NS sector Λനʹ T ରෆมఆԽཧͰΔ [8]ɽ

ͷཧɼmassless ͱܭ gmicroνɼKalb-Ramond (B ) Bmicroνɼdilaton φ ΛΉɽT

ରͷԼͰɼܭͷҰൠඪมͱɼB ͷ U(1) ήʔδมΔɽΕΛزԿత

ʹදݱΔͱɼ௨ৗͷ (ͷӡಈϞʔυݭ) ͱ༨ (winding Ϟʔυ) ΛΈΘ

ɼҰൠԽͷಋೖΛଅɽ

ҰൠԽଋ (generalized tangent bundle) E ɼଟମ M ͷଋ TM ͱ ༨ଋ T lowastM ͷ

ʹΑߏΕΔɽ

E = TM oplus T lowastM (21)

ҰൠԽଋҰൠԽزԿͰऔΓѻΘΕΔɽҰൠԽزԿɼ2004 ʹ Hitchin [11] ͱ Gualtieri

[16] ʹΑఏҊΕɽϕΫτϧΛଋ TM ͷஅ Γ(TM) ͱΔͱɼඍ 1 ༨ଋܗ

T lowastM ͷஅ Γ(T lowastM) ͰΔɽɼΓ(E) ϕΫτϧͱඍ 1 తʹͱܗΛܗ

ͳΓɼ ΕΛҰൠԽϕΫτϧͱɽҰൠԽزԿͰɼඍಉ૬ (diffeomorphism) ͱήʔδ

มΛΈΘɼҰൠԽʹରΔҰൠԽඍಉ૬ΛಋೖΔɽͷՌɼCourant

ѥ [17] ͱɼରশΛهड़ΔͰͷڧͳಓΛఏڙΔɽจݙ [17] ͷ Courant ѥ

ɼҎԼͷހͰ༩ΒΕΔߏͷҰछͰΔɽXi isin Γ(TM) ξi isin Γ(T lowastM) ͱͱ

ɼҎԼͷ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (22)

ʹΑ༩ΒΕΔɽCourant ѥʹ 3 ষͰΑΓࡉʹड़Δɽ

Βʹ ͳมڞͱרͷݭ x Λಋೖɼ௨ৗͷඪ x ͱಉʹѻͱͰɼඪ

త (ଟମ) ͷݩ 2 ഒʹͳΓɼഒԽزԿ (doubled geometry) ʹ౸ୡΔɽഒԽز

10

Կͷతͳମతߏʹߦࢼஈ֊ʹΔɽۂͳͲͷزԿతͳΛఆɼ

తʹదͳഒԽزԿΛߏஙΔࢼΈ [18 19] ͳͲͰߦΘΕΔɽ

ҎԼɼഒԽͷඪΛ༩ɼDFT ͰѻܭͷఆΛ༩ΔɽҎɼ௨ৗͷॏཧ

ఆΕΔͷݩΛ D ͱɼDFTݩ ͷഒԽ 2D ͰΔͱΔɽҰൠʹɼݩ

ཧఆΕΔݭ 10 ͰΔͱΒɼഒԽݩ 20 ΒΕΔɽഒߟͰΔͱݩ

Խͷඪ (ഒԽඪ) xM ҎԼͷΑʹॻΔͷͱΔɽxmicro ӡಈͱ Fourier ͳڞ

ඪɼxmicro רͱ Fourier ඪͰΔɽͳڞ

xM =

xmicro

xmicro

$ (M = 1 2D micro = 1 D) (23)

T ରΛͱΔͱɼxmicro ͱ xmicro ͷೖΕସʹରԠΔɽɼഒԽͰͷඍԋࢠҎԼͷ

Αʹදɽ

partM =part

partxM=

partmicro

partmicro

$ partmicro =

part

partxmicro partmicro =

part

partxmicro (24)

ҰൠʹɼD ίϯύΫτʹΔݩ T ରม (T-duality ( O(DD) Ͱ༩ΒΕΔɽ

ഒԽ O(DD) ෆมͳߏ η Λɽड़ͷ xM ͷఆΒɼη ͱͷٯҎԼͷΑʹॻ

Δɽ

ηMN =

0 δmicroν

δmicroν 0

$ ηMN =

0 δmicroν

δmicroν 0

$ (25)

ഒԽඪͷҰൠԽςϯιϧͷͷԼɼ η ʹΑߦΘΕΔɽ

؆୯ͳ DFT ɼtype II ॏཧͷ NS-NS sector ΛݩʹఆԽΕΔɽͷཧʹݱ

ΕΔجຊతͳཁૉɼܭ gmicroνɼKalb-Ramond Bmicroνɼdilaton φ ͷ 3 ͰΔɽΕΒͷΛ

T ରനͳཧͰѻʹɼO(DD) มͱͳΔΑʹΈඞཁΔɽΕΛҰൠڞ

Խܭ (generalized metric) HMN ͱɽ

HMN =

gmicroν minusgmicroρBρν

Bmicroρgρν gmicroν minusBmicroρgρσBσν

$ (26)

ɼHMN ҎԼͷΛຬɽ

HMN = ηMKηNLHKL (27)

Ճɼdilaton ҰൠԽΕΔɽDFT dilaton (ҰൠԽ dilaton) d ҎԼͷΑʹͳΔɽ

eminus2d =radicminusgeminus2φ (28)

ɼg ܭͷߦͰΔɽͷΈʹΑΓɼHMN d DFT ʹ O(DD)

มΛडΔɽɼഒԽͷ HMN (xM ) d(xM ) ͱͳΔɽ

11

22 DFTͷ༻ͱ Cހ

લઅͰఏΛ༻ɼO(DD) ෆมͳ DFT ͷ༻ɼҰൠԽ Ricci εΧϥʔͱ

ʹΑҎԼͷΑʹ༩ΒΕΔ [18]

SDFT =

d2Dx eminus2dR(H d) (29)

R =1

8HMNpartMHKLpartNHKL minus

1

2HMNpartMHKLpartKHNL

+ 4HMNpartMpartNdminus partMpartNHMN minus 4HMNpartMd partNd+ 4partMHMNpartNd (210)

ͷ༻ O(DD) εΧϥʔҰൠԽήʔδεΧϥʔͰΔͱɼ[18] ʹΕɽ

༺ (29) ɼେҬతͳ O(DD) ରশ

HMN rarr HLKM ML M N

K xM rarr xNM MN (211)

ʹରനʹෆมͰΔɽͰɼΕΛͱʹہॴରশͲͷΑʹنఆΕΔΖʁ

ҰൠԽϕΫτϧ V ʹରΔҰൠԽඍಉ૬ͷແݶখ൛ɼͳΘഒԽඪͷແݶখมҎ

ԼͷΑʹఆΕΔ [8]ɽ

V M rarr V M + δΞVM xM rarr xM minus ΞM (212)

ɼδΞ ҎԼͷΑͳҰൠԽ Lie ඍͰ༩ΒΕΔɽॏΈ w(V ) ΛҰൠԽϕΫτϧ V M

ͷҰൠԽ Lie ඍҎԼͷΑʹఆΕΔɽ

ampLΞVM = ΞKpartKV M + (partMΞK minus partKΞM )V K + w(V )V MpartKΞK (213)

Ұ௨ৗͷ Lie ඍͱಉҠಈΛදɼೋͷހͷதͷ O(DD) มΛදɽHͷॏΈ w(H) = 0 ͰΓɼҰൠԽ dilaton ͷॏΈ w(eminus2d) = 1 ͰΔɽɼҰൠԽ Lie ඍ

O(DD) ΔɽΛอɽͳΘɼҎԼͷߏ

ampLΞηMN = ampLΞηMN = 0 (214)

ҰൠԽ Lie ඍɼDFT ʹΔήʔδରশͷߏΛఆΔɽͷߏ C

[ ampLΞ1 ampLΞ2 ] = ampL[Ξ1Ξ2]C + F (Ξ1Ξ2 middot) (215)

ΛͳɼC ހ

[Ξ1Ξ2]MC = 2ΞK

[1partKΞM2] minus ηKLΞ

K[1part

MΞL2]

= ΞK1 partKΞM

2 minus ΞK2 partKΞM

1 minus1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 ) (216)

ʹΑࢧΕΔɽɼΞ[1Ξ2] = (Ξ1Ξ2 minus Ξ2Ξ1)2 ͰΔɽɼF ҎԼʹΑͳ

༨ͳͰΔɽ

F (Ξ1Ξ2 V )M =1

2ηKL(Ξ

K1 partPΞL

2 minus ΞK2 partPΞL

1 )partPVM minus (partPΞM

1 partPΞK2 minus partPΞM

2 partPΞK1 )VK

(217)

F ଘࡏΔͱΒɼҰൠԽ Lie ඍͷࢠ C ͰดͳɽɼCހ ހ Jacobi

ΛຬͳɽF ΛফڈɼC ଋ߆ڧઅͰઆΔఆΔΛดΔʹɼنހ

(strong constraint) ΛͰΔɽ

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

3

ୈ 1ষ Ίʹ 5

ୈ 2ষ Double Field Theory 9

21 DFT ԿزΕΔݱʹ 9

22 DFT ͷ༻ͱ C ހ 11

23 ཧత߆ଋ 12

ୈ 3ষ ѥ (Algebroid)ͷಋೖ 15

31 Lieʢʣͱ Drinfelrsquod double 15

32 Lie ѥ 18

33 Courant ѥ 20

34 Double ʹΑΔ Vaisman ѥͷߏ 23

ୈ 4ষ Vaisman algebroidݱΕΔزԿ 31

41 ύϥෳૉߏ 31

42 ύϥΤϧϛʔτଟମ 32

43 ύϥυϧϘʔίϗϞϩδʔ 35

44 DFT ʹΔ Vaisman ѥ 37

45 Derivaiton ͷΕ 39

46 ҰൠزԿͱͷ 41

ୈ 5ষ ͱΊ 43

A ϊʔτܭ 46

A1 C ͷހ Jacobiator ͷಋग़ 46

A2 C Βހ Courant ͷހ reduction 48

A3 T (e1 e2 e3) ͷ (336) ͷಋग़ 49

A4 T (e1 e2 e3) ͷ (337) ͷಋग़ 51

A5 Axiom C1 ͷ 52

A6 (A45) ͷಋग़ 54

A7 (A46) ͷಋग़ 55

A8 Axiom C2 ͷ 56

A9 Axiom C3 ͷ 58

4

A10 Axiom C4 ͷ 59

A11 Axiom C5 ͷ 62

A12 NK = NP +NP ͰΔͱͷ 63

ݙจߟ 65

5

ୈ 1ষ

Ίʹ

ΛѻΊͷཧͱ༺ޓ૬ڧཧɼϋυϩϯΛΔͱΛతʹɼݭ 1960

ʹੜΕɽݭཧͰɼͷݯతͳ୯ҐͱͷΘΓʹ 1 ݭͷݩ (string) Λ༻

ཧΛߏஙΔɽݭͷৼಈϞʔυͷҧʹΑɼҟͳΔछͷϋυϩϯΛҰهड़Ͱ

ΔɼͱͷͰɽɼ1970 ʹͳΔͱࢠ৭ (quantum chromodynamics

QCD) Εɼڧ૬ޓ༻Λهड़Δཧͱਖ਼ͳͷ QCD ͰΔͱΕɽ

ɼดݭͷىঢ়ଶॏʹΔͱࢦఠΕɼݭཧɼࠓࢠॏཧ

ͷͱͼͷΛΈΔͱͱͳɽॏΛهड़ΔཧͱݭཧΛղऍΔʹ

ΓɼҎԼͷΑͳΑಛΔͱΘɽ

bull ॏࢠ (graviton) ཧͷεϖΫτϧʹݱΕΔɽ

bull ݭཧͰɼͷݩ 10 ఆΔɽʹݩ

bull ͷுݭཧʹΔਓҝతͳύϥϝʔλʔݭ T ͷΈͰΔɽ

ͷఆΔҰɼզʑΔݩͷ 4 ͷॲཧݩͰΔɽɼ༨ݩ

ΛߟͳͳΒͳɽ༨ݩͷରॲͷͻͱɼΤωϧΪʔͰݕग़ΕͳΑ

ʹɼΛίϯύΫτԽΔͱͰΔɽίϯύΫτԽͷ೦ɼKaluza-Klein (KK) ཧ [12]

ͰߟҊΕɽݭཧͷಛɼ૬ޓ༻ͰͳͷுҬͰىΔͱͰΔɽ

ͷࢠ (Quantum Field Theory QFT) ͰɼॏͷಈܭΛߦͱىɽ

ΕɼQFT Ͱͷখ୯ҐʹମΛͳࢠΛ༻ΔΊʹੜΔͰɼ

ΕݴॏཧͰΔɽݶۃཧͷΤωϧΪʔݭཧͰຊతʹճආͰΔɽݭ

ɼݭཧطଘͷཧͱॏཧΛแΔ [3]ɽ

ݟΘΔɽΕΒͷʹذͷͷͷʹଟɼݟཧզʑʹΒݭ

ͷݩͱͳͷ ʮରʯͰΔɽͳରɼAdSCFT ରԠͰΖ [4]ɽΕ

ɼʮD ͷυɾδολʔݩ (Anti-de Sitter AdS) ͰఆΕॏཧɼDܠഎ minus 1

ཧܗڞͷݩ (Conformal Field Theory CFT) ͱରͰΔʯͱ༧ͰΔɽͰ

ʮରͰΔʯͱɼʮ2 ͷҟͳΔཧಉཧΛදͱʯΛҙຯΔɽͷ༧ͷ༻ͳ

ɼҰͷཧͷڧͱɼରͳยͷཧͰऑͱʹΓɼݭཧͷ

ඇಈతͳཧղΛΊΔͱͱͳɽݭཧʹݱΕΔରɼଞʹ S ର [5]ɼU ର

[6]ɼຊमจͷதͱͳΔ T ର [7] ΔɽU ରɼT ରͱ S ରΛ

6

ΘΑͳରশͰΔɽݭཧɼՁͳཧ 5 ΓɼΕΒͷରΛ௨

ҠΓͱΒΕΔɽಛʹ T ରɼݭͷʹདྷΔɼݭཧʹಛͳର

ͰΔɽT ରݱΕΔ؆୯ͳɼ1 ݩ S1 ʹίϯύΫτԽͰΔɽ

ݭ 1 ͷମͰΔͱΒɼίϯύΫτԽΕݩ S1 ͷΤרɽͷרʹ

ωϧΪʔ S1 ͷܘΛ R ͱɼݭͷுΛઃఆΕܭͰΔɽͷՌݭͷεϖΫ

τϧʹد༩ɼݭͷεϖΫτϧ M2 ͶҎԼͷΑʹॻͱͰΔɽ

M2 prop n

R

2+

mR

αprime

$2

nm isin Z (11)

ҰݭͷӡಈʹɼೋݭͷுʹདྷΔΤωϧΪʔΛදɽɼαprime ݭͷு

ΛΊΔύϥϝʔλͰΔɽͷݭͷεϖΫτϧܘ RhArr αprimeR ͷೖΕସͷԼͰɼݭͷӡಈ

n ͱר m ͷΛΔɽͷೖΕସͷԼͰཧෆมͱͳΔͱΛ T ର

ͱɽT ରʹΑΓɼݭཧͰרͱݱΕɼݭࢠͱҟͳΔͰ

ʹཧݭड़ΔͱಡΈऔΕΔɽɼهΛ T ରӅΕରশͰΔɽT

ରΛനͳରশͱΔཧߟΒΕͳΖʁ

Double Field Theory (DFT) ɼT ରΛཧͷରশͱΔॏཧͰΔ [8]ɽ

DFT ͷಛɼݭͷӡಈʹ Fourier ʹרͷݭඪͱͳڞ Fourier ඪΛՁͳڞ

ʹѻͱͰɼඪΛ double ʹுɼΛ O(DD) ΜͱʹΔɽΕʹܗมͳڞ

ʹΑΓɼT ରΛനͳରশͱݱΔɽDFT ுΕͰఆΕΔҰͰɼ

ݭཧͰͷݩ 10 ͱΔɽDFTݩ ཧతʹਖ਼Λɼཧ

ͱͳΑʹΔΊʹɼԿΒͷ߆ଋΛ༩ͷݩͷΛΘΔඞ

ཁΔɽͷΛཧ section ͱɽͷʹΑΓɼுͷதΒཧతʹҙຯ

Δ෦ΛࢦఆΔɽDFT Ͱɼ2D ͷഒԽݩ (doubled) ͷதΒɼD ͷ෦ݩ

ΛબͿͱʹͳΔɽDFT Ͱ section ΛຬͻͱͷɼഒԽʹఆΕ

ΒΏΔύϥϝʔλΒɼרඪͷґଘΛऔΓআͱͰΔɽ

ͷΑͳɼରʹجཧҰݟඒݟɼΕͰѻͳߴΤωϧΪʔε

έʔϧͰͷͷࢠΛهड़ΔΑʹࢥΔɽɼݭཧͷΤωϧΪʔεέʔϧݱ

ݕՄͳϨϕϧΛͱΖʹΔΊɼཧਖ਼ͲΛݕͰݧͷज़Ͱࡏ

ΔͱɽͰɼগͳͱߏஙཧʹͳͱΛɼͷΛआΓ

ΊɽΛѻɼͷΛௐΔͱɼزԿͰΔɽ

Ұൠ૬ରཧ Einstein ʹΑΕͷɼ1915 ͷͱͰΔ [10]ɽͷཧʹ

ΑΔॏཁͳओுɼʮॏͱͷΈͰΔʯͱͱͰΔɽɼҰൠ૬

ରཧΛఆԽΔʹɼۂ () Λهड़ΔزԿඞཁෆՄͰɽͷ

Ίɼ() ϦʔϚϯزԿॏΛهड़ΔπʔϧͱɼҰൠ૬ରཧͷͱͱʹ

ΕΔΑʹͳɽҰൠ૬ରཧͱϦʔϚϯزԿͷΈΘɼॏཧΛߏஙɼ

ΔΖɽݴͱͰΔͱΛثͳڧԿزʹΛཧղΔߏ

७ਮͱͷزԿɼݱలΛଓΔɽදతͳɼ2004 ʹ Hitchin Βʹ

ΑఏҊΕɼҰൠԽزԿ (generalized geometry) ͰΔ [11]ɽҰൠԽزԿͰɼଟ

ମͷଋ (tangent bundle) Λɼଋͱ༨ଋ (cotangent bundle) ͷʹுΔɽҰൠԽزԿ

ɼΔछͷॏཧΛهड़ΔπʔϧͱΔɽҰൠ૬ରཧͷରশΛ

7

ΊுͱɼॏཧଘࡏΔɽҰൠԽزԿɼݭཧΒಘΒΕΔॏཧ

ͷɼಛʹ NS-NS sector ʹରͷزԿతͳඳΛ༩Δɽ

ΕΒͷલʹΑߟΔͱɼDFT ॏཧͷ T ରෆมͳఆԽΛతͱߏங

ΕॏཧͰΔΒɼͷഎܠʹΔزԿతͳඳʹΑఆԽΕΔΖɽDFT

ͷഎܠʹΔزԿͱΕΔͷɼഒԽزԿ (doubled geometry) [12] ͰΔɽഒ

ԽزԿड़ͷϦʔϚϯزԿҰൠԽزԿͱҟͳΔͷɼతʹਖ਼ʹߏஙΕ

ͳͰΔɽͷҙຯͰɼഒԽزԿݱஈ֊Ͱ DFT ݴͷ૯শͱߏΕΔతͳݱʹ

Αɽͱ ഒԽزԿΛඋΔͱͰɼDFT ͷཧߏΛཧղͰΔͱظΕ

ΔɽΕɼT ରΛ௨ɼݟݭΔߴΤωϧΪʔͷΈΛඥղେͳख

ΓʹͳΔΖɽ

ຊमจɼ2020 1 ʹग़൛จ [13] ͷ༰ΛϕʔεʹɼDFT ͷزԿతͳඳʹ

ணҎԼͷΑʹߏͷͰΔɽ

ୈ 2ষ ͷষͰɼDFT ͱɼͷഎܠʹΔ ഒԽزԿΛಋೖΔɽॳΊʹɼDFT Ͱ༻Δ

ഒԽඪΛఏΔɽʹɼҰൠԽ Lie ඍΛఆɼDFT ͷήʔδରশΛهड़Δ

C ଋͱԿʹड़߆ڧͷষͰͷओͱͳΔޙΛ༩Δɽɼހ

Δɽ

ୈ 3ষ લͷষͰड़ɼC ΒΔɽCʹɼతͳߏఆΔنހ

ߏఆΔنހ Vaisman ѥ (algebroid) ͱݺΕΔɽLie (algebra)

Λग़ͱɼߏΛঃʑʹҰൠԽͱͰ Vaisman ѥͷఆΛ༩Δɽ

ɼͷͰΑΒΕΔ Lie ͷ Drinfelrsquod double ͱΛݩʹɼ

ͷ Lie ѥΒ Vaisman ѥΛߏஙͰΔͱΛΔɽ

ୈ 4ষ ୈ 3 ষͷՌΛɼDFT ԽύϥΤϧϛʔτݱԿͷతͳزΔɽഒԽݩʹ

(para-Hermitian) ଟମʹΑͳΕΔͱࢦఠΔ [1415]ɽͷՌʹج

ɼύϥΤϧϛʔτଟମʹɼલͷষͰఆɼdouble ʹΑಘΒΕΔ Vaisman

ѥΛߏஙΔɽɼܭͷՌΒɼ߆ڧଋͷతͳݯىʹ

ΔɽՃɼഒԽزԿͱҰൠԽزԿͷʹΔɽ

ୈ 5ষ ҎͷΛͱΊΔɽɼޙࠓͷలʹड़Δɽ

ຊจதʹݱΕʹΔܭͷࡉɼAppendix ʹճɽಛʹɼୈ 3 ষͰߦ

Vaisman ѥͷߏʹඞཁͳܭશࡌهɽஅΓͳݶΓɼຊจதͷهҎԼͷҙ

ຯͰ༻Δɽ༻ޠͷఆඞཁʹԠຊจதͰ༩Δɽ

8

M ଟମ

M ഒԽ (ύϥΤϧϛʔτଟମ)

g Lie algebra

E (L L) Lie ѥ ( Lagrangian ෦ Dirac (ߏ

Elowast E ͷରɼରଋ

ρbull bullrarr TM Anchor map

C Courant ѥ

V Vaisman ѥ

[middot middot] TM ͷ Lie ހ

[middot middot]ELL E L L ͷ Lie ހ

[middot middot]S Schouten(-Nijenhuis) ހ

[middot middot]c C ͷ Courant ހ

[middot middot]V V ͷ Vaisman ހ

[middot middot]C C ހ

⟨middot middot⟩

(middot middot) ઢܗܕ

9

ୈ 2ষ

Double Field Theory

ͷষͰɼDouble Field Theory ͷجతͳ෦ʹ؆୯ʹड़Δɽಛʹɼ3 ষҎ

ͰඞཁͱͳΔɼDFT ʹΔήʔδʹݟʹࡉɽɼDFT ΕΔഒԽݱʹ

ɼDFTʹԿͷཁૉΛ༩Δɽز ʹΔ༻ͱήʔδมΛ༩ɼΕΛఆΔހͱ

ɼC ଋΛಋೖΔɽ߆ڧͷดΔΑʹɼΛಋೖΔɽήʔδมހ

21 DFTʹݱΕΔزԿ

DFT ɼtype II ॏཧͷ NS-NS sector Λനʹ T ରෆมఆԽཧͰΔ [8]ɽ

ͷཧɼmassless ͱܭ gmicroνɼKalb-Ramond (B ) Bmicroνɼdilaton φ ΛΉɽT

ରͷԼͰɼܭͷҰൠඪมͱɼB ͷ U(1) ήʔδมΔɽΕΛزԿత

ʹදݱΔͱɼ௨ৗͷ (ͷӡಈϞʔυݭ) ͱ༨ (winding Ϟʔυ) ΛΈΘ

ɼҰൠԽͷಋೖΛଅɽ

ҰൠԽଋ (generalized tangent bundle) E ɼଟମ M ͷଋ TM ͱ ༨ଋ T lowastM ͷ

ʹΑߏΕΔɽ

E = TM oplus T lowastM (21)

ҰൠԽଋҰൠԽزԿͰऔΓѻΘΕΔɽҰൠԽزԿɼ2004 ʹ Hitchin [11] ͱ Gualtieri

[16] ʹΑఏҊΕɽϕΫτϧΛଋ TM ͷஅ Γ(TM) ͱΔͱɼඍ 1 ༨ଋܗ

T lowastM ͷஅ Γ(T lowastM) ͰΔɽɼΓ(E) ϕΫτϧͱඍ 1 తʹͱܗΛܗ

ͳΓɼ ΕΛҰൠԽϕΫτϧͱɽҰൠԽزԿͰɼඍಉ૬ (diffeomorphism) ͱήʔδ

มΛΈΘɼҰൠԽʹରΔҰൠԽඍಉ૬ΛಋೖΔɽͷՌɼCourant

ѥ [17] ͱɼରশΛهड़ΔͰͷڧͳಓΛఏڙΔɽจݙ [17] ͷ Courant ѥ

ɼҎԼͷހͰ༩ΒΕΔߏͷҰछͰΔɽXi isin Γ(TM) ξi isin Γ(T lowastM) ͱͱ

ɼҎԼͷ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (22)

ʹΑ༩ΒΕΔɽCourant ѥʹ 3 ষͰΑΓࡉʹड़Δɽ

Βʹ ͳมڞͱרͷݭ x Λಋೖɼ௨ৗͷඪ x ͱಉʹѻͱͰɼඪ

త (ଟମ) ͷݩ 2 ഒʹͳΓɼഒԽزԿ (doubled geometry) ʹ౸ୡΔɽഒԽز

10

Կͷతͳମతߏʹߦࢼஈ֊ʹΔɽۂͳͲͷزԿతͳΛఆɼ

తʹదͳഒԽزԿΛߏஙΔࢼΈ [18 19] ͳͲͰߦΘΕΔɽ

ҎԼɼഒԽͷඪΛ༩ɼDFT ͰѻܭͷఆΛ༩ΔɽҎɼ௨ৗͷॏཧ

ఆΕΔͷݩΛ D ͱɼDFTݩ ͷഒԽ 2D ͰΔͱΔɽҰൠʹɼݩ

ཧఆΕΔݭ 10 ͰΔͱΒɼഒԽݩ 20 ΒΕΔɽഒߟͰΔͱݩ

Խͷඪ (ഒԽඪ) xM ҎԼͷΑʹॻΔͷͱΔɽxmicro ӡಈͱ Fourier ͳڞ

ඪɼxmicro רͱ Fourier ඪͰΔɽͳڞ

xM =

xmicro

xmicro

$ (M = 1 2D micro = 1 D) (23)

T ରΛͱΔͱɼxmicro ͱ xmicro ͷೖΕସʹରԠΔɽɼഒԽͰͷඍԋࢠҎԼͷ

Αʹදɽ

partM =part

partxM=

partmicro

partmicro

$ partmicro =

part

partxmicro partmicro =

part

partxmicro (24)

ҰൠʹɼD ίϯύΫτʹΔݩ T ରม (T-duality ( O(DD) Ͱ༩ΒΕΔɽ

ഒԽ O(DD) ෆมͳߏ η Λɽड़ͷ xM ͷఆΒɼη ͱͷٯҎԼͷΑʹॻ

Δɽ

ηMN =

0 δmicroν

δmicroν 0

$ ηMN =

0 δmicroν

δmicroν 0

$ (25)

ഒԽඪͷҰൠԽςϯιϧͷͷԼɼ η ʹΑߦΘΕΔɽ

؆୯ͳ DFT ɼtype II ॏཧͷ NS-NS sector ΛݩʹఆԽΕΔɽͷཧʹݱ

ΕΔجຊతͳཁૉɼܭ gmicroνɼKalb-Ramond Bmicroνɼdilaton φ ͷ 3 ͰΔɽΕΒͷΛ

T ରനͳཧͰѻʹɼO(DD) มͱͳΔΑʹΈඞཁΔɽΕΛҰൠڞ

Խܭ (generalized metric) HMN ͱɽ

HMN =

gmicroν minusgmicroρBρν

Bmicroρgρν gmicroν minusBmicroρgρσBσν

$ (26)

ɼHMN ҎԼͷΛຬɽ

HMN = ηMKηNLHKL (27)

Ճɼdilaton ҰൠԽΕΔɽDFT dilaton (ҰൠԽ dilaton) d ҎԼͷΑʹͳΔɽ

eminus2d =radicminusgeminus2φ (28)

ɼg ܭͷߦͰΔɽͷΈʹΑΓɼHMN d DFT ʹ O(DD)

มΛडΔɽɼഒԽͷ HMN (xM ) d(xM ) ͱͳΔɽ

11

22 DFTͷ༻ͱ Cހ

લઅͰఏΛ༻ɼO(DD) ෆมͳ DFT ͷ༻ɼҰൠԽ Ricci εΧϥʔͱ

ʹΑҎԼͷΑʹ༩ΒΕΔ [18]

SDFT =

d2Dx eminus2dR(H d) (29)

R =1

8HMNpartMHKLpartNHKL minus

1

2HMNpartMHKLpartKHNL

+ 4HMNpartMpartNdminus partMpartNHMN minus 4HMNpartMd partNd+ 4partMHMNpartNd (210)

ͷ༻ O(DD) εΧϥʔҰൠԽήʔδεΧϥʔͰΔͱɼ[18] ʹΕɽ

༺ (29) ɼେҬతͳ O(DD) ରশ

HMN rarr HLKM ML M N

K xM rarr xNM MN (211)

ʹରനʹෆมͰΔɽͰɼΕΛͱʹہॴରশͲͷΑʹنఆΕΔΖʁ

ҰൠԽϕΫτϧ V ʹରΔҰൠԽඍಉ૬ͷແݶখ൛ɼͳΘഒԽඪͷແݶখมҎ

ԼͷΑʹఆΕΔ [8]ɽ

V M rarr V M + δΞVM xM rarr xM minus ΞM (212)

ɼδΞ ҎԼͷΑͳҰൠԽ Lie ඍͰ༩ΒΕΔɽॏΈ w(V ) ΛҰൠԽϕΫτϧ V M

ͷҰൠԽ Lie ඍҎԼͷΑʹఆΕΔɽ

ampLΞVM = ΞKpartKV M + (partMΞK minus partKΞM )V K + w(V )V MpartKΞK (213)

Ұ௨ৗͷ Lie ඍͱಉҠಈΛදɼೋͷހͷதͷ O(DD) มΛදɽHͷॏΈ w(H) = 0 ͰΓɼҰൠԽ dilaton ͷॏΈ w(eminus2d) = 1 ͰΔɽɼҰൠԽ Lie ඍ

O(DD) ΔɽΛอɽͳΘɼҎԼͷߏ

ampLΞηMN = ampLΞηMN = 0 (214)

ҰൠԽ Lie ඍɼDFT ʹΔήʔδରশͷߏΛఆΔɽͷߏ C

[ ampLΞ1 ampLΞ2 ] = ampL[Ξ1Ξ2]C + F (Ξ1Ξ2 middot) (215)

ΛͳɼC ހ

[Ξ1Ξ2]MC = 2ΞK

[1partKΞM2] minus ηKLΞ

K[1part

MΞL2]

= ΞK1 partKΞM

2 minus ΞK2 partKΞM

1 minus1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 ) (216)

ʹΑࢧΕΔɽɼΞ[1Ξ2] = (Ξ1Ξ2 minus Ξ2Ξ1)2 ͰΔɽɼF ҎԼʹΑͳ

༨ͳͰΔɽ

F (Ξ1Ξ2 V )M =1

2ηKL(Ξ

K1 partPΞL

2 minus ΞK2 partPΞL

1 )partPVM minus (partPΞM

1 partPΞK2 minus partPΞM

2 partPΞK1 )VK

(217)

F ଘࡏΔͱΒɼҰൠԽ Lie ඍͷࢠ C ͰดͳɽɼCހ ހ Jacobi

ΛຬͳɽF ΛফڈɼC ଋ߆ڧઅͰઆΔఆΔΛดΔʹɼنހ

(strong constraint) ΛͰΔɽ

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

4

A10 Axiom C4 ͷ 59

A11 Axiom C5 ͷ 62

A12 NK = NP +NP ͰΔͱͷ 63

ݙจߟ 65

5

ୈ 1ষ

Ίʹ

ΛѻΊͷཧͱ༺ޓ૬ڧཧɼϋυϩϯΛΔͱΛతʹɼݭ 1960

ʹੜΕɽݭཧͰɼͷݯతͳ୯ҐͱͷΘΓʹ 1 ݭͷݩ (string) Λ༻

ཧΛߏஙΔɽݭͷৼಈϞʔυͷҧʹΑɼҟͳΔछͷϋυϩϯΛҰهड़Ͱ

ΔɼͱͷͰɽɼ1970 ʹͳΔͱࢠ৭ (quantum chromodynamics

QCD) Εɼڧ૬ޓ༻Λهड़Δཧͱਖ਼ͳͷ QCD ͰΔͱΕɽ

ɼดݭͷىঢ়ଶॏʹΔͱࢦఠΕɼݭཧɼࠓࢠॏཧ

ͷͱͼͷΛΈΔͱͱͳɽॏΛهड़ΔཧͱݭཧΛղऍΔʹ

ΓɼҎԼͷΑͳΑಛΔͱΘɽ

bull ॏࢠ (graviton) ཧͷεϖΫτϧʹݱΕΔɽ

bull ݭཧͰɼͷݩ 10 ఆΔɽʹݩ

bull ͷுݭཧʹΔਓҝతͳύϥϝʔλʔݭ T ͷΈͰΔɽ

ͷఆΔҰɼզʑΔݩͷ 4 ͷॲཧݩͰΔɽɼ༨ݩ

ΛߟͳͳΒͳɽ༨ݩͷରॲͷͻͱɼΤωϧΪʔͰݕग़ΕͳΑ

ʹɼΛίϯύΫτԽΔͱͰΔɽίϯύΫτԽͷ೦ɼKaluza-Klein (KK) ཧ [12]

ͰߟҊΕɽݭཧͷಛɼ૬ޓ༻ͰͳͷுҬͰىΔͱͰΔɽ

ͷࢠ (Quantum Field Theory QFT) ͰɼॏͷಈܭΛߦͱىɽ

ΕɼQFT Ͱͷখ୯ҐʹମΛͳࢠΛ༻ΔΊʹੜΔͰɼ

ΕݴॏཧͰΔɽݶۃཧͷΤωϧΪʔݭཧͰຊతʹճආͰΔɽݭ

ɼݭཧطଘͷཧͱॏཧΛแΔ [3]ɽ

ݟΘΔɽΕΒͷʹذͷͷͷʹଟɼݟཧզʑʹΒݭ

ͷݩͱͳͷ ʮରʯͰΔɽͳରɼAdSCFT ରԠͰΖ [4]ɽΕ

ɼʮD ͷυɾδολʔݩ (Anti-de Sitter AdS) ͰఆΕॏཧɼDܠഎ minus 1

ཧܗڞͷݩ (Conformal Field Theory CFT) ͱରͰΔʯͱ༧ͰΔɽͰ

ʮରͰΔʯͱɼʮ2 ͷҟͳΔཧಉཧΛදͱʯΛҙຯΔɽͷ༧ͷ༻ͳ

ɼҰͷཧͷڧͱɼରͳยͷཧͰऑͱʹΓɼݭཧͷ

ඇಈతͳཧղΛΊΔͱͱͳɽݭཧʹݱΕΔରɼଞʹ S ର [5]ɼU ର

[6]ɼຊमจͷதͱͳΔ T ର [7] ΔɽU ରɼT ରͱ S ରΛ

6

ΘΑͳରশͰΔɽݭཧɼՁͳཧ 5 ΓɼΕΒͷରΛ௨

ҠΓͱΒΕΔɽಛʹ T ରɼݭͷʹདྷΔɼݭཧʹಛͳର

ͰΔɽT ରݱΕΔ؆୯ͳɼ1 ݩ S1 ʹίϯύΫτԽͰΔɽ

ݭ 1 ͷମͰΔͱΒɼίϯύΫτԽΕݩ S1 ͷΤרɽͷרʹ

ωϧΪʔ S1 ͷܘΛ R ͱɼݭͷுΛઃఆΕܭͰΔɽͷՌݭͷεϖΫ

τϧʹد༩ɼݭͷεϖΫτϧ M2 ͶҎԼͷΑʹॻͱͰΔɽ

M2 prop n

R

2+

mR

αprime

$2

nm isin Z (11)

ҰݭͷӡಈʹɼೋݭͷுʹདྷΔΤωϧΪʔΛදɽɼαprime ݭͷு

ΛΊΔύϥϝʔλͰΔɽͷݭͷεϖΫτϧܘ RhArr αprimeR ͷೖΕସͷԼͰɼݭͷӡಈ

n ͱר m ͷΛΔɽͷೖΕସͷԼͰཧෆมͱͳΔͱΛ T ର

ͱɽT ରʹΑΓɼݭཧͰרͱݱΕɼݭࢠͱҟͳΔͰ

ʹཧݭड़ΔͱಡΈऔΕΔɽɼهΛ T ରӅΕରশͰΔɽT

ରΛനͳରশͱΔཧߟΒΕͳΖʁ

Double Field Theory (DFT) ɼT ରΛཧͷରশͱΔॏཧͰΔ [8]ɽ

DFT ͷಛɼݭͷӡಈʹ Fourier ʹרͷݭඪͱͳڞ Fourier ඪΛՁͳڞ

ʹѻͱͰɼඪΛ double ʹுɼΛ O(DD) ΜͱʹΔɽΕʹܗมͳڞ

ʹΑΓɼT ରΛനͳରশͱݱΔɽDFT ுΕͰఆΕΔҰͰɼ

ݭཧͰͷݩ 10 ͱΔɽDFTݩ ཧతʹਖ਼Λɼཧ

ͱͳΑʹΔΊʹɼԿΒͷ߆ଋΛ༩ͷݩͷΛΘΔඞ

ཁΔɽͷΛཧ section ͱɽͷʹΑΓɼுͷதΒཧతʹҙຯ

Δ෦ΛࢦఆΔɽDFT Ͱɼ2D ͷഒԽݩ (doubled) ͷதΒɼD ͷ෦ݩ

ΛબͿͱʹͳΔɽDFT Ͱ section ΛຬͻͱͷɼഒԽʹఆΕ

ΒΏΔύϥϝʔλΒɼרඪͷґଘΛऔΓআͱͰΔɽ

ͷΑͳɼରʹجཧҰݟඒݟɼΕͰѻͳߴΤωϧΪʔε

έʔϧͰͷͷࢠΛهड़ΔΑʹࢥΔɽɼݭཧͷΤωϧΪʔεέʔϧݱ

ݕՄͳϨϕϧΛͱΖʹΔΊɼཧਖ਼ͲΛݕͰݧͷज़Ͱࡏ

ΔͱɽͰɼগͳͱߏஙཧʹͳͱΛɼͷΛआΓ

ΊɽΛѻɼͷΛௐΔͱɼزԿͰΔɽ

Ұൠ૬ରཧ Einstein ʹΑΕͷɼ1915 ͷͱͰΔ [10]ɽͷཧʹ

ΑΔॏཁͳओுɼʮॏͱͷΈͰΔʯͱͱͰΔɽɼҰൠ૬

ରཧΛఆԽΔʹɼۂ () Λهड़ΔزԿඞཁෆՄͰɽͷ

Ίɼ() ϦʔϚϯزԿॏΛهड़ΔπʔϧͱɼҰൠ૬ରཧͷͱͱʹ

ΕΔΑʹͳɽҰൠ૬ରཧͱϦʔϚϯزԿͷΈΘɼॏཧΛߏஙɼ

ΔΖɽݴͱͰΔͱΛثͳڧԿزʹΛཧղΔߏ

७ਮͱͷزԿɼݱలΛଓΔɽදతͳɼ2004 ʹ Hitchin Βʹ

ΑఏҊΕɼҰൠԽزԿ (generalized geometry) ͰΔ [11]ɽҰൠԽزԿͰɼଟ

ମͷଋ (tangent bundle) Λɼଋͱ༨ଋ (cotangent bundle) ͷʹுΔɽҰൠԽزԿ

ɼΔछͷॏཧΛهड़ΔπʔϧͱΔɽҰൠ૬ରཧͷରশΛ

7

ΊுͱɼॏཧଘࡏΔɽҰൠԽزԿɼݭཧΒಘΒΕΔॏཧ

ͷɼಛʹ NS-NS sector ʹରͷزԿతͳඳΛ༩Δɽ

ΕΒͷલʹΑߟΔͱɼDFT ॏཧͷ T ରෆมͳఆԽΛతͱߏங

ΕॏཧͰΔΒɼͷഎܠʹΔزԿతͳඳʹΑఆԽΕΔΖɽDFT

ͷഎܠʹΔزԿͱΕΔͷɼഒԽزԿ (doubled geometry) [12] ͰΔɽഒ

ԽزԿड़ͷϦʔϚϯزԿҰൠԽزԿͱҟͳΔͷɼతʹਖ਼ʹߏஙΕ

ͳͰΔɽͷҙຯͰɼഒԽزԿݱஈ֊Ͱ DFT ݴͷ૯শͱߏΕΔతͳݱʹ

Αɽͱ ഒԽزԿΛඋΔͱͰɼDFT ͷཧߏΛཧղͰΔͱظΕ

ΔɽΕɼT ରΛ௨ɼݟݭΔߴΤωϧΪʔͷΈΛඥղେͳख

ΓʹͳΔΖɽ

ຊमจɼ2020 1 ʹग़൛จ [13] ͷ༰ΛϕʔεʹɼDFT ͷزԿతͳඳʹ

ணҎԼͷΑʹߏͷͰΔɽ

ୈ 2ষ ͷষͰɼDFT ͱɼͷഎܠʹΔ ഒԽزԿΛಋೖΔɽॳΊʹɼDFT Ͱ༻Δ

ഒԽඪΛఏΔɽʹɼҰൠԽ Lie ඍΛఆɼDFT ͷήʔδରশΛهड़Δ

C ଋͱԿʹड़߆ڧͷষͰͷओͱͳΔޙΛ༩Δɽɼހ

Δɽ

ୈ 3ষ લͷষͰड़ɼC ΒΔɽCʹɼతͳߏఆΔنހ

ߏఆΔنހ Vaisman ѥ (algebroid) ͱݺΕΔɽLie (algebra)

Λग़ͱɼߏΛঃʑʹҰൠԽͱͰ Vaisman ѥͷఆΛ༩Δɽ

ɼͷͰΑΒΕΔ Lie ͷ Drinfelrsquod double ͱΛݩʹɼ

ͷ Lie ѥΒ Vaisman ѥΛߏஙͰΔͱΛΔɽ

ୈ 4ষ ୈ 3 ষͷՌΛɼDFT ԽύϥΤϧϛʔτݱԿͷతͳزΔɽഒԽݩʹ

(para-Hermitian) ଟମʹΑͳΕΔͱࢦఠΔ [1415]ɽͷՌʹج

ɼύϥΤϧϛʔτଟମʹɼલͷষͰఆɼdouble ʹΑಘΒΕΔ Vaisman

ѥΛߏஙΔɽɼܭͷՌΒɼ߆ڧଋͷతͳݯىʹ

ΔɽՃɼഒԽزԿͱҰൠԽزԿͷʹΔɽ

ୈ 5ষ ҎͷΛͱΊΔɽɼޙࠓͷలʹड़Δɽ

ຊจதʹݱΕʹΔܭͷࡉɼAppendix ʹճɽಛʹɼୈ 3 ষͰߦ

Vaisman ѥͷߏʹඞཁͳܭશࡌهɽஅΓͳݶΓɼຊจதͷهҎԼͷҙ

ຯͰ༻Δɽ༻ޠͷఆඞཁʹԠຊจதͰ༩Δɽ

8

M ଟମ

M ഒԽ (ύϥΤϧϛʔτଟମ)

g Lie algebra

E (L L) Lie ѥ ( Lagrangian ෦ Dirac (ߏ

Elowast E ͷରɼରଋ

ρbull bullrarr TM Anchor map

C Courant ѥ

V Vaisman ѥ

[middot middot] TM ͷ Lie ހ

[middot middot]ELL E L L ͷ Lie ހ

[middot middot]S Schouten(-Nijenhuis) ހ

[middot middot]c C ͷ Courant ހ

[middot middot]V V ͷ Vaisman ހ

[middot middot]C C ހ

⟨middot middot⟩

(middot middot) ઢܗܕ

9

ୈ 2ষ

Double Field Theory

ͷষͰɼDouble Field Theory ͷجతͳ෦ʹ؆୯ʹड़Δɽಛʹɼ3 ষҎ

ͰඞཁͱͳΔɼDFT ʹΔήʔδʹݟʹࡉɽɼDFT ΕΔഒԽݱʹ

ɼDFTʹԿͷཁૉΛ༩Δɽز ʹΔ༻ͱήʔδมΛ༩ɼΕΛఆΔހͱ

ɼC ଋΛಋೖΔɽ߆ڧͷดΔΑʹɼΛಋೖΔɽήʔδมހ

21 DFTʹݱΕΔزԿ

DFT ɼtype II ॏཧͷ NS-NS sector Λനʹ T ରෆมఆԽཧͰΔ [8]ɽ

ͷཧɼmassless ͱܭ gmicroνɼKalb-Ramond (B ) Bmicroνɼdilaton φ ΛΉɽT

ରͷԼͰɼܭͷҰൠඪมͱɼB ͷ U(1) ήʔδมΔɽΕΛزԿత

ʹදݱΔͱɼ௨ৗͷ (ͷӡಈϞʔυݭ) ͱ༨ (winding Ϟʔυ) ΛΈΘ

ɼҰൠԽͷಋೖΛଅɽ

ҰൠԽଋ (generalized tangent bundle) E ɼଟମ M ͷଋ TM ͱ ༨ଋ T lowastM ͷ

ʹΑߏΕΔɽ

E = TM oplus T lowastM (21)

ҰൠԽଋҰൠԽزԿͰऔΓѻΘΕΔɽҰൠԽزԿɼ2004 ʹ Hitchin [11] ͱ Gualtieri

[16] ʹΑఏҊΕɽϕΫτϧΛଋ TM ͷஅ Γ(TM) ͱΔͱɼඍ 1 ༨ଋܗ

T lowastM ͷஅ Γ(T lowastM) ͰΔɽɼΓ(E) ϕΫτϧͱඍ 1 తʹͱܗΛܗ

ͳΓɼ ΕΛҰൠԽϕΫτϧͱɽҰൠԽزԿͰɼඍಉ૬ (diffeomorphism) ͱήʔδ

มΛΈΘɼҰൠԽʹରΔҰൠԽඍಉ૬ΛಋೖΔɽͷՌɼCourant

ѥ [17] ͱɼରশΛهड़ΔͰͷڧͳಓΛఏڙΔɽจݙ [17] ͷ Courant ѥ

ɼҎԼͷހͰ༩ΒΕΔߏͷҰछͰΔɽXi isin Γ(TM) ξi isin Γ(T lowastM) ͱͱ

ɼҎԼͷ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (22)

ʹΑ༩ΒΕΔɽCourant ѥʹ 3 ষͰΑΓࡉʹड़Δɽ

Βʹ ͳมڞͱרͷݭ x Λಋೖɼ௨ৗͷඪ x ͱಉʹѻͱͰɼඪ

త (ଟମ) ͷݩ 2 ഒʹͳΓɼഒԽزԿ (doubled geometry) ʹ౸ୡΔɽഒԽز

10

Կͷతͳମతߏʹߦࢼஈ֊ʹΔɽۂͳͲͷزԿతͳΛఆɼ

తʹదͳഒԽزԿΛߏஙΔࢼΈ [18 19] ͳͲͰߦΘΕΔɽ

ҎԼɼഒԽͷඪΛ༩ɼDFT ͰѻܭͷఆΛ༩ΔɽҎɼ௨ৗͷॏཧ

ఆΕΔͷݩΛ D ͱɼDFTݩ ͷഒԽ 2D ͰΔͱΔɽҰൠʹɼݩ

ཧఆΕΔݭ 10 ͰΔͱΒɼഒԽݩ 20 ΒΕΔɽഒߟͰΔͱݩ

Խͷඪ (ഒԽඪ) xM ҎԼͷΑʹॻΔͷͱΔɽxmicro ӡಈͱ Fourier ͳڞ

ඪɼxmicro רͱ Fourier ඪͰΔɽͳڞ

xM =

xmicro

xmicro

$ (M = 1 2D micro = 1 D) (23)

T ରΛͱΔͱɼxmicro ͱ xmicro ͷೖΕସʹରԠΔɽɼഒԽͰͷඍԋࢠҎԼͷ

Αʹදɽ

partM =part

partxM=

partmicro

partmicro

$ partmicro =

part

partxmicro partmicro =

part

partxmicro (24)

ҰൠʹɼD ίϯύΫτʹΔݩ T ରม (T-duality ( O(DD) Ͱ༩ΒΕΔɽ

ഒԽ O(DD) ෆมͳߏ η Λɽड़ͷ xM ͷఆΒɼη ͱͷٯҎԼͷΑʹॻ

Δɽ

ηMN =

0 δmicroν

δmicroν 0

$ ηMN =

0 δmicroν

δmicroν 0

$ (25)

ഒԽඪͷҰൠԽςϯιϧͷͷԼɼ η ʹΑߦΘΕΔɽ

؆୯ͳ DFT ɼtype II ॏཧͷ NS-NS sector ΛݩʹఆԽΕΔɽͷཧʹݱ

ΕΔجຊతͳཁૉɼܭ gmicroνɼKalb-Ramond Bmicroνɼdilaton φ ͷ 3 ͰΔɽΕΒͷΛ

T ରനͳཧͰѻʹɼO(DD) มͱͳΔΑʹΈඞཁΔɽΕΛҰൠڞ

Խܭ (generalized metric) HMN ͱɽ

HMN =

gmicroν minusgmicroρBρν

Bmicroρgρν gmicroν minusBmicroρgρσBσν

$ (26)

ɼHMN ҎԼͷΛຬɽ

HMN = ηMKηNLHKL (27)

Ճɼdilaton ҰൠԽΕΔɽDFT dilaton (ҰൠԽ dilaton) d ҎԼͷΑʹͳΔɽ

eminus2d =radicminusgeminus2φ (28)

ɼg ܭͷߦͰΔɽͷΈʹΑΓɼHMN d DFT ʹ O(DD)

มΛडΔɽɼഒԽͷ HMN (xM ) d(xM ) ͱͳΔɽ

11

22 DFTͷ༻ͱ Cހ

લઅͰఏΛ༻ɼO(DD) ෆมͳ DFT ͷ༻ɼҰൠԽ Ricci εΧϥʔͱ

ʹΑҎԼͷΑʹ༩ΒΕΔ [18]

SDFT =

d2Dx eminus2dR(H d) (29)

R =1

8HMNpartMHKLpartNHKL minus

1

2HMNpartMHKLpartKHNL

+ 4HMNpartMpartNdminus partMpartNHMN minus 4HMNpartMd partNd+ 4partMHMNpartNd (210)

ͷ༻ O(DD) εΧϥʔҰൠԽήʔδεΧϥʔͰΔͱɼ[18] ʹΕɽ

༺ (29) ɼେҬతͳ O(DD) ରশ

HMN rarr HLKM ML M N

K xM rarr xNM MN (211)

ʹରനʹෆมͰΔɽͰɼΕΛͱʹہॴରশͲͷΑʹنఆΕΔΖʁ

ҰൠԽϕΫτϧ V ʹରΔҰൠԽඍಉ૬ͷແݶখ൛ɼͳΘഒԽඪͷແݶখมҎ

ԼͷΑʹఆΕΔ [8]ɽ

V M rarr V M + δΞVM xM rarr xM minus ΞM (212)

ɼδΞ ҎԼͷΑͳҰൠԽ Lie ඍͰ༩ΒΕΔɽॏΈ w(V ) ΛҰൠԽϕΫτϧ V M

ͷҰൠԽ Lie ඍҎԼͷΑʹఆΕΔɽ

ampLΞVM = ΞKpartKV M + (partMΞK minus partKΞM )V K + w(V )V MpartKΞK (213)

Ұ௨ৗͷ Lie ඍͱಉҠಈΛදɼೋͷހͷதͷ O(DD) มΛදɽHͷॏΈ w(H) = 0 ͰΓɼҰൠԽ dilaton ͷॏΈ w(eminus2d) = 1 ͰΔɽɼҰൠԽ Lie ඍ

O(DD) ΔɽΛอɽͳΘɼҎԼͷߏ

ampLΞηMN = ampLΞηMN = 0 (214)

ҰൠԽ Lie ඍɼDFT ʹΔήʔδରশͷߏΛఆΔɽͷߏ C

[ ampLΞ1 ampLΞ2 ] = ampL[Ξ1Ξ2]C + F (Ξ1Ξ2 middot) (215)

ΛͳɼC ހ

[Ξ1Ξ2]MC = 2ΞK

[1partKΞM2] minus ηKLΞ

K[1part

MΞL2]

= ΞK1 partKΞM

2 minus ΞK2 partKΞM

1 minus1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 ) (216)

ʹΑࢧΕΔɽɼΞ[1Ξ2] = (Ξ1Ξ2 minus Ξ2Ξ1)2 ͰΔɽɼF ҎԼʹΑͳ

༨ͳͰΔɽ

F (Ξ1Ξ2 V )M =1

2ηKL(Ξ

K1 partPΞL

2 minus ΞK2 partPΞL

1 )partPVM minus (partPΞM

1 partPΞK2 minus partPΞM

2 partPΞK1 )VK

(217)

F ଘࡏΔͱΒɼҰൠԽ Lie ඍͷࢠ C ͰดͳɽɼCހ ހ Jacobi

ΛຬͳɽF ΛফڈɼC ଋ߆ڧઅͰઆΔఆΔΛดΔʹɼنހ

(strong constraint) ΛͰΔɽ

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

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[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

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mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

5

ୈ 1ষ

Ίʹ

ΛѻΊͷཧͱ༺ޓ૬ڧཧɼϋυϩϯΛΔͱΛతʹɼݭ 1960

ʹੜΕɽݭཧͰɼͷݯతͳ୯ҐͱͷΘΓʹ 1 ݭͷݩ (string) Λ༻

ཧΛߏஙΔɽݭͷৼಈϞʔυͷҧʹΑɼҟͳΔछͷϋυϩϯΛҰهड़Ͱ

ΔɼͱͷͰɽɼ1970 ʹͳΔͱࢠ৭ (quantum chromodynamics

QCD) Εɼڧ૬ޓ༻Λهड़Δཧͱਖ਼ͳͷ QCD ͰΔͱΕɽ

ɼดݭͷىঢ়ଶॏʹΔͱࢦఠΕɼݭཧɼࠓࢠॏཧ

ͷͱͼͷΛΈΔͱͱͳɽॏΛهड़ΔཧͱݭཧΛղऍΔʹ

ΓɼҎԼͷΑͳΑಛΔͱΘɽ

bull ॏࢠ (graviton) ཧͷεϖΫτϧʹݱΕΔɽ

bull ݭཧͰɼͷݩ 10 ఆΔɽʹݩ

bull ͷுݭཧʹΔਓҝతͳύϥϝʔλʔݭ T ͷΈͰΔɽ

ͷఆΔҰɼզʑΔݩͷ 4 ͷॲཧݩͰΔɽɼ༨ݩ

ΛߟͳͳΒͳɽ༨ݩͷରॲͷͻͱɼΤωϧΪʔͰݕग़ΕͳΑ

ʹɼΛίϯύΫτԽΔͱͰΔɽίϯύΫτԽͷ೦ɼKaluza-Klein (KK) ཧ [12]

ͰߟҊΕɽݭཧͷಛɼ૬ޓ༻ͰͳͷுҬͰىΔͱͰΔɽ

ͷࢠ (Quantum Field Theory QFT) ͰɼॏͷಈܭΛߦͱىɽ

ΕɼQFT Ͱͷখ୯ҐʹମΛͳࢠΛ༻ΔΊʹੜΔͰɼ

ΕݴॏཧͰΔɽݶۃཧͷΤωϧΪʔݭཧͰຊతʹճආͰΔɽݭ

ɼݭཧطଘͷཧͱॏཧΛแΔ [3]ɽ

ݟΘΔɽΕΒͷʹذͷͷͷʹଟɼݟཧզʑʹΒݭ

ͷݩͱͳͷ ʮରʯͰΔɽͳରɼAdSCFT ରԠͰΖ [4]ɽΕ

ɼʮD ͷυɾδολʔݩ (Anti-de Sitter AdS) ͰఆΕॏཧɼDܠഎ minus 1

ཧܗڞͷݩ (Conformal Field Theory CFT) ͱରͰΔʯͱ༧ͰΔɽͰ

ʮରͰΔʯͱɼʮ2 ͷҟͳΔཧಉཧΛදͱʯΛҙຯΔɽͷ༧ͷ༻ͳ

ɼҰͷཧͷڧͱɼରͳยͷཧͰऑͱʹΓɼݭཧͷ

ඇಈతͳཧղΛΊΔͱͱͳɽݭཧʹݱΕΔରɼଞʹ S ର [5]ɼU ର

[6]ɼຊमจͷதͱͳΔ T ର [7] ΔɽU ରɼT ରͱ S ରΛ

6

ΘΑͳରশͰΔɽݭཧɼՁͳཧ 5 ΓɼΕΒͷରΛ௨

ҠΓͱΒΕΔɽಛʹ T ରɼݭͷʹདྷΔɼݭཧʹಛͳର

ͰΔɽT ରݱΕΔ؆୯ͳɼ1 ݩ S1 ʹίϯύΫτԽͰΔɽ

ݭ 1 ͷମͰΔͱΒɼίϯύΫτԽΕݩ S1 ͷΤרɽͷרʹ

ωϧΪʔ S1 ͷܘΛ R ͱɼݭͷுΛઃఆΕܭͰΔɽͷՌݭͷεϖΫ

τϧʹد༩ɼݭͷεϖΫτϧ M2 ͶҎԼͷΑʹॻͱͰΔɽ

M2 prop n

R

2+

mR

αprime

$2

nm isin Z (11)

ҰݭͷӡಈʹɼೋݭͷுʹདྷΔΤωϧΪʔΛදɽɼαprime ݭͷு

ΛΊΔύϥϝʔλͰΔɽͷݭͷεϖΫτϧܘ RhArr αprimeR ͷೖΕସͷԼͰɼݭͷӡಈ

n ͱר m ͷΛΔɽͷೖΕସͷԼͰཧෆมͱͳΔͱΛ T ର

ͱɽT ରʹΑΓɼݭཧͰרͱݱΕɼݭࢠͱҟͳΔͰ

ʹཧݭड़ΔͱಡΈऔΕΔɽɼهΛ T ରӅΕରশͰΔɽT

ରΛനͳରশͱΔཧߟΒΕͳΖʁ

Double Field Theory (DFT) ɼT ରΛཧͷରশͱΔॏཧͰΔ [8]ɽ

DFT ͷಛɼݭͷӡಈʹ Fourier ʹרͷݭඪͱͳڞ Fourier ඪΛՁͳڞ

ʹѻͱͰɼඪΛ double ʹுɼΛ O(DD) ΜͱʹΔɽΕʹܗมͳڞ

ʹΑΓɼT ରΛനͳରশͱݱΔɽDFT ுΕͰఆΕΔҰͰɼ

ݭཧͰͷݩ 10 ͱΔɽDFTݩ ཧతʹਖ਼Λɼཧ

ͱͳΑʹΔΊʹɼԿΒͷ߆ଋΛ༩ͷݩͷΛΘΔඞ

ཁΔɽͷΛཧ section ͱɽͷʹΑΓɼுͷதΒཧతʹҙຯ

Δ෦ΛࢦఆΔɽDFT Ͱɼ2D ͷഒԽݩ (doubled) ͷதΒɼD ͷ෦ݩ

ΛબͿͱʹͳΔɽDFT Ͱ section ΛຬͻͱͷɼഒԽʹఆΕ

ΒΏΔύϥϝʔλΒɼרඪͷґଘΛऔΓআͱͰΔɽ

ͷΑͳɼରʹجཧҰݟඒݟɼΕͰѻͳߴΤωϧΪʔε

έʔϧͰͷͷࢠΛهड़ΔΑʹࢥΔɽɼݭཧͷΤωϧΪʔεέʔϧݱ

ݕՄͳϨϕϧΛͱΖʹΔΊɼཧਖ਼ͲΛݕͰݧͷज़Ͱࡏ

ΔͱɽͰɼগͳͱߏஙཧʹͳͱΛɼͷΛआΓ

ΊɽΛѻɼͷΛௐΔͱɼزԿͰΔɽ

Ұൠ૬ରཧ Einstein ʹΑΕͷɼ1915 ͷͱͰΔ [10]ɽͷཧʹ

ΑΔॏཁͳओுɼʮॏͱͷΈͰΔʯͱͱͰΔɽɼҰൠ૬

ରཧΛఆԽΔʹɼۂ () Λهड़ΔزԿඞཁෆՄͰɽͷ

Ίɼ() ϦʔϚϯزԿॏΛهड़ΔπʔϧͱɼҰൠ૬ରཧͷͱͱʹ

ΕΔΑʹͳɽҰൠ૬ରཧͱϦʔϚϯزԿͷΈΘɼॏཧΛߏஙɼ

ΔΖɽݴͱͰΔͱΛثͳڧԿزʹΛཧղΔߏ

७ਮͱͷزԿɼݱలΛଓΔɽදతͳɼ2004 ʹ Hitchin Βʹ

ΑఏҊΕɼҰൠԽزԿ (generalized geometry) ͰΔ [11]ɽҰൠԽزԿͰɼଟ

ମͷଋ (tangent bundle) Λɼଋͱ༨ଋ (cotangent bundle) ͷʹுΔɽҰൠԽزԿ

ɼΔछͷॏཧΛهड़ΔπʔϧͱΔɽҰൠ૬ରཧͷରশΛ

7

ΊுͱɼॏཧଘࡏΔɽҰൠԽزԿɼݭཧΒಘΒΕΔॏཧ

ͷɼಛʹ NS-NS sector ʹରͷزԿతͳඳΛ༩Δɽ

ΕΒͷલʹΑߟΔͱɼDFT ॏཧͷ T ରෆมͳఆԽΛతͱߏங

ΕॏཧͰΔΒɼͷഎܠʹΔزԿతͳඳʹΑఆԽΕΔΖɽDFT

ͷഎܠʹΔزԿͱΕΔͷɼഒԽزԿ (doubled geometry) [12] ͰΔɽഒ

ԽزԿड़ͷϦʔϚϯزԿҰൠԽزԿͱҟͳΔͷɼతʹਖ਼ʹߏஙΕ

ͳͰΔɽͷҙຯͰɼഒԽزԿݱஈ֊Ͱ DFT ݴͷ૯শͱߏΕΔతͳݱʹ

Αɽͱ ഒԽزԿΛඋΔͱͰɼDFT ͷཧߏΛཧղͰΔͱظΕ

ΔɽΕɼT ରΛ௨ɼݟݭΔߴΤωϧΪʔͷΈΛඥղେͳख

ΓʹͳΔΖɽ

ຊमจɼ2020 1 ʹग़൛จ [13] ͷ༰ΛϕʔεʹɼDFT ͷزԿతͳඳʹ

ணҎԼͷΑʹߏͷͰΔɽ

ୈ 2ষ ͷষͰɼDFT ͱɼͷഎܠʹΔ ഒԽزԿΛಋೖΔɽॳΊʹɼDFT Ͱ༻Δ

ഒԽඪΛఏΔɽʹɼҰൠԽ Lie ඍΛఆɼDFT ͷήʔδରশΛهड़Δ

C ଋͱԿʹड़߆ڧͷষͰͷओͱͳΔޙΛ༩Δɽɼހ

Δɽ

ୈ 3ষ લͷষͰड़ɼC ΒΔɽCʹɼతͳߏఆΔنހ

ߏఆΔنހ Vaisman ѥ (algebroid) ͱݺΕΔɽLie (algebra)

Λग़ͱɼߏΛঃʑʹҰൠԽͱͰ Vaisman ѥͷఆΛ༩Δɽ

ɼͷͰΑΒΕΔ Lie ͷ Drinfelrsquod double ͱΛݩʹɼ

ͷ Lie ѥΒ Vaisman ѥΛߏஙͰΔͱΛΔɽ

ୈ 4ষ ୈ 3 ষͷՌΛɼDFT ԽύϥΤϧϛʔτݱԿͷతͳزΔɽഒԽݩʹ

(para-Hermitian) ଟମʹΑͳΕΔͱࢦఠΔ [1415]ɽͷՌʹج

ɼύϥΤϧϛʔτଟମʹɼલͷষͰఆɼdouble ʹΑಘΒΕΔ Vaisman

ѥΛߏஙΔɽɼܭͷՌΒɼ߆ڧଋͷతͳݯىʹ

ΔɽՃɼഒԽزԿͱҰൠԽزԿͷʹΔɽ

ୈ 5ষ ҎͷΛͱΊΔɽɼޙࠓͷలʹड़Δɽ

ຊจதʹݱΕʹΔܭͷࡉɼAppendix ʹճɽಛʹɼୈ 3 ষͰߦ

Vaisman ѥͷߏʹඞཁͳܭશࡌهɽஅΓͳݶΓɼຊจதͷهҎԼͷҙ

ຯͰ༻Δɽ༻ޠͷఆඞཁʹԠຊจதͰ༩Δɽ

8

M ଟମ

M ഒԽ (ύϥΤϧϛʔτଟମ)

g Lie algebra

E (L L) Lie ѥ ( Lagrangian ෦ Dirac (ߏ

Elowast E ͷରɼରଋ

ρbull bullrarr TM Anchor map

C Courant ѥ

V Vaisman ѥ

[middot middot] TM ͷ Lie ހ

[middot middot]ELL E L L ͷ Lie ހ

[middot middot]S Schouten(-Nijenhuis) ހ

[middot middot]c C ͷ Courant ހ

[middot middot]V V ͷ Vaisman ހ

[middot middot]C C ހ

⟨middot middot⟩

(middot middot) ઢܗܕ

9

ୈ 2ষ

Double Field Theory

ͷষͰɼDouble Field Theory ͷجతͳ෦ʹ؆୯ʹड़Δɽಛʹɼ3 ষҎ

ͰඞཁͱͳΔɼDFT ʹΔήʔδʹݟʹࡉɽɼDFT ΕΔഒԽݱʹ

ɼDFTʹԿͷཁૉΛ༩Δɽز ʹΔ༻ͱήʔδมΛ༩ɼΕΛఆΔހͱ

ɼC ଋΛಋೖΔɽ߆ڧͷดΔΑʹɼΛಋೖΔɽήʔδมހ

21 DFTʹݱΕΔزԿ

DFT ɼtype II ॏཧͷ NS-NS sector Λനʹ T ରෆมఆԽཧͰΔ [8]ɽ

ͷཧɼmassless ͱܭ gmicroνɼKalb-Ramond (B ) Bmicroνɼdilaton φ ΛΉɽT

ରͷԼͰɼܭͷҰൠඪมͱɼB ͷ U(1) ήʔδมΔɽΕΛزԿత

ʹදݱΔͱɼ௨ৗͷ (ͷӡಈϞʔυݭ) ͱ༨ (winding Ϟʔυ) ΛΈΘ

ɼҰൠԽͷಋೖΛଅɽ

ҰൠԽଋ (generalized tangent bundle) E ɼଟମ M ͷଋ TM ͱ ༨ଋ T lowastM ͷ

ʹΑߏΕΔɽ

E = TM oplus T lowastM (21)

ҰൠԽଋҰൠԽزԿͰऔΓѻΘΕΔɽҰൠԽزԿɼ2004 ʹ Hitchin [11] ͱ Gualtieri

[16] ʹΑఏҊΕɽϕΫτϧΛଋ TM ͷஅ Γ(TM) ͱΔͱɼඍ 1 ༨ଋܗ

T lowastM ͷஅ Γ(T lowastM) ͰΔɽɼΓ(E) ϕΫτϧͱඍ 1 తʹͱܗΛܗ

ͳΓɼ ΕΛҰൠԽϕΫτϧͱɽҰൠԽزԿͰɼඍಉ૬ (diffeomorphism) ͱήʔδ

มΛΈΘɼҰൠԽʹରΔҰൠԽඍಉ૬ΛಋೖΔɽͷՌɼCourant

ѥ [17] ͱɼରশΛهड़ΔͰͷڧͳಓΛఏڙΔɽจݙ [17] ͷ Courant ѥ

ɼҎԼͷހͰ༩ΒΕΔߏͷҰछͰΔɽXi isin Γ(TM) ξi isin Γ(T lowastM) ͱͱ

ɼҎԼͷ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (22)

ʹΑ༩ΒΕΔɽCourant ѥʹ 3 ষͰΑΓࡉʹड़Δɽ

Βʹ ͳมڞͱרͷݭ x Λಋೖɼ௨ৗͷඪ x ͱಉʹѻͱͰɼඪ

త (ଟମ) ͷݩ 2 ഒʹͳΓɼഒԽزԿ (doubled geometry) ʹ౸ୡΔɽഒԽز

10

Կͷతͳମతߏʹߦࢼஈ֊ʹΔɽۂͳͲͷزԿతͳΛఆɼ

తʹదͳഒԽزԿΛߏஙΔࢼΈ [18 19] ͳͲͰߦΘΕΔɽ

ҎԼɼഒԽͷඪΛ༩ɼDFT ͰѻܭͷఆΛ༩ΔɽҎɼ௨ৗͷॏཧ

ఆΕΔͷݩΛ D ͱɼDFTݩ ͷഒԽ 2D ͰΔͱΔɽҰൠʹɼݩ

ཧఆΕΔݭ 10 ͰΔͱΒɼഒԽݩ 20 ΒΕΔɽഒߟͰΔͱݩ

Խͷඪ (ഒԽඪ) xM ҎԼͷΑʹॻΔͷͱΔɽxmicro ӡಈͱ Fourier ͳڞ

ඪɼxmicro רͱ Fourier ඪͰΔɽͳڞ

xM =

xmicro

xmicro

$ (M = 1 2D micro = 1 D) (23)

T ରΛͱΔͱɼxmicro ͱ xmicro ͷೖΕସʹରԠΔɽɼഒԽͰͷඍԋࢠҎԼͷ

Αʹදɽ

partM =part

partxM=

partmicro

partmicro

$ partmicro =

part

partxmicro partmicro =

part

partxmicro (24)

ҰൠʹɼD ίϯύΫτʹΔݩ T ରม (T-duality ( O(DD) Ͱ༩ΒΕΔɽ

ഒԽ O(DD) ෆมͳߏ η Λɽड़ͷ xM ͷఆΒɼη ͱͷٯҎԼͷΑʹॻ

Δɽ

ηMN =

0 δmicroν

δmicroν 0

$ ηMN =

0 δmicroν

δmicroν 0

$ (25)

ഒԽඪͷҰൠԽςϯιϧͷͷԼɼ η ʹΑߦΘΕΔɽ

؆୯ͳ DFT ɼtype II ॏཧͷ NS-NS sector ΛݩʹఆԽΕΔɽͷཧʹݱ

ΕΔجຊతͳཁૉɼܭ gmicroνɼKalb-Ramond Bmicroνɼdilaton φ ͷ 3 ͰΔɽΕΒͷΛ

T ରനͳཧͰѻʹɼO(DD) มͱͳΔΑʹΈඞཁΔɽΕΛҰൠڞ

Խܭ (generalized metric) HMN ͱɽ

HMN =

gmicroν minusgmicroρBρν

Bmicroρgρν gmicroν minusBmicroρgρσBσν

$ (26)

ɼHMN ҎԼͷΛຬɽ

HMN = ηMKηNLHKL (27)

Ճɼdilaton ҰൠԽΕΔɽDFT dilaton (ҰൠԽ dilaton) d ҎԼͷΑʹͳΔɽ

eminus2d =radicminusgeminus2φ (28)

ɼg ܭͷߦͰΔɽͷΈʹΑΓɼHMN d DFT ʹ O(DD)

มΛडΔɽɼഒԽͷ HMN (xM ) d(xM ) ͱͳΔɽ

11

22 DFTͷ༻ͱ Cހ

લઅͰఏΛ༻ɼO(DD) ෆมͳ DFT ͷ༻ɼҰൠԽ Ricci εΧϥʔͱ

ʹΑҎԼͷΑʹ༩ΒΕΔ [18]

SDFT =

d2Dx eminus2dR(H d) (29)

R =1

8HMNpartMHKLpartNHKL minus

1

2HMNpartMHKLpartKHNL

+ 4HMNpartMpartNdminus partMpartNHMN minus 4HMNpartMd partNd+ 4partMHMNpartNd (210)

ͷ༻ O(DD) εΧϥʔҰൠԽήʔδεΧϥʔͰΔͱɼ[18] ʹΕɽ

༺ (29) ɼେҬతͳ O(DD) ରশ

HMN rarr HLKM ML M N

K xM rarr xNM MN (211)

ʹରനʹෆมͰΔɽͰɼΕΛͱʹہॴରশͲͷΑʹنఆΕΔΖʁ

ҰൠԽϕΫτϧ V ʹରΔҰൠԽඍಉ૬ͷແݶখ൛ɼͳΘഒԽඪͷແݶখมҎ

ԼͷΑʹఆΕΔ [8]ɽ

V M rarr V M + δΞVM xM rarr xM minus ΞM (212)

ɼδΞ ҎԼͷΑͳҰൠԽ Lie ඍͰ༩ΒΕΔɽॏΈ w(V ) ΛҰൠԽϕΫτϧ V M

ͷҰൠԽ Lie ඍҎԼͷΑʹఆΕΔɽ

ampLΞVM = ΞKpartKV M + (partMΞK minus partKΞM )V K + w(V )V MpartKΞK (213)

Ұ௨ৗͷ Lie ඍͱಉҠಈΛදɼೋͷހͷதͷ O(DD) มΛදɽHͷॏΈ w(H) = 0 ͰΓɼҰൠԽ dilaton ͷॏΈ w(eminus2d) = 1 ͰΔɽɼҰൠԽ Lie ඍ

O(DD) ΔɽΛอɽͳΘɼҎԼͷߏ

ampLΞηMN = ampLΞηMN = 0 (214)

ҰൠԽ Lie ඍɼDFT ʹΔήʔδରশͷߏΛఆΔɽͷߏ C

[ ampLΞ1 ampLΞ2 ] = ampL[Ξ1Ξ2]C + F (Ξ1Ξ2 middot) (215)

ΛͳɼC ހ

[Ξ1Ξ2]MC = 2ΞK

[1partKΞM2] minus ηKLΞ

K[1part

MΞL2]

= ΞK1 partKΞM

2 minus ΞK2 partKΞM

1 minus1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 ) (216)

ʹΑࢧΕΔɽɼΞ[1Ξ2] = (Ξ1Ξ2 minus Ξ2Ξ1)2 ͰΔɽɼF ҎԼʹΑͳ

༨ͳͰΔɽ

F (Ξ1Ξ2 V )M =1

2ηKL(Ξ

K1 partPΞL

2 minus ΞK2 partPΞL

1 )partPVM minus (partPΞM

1 partPΞK2 minus partPΞM

2 partPΞK1 )VK

(217)

F ଘࡏΔͱΒɼҰൠԽ Lie ඍͷࢠ C ͰดͳɽɼCހ ހ Jacobi

ΛຬͳɽF ΛফڈɼC ଋ߆ڧઅͰઆΔఆΔΛดΔʹɼنހ

(strong constraint) ΛͰΔɽ

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

6

ΘΑͳରশͰΔɽݭཧɼՁͳཧ 5 ΓɼΕΒͷରΛ௨

ҠΓͱΒΕΔɽಛʹ T ରɼݭͷʹདྷΔɼݭཧʹಛͳର

ͰΔɽT ରݱΕΔ؆୯ͳɼ1 ݩ S1 ʹίϯύΫτԽͰΔɽ

ݭ 1 ͷମͰΔͱΒɼίϯύΫτԽΕݩ S1 ͷΤרɽͷרʹ

ωϧΪʔ S1 ͷܘΛ R ͱɼݭͷுΛઃఆΕܭͰΔɽͷՌݭͷεϖΫ

τϧʹد༩ɼݭͷεϖΫτϧ M2 ͶҎԼͷΑʹॻͱͰΔɽ

M2 prop n

R

2+

mR

αprime

$2

nm isin Z (11)

ҰݭͷӡಈʹɼೋݭͷுʹདྷΔΤωϧΪʔΛදɽɼαprime ݭͷு

ΛΊΔύϥϝʔλͰΔɽͷݭͷεϖΫτϧܘ RhArr αprimeR ͷೖΕସͷԼͰɼݭͷӡಈ

n ͱר m ͷΛΔɽͷೖΕସͷԼͰཧෆมͱͳΔͱΛ T ର

ͱɽT ରʹΑΓɼݭཧͰרͱݱΕɼݭࢠͱҟͳΔͰ

ʹཧݭड़ΔͱಡΈऔΕΔɽɼهΛ T ରӅΕରশͰΔɽT

ରΛനͳରশͱΔཧߟΒΕͳΖʁ

Double Field Theory (DFT) ɼT ରΛཧͷରশͱΔॏཧͰΔ [8]ɽ

DFT ͷಛɼݭͷӡಈʹ Fourier ʹרͷݭඪͱͳڞ Fourier ඪΛՁͳڞ

ʹѻͱͰɼඪΛ double ʹுɼΛ O(DD) ΜͱʹΔɽΕʹܗมͳڞ

ʹΑΓɼT ରΛനͳରশͱݱΔɽDFT ுΕͰఆΕΔҰͰɼ

ݭཧͰͷݩ 10 ͱΔɽDFTݩ ཧతʹਖ਼Λɼཧ

ͱͳΑʹΔΊʹɼԿΒͷ߆ଋΛ༩ͷݩͷΛΘΔඞ

ཁΔɽͷΛཧ section ͱɽͷʹΑΓɼுͷதΒཧతʹҙຯ

Δ෦ΛࢦఆΔɽDFT Ͱɼ2D ͷഒԽݩ (doubled) ͷதΒɼD ͷ෦ݩ

ΛબͿͱʹͳΔɽDFT Ͱ section ΛຬͻͱͷɼഒԽʹఆΕ

ΒΏΔύϥϝʔλΒɼרඪͷґଘΛऔΓআͱͰΔɽ

ͷΑͳɼରʹجཧҰݟඒݟɼΕͰѻͳߴΤωϧΪʔε

έʔϧͰͷͷࢠΛهड़ΔΑʹࢥΔɽɼݭཧͷΤωϧΪʔεέʔϧݱ

ݕՄͳϨϕϧΛͱΖʹΔΊɼཧਖ਼ͲΛݕͰݧͷज़Ͱࡏ

ΔͱɽͰɼগͳͱߏஙཧʹͳͱΛɼͷΛआΓ

ΊɽΛѻɼͷΛௐΔͱɼزԿͰΔɽ

Ұൠ૬ରཧ Einstein ʹΑΕͷɼ1915 ͷͱͰΔ [10]ɽͷཧʹ

ΑΔॏཁͳओுɼʮॏͱͷΈͰΔʯͱͱͰΔɽɼҰൠ૬

ରཧΛఆԽΔʹɼۂ () Λهड़ΔزԿඞཁෆՄͰɽͷ

Ίɼ() ϦʔϚϯزԿॏΛهड़ΔπʔϧͱɼҰൠ૬ରཧͷͱͱʹ

ΕΔΑʹͳɽҰൠ૬ରཧͱϦʔϚϯزԿͷΈΘɼॏཧΛߏஙɼ

ΔΖɽݴͱͰΔͱΛثͳڧԿزʹΛཧղΔߏ

७ਮͱͷزԿɼݱలΛଓΔɽදతͳɼ2004 ʹ Hitchin Βʹ

ΑఏҊΕɼҰൠԽزԿ (generalized geometry) ͰΔ [11]ɽҰൠԽزԿͰɼଟ

ମͷଋ (tangent bundle) Λɼଋͱ༨ଋ (cotangent bundle) ͷʹுΔɽҰൠԽزԿ

ɼΔछͷॏཧΛهड़ΔπʔϧͱΔɽҰൠ૬ରཧͷରশΛ

7

ΊுͱɼॏཧଘࡏΔɽҰൠԽزԿɼݭཧΒಘΒΕΔॏཧ

ͷɼಛʹ NS-NS sector ʹରͷزԿతͳඳΛ༩Δɽ

ΕΒͷલʹΑߟΔͱɼDFT ॏཧͷ T ରෆมͳఆԽΛతͱߏங

ΕॏཧͰΔΒɼͷഎܠʹΔزԿతͳඳʹΑఆԽΕΔΖɽDFT

ͷഎܠʹΔزԿͱΕΔͷɼഒԽزԿ (doubled geometry) [12] ͰΔɽഒ

ԽزԿड़ͷϦʔϚϯزԿҰൠԽزԿͱҟͳΔͷɼతʹਖ਼ʹߏஙΕ

ͳͰΔɽͷҙຯͰɼഒԽزԿݱஈ֊Ͱ DFT ݴͷ૯শͱߏΕΔతͳݱʹ

Αɽͱ ഒԽزԿΛඋΔͱͰɼDFT ͷཧߏΛཧղͰΔͱظΕ

ΔɽΕɼT ରΛ௨ɼݟݭΔߴΤωϧΪʔͷΈΛඥղେͳख

ΓʹͳΔΖɽ

ຊमจɼ2020 1 ʹग़൛จ [13] ͷ༰ΛϕʔεʹɼDFT ͷزԿతͳඳʹ

ணҎԼͷΑʹߏͷͰΔɽ

ୈ 2ষ ͷষͰɼDFT ͱɼͷഎܠʹΔ ഒԽزԿΛಋೖΔɽॳΊʹɼDFT Ͱ༻Δ

ഒԽඪΛఏΔɽʹɼҰൠԽ Lie ඍΛఆɼDFT ͷήʔδରশΛهड़Δ

C ଋͱԿʹड़߆ڧͷষͰͷओͱͳΔޙΛ༩Δɽɼހ

Δɽ

ୈ 3ষ લͷষͰड़ɼC ΒΔɽCʹɼతͳߏఆΔنހ

ߏఆΔنހ Vaisman ѥ (algebroid) ͱݺΕΔɽLie (algebra)

Λग़ͱɼߏΛঃʑʹҰൠԽͱͰ Vaisman ѥͷఆΛ༩Δɽ

ɼͷͰΑΒΕΔ Lie ͷ Drinfelrsquod double ͱΛݩʹɼ

ͷ Lie ѥΒ Vaisman ѥΛߏஙͰΔͱΛΔɽ

ୈ 4ষ ୈ 3 ষͷՌΛɼDFT ԽύϥΤϧϛʔτݱԿͷతͳزΔɽഒԽݩʹ

(para-Hermitian) ଟମʹΑͳΕΔͱࢦఠΔ [1415]ɽͷՌʹج

ɼύϥΤϧϛʔτଟମʹɼલͷষͰఆɼdouble ʹΑಘΒΕΔ Vaisman

ѥΛߏஙΔɽɼܭͷՌΒɼ߆ڧଋͷతͳݯىʹ

ΔɽՃɼഒԽزԿͱҰൠԽزԿͷʹΔɽ

ୈ 5ষ ҎͷΛͱΊΔɽɼޙࠓͷలʹड़Δɽ

ຊจதʹݱΕʹΔܭͷࡉɼAppendix ʹճɽಛʹɼୈ 3 ষͰߦ

Vaisman ѥͷߏʹඞཁͳܭશࡌهɽஅΓͳݶΓɼຊจதͷهҎԼͷҙ

ຯͰ༻Δɽ༻ޠͷఆඞཁʹԠຊจதͰ༩Δɽ

8

M ଟମ

M ഒԽ (ύϥΤϧϛʔτଟମ)

g Lie algebra

E (L L) Lie ѥ ( Lagrangian ෦ Dirac (ߏ

Elowast E ͷରɼରଋ

ρbull bullrarr TM Anchor map

C Courant ѥ

V Vaisman ѥ

[middot middot] TM ͷ Lie ހ

[middot middot]ELL E L L ͷ Lie ހ

[middot middot]S Schouten(-Nijenhuis) ހ

[middot middot]c C ͷ Courant ހ

[middot middot]V V ͷ Vaisman ހ

[middot middot]C C ހ

⟨middot middot⟩

(middot middot) ઢܗܕ

9

ୈ 2ষ

Double Field Theory

ͷষͰɼDouble Field Theory ͷجతͳ෦ʹ؆୯ʹड़Δɽಛʹɼ3 ষҎ

ͰඞཁͱͳΔɼDFT ʹΔήʔδʹݟʹࡉɽɼDFT ΕΔഒԽݱʹ

ɼDFTʹԿͷཁૉΛ༩Δɽز ʹΔ༻ͱήʔδมΛ༩ɼΕΛఆΔހͱ

ɼC ଋΛಋೖΔɽ߆ڧͷดΔΑʹɼΛಋೖΔɽήʔδมހ

21 DFTʹݱΕΔزԿ

DFT ɼtype II ॏཧͷ NS-NS sector Λനʹ T ରෆมఆԽཧͰΔ [8]ɽ

ͷཧɼmassless ͱܭ gmicroνɼKalb-Ramond (B ) Bmicroνɼdilaton φ ΛΉɽT

ରͷԼͰɼܭͷҰൠඪมͱɼB ͷ U(1) ήʔδมΔɽΕΛزԿత

ʹදݱΔͱɼ௨ৗͷ (ͷӡಈϞʔυݭ) ͱ༨ (winding Ϟʔυ) ΛΈΘ

ɼҰൠԽͷಋೖΛଅɽ

ҰൠԽଋ (generalized tangent bundle) E ɼଟମ M ͷଋ TM ͱ ༨ଋ T lowastM ͷ

ʹΑߏΕΔɽ

E = TM oplus T lowastM (21)

ҰൠԽଋҰൠԽزԿͰऔΓѻΘΕΔɽҰൠԽزԿɼ2004 ʹ Hitchin [11] ͱ Gualtieri

[16] ʹΑఏҊΕɽϕΫτϧΛଋ TM ͷஅ Γ(TM) ͱΔͱɼඍ 1 ༨ଋܗ

T lowastM ͷஅ Γ(T lowastM) ͰΔɽɼΓ(E) ϕΫτϧͱඍ 1 తʹͱܗΛܗ

ͳΓɼ ΕΛҰൠԽϕΫτϧͱɽҰൠԽزԿͰɼඍಉ૬ (diffeomorphism) ͱήʔδ

มΛΈΘɼҰൠԽʹରΔҰൠԽඍಉ૬ΛಋೖΔɽͷՌɼCourant

ѥ [17] ͱɼରশΛهड़ΔͰͷڧͳಓΛఏڙΔɽจݙ [17] ͷ Courant ѥ

ɼҎԼͷހͰ༩ΒΕΔߏͷҰछͰΔɽXi isin Γ(TM) ξi isin Γ(T lowastM) ͱͱ

ɼҎԼͷ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (22)

ʹΑ༩ΒΕΔɽCourant ѥʹ 3 ষͰΑΓࡉʹड़Δɽ

Βʹ ͳมڞͱרͷݭ x Λಋೖɼ௨ৗͷඪ x ͱಉʹѻͱͰɼඪ

త (ଟମ) ͷݩ 2 ഒʹͳΓɼഒԽزԿ (doubled geometry) ʹ౸ୡΔɽഒԽز

10

Կͷతͳମతߏʹߦࢼஈ֊ʹΔɽۂͳͲͷزԿతͳΛఆɼ

తʹదͳഒԽزԿΛߏஙΔࢼΈ [18 19] ͳͲͰߦΘΕΔɽ

ҎԼɼഒԽͷඪΛ༩ɼDFT ͰѻܭͷఆΛ༩ΔɽҎɼ௨ৗͷॏཧ

ఆΕΔͷݩΛ D ͱɼDFTݩ ͷഒԽ 2D ͰΔͱΔɽҰൠʹɼݩ

ཧఆΕΔݭ 10 ͰΔͱΒɼഒԽݩ 20 ΒΕΔɽഒߟͰΔͱݩ

Խͷඪ (ഒԽඪ) xM ҎԼͷΑʹॻΔͷͱΔɽxmicro ӡಈͱ Fourier ͳڞ

ඪɼxmicro רͱ Fourier ඪͰΔɽͳڞ

xM =

xmicro

xmicro

$ (M = 1 2D micro = 1 D) (23)

T ରΛͱΔͱɼxmicro ͱ xmicro ͷೖΕସʹରԠΔɽɼഒԽͰͷඍԋࢠҎԼͷ

Αʹදɽ

partM =part

partxM=

partmicro

partmicro

$ partmicro =

part

partxmicro partmicro =

part

partxmicro (24)

ҰൠʹɼD ίϯύΫτʹΔݩ T ରม (T-duality ( O(DD) Ͱ༩ΒΕΔɽ

ഒԽ O(DD) ෆมͳߏ η Λɽड़ͷ xM ͷఆΒɼη ͱͷٯҎԼͷΑʹॻ

Δɽ

ηMN =

0 δmicroν

δmicroν 0

$ ηMN =

0 δmicroν

δmicroν 0

$ (25)

ഒԽඪͷҰൠԽςϯιϧͷͷԼɼ η ʹΑߦΘΕΔɽ

؆୯ͳ DFT ɼtype II ॏཧͷ NS-NS sector ΛݩʹఆԽΕΔɽͷཧʹݱ

ΕΔجຊతͳཁૉɼܭ gmicroνɼKalb-Ramond Bmicroνɼdilaton φ ͷ 3 ͰΔɽΕΒͷΛ

T ରനͳཧͰѻʹɼO(DD) มͱͳΔΑʹΈඞཁΔɽΕΛҰൠڞ

Խܭ (generalized metric) HMN ͱɽ

HMN =

gmicroν minusgmicroρBρν

Bmicroρgρν gmicroν minusBmicroρgρσBσν

$ (26)

ɼHMN ҎԼͷΛຬɽ

HMN = ηMKηNLHKL (27)

Ճɼdilaton ҰൠԽΕΔɽDFT dilaton (ҰൠԽ dilaton) d ҎԼͷΑʹͳΔɽ

eminus2d =radicminusgeminus2φ (28)

ɼg ܭͷߦͰΔɽͷΈʹΑΓɼHMN d DFT ʹ O(DD)

มΛडΔɽɼഒԽͷ HMN (xM ) d(xM ) ͱͳΔɽ

11

22 DFTͷ༻ͱ Cހ

લઅͰఏΛ༻ɼO(DD) ෆมͳ DFT ͷ༻ɼҰൠԽ Ricci εΧϥʔͱ

ʹΑҎԼͷΑʹ༩ΒΕΔ [18]

SDFT =

d2Dx eminus2dR(H d) (29)

R =1

8HMNpartMHKLpartNHKL minus

1

2HMNpartMHKLpartKHNL

+ 4HMNpartMpartNdminus partMpartNHMN minus 4HMNpartMd partNd+ 4partMHMNpartNd (210)

ͷ༻ O(DD) εΧϥʔҰൠԽήʔδεΧϥʔͰΔͱɼ[18] ʹΕɽ

༺ (29) ɼେҬతͳ O(DD) ରশ

HMN rarr HLKM ML M N

K xM rarr xNM MN (211)

ʹରനʹෆมͰΔɽͰɼΕΛͱʹہॴରশͲͷΑʹنఆΕΔΖʁ

ҰൠԽϕΫτϧ V ʹରΔҰൠԽඍಉ૬ͷແݶখ൛ɼͳΘഒԽඪͷແݶখมҎ

ԼͷΑʹఆΕΔ [8]ɽ

V M rarr V M + δΞVM xM rarr xM minus ΞM (212)

ɼδΞ ҎԼͷΑͳҰൠԽ Lie ඍͰ༩ΒΕΔɽॏΈ w(V ) ΛҰൠԽϕΫτϧ V M

ͷҰൠԽ Lie ඍҎԼͷΑʹఆΕΔɽ

ampLΞVM = ΞKpartKV M + (partMΞK minus partKΞM )V K + w(V )V MpartKΞK (213)

Ұ௨ৗͷ Lie ඍͱಉҠಈΛදɼೋͷހͷதͷ O(DD) มΛදɽHͷॏΈ w(H) = 0 ͰΓɼҰൠԽ dilaton ͷॏΈ w(eminus2d) = 1 ͰΔɽɼҰൠԽ Lie ඍ

O(DD) ΔɽΛอɽͳΘɼҎԼͷߏ

ampLΞηMN = ampLΞηMN = 0 (214)

ҰൠԽ Lie ඍɼDFT ʹΔήʔδରশͷߏΛఆΔɽͷߏ C

[ ampLΞ1 ampLΞ2 ] = ampL[Ξ1Ξ2]C + F (Ξ1Ξ2 middot) (215)

ΛͳɼC ހ

[Ξ1Ξ2]MC = 2ΞK

[1partKΞM2] minus ηKLΞ

K[1part

MΞL2]

= ΞK1 partKΞM

2 minus ΞK2 partKΞM

1 minus1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 ) (216)

ʹΑࢧΕΔɽɼΞ[1Ξ2] = (Ξ1Ξ2 minus Ξ2Ξ1)2 ͰΔɽɼF ҎԼʹΑͳ

༨ͳͰΔɽ

F (Ξ1Ξ2 V )M =1

2ηKL(Ξ

K1 partPΞL

2 minus ΞK2 partPΞL

1 )partPVM minus (partPΞM

1 partPΞK2 minus partPΞM

2 partPΞK1 )VK

(217)

F ଘࡏΔͱΒɼҰൠԽ Lie ඍͷࢠ C ͰดͳɽɼCހ ހ Jacobi

ΛຬͳɽF ΛফڈɼC ଋ߆ڧઅͰઆΔఆΔΛดΔʹɼنހ

(strong constraint) ΛͰΔɽ

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

7

ΊுͱɼॏཧଘࡏΔɽҰൠԽزԿɼݭཧΒಘΒΕΔॏཧ

ͷɼಛʹ NS-NS sector ʹରͷزԿతͳඳΛ༩Δɽ

ΕΒͷલʹΑߟΔͱɼDFT ॏཧͷ T ରෆมͳఆԽΛతͱߏங

ΕॏཧͰΔΒɼͷഎܠʹΔزԿతͳඳʹΑఆԽΕΔΖɽDFT

ͷഎܠʹΔزԿͱΕΔͷɼഒԽزԿ (doubled geometry) [12] ͰΔɽഒ

ԽزԿड़ͷϦʔϚϯزԿҰൠԽزԿͱҟͳΔͷɼతʹਖ਼ʹߏஙΕ

ͳͰΔɽͷҙຯͰɼഒԽزԿݱஈ֊Ͱ DFT ݴͷ૯শͱߏΕΔతͳݱʹ

Αɽͱ ഒԽزԿΛඋΔͱͰɼDFT ͷཧߏΛཧղͰΔͱظΕ

ΔɽΕɼT ରΛ௨ɼݟݭΔߴΤωϧΪʔͷΈΛඥղେͳख

ΓʹͳΔΖɽ

ຊमจɼ2020 1 ʹग़൛จ [13] ͷ༰ΛϕʔεʹɼDFT ͷزԿతͳඳʹ

ணҎԼͷΑʹߏͷͰΔɽ

ୈ 2ষ ͷষͰɼDFT ͱɼͷഎܠʹΔ ഒԽزԿΛಋೖΔɽॳΊʹɼDFT Ͱ༻Δ

ഒԽඪΛఏΔɽʹɼҰൠԽ Lie ඍΛఆɼDFT ͷήʔδରশΛهड़Δ

C ଋͱԿʹड़߆ڧͷষͰͷओͱͳΔޙΛ༩Δɽɼހ

Δɽ

ୈ 3ষ લͷষͰड़ɼC ΒΔɽCʹɼతͳߏఆΔنހ

ߏఆΔنހ Vaisman ѥ (algebroid) ͱݺΕΔɽLie (algebra)

Λग़ͱɼߏΛঃʑʹҰൠԽͱͰ Vaisman ѥͷఆΛ༩Δɽ

ɼͷͰΑΒΕΔ Lie ͷ Drinfelrsquod double ͱΛݩʹɼ

ͷ Lie ѥΒ Vaisman ѥΛߏஙͰΔͱΛΔɽ

ୈ 4ষ ୈ 3 ষͷՌΛɼDFT ԽύϥΤϧϛʔτݱԿͷతͳزΔɽഒԽݩʹ

(para-Hermitian) ଟମʹΑͳΕΔͱࢦఠΔ [1415]ɽͷՌʹج

ɼύϥΤϧϛʔτଟମʹɼલͷষͰఆɼdouble ʹΑಘΒΕΔ Vaisman

ѥΛߏஙΔɽɼܭͷՌΒɼ߆ڧଋͷతͳݯىʹ

ΔɽՃɼഒԽزԿͱҰൠԽزԿͷʹΔɽ

ୈ 5ষ ҎͷΛͱΊΔɽɼޙࠓͷలʹड़Δɽ

ຊจதʹݱΕʹΔܭͷࡉɼAppendix ʹճɽಛʹɼୈ 3 ষͰߦ

Vaisman ѥͷߏʹඞཁͳܭશࡌهɽஅΓͳݶΓɼຊจதͷهҎԼͷҙ

ຯͰ༻Δɽ༻ޠͷఆඞཁʹԠຊจதͰ༩Δɽ

8

M ଟମ

M ഒԽ (ύϥΤϧϛʔτଟମ)

g Lie algebra

E (L L) Lie ѥ ( Lagrangian ෦ Dirac (ߏ

Elowast E ͷରɼରଋ

ρbull bullrarr TM Anchor map

C Courant ѥ

V Vaisman ѥ

[middot middot] TM ͷ Lie ހ

[middot middot]ELL E L L ͷ Lie ހ

[middot middot]S Schouten(-Nijenhuis) ހ

[middot middot]c C ͷ Courant ހ

[middot middot]V V ͷ Vaisman ހ

[middot middot]C C ހ

⟨middot middot⟩

(middot middot) ઢܗܕ

9

ୈ 2ষ

Double Field Theory

ͷষͰɼDouble Field Theory ͷجతͳ෦ʹ؆୯ʹड़Δɽಛʹɼ3 ষҎ

ͰඞཁͱͳΔɼDFT ʹΔήʔδʹݟʹࡉɽɼDFT ΕΔഒԽݱʹ

ɼDFTʹԿͷཁૉΛ༩Δɽز ʹΔ༻ͱήʔδมΛ༩ɼΕΛఆΔހͱ

ɼC ଋΛಋೖΔɽ߆ڧͷดΔΑʹɼΛಋೖΔɽήʔδมހ

21 DFTʹݱΕΔزԿ

DFT ɼtype II ॏཧͷ NS-NS sector Λനʹ T ରෆมఆԽཧͰΔ [8]ɽ

ͷཧɼmassless ͱܭ gmicroνɼKalb-Ramond (B ) Bmicroνɼdilaton φ ΛΉɽT

ରͷԼͰɼܭͷҰൠඪมͱɼB ͷ U(1) ήʔδมΔɽΕΛزԿత

ʹදݱΔͱɼ௨ৗͷ (ͷӡಈϞʔυݭ) ͱ༨ (winding Ϟʔυ) ΛΈΘ

ɼҰൠԽͷಋೖΛଅɽ

ҰൠԽଋ (generalized tangent bundle) E ɼଟମ M ͷଋ TM ͱ ༨ଋ T lowastM ͷ

ʹΑߏΕΔɽ

E = TM oplus T lowastM (21)

ҰൠԽଋҰൠԽزԿͰऔΓѻΘΕΔɽҰൠԽزԿɼ2004 ʹ Hitchin [11] ͱ Gualtieri

[16] ʹΑఏҊΕɽϕΫτϧΛଋ TM ͷஅ Γ(TM) ͱΔͱɼඍ 1 ༨ଋܗ

T lowastM ͷஅ Γ(T lowastM) ͰΔɽɼΓ(E) ϕΫτϧͱඍ 1 తʹͱܗΛܗ

ͳΓɼ ΕΛҰൠԽϕΫτϧͱɽҰൠԽزԿͰɼඍಉ૬ (diffeomorphism) ͱήʔδ

มΛΈΘɼҰൠԽʹରΔҰൠԽඍಉ૬ΛಋೖΔɽͷՌɼCourant

ѥ [17] ͱɼରশΛهड़ΔͰͷڧͳಓΛఏڙΔɽจݙ [17] ͷ Courant ѥ

ɼҎԼͷހͰ༩ΒΕΔߏͷҰछͰΔɽXi isin Γ(TM) ξi isin Γ(T lowastM) ͱͱ

ɼҎԼͷ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (22)

ʹΑ༩ΒΕΔɽCourant ѥʹ 3 ষͰΑΓࡉʹड़Δɽ

Βʹ ͳมڞͱרͷݭ x Λಋೖɼ௨ৗͷඪ x ͱಉʹѻͱͰɼඪ

త (ଟମ) ͷݩ 2 ഒʹͳΓɼഒԽزԿ (doubled geometry) ʹ౸ୡΔɽഒԽز

10

Կͷతͳମతߏʹߦࢼஈ֊ʹΔɽۂͳͲͷزԿతͳΛఆɼ

తʹదͳഒԽزԿΛߏஙΔࢼΈ [18 19] ͳͲͰߦΘΕΔɽ

ҎԼɼഒԽͷඪΛ༩ɼDFT ͰѻܭͷఆΛ༩ΔɽҎɼ௨ৗͷॏཧ

ఆΕΔͷݩΛ D ͱɼDFTݩ ͷഒԽ 2D ͰΔͱΔɽҰൠʹɼݩ

ཧఆΕΔݭ 10 ͰΔͱΒɼഒԽݩ 20 ΒΕΔɽഒߟͰΔͱݩ

Խͷඪ (ഒԽඪ) xM ҎԼͷΑʹॻΔͷͱΔɽxmicro ӡಈͱ Fourier ͳڞ

ඪɼxmicro רͱ Fourier ඪͰΔɽͳڞ

xM =

xmicro

xmicro

$ (M = 1 2D micro = 1 D) (23)

T ରΛͱΔͱɼxmicro ͱ xmicro ͷೖΕସʹରԠΔɽɼഒԽͰͷඍԋࢠҎԼͷ

Αʹදɽ

partM =part

partxM=

partmicro

partmicro

$ partmicro =

part

partxmicro partmicro =

part

partxmicro (24)

ҰൠʹɼD ίϯύΫτʹΔݩ T ରม (T-duality ( O(DD) Ͱ༩ΒΕΔɽ

ഒԽ O(DD) ෆมͳߏ η Λɽड़ͷ xM ͷఆΒɼη ͱͷٯҎԼͷΑʹॻ

Δɽ

ηMN =

0 δmicroν

δmicroν 0

$ ηMN =

0 δmicroν

δmicroν 0

$ (25)

ഒԽඪͷҰൠԽςϯιϧͷͷԼɼ η ʹΑߦΘΕΔɽ

؆୯ͳ DFT ɼtype II ॏཧͷ NS-NS sector ΛݩʹఆԽΕΔɽͷཧʹݱ

ΕΔجຊతͳཁૉɼܭ gmicroνɼKalb-Ramond Bmicroνɼdilaton φ ͷ 3 ͰΔɽΕΒͷΛ

T ରനͳཧͰѻʹɼO(DD) มͱͳΔΑʹΈඞཁΔɽΕΛҰൠڞ

Խܭ (generalized metric) HMN ͱɽ

HMN =

gmicroν minusgmicroρBρν

Bmicroρgρν gmicroν minusBmicroρgρσBσν

$ (26)

ɼHMN ҎԼͷΛຬɽ

HMN = ηMKηNLHKL (27)

Ճɼdilaton ҰൠԽΕΔɽDFT dilaton (ҰൠԽ dilaton) d ҎԼͷΑʹͳΔɽ

eminus2d =radicminusgeminus2φ (28)

ɼg ܭͷߦͰΔɽͷΈʹΑΓɼHMN d DFT ʹ O(DD)

มΛडΔɽɼഒԽͷ HMN (xM ) d(xM ) ͱͳΔɽ

11

22 DFTͷ༻ͱ Cހ

લઅͰఏΛ༻ɼO(DD) ෆมͳ DFT ͷ༻ɼҰൠԽ Ricci εΧϥʔͱ

ʹΑҎԼͷΑʹ༩ΒΕΔ [18]

SDFT =

d2Dx eminus2dR(H d) (29)

R =1

8HMNpartMHKLpartNHKL minus

1

2HMNpartMHKLpartKHNL

+ 4HMNpartMpartNdminus partMpartNHMN minus 4HMNpartMd partNd+ 4partMHMNpartNd (210)

ͷ༻ O(DD) εΧϥʔҰൠԽήʔδεΧϥʔͰΔͱɼ[18] ʹΕɽ

༺ (29) ɼେҬతͳ O(DD) ରশ

HMN rarr HLKM ML M N

K xM rarr xNM MN (211)

ʹରനʹෆมͰΔɽͰɼΕΛͱʹہॴରশͲͷΑʹنఆΕΔΖʁ

ҰൠԽϕΫτϧ V ʹରΔҰൠԽඍಉ૬ͷແݶখ൛ɼͳΘഒԽඪͷແݶখมҎ

ԼͷΑʹఆΕΔ [8]ɽ

V M rarr V M + δΞVM xM rarr xM minus ΞM (212)

ɼδΞ ҎԼͷΑͳҰൠԽ Lie ඍͰ༩ΒΕΔɽॏΈ w(V ) ΛҰൠԽϕΫτϧ V M

ͷҰൠԽ Lie ඍҎԼͷΑʹఆΕΔɽ

ampLΞVM = ΞKpartKV M + (partMΞK minus partKΞM )V K + w(V )V MpartKΞK (213)

Ұ௨ৗͷ Lie ඍͱಉҠಈΛදɼೋͷހͷதͷ O(DD) มΛදɽHͷॏΈ w(H) = 0 ͰΓɼҰൠԽ dilaton ͷॏΈ w(eminus2d) = 1 ͰΔɽɼҰൠԽ Lie ඍ

O(DD) ΔɽΛอɽͳΘɼҎԼͷߏ

ampLΞηMN = ampLΞηMN = 0 (214)

ҰൠԽ Lie ඍɼDFT ʹΔήʔδରশͷߏΛఆΔɽͷߏ C

[ ampLΞ1 ampLΞ2 ] = ampL[Ξ1Ξ2]C + F (Ξ1Ξ2 middot) (215)

ΛͳɼC ހ

[Ξ1Ξ2]MC = 2ΞK

[1partKΞM2] minus ηKLΞ

K[1part

MΞL2]

= ΞK1 partKΞM

2 minus ΞK2 partKΞM

1 minus1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 ) (216)

ʹΑࢧΕΔɽɼΞ[1Ξ2] = (Ξ1Ξ2 minus Ξ2Ξ1)2 ͰΔɽɼF ҎԼʹΑͳ

༨ͳͰΔɽ

F (Ξ1Ξ2 V )M =1

2ηKL(Ξ

K1 partPΞL

2 minus ΞK2 partPΞL

1 )partPVM minus (partPΞM

1 partPΞK2 minus partPΞM

2 partPΞK1 )VK

(217)

F ଘࡏΔͱΒɼҰൠԽ Lie ඍͷࢠ C ͰดͳɽɼCހ ހ Jacobi

ΛຬͳɽF ΛফڈɼC ଋ߆ڧઅͰઆΔఆΔΛดΔʹɼنހ

(strong constraint) ΛͰΔɽ

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

8

M ଟମ

M ഒԽ (ύϥΤϧϛʔτଟମ)

g Lie algebra

E (L L) Lie ѥ ( Lagrangian ෦ Dirac (ߏ

Elowast E ͷରɼରଋ

ρbull bullrarr TM Anchor map

C Courant ѥ

V Vaisman ѥ

[middot middot] TM ͷ Lie ހ

[middot middot]ELL E L L ͷ Lie ހ

[middot middot]S Schouten(-Nijenhuis) ހ

[middot middot]c C ͷ Courant ހ

[middot middot]V V ͷ Vaisman ހ

[middot middot]C C ހ

⟨middot middot⟩

(middot middot) ઢܗܕ

9

ୈ 2ষ

Double Field Theory

ͷষͰɼDouble Field Theory ͷجతͳ෦ʹ؆୯ʹड़Δɽಛʹɼ3 ষҎ

ͰඞཁͱͳΔɼDFT ʹΔήʔδʹݟʹࡉɽɼDFT ΕΔഒԽݱʹ

ɼDFTʹԿͷཁૉΛ༩Δɽز ʹΔ༻ͱήʔδมΛ༩ɼΕΛఆΔހͱ

ɼC ଋΛಋೖΔɽ߆ڧͷดΔΑʹɼΛಋೖΔɽήʔδมހ

21 DFTʹݱΕΔزԿ

DFT ɼtype II ॏཧͷ NS-NS sector Λനʹ T ରෆมఆԽཧͰΔ [8]ɽ

ͷཧɼmassless ͱܭ gmicroνɼKalb-Ramond (B ) Bmicroνɼdilaton φ ΛΉɽT

ରͷԼͰɼܭͷҰൠඪมͱɼB ͷ U(1) ήʔδมΔɽΕΛزԿత

ʹදݱΔͱɼ௨ৗͷ (ͷӡಈϞʔυݭ) ͱ༨ (winding Ϟʔυ) ΛΈΘ

ɼҰൠԽͷಋೖΛଅɽ

ҰൠԽଋ (generalized tangent bundle) E ɼଟମ M ͷଋ TM ͱ ༨ଋ T lowastM ͷ

ʹΑߏΕΔɽ

E = TM oplus T lowastM (21)

ҰൠԽଋҰൠԽزԿͰऔΓѻΘΕΔɽҰൠԽزԿɼ2004 ʹ Hitchin [11] ͱ Gualtieri

[16] ʹΑఏҊΕɽϕΫτϧΛଋ TM ͷஅ Γ(TM) ͱΔͱɼඍ 1 ༨ଋܗ

T lowastM ͷஅ Γ(T lowastM) ͰΔɽɼΓ(E) ϕΫτϧͱඍ 1 తʹͱܗΛܗ

ͳΓɼ ΕΛҰൠԽϕΫτϧͱɽҰൠԽزԿͰɼඍಉ૬ (diffeomorphism) ͱήʔδ

มΛΈΘɼҰൠԽʹରΔҰൠԽඍಉ૬ΛಋೖΔɽͷՌɼCourant

ѥ [17] ͱɼରশΛهड़ΔͰͷڧͳಓΛఏڙΔɽจݙ [17] ͷ Courant ѥ

ɼҎԼͷހͰ༩ΒΕΔߏͷҰछͰΔɽXi isin Γ(TM) ξi isin Γ(T lowastM) ͱͱ

ɼҎԼͷ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (22)

ʹΑ༩ΒΕΔɽCourant ѥʹ 3 ষͰΑΓࡉʹड़Δɽ

Βʹ ͳมڞͱרͷݭ x Λಋೖɼ௨ৗͷඪ x ͱಉʹѻͱͰɼඪ

త (ଟମ) ͷݩ 2 ഒʹͳΓɼഒԽزԿ (doubled geometry) ʹ౸ୡΔɽഒԽز

10

Կͷతͳମతߏʹߦࢼஈ֊ʹΔɽۂͳͲͷزԿతͳΛఆɼ

తʹదͳഒԽزԿΛߏஙΔࢼΈ [18 19] ͳͲͰߦΘΕΔɽ

ҎԼɼഒԽͷඪΛ༩ɼDFT ͰѻܭͷఆΛ༩ΔɽҎɼ௨ৗͷॏཧ

ఆΕΔͷݩΛ D ͱɼDFTݩ ͷഒԽ 2D ͰΔͱΔɽҰൠʹɼݩ

ཧఆΕΔݭ 10 ͰΔͱΒɼഒԽݩ 20 ΒΕΔɽഒߟͰΔͱݩ

Խͷඪ (ഒԽඪ) xM ҎԼͷΑʹॻΔͷͱΔɽxmicro ӡಈͱ Fourier ͳڞ

ඪɼxmicro רͱ Fourier ඪͰΔɽͳڞ

xM =

xmicro

xmicro

$ (M = 1 2D micro = 1 D) (23)

T ରΛͱΔͱɼxmicro ͱ xmicro ͷೖΕସʹରԠΔɽɼഒԽͰͷඍԋࢠҎԼͷ

Αʹදɽ

partM =part

partxM=

partmicro

partmicro

$ partmicro =

part

partxmicro partmicro =

part

partxmicro (24)

ҰൠʹɼD ίϯύΫτʹΔݩ T ରม (T-duality ( O(DD) Ͱ༩ΒΕΔɽ

ഒԽ O(DD) ෆมͳߏ η Λɽड़ͷ xM ͷఆΒɼη ͱͷٯҎԼͷΑʹॻ

Δɽ

ηMN =

0 δmicroν

δmicroν 0

$ ηMN =

0 δmicroν

δmicroν 0

$ (25)

ഒԽඪͷҰൠԽςϯιϧͷͷԼɼ η ʹΑߦΘΕΔɽ

؆୯ͳ DFT ɼtype II ॏཧͷ NS-NS sector ΛݩʹఆԽΕΔɽͷཧʹݱ

ΕΔجຊతͳཁૉɼܭ gmicroνɼKalb-Ramond Bmicroνɼdilaton φ ͷ 3 ͰΔɽΕΒͷΛ

T ରനͳཧͰѻʹɼO(DD) มͱͳΔΑʹΈඞཁΔɽΕΛҰൠڞ

Խܭ (generalized metric) HMN ͱɽ

HMN =

gmicroν minusgmicroρBρν

Bmicroρgρν gmicroν minusBmicroρgρσBσν

$ (26)

ɼHMN ҎԼͷΛຬɽ

HMN = ηMKηNLHKL (27)

Ճɼdilaton ҰൠԽΕΔɽDFT dilaton (ҰൠԽ dilaton) d ҎԼͷΑʹͳΔɽ

eminus2d =radicminusgeminus2φ (28)

ɼg ܭͷߦͰΔɽͷΈʹΑΓɼHMN d DFT ʹ O(DD)

มΛडΔɽɼഒԽͷ HMN (xM ) d(xM ) ͱͳΔɽ

11

22 DFTͷ༻ͱ Cހ

લઅͰఏΛ༻ɼO(DD) ෆมͳ DFT ͷ༻ɼҰൠԽ Ricci εΧϥʔͱ

ʹΑҎԼͷΑʹ༩ΒΕΔ [18]

SDFT =

d2Dx eminus2dR(H d) (29)

R =1

8HMNpartMHKLpartNHKL minus

1

2HMNpartMHKLpartKHNL

+ 4HMNpartMpartNdminus partMpartNHMN minus 4HMNpartMd partNd+ 4partMHMNpartNd (210)

ͷ༻ O(DD) εΧϥʔҰൠԽήʔδεΧϥʔͰΔͱɼ[18] ʹΕɽ

༺ (29) ɼେҬతͳ O(DD) ରশ

HMN rarr HLKM ML M N

K xM rarr xNM MN (211)

ʹରനʹෆมͰΔɽͰɼΕΛͱʹہॴରশͲͷΑʹنఆΕΔΖʁ

ҰൠԽϕΫτϧ V ʹରΔҰൠԽඍಉ૬ͷແݶখ൛ɼͳΘഒԽඪͷແݶখมҎ

ԼͷΑʹఆΕΔ [8]ɽ

V M rarr V M + δΞVM xM rarr xM minus ΞM (212)

ɼδΞ ҎԼͷΑͳҰൠԽ Lie ඍͰ༩ΒΕΔɽॏΈ w(V ) ΛҰൠԽϕΫτϧ V M

ͷҰൠԽ Lie ඍҎԼͷΑʹఆΕΔɽ

ampLΞVM = ΞKpartKV M + (partMΞK minus partKΞM )V K + w(V )V MpartKΞK (213)

Ұ௨ৗͷ Lie ඍͱಉҠಈΛදɼೋͷހͷதͷ O(DD) มΛදɽHͷॏΈ w(H) = 0 ͰΓɼҰൠԽ dilaton ͷॏΈ w(eminus2d) = 1 ͰΔɽɼҰൠԽ Lie ඍ

O(DD) ΔɽΛอɽͳΘɼҎԼͷߏ

ampLΞηMN = ampLΞηMN = 0 (214)

ҰൠԽ Lie ඍɼDFT ʹΔήʔδରশͷߏΛఆΔɽͷߏ C

[ ampLΞ1 ampLΞ2 ] = ampL[Ξ1Ξ2]C + F (Ξ1Ξ2 middot) (215)

ΛͳɼC ހ

[Ξ1Ξ2]MC = 2ΞK

[1partKΞM2] minus ηKLΞ

K[1part

MΞL2]

= ΞK1 partKΞM

2 minus ΞK2 partKΞM

1 minus1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 ) (216)

ʹΑࢧΕΔɽɼΞ[1Ξ2] = (Ξ1Ξ2 minus Ξ2Ξ1)2 ͰΔɽɼF ҎԼʹΑͳ

༨ͳͰΔɽ

F (Ξ1Ξ2 V )M =1

2ηKL(Ξ

K1 partPΞL

2 minus ΞK2 partPΞL

1 )partPVM minus (partPΞM

1 partPΞK2 minus partPΞM

2 partPΞK1 )VK

(217)

F ଘࡏΔͱΒɼҰൠԽ Lie ඍͷࢠ C ͰดͳɽɼCހ ހ Jacobi

ΛຬͳɽF ΛফڈɼC ଋ߆ڧઅͰઆΔఆΔΛดΔʹɼنހ

(strong constraint) ΛͰΔɽ

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

9

ୈ 2ষ

Double Field Theory

ͷষͰɼDouble Field Theory ͷجతͳ෦ʹ؆୯ʹड़Δɽಛʹɼ3 ষҎ

ͰඞཁͱͳΔɼDFT ʹΔήʔδʹݟʹࡉɽɼDFT ΕΔഒԽݱʹ

ɼDFTʹԿͷཁૉΛ༩Δɽز ʹΔ༻ͱήʔδมΛ༩ɼΕΛఆΔހͱ

ɼC ଋΛಋೖΔɽ߆ڧͷดΔΑʹɼΛಋೖΔɽήʔδมހ

21 DFTʹݱΕΔزԿ

DFT ɼtype II ॏཧͷ NS-NS sector Λനʹ T ରෆมఆԽཧͰΔ [8]ɽ

ͷཧɼmassless ͱܭ gmicroνɼKalb-Ramond (B ) Bmicroνɼdilaton φ ΛΉɽT

ରͷԼͰɼܭͷҰൠඪมͱɼB ͷ U(1) ήʔδมΔɽΕΛزԿత

ʹදݱΔͱɼ௨ৗͷ (ͷӡಈϞʔυݭ) ͱ༨ (winding Ϟʔυ) ΛΈΘ

ɼҰൠԽͷಋೖΛଅɽ

ҰൠԽଋ (generalized tangent bundle) E ɼଟମ M ͷଋ TM ͱ ༨ଋ T lowastM ͷ

ʹΑߏΕΔɽ

E = TM oplus T lowastM (21)

ҰൠԽଋҰൠԽزԿͰऔΓѻΘΕΔɽҰൠԽزԿɼ2004 ʹ Hitchin [11] ͱ Gualtieri

[16] ʹΑఏҊΕɽϕΫτϧΛଋ TM ͷஅ Γ(TM) ͱΔͱɼඍ 1 ༨ଋܗ

T lowastM ͷஅ Γ(T lowastM) ͰΔɽɼΓ(E) ϕΫτϧͱඍ 1 తʹͱܗΛܗ

ͳΓɼ ΕΛҰൠԽϕΫτϧͱɽҰൠԽزԿͰɼඍಉ૬ (diffeomorphism) ͱήʔδ

มΛΈΘɼҰൠԽʹରΔҰൠԽඍಉ૬ΛಋೖΔɽͷՌɼCourant

ѥ [17] ͱɼରশΛهड़ΔͰͷڧͳಓΛఏڙΔɽจݙ [17] ͷ Courant ѥ

ɼҎԼͷހͰ༩ΒΕΔߏͷҰछͰΔɽXi isin Γ(TM) ξi isin Γ(T lowastM) ͱͱ

ɼҎԼͷ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (22)

ʹΑ༩ΒΕΔɽCourant ѥʹ 3 ষͰΑΓࡉʹड़Δɽ

Βʹ ͳมڞͱרͷݭ x Λಋೖɼ௨ৗͷඪ x ͱಉʹѻͱͰɼඪ

త (ଟମ) ͷݩ 2 ഒʹͳΓɼഒԽزԿ (doubled geometry) ʹ౸ୡΔɽഒԽز

10

Կͷతͳମతߏʹߦࢼஈ֊ʹΔɽۂͳͲͷزԿతͳΛఆɼ

తʹదͳഒԽزԿΛߏஙΔࢼΈ [18 19] ͳͲͰߦΘΕΔɽ

ҎԼɼഒԽͷඪΛ༩ɼDFT ͰѻܭͷఆΛ༩ΔɽҎɼ௨ৗͷॏཧ

ఆΕΔͷݩΛ D ͱɼDFTݩ ͷഒԽ 2D ͰΔͱΔɽҰൠʹɼݩ

ཧఆΕΔݭ 10 ͰΔͱΒɼഒԽݩ 20 ΒΕΔɽഒߟͰΔͱݩ

Խͷඪ (ഒԽඪ) xM ҎԼͷΑʹॻΔͷͱΔɽxmicro ӡಈͱ Fourier ͳڞ

ඪɼxmicro רͱ Fourier ඪͰΔɽͳڞ

xM =

xmicro

xmicro

$ (M = 1 2D micro = 1 D) (23)

T ରΛͱΔͱɼxmicro ͱ xmicro ͷೖΕସʹରԠΔɽɼഒԽͰͷඍԋࢠҎԼͷ

Αʹදɽ

partM =part

partxM=

partmicro

partmicro

$ partmicro =

part

partxmicro partmicro =

part

partxmicro (24)

ҰൠʹɼD ίϯύΫτʹΔݩ T ରม (T-duality ( O(DD) Ͱ༩ΒΕΔɽ

ഒԽ O(DD) ෆมͳߏ η Λɽड़ͷ xM ͷఆΒɼη ͱͷٯҎԼͷΑʹॻ

Δɽ

ηMN =

0 δmicroν

δmicroν 0

$ ηMN =

0 δmicroν

δmicroν 0

$ (25)

ഒԽඪͷҰൠԽςϯιϧͷͷԼɼ η ʹΑߦΘΕΔɽ

؆୯ͳ DFT ɼtype II ॏཧͷ NS-NS sector ΛݩʹఆԽΕΔɽͷཧʹݱ

ΕΔجຊతͳཁૉɼܭ gmicroνɼKalb-Ramond Bmicroνɼdilaton φ ͷ 3 ͰΔɽΕΒͷΛ

T ରനͳཧͰѻʹɼO(DD) มͱͳΔΑʹΈඞཁΔɽΕΛҰൠڞ

Խܭ (generalized metric) HMN ͱɽ

HMN =

gmicroν minusgmicroρBρν

Bmicroρgρν gmicroν minusBmicroρgρσBσν

$ (26)

ɼHMN ҎԼͷΛຬɽ

HMN = ηMKηNLHKL (27)

Ճɼdilaton ҰൠԽΕΔɽDFT dilaton (ҰൠԽ dilaton) d ҎԼͷΑʹͳΔɽ

eminus2d =radicminusgeminus2φ (28)

ɼg ܭͷߦͰΔɽͷΈʹΑΓɼHMN d DFT ʹ O(DD)

มΛडΔɽɼഒԽͷ HMN (xM ) d(xM ) ͱͳΔɽ

11

22 DFTͷ༻ͱ Cހ

લઅͰఏΛ༻ɼO(DD) ෆมͳ DFT ͷ༻ɼҰൠԽ Ricci εΧϥʔͱ

ʹΑҎԼͷΑʹ༩ΒΕΔ [18]

SDFT =

d2Dx eminus2dR(H d) (29)

R =1

8HMNpartMHKLpartNHKL minus

1

2HMNpartMHKLpartKHNL

+ 4HMNpartMpartNdminus partMpartNHMN minus 4HMNpartMd partNd+ 4partMHMNpartNd (210)

ͷ༻ O(DD) εΧϥʔҰൠԽήʔδεΧϥʔͰΔͱɼ[18] ʹΕɽ

༺ (29) ɼେҬతͳ O(DD) ରশ

HMN rarr HLKM ML M N

K xM rarr xNM MN (211)

ʹରനʹෆมͰΔɽͰɼΕΛͱʹہॴରশͲͷΑʹنఆΕΔΖʁ

ҰൠԽϕΫτϧ V ʹରΔҰൠԽඍಉ૬ͷແݶখ൛ɼͳΘഒԽඪͷແݶখมҎ

ԼͷΑʹఆΕΔ [8]ɽ

V M rarr V M + δΞVM xM rarr xM minus ΞM (212)

ɼδΞ ҎԼͷΑͳҰൠԽ Lie ඍͰ༩ΒΕΔɽॏΈ w(V ) ΛҰൠԽϕΫτϧ V M

ͷҰൠԽ Lie ඍҎԼͷΑʹఆΕΔɽ

ampLΞVM = ΞKpartKV M + (partMΞK minus partKΞM )V K + w(V )V MpartKΞK (213)

Ұ௨ৗͷ Lie ඍͱಉҠಈΛදɼೋͷހͷதͷ O(DD) มΛදɽHͷॏΈ w(H) = 0 ͰΓɼҰൠԽ dilaton ͷॏΈ w(eminus2d) = 1 ͰΔɽɼҰൠԽ Lie ඍ

O(DD) ΔɽΛอɽͳΘɼҎԼͷߏ

ampLΞηMN = ampLΞηMN = 0 (214)

ҰൠԽ Lie ඍɼDFT ʹΔήʔδରশͷߏΛఆΔɽͷߏ C

[ ampLΞ1 ampLΞ2 ] = ampL[Ξ1Ξ2]C + F (Ξ1Ξ2 middot) (215)

ΛͳɼC ހ

[Ξ1Ξ2]MC = 2ΞK

[1partKΞM2] minus ηKLΞ

K[1part

MΞL2]

= ΞK1 partKΞM

2 minus ΞK2 partKΞM

1 minus1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 ) (216)

ʹΑࢧΕΔɽɼΞ[1Ξ2] = (Ξ1Ξ2 minus Ξ2Ξ1)2 ͰΔɽɼF ҎԼʹΑͳ

༨ͳͰΔɽ

F (Ξ1Ξ2 V )M =1

2ηKL(Ξ

K1 partPΞL

2 minus ΞK2 partPΞL

1 )partPVM minus (partPΞM

1 partPΞK2 minus partPΞM

2 partPΞK1 )VK

(217)

F ଘࡏΔͱΒɼҰൠԽ Lie ඍͷࢠ C ͰดͳɽɼCހ ހ Jacobi

ΛຬͳɽF ΛফڈɼC ଋ߆ڧઅͰઆΔఆΔΛดΔʹɼنހ

(strong constraint) ΛͰΔɽ

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

10

Կͷతͳମతߏʹߦࢼஈ֊ʹΔɽۂͳͲͷزԿతͳΛఆɼ

తʹదͳഒԽزԿΛߏஙΔࢼΈ [18 19] ͳͲͰߦΘΕΔɽ

ҎԼɼഒԽͷඪΛ༩ɼDFT ͰѻܭͷఆΛ༩ΔɽҎɼ௨ৗͷॏཧ

ఆΕΔͷݩΛ D ͱɼDFTݩ ͷഒԽ 2D ͰΔͱΔɽҰൠʹɼݩ

ཧఆΕΔݭ 10 ͰΔͱΒɼഒԽݩ 20 ΒΕΔɽഒߟͰΔͱݩ

Խͷඪ (ഒԽඪ) xM ҎԼͷΑʹॻΔͷͱΔɽxmicro ӡಈͱ Fourier ͳڞ

ඪɼxmicro רͱ Fourier ඪͰΔɽͳڞ

xM =

xmicro

xmicro

$ (M = 1 2D micro = 1 D) (23)

T ରΛͱΔͱɼxmicro ͱ xmicro ͷೖΕସʹରԠΔɽɼഒԽͰͷඍԋࢠҎԼͷ

Αʹදɽ

partM =part

partxM=

partmicro

partmicro

$ partmicro =

part

partxmicro partmicro =

part

partxmicro (24)

ҰൠʹɼD ίϯύΫτʹΔݩ T ରม (T-duality ( O(DD) Ͱ༩ΒΕΔɽ

ഒԽ O(DD) ෆมͳߏ η Λɽड़ͷ xM ͷఆΒɼη ͱͷٯҎԼͷΑʹॻ

Δɽ

ηMN =

0 δmicroν

δmicroν 0

$ ηMN =

0 δmicroν

δmicroν 0

$ (25)

ഒԽඪͷҰൠԽςϯιϧͷͷԼɼ η ʹΑߦΘΕΔɽ

؆୯ͳ DFT ɼtype II ॏཧͷ NS-NS sector ΛݩʹఆԽΕΔɽͷཧʹݱ

ΕΔجຊతͳཁૉɼܭ gmicroνɼKalb-Ramond Bmicroνɼdilaton φ ͷ 3 ͰΔɽΕΒͷΛ

T ରനͳཧͰѻʹɼO(DD) มͱͳΔΑʹΈඞཁΔɽΕΛҰൠڞ

Խܭ (generalized metric) HMN ͱɽ

HMN =

gmicroν minusgmicroρBρν

Bmicroρgρν gmicroν minusBmicroρgρσBσν

$ (26)

ɼHMN ҎԼͷΛຬɽ

HMN = ηMKηNLHKL (27)

Ճɼdilaton ҰൠԽΕΔɽDFT dilaton (ҰൠԽ dilaton) d ҎԼͷΑʹͳΔɽ

eminus2d =radicminusgeminus2φ (28)

ɼg ܭͷߦͰΔɽͷΈʹΑΓɼHMN d DFT ʹ O(DD)

มΛडΔɽɼഒԽͷ HMN (xM ) d(xM ) ͱͳΔɽ

11

22 DFTͷ༻ͱ Cހ

લઅͰఏΛ༻ɼO(DD) ෆมͳ DFT ͷ༻ɼҰൠԽ Ricci εΧϥʔͱ

ʹΑҎԼͷΑʹ༩ΒΕΔ [18]

SDFT =

d2Dx eminus2dR(H d) (29)

R =1

8HMNpartMHKLpartNHKL minus

1

2HMNpartMHKLpartKHNL

+ 4HMNpartMpartNdminus partMpartNHMN minus 4HMNpartMd partNd+ 4partMHMNpartNd (210)

ͷ༻ O(DD) εΧϥʔҰൠԽήʔδεΧϥʔͰΔͱɼ[18] ʹΕɽ

༺ (29) ɼେҬతͳ O(DD) ରশ

HMN rarr HLKM ML M N

K xM rarr xNM MN (211)

ʹରനʹෆมͰΔɽͰɼΕΛͱʹہॴରশͲͷΑʹنఆΕΔΖʁ

ҰൠԽϕΫτϧ V ʹରΔҰൠԽඍಉ૬ͷແݶখ൛ɼͳΘഒԽඪͷແݶখมҎ

ԼͷΑʹఆΕΔ [8]ɽ

V M rarr V M + δΞVM xM rarr xM minus ΞM (212)

ɼδΞ ҎԼͷΑͳҰൠԽ Lie ඍͰ༩ΒΕΔɽॏΈ w(V ) ΛҰൠԽϕΫτϧ V M

ͷҰൠԽ Lie ඍҎԼͷΑʹఆΕΔɽ

ampLΞVM = ΞKpartKV M + (partMΞK minus partKΞM )V K + w(V )V MpartKΞK (213)

Ұ௨ৗͷ Lie ඍͱಉҠಈΛදɼೋͷހͷதͷ O(DD) มΛදɽHͷॏΈ w(H) = 0 ͰΓɼҰൠԽ dilaton ͷॏΈ w(eminus2d) = 1 ͰΔɽɼҰൠԽ Lie ඍ

O(DD) ΔɽΛอɽͳΘɼҎԼͷߏ

ampLΞηMN = ampLΞηMN = 0 (214)

ҰൠԽ Lie ඍɼDFT ʹΔήʔδରশͷߏΛఆΔɽͷߏ C

[ ampLΞ1 ampLΞ2 ] = ampL[Ξ1Ξ2]C + F (Ξ1Ξ2 middot) (215)

ΛͳɼC ހ

[Ξ1Ξ2]MC = 2ΞK

[1partKΞM2] minus ηKLΞ

K[1part

MΞL2]

= ΞK1 partKΞM

2 minus ΞK2 partKΞM

1 minus1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 ) (216)

ʹΑࢧΕΔɽɼΞ[1Ξ2] = (Ξ1Ξ2 minus Ξ2Ξ1)2 ͰΔɽɼF ҎԼʹΑͳ

༨ͳͰΔɽ

F (Ξ1Ξ2 V )M =1

2ηKL(Ξ

K1 partPΞL

2 minus ΞK2 partPΞL

1 )partPVM minus (partPΞM

1 partPΞK2 minus partPΞM

2 partPΞK1 )VK

(217)

F ଘࡏΔͱΒɼҰൠԽ Lie ඍͷࢠ C ͰดͳɽɼCހ ހ Jacobi

ΛຬͳɽF ΛফڈɼC ଋ߆ڧઅͰઆΔఆΔΛดΔʹɼنހ

(strong constraint) ΛͰΔɽ

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

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calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

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prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

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Physique theorique 53 (1990) 35

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Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

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no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

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Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

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[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

11

22 DFTͷ༻ͱ Cހ

લઅͰఏΛ༻ɼO(DD) ෆมͳ DFT ͷ༻ɼҰൠԽ Ricci εΧϥʔͱ

ʹΑҎԼͷΑʹ༩ΒΕΔ [18]

SDFT =

d2Dx eminus2dR(H d) (29)

R =1

8HMNpartMHKLpartNHKL minus

1

2HMNpartMHKLpartKHNL

+ 4HMNpartMpartNdminus partMpartNHMN minus 4HMNpartMd partNd+ 4partMHMNpartNd (210)

ͷ༻ O(DD) εΧϥʔҰൠԽήʔδεΧϥʔͰΔͱɼ[18] ʹΕɽ

༺ (29) ɼେҬతͳ O(DD) ରশ

HMN rarr HLKM ML M N

K xM rarr xNM MN (211)

ʹରനʹෆมͰΔɽͰɼΕΛͱʹہॴରশͲͷΑʹنఆΕΔΖʁ

ҰൠԽϕΫτϧ V ʹରΔҰൠԽඍಉ૬ͷແݶখ൛ɼͳΘഒԽඪͷແݶখมҎ

ԼͷΑʹఆΕΔ [8]ɽ

V M rarr V M + δΞVM xM rarr xM minus ΞM (212)

ɼδΞ ҎԼͷΑͳҰൠԽ Lie ඍͰ༩ΒΕΔɽॏΈ w(V ) ΛҰൠԽϕΫτϧ V M

ͷҰൠԽ Lie ඍҎԼͷΑʹఆΕΔɽ

ampLΞVM = ΞKpartKV M + (partMΞK minus partKΞM )V K + w(V )V MpartKΞK (213)

Ұ௨ৗͷ Lie ඍͱಉҠಈΛදɼೋͷހͷதͷ O(DD) มΛදɽHͷॏΈ w(H) = 0 ͰΓɼҰൠԽ dilaton ͷॏΈ w(eminus2d) = 1 ͰΔɽɼҰൠԽ Lie ඍ

O(DD) ΔɽΛอɽͳΘɼҎԼͷߏ

ampLΞηMN = ampLΞηMN = 0 (214)

ҰൠԽ Lie ඍɼDFT ʹΔήʔδରশͷߏΛఆΔɽͷߏ C

[ ampLΞ1 ampLΞ2 ] = ampL[Ξ1Ξ2]C + F (Ξ1Ξ2 middot) (215)

ΛͳɼC ހ

[Ξ1Ξ2]MC = 2ΞK

[1partKΞM2] minus ηKLΞ

K[1part

MΞL2]

= ΞK1 partKΞM

2 minus ΞK2 partKΞM

1 minus1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 ) (216)

ʹΑࢧΕΔɽɼΞ[1Ξ2] = (Ξ1Ξ2 minus Ξ2Ξ1)2 ͰΔɽɼF ҎԼʹΑͳ

༨ͳͰΔɽ

F (Ξ1Ξ2 V )M =1

2ηKL(Ξ

K1 partPΞL

2 minus ΞK2 partPΞL

1 )partPVM minus (partPΞM

1 partPΞK2 minus partPΞM

2 partPΞK1 )VK

(217)

F ଘࡏΔͱΒɼҰൠԽ Lie ඍͷࢠ C ͰดͳɽɼCހ ހ Jacobi

ΛຬͳɽF ΛফڈɼC ଋ߆ڧઅͰઆΔఆΔΛดΔʹɼنހ

(strong constraint) ΛͰΔɽ

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

12

23 ཧత߆ଋ

ཧͷݭ level-matching ʹདྷɼDFT ଋ߆ΕΔɼҎԼͷऑݱʹ (weak

constraint) ΛຬΔඞཁΔɽͳΘɼҙͷͱήʔδύϥϝʔλΛ Φ ʹରɼ

partmicropartmicroΦ = 0 (218)

ͱΕΔɽҰൠʹɼͷͷΛຬͳɼӨԋࢠΛಋೖΔͱ

ͰΕΛਖ਼Δɽࡍʹɼऑ߆ଋʹج DFT ɼओʹ [22] ͰΕΔɽ

ɼӨԋࢠΛ༻ܭΓʹΔͱͰɽ

ͷظॳ DFT ɼݭͷͷཧΛ༻ߏஙΕ [8]ɽΕɼmassless ͷΏΒͱ 2 ֊

ͰͷඍΛΉɼͷ 3 ͷظॳཧͰΔɽͰͷήʔδෆมΛ DFT ΛɼΑΓߴ

ͱͳɽࠔΑΓʹࡏͷଘࢠཧʹలΔͱɼड़ͷӨԋͷήʔδෆมΛ

ͷঢ়گΛଧΔʹɼDFT ͱήʔδύϥϝʔλͱʹՃɼΕΒͷҙͷʹରΔ

Λ

partmicropartmicro(ΦΨ) = 0 (219)

Ͱফ໓ΔඞཁΔɽ(ΦΨ) ɼDFT ͱήʔδύϥϝʔλͷҙͷΛදɽ(219) Λڧ

ଋ߆ (strong constraint) ͱɽ߆ڧଋɼশͷ௨ΓʹରڧΛ༩

ΔͱΒɼҎԼͷͱͰΔ [23]ɽ߆ڧଋΛຬͷ༩ΒΕͳΒɼ

x ඪඪͷΈʹґଘΔΑͳഒԽ (x x) ͷऔΓඞଘࡏΔɽݴΔͱɼ߆ڧ

ଋΛͱཧਅʹഒԽΕͳɽ

ଋΛΘɼཧత߆ଋͱऑ߆ڧ section ͱݺͿɽΕΒͷΛղ

؆୯ͳɼඪͷʹґଘΔΑʹΔͱͰΔɽಛʹɼඪ xmicro ʹґଘ

ɼרඪ xmicro ʹґଘͳ (partmicroΦ = 0) ɼDFT ҰൠԽزԿʹԊॏཧͷ

ఆԽʹࢸΔɽxmicro ͱ xmicro ͷࡏΛߟΔͱՄͰΔɼηMN ͰΛܗΔඪͱ

Δͷɼxmicroݱ xmicro ͷΕҰʹґଘΔͰΔɽ

Ͱɼཧత section ʹతͳଆ໘ΒগߟɽɼഒԽز

ԿͷతʹશͳݱԽݱͳͱʹҙΔɽ߆ڧଋΛͱͰɼཧ

ͷతഎܠഒԽزԿΒ ҰൠԽزԿҠߦΔɽͷҙຯͰɼഒԽزԿݱΔ

ͳΒɼΕطଘͷҰൠԽزԿΛแΔͰΔɽɼ2D ͷඪతݩ )

ଟମ) η ʹରେ (maximally isotropic) ͳೋͷ෦ʹΕΔɽ

ɼͲΒΛཧతͱΈͳ (xmicro ଆͱΈͳ) ͰΔɽO(DD) ܭͷߏ η ʹ

ΑΒΕΔ෦ͱͷͷಛఆɼ[24] ͰٴݴΔɽഒԽزԿͷతͳඳ

ͱΕΔͷɼύϥΤϧϛʔτزԿ (para-Hermitian geometry) [14 15 25]

ɼͷுͰΔϘϧϯزԿ (Born geometry) [26 27] ͰΔɽΕΒͷزԿɼύϥෳ

ૉ (para-complex) ʹʹΑવߏ 2D ͷఈݩ (ଟମ) ͷͷΛ༩Δ

ͱɼഒԽزԿͷهड़ʹΓඇৗʹΑΛɽࡉ 4 ষͰΔɽ

ɼήʔδʹٴݴɽऑ߆ଋɼO(DD) ͰҎԼͷܗ

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

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[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

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[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

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[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

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[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

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[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

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Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

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[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

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[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

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meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

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[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

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tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

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[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

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[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

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[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

13

ΑʹදΕΔɽΦ Λҙͷήʔδύϥϝʔλͱ

ηMNpartMpartNΦ = 0 (220)

ಉʹɼ߆ڧଋҎԼͷΑʹදΕΔɽΦi Λҙͷήʔδύϥϝʔλͱ

ηMNpartMΦ1partNΦ2 = 0 (221)

લઅͰड़௨Γɼ߆ڧଋΛͱɼ(217) ͷ F ΛফڈɼDFT ͷήʔδด

ΔΊͷͰΔɽ

ҙͷހ [middot middot] ͷ Jacobiator ɼҎԼͷΑʹఆΕΔɽ

Jac(Ξ1Ξ2Ξ3) = [[Ξ1Ξ2]Ξ3] + [[Ξ2Ξ3]Ξ1] + [[Ξ3Ξ1]Ξ2] (222)

ͷఆʹͷͱΓܭΔͱɼҰൠʹ C ͷހ Jacobiator JC ҎԼͷΑʹͳΔɽಋग़

Appendix A1 Λরͷͱɽ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (223)

ɼpartbull ɼηMN ͰඍԋࢠͰΔɽɼNC Ξi શରশͳʹ

ҎԼͷ

NC(Ξ1Ξ2Ξ3) =1

3

ηMN [Ξ1Ξ2]

MC ΞN

3 + cp(

(224)

ͰΓɼSCC ߆ڧଋΛফΔͰΔɽ߆ڧଋΛޙɼ(223) ͷӈล

Ұൠʹ 0 ͰͳɽC ހ Jacobi Λຬɼগͳͱ DFT ͷήʔδ Lie

ͰͳͱʹΘΔɽɼ௨ৗͷ Lie ඍͷࢠͷுͱɼҎԼͷΑʹ D

ހ [18] ΛಋೖɼC ಘΒΕΔɽߏͱಉހ

[Ξ1Ξ2]MD = ampLΞ1Ξ

M2 (225)

D ͱހ C ΔɽʹҎԼͷΑހ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (226)

D ରশͰͳɼJacobiހ ΛຬҰɼC ରশͰހ Jacobi Λຬͳɽ

D ͱހ C จͰɼCΔɽຊमʹతʹදཪͷހ Δɽ༺ΑΔఆΛʹހ

C ఆΔήʔδɼنͷހ ॳظͷจݙͰ [2930] ͰѻΘΕΔ΄ɼ[19ndash21] ͳ

ͲͰΕΔɽC ରʹހ partmicroΦ = 0 Λɼ߆ڧଋΛʹղͱɼC ހ

(22) ͷ Courant ހ [17] ʹมԽΔɽಉʹɼD ހ Dorfman bracket operator [333453]

ʹมԽΔɽC Βހ Courant ΛಘΔաఔɼAppendixހ A2 ʹɽͷΑʹಘΒ

ΕΔ Courant ހ Dorfman ɼހ Ε Courant ѥͱݺΕΔߏΛఆΔ [34]ɽ

ΑɼC ԿزҰൠԽހ [11 31] ΕΔݱʹ Courant ހ [17] Λ O(DD) มԽͷڞ

ͱΔ [32]ɽɼ(216) Ͱ C ɼ[35]ܗͷހ ͰఆΕ Courant తܗͱހ

ʹಉܗͰΔɽɼ[35] ͷ Courant ͷހ Jacobiator ͷܭՌ (223) ͱҟͳΔ

ΊɼC ހ Courant ѥΛఆͳɽɼഒԽΛύϥΤϧϛʔτزԿʹ

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

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[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

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mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

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[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

14

ΑతʹݱΔࢼΈΓɼͷաఔͰ C ߏఆΔنހ metric ѥͰ

ΔͱࢦఠͳΕ [14 15]ɽɼmetric ѥͱຊతʹಉߏɼ[36] ʹ

pre-DFT ѥͱ༩ΒΕΔɽཚΛආΔΊɼຊमจͰ metric ѥͷ

ͱΛɼݟͷΛͱ Vaisman ѥͱݺশͰҰΔɽVaisman ѥ Courant

ѥΛΒʹҰൠԽ೦ͰΔɽষҎͰɼVaisman ѥแΔߏʹ

ड़ɼ߆ڧଋͷతͳग़ʹɽ

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

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[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

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mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

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Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

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no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

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Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

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[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

15

ୈ 3ষ

ѥ (Algebroid)ͷಋೖ

ͷষͰɼछʑͷߏͱɼ(classical) Drinfelrsquod double ͱʹհΔɽ

Drinfelrsquod double ݩʑ Hopf ʹରߟҊΕɼରͳ Hopf ͷʹΑɼͳ

Hopf ಘΒΕΔͱΛࢦɽHopf ҰݴͰݴ Lie Λύϥϝʔλมܗͷ

ͰɼύϥϝʔλʔʹରΔݶۃΛͱΔͱ Lie ಘΒΕΔɽHopf Β Lie ΛಘΔ

Λ యݹ (classical) ΛͱΔͱɽHopfݶۃ ͷ Drinfelrsquod double ͷݹయݶۃͱɼLie

ͷ (classical) Drinfelrsquod double ΛߟΒΕΔɽΕʹɼจݙ [37 38] ʹɽ

ͷɼDFT ͷ߆ڧଋͷతͳݯىΛΔΊͷجͱͳΔɽLie ͷ Drinfelrsquod

double ͷுͱɼจݙ [35] ʹ Lie ѥ (algebroid) ͷ Drinfelrsquod double ҊΕߟ

ɽզʑɼͷΒʹҰൠԽͱ C نఆΔ Vaisman ѥɼLie ѥʹର

Drinfelrsquod double ͱͰಘΒΕΔͱΛߦΛʹ [13]ɽͷΛɼจதͰ

୯ʹ ldquodoublerdquo ͱݺͿɽ

31 Lieʢʣͱ Drinfelrsquod double

ॳΊʹɼLie ΛҎԼͷΑʹఆΔɽ

Lie g V Λ ମ K ͷϕΫτϧͱΔɽV ͷݩΛҾͱΔରশઢܕͳೋԋͱ

ɼLie ހ (bracket) [middot middot] V times V rarr V ༩ΒΕΔͱɼV ͱ Lie ͷހ (V [middot middot]) Λ

Lie ͱɼg ͱදهΔɽLie ހ Jacobi ΛຬɽͳΘɼx y z isin g Ͱ

Δͱ

[[x y] z] + [[y z] x] + [[z x] y] = 0 (31) V ͷର (dual) ϕΫτϧ V lowast ʹରɼಉʹ Lie ΛఆͰΔɽV lowast ͱ V lowast ͷݩ

ΛҾͱΔ Lie ހ [middot middot]lowast V lowast times V lowast rarr V lowast ͷ (V lowast [middot middot]lowast) Λɼg ʹରΔରͳ Lie glowast

ͱɽg ͱ glowast ͷʹવʹ ⟨middot middot⟩ఆΕΔɽ⟨middot middot⟩ K ͷΛͱΔɽ

ɼϕΫτϧʹ R ʹΑΔ Lie g ͷදݱ ϱ ΛߟΔɽx isin gm isin R ͱΔͱɼx Ұ

ൠʹɼ m ʹର ϱ(x) middotm ͱ༻Δɽಛʹɼg g ༻ΔΛߟΔͱͰɼg ͷ

ਵ (adjoint) දݱ ad x isin g +rarr adx isin End g ΛنఆͰΔɽਵදݱɼLie Λ༻Δͱހ

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

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[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

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Trans Am Math Soc 63 (1948) 85

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Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

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prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

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[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

16

ҎԼͷΑʹॻΔɽ

adx(y) = [x y] x y isin g (32)

x isin g ҙͷςϯιϧ otimespg = gotimes gotimes middot middot middototimes g ɼҎԼͷΑʹ༻Δɽ

ϱ(x) middot (y1 otimes middot middot middototimes yp) = ad(p)x (y1 otimes middot middot middototimes yp)

= adx(y1)otimes y2 otimes middot middot middototimes yp + y1 otimes adx(y2)otimes middot middot middototimes yp + middot middot middotmiddot middot middot+ y1 otimes middot middot middototimes adx(yp) (33)

ɼad Leibniz ଇΛຬɽLie ͷހ Jacobi (32) ɼad ͷ Leibniz ଇͱ

ղऍͰΔɽ

adz([x y]) = [adz(x) y] + [x adz(y)] (34)

ಉʹɼp ͷ andpg ͷ x isin g ͷ༻ΛߟΔɽશରশͳςϯιϧ otimespg ʹରΔ

ΕΑɼҎԼͷΑʹఆͰΔɽߟΛ༺

ϱ(x) middot y1 and y2 = [x y1] and y2 + y1 and [x y2] (35)

ͷͱɼଟମͷ༨ଋͷඍԋࢠ d ͱಉΑʹߟΕɼԋࢠ d andpglowast rarr andp+1glowast

ΛఆͰΔɽd2 = 0 ͰΔɽಉͷखॱͰɼglowast ͷදݱͱͷ༻ΛߟͱͰɼd ͱ

ରͳඍ༻ૉ dlowast andpg rarr andp+1g ఆͰΔɽd2lowast = 0 ͰΔɽΕΒΛ༻ΔͱͰɼg

ʹରɼLie ίϗϞϩδʔΛఆͰΔ [39]ɽ

Lie ߴɼҾހ andpg ͷͱுͰΔɽΕΛ Schouten-Nijenhuis ހ [middot middot]S ͱ

[40]ɽ[middot middot]S ҎԼͷΑʹఆΕΔɽɼa isin andpg b isin andqg c isin andrg ͰΔɽ

(i) [a b]S = minus(minus)(pminus1)(qminus1)[b a]S

(ii) [a b and c]S = [a b]S and c+ (minus)(pminus1)qb and [a c]S

(iii) (minus)(pminus1)(rminus1)[a [b c]S]S + (minus)(qminus1)(rminus1)[b [c a]S]S + (minus)(rminus1)(qminus1)[c [a b]S]S = 0

(iv) [middot middot]S ͷҾͷɼগͳͱͲΒҰ and0g = K ͰΔͱɼͷՌ 0 ʹͳΔ

Schouten-Nijenhuis ɼGerstenhaberހ Λ [41]ɽ

ɼgʹ ͱ glowast ͷΛௐΔɽҰൠʹ Lie ހ [middot middot] ɼg ͷͷݩΒผͷ g ͷݩΛಘ

ΔͷͰɼઢܗ (bilinear map) ͱΈͳΔɽLie middot]ɼߟରশͳͱΛހ middot] Λ

micro and2grarr g ͱΔͱɼ Εʹରͳ Lie ހ [middot middot]lowast microlowast and2glowast rarr glowast ͱॻΔɽmicro ͷ co-bracket

Λ δ grarr and2g Δɽx isin g ξ isin and2glowast ͱΔͱɼmicrolowast ͷਵ microlowastlowast grarr and2g Λ௨

⟨x microlowast(ξ)⟩ = ⟨microlowastlowast(x) ξ⟩ (36)

ͱ༩ΒΕΔͱΒɼδ microlowastlowast ͰఆͰΔɽରͳ Lie ހ microlowast Jacobi Λຬͱ

Βɼδ Jacobi ΛຬɽݴΔͱɼରͳ Lie ހ microlowast ɼδlowast ʹΑఆͰΔɽ

ɼLie g = (V [middot middot]) ͱɼ[middot middot] ͷ co-bracket δ Λఆɼରͳ Lie

glowast = (V lowast [middot middot]lowast) ʹΑવʹ༠ಋͰΔɽ

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

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(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

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[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

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[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

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(2015) 002 [arXiv150405913 [hep-th]]

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[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

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[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

17

Lie Lie (V [middot middot]) Λఆɼ[middot middot] ͷ co-bracket Λ δ ͱΔɽಛʹɼco-bracket δ ͷ

1-cocycle Λຬɼ

δ([x y]) = ad(2)x δ(y)minus ad(2)y δ(x) x y isin g (37)

(g micro δ) ͷͷͱΛ Lie ͱɽ (g micro δ) Lie ͳΒɼ(glowast microlowast δlowast) ಉ Lie Λ༩ΔɽΑɼޙࠓ Lie

ͷͱΛ (g glowast) ͱදهΔɽ

ɼLieʹ ʹର Drinfelrsquod double ΛߦɽLie (g glowast) ΛߏΔ g glowast ʹ

ରɼͷ d = g oplus glowast ΛߟΔͱɼd ʹඇୀԽ (non-degenerate) Ͱରশͳઢܗܕ

(bilinear form) (middot middot) ΛҎԼͷΑʹఆͰΔɽ

(x y) = (ξ η) = 0 (x ξ) = ⟨ξ x⟩ x y isin g ξ η isin glowast (38)

ଓɼड़ͷઢܗܕΛෆมʹอΑʹ d ހʹ [middot middot]d ΛఆΔɽͰɼઢܕ

ɽࢦΔͱΛΛෆมʹอͱҎԼͷܗ

(y [x ξ]d) = ([y x]d ξ) (39)

ɼg glowast ΕΕ d ͷ෦ (subalgebra) ͰΔͱΒɼ

[x y]d = [x y] [ξ η]d = [ξ η]lowast x y isin g ξ η isin glowast (310)

ͱΔͷવͰΔɽʹɼg ͱ glowast ΫϩεΔ [x ξ]d ΛߟΔɽ

(y [x ξ]d) = ([y x]d ξ)

= ([y x] ξ) = ⟨ξ [y x]⟩ = ⟨ξminusadx(y)⟩ = ⟨adlowastxξ y⟩ = (y adlowastxξ) (311)

Ͱɼ1 ஈͷހ [middot middot]d ઢܗܕΛෆมʹอͱཁΒΔɽɼ

2 ஈͷࠨͷ g Lie ෦ͳͷͰɼ(310) ʹΑΓΔɽɼ2 ஈͷӈΒ

2 ൪ͷɼad ͷ co-adjoint Λ adlowastx = minus(adx)lowast ͱఆͱʹΑΓΔɽಉͷ

Βɼ(η [x ξ]d) = minus(η adlowastξx) ͰΔͱಋΔɽΑɼ[x ξ]d ҎԼͷΑʹॻΔɽ

[x ξ]d = minusadlowastξx+ adlowastxξ (312)

(37) (310) (312) Βɼ[middot middot]d Jacobi ΛຬͱΘΔɽΑɼg ͱ glowast ͷʹ

1-cocycle (37) ɼ(g glowast) Lie Λͳͱɼదʹ [middot middot]d ΛఆΊΕ

(d = g oplus glowast [middot middot]d) Lie ͱͳΔɽͷΑʹɼͳ Lie d ΛߏஙΔͷͱΛ

Lie ͷ (classical) Drinfelrsquod double ͱɽඇୀԽͳઢܗܕͱɼΕΛෆมʹอ

Αͳ Lie ΑΔʹހ Lie ͱͷΛɼಛʹ quadratic Lie ͱɽd વͱ quadratic

Lie ͱͳΔɽ

Drinfelrsquod double ͷٯΛߟΔͱͰΔɽp Λ quadratic Lie ͱΔɽa b p ͷ෦

ͰΓɼ p ͷઢܗܕʹରత (isotropic) ͰΔ1ͱɼ(p a b) ͷΛ

Manin triple ͱ [42]ɽड़ͷ (d g glowast) ͷɼવ Manin triple ͰΔɽ

1 g (middot middot) ʹରతͱɼҙͷ x y isin g Ͱ (x y) = 0 ͰΔͱΛɽ

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

18

32 Lieѥ

ͰɼLie ͷҰൠԽͱଟମͷϕΫτϧଋΛ༻ Lie ѥͷఆ [43] Λ༩

ΔɽͷޙɼLie ͱͷେͳҧʹड़Δɽ

Lie ѥ E ҎԼͷߏͷ (E [middot middot]E ρ) ΛɼLie ѥͱɽ

bull M ͷϕΫτϧଋ EπminusrarrM

bull E ͷஅ (section) Γ(E) ΛҾͱΔ Lie ހ [middot middot]E Γ(E)times Γ(E)rarr Γ(E)

bull E Βଋ TM ͷଋ (bundle map) ρ E rarr TM

ρ ͱ [middot middot]E ͷʹɼҎԼͷΔͱΛཁٻΔɽ

ρ([XY ]E) = [ρ(X) ρ(Y )] X Y isin Γ(E) (313)

ρ Λ anchor map ͱݺͿɽ

Lieހ [middot middot]E JacobiΛຬɽɼLeibnizଇΛຬɽͳΘɼ f isin Cinfin(M)

ʹରɼ

[X fY ]E = (ρ(X) middot f)Y + f [XY ]E (314)

(ρ(X) middot f) ɼρ(X) f ͷඍͱ༻ΔͱΛҙຯΔɽ E ͷஅͰಘΒΕΔϕΫτϧ Γ(E) ɼM ͷͱϕΫτϧҰରҰͰରԠͷͱΈ

ͳΔɽҰͰɼલઅͰड़௨ΓɼLie ϕΫτϧʹରఆΕΔɽ

Lie ѥɼ M ͷͱ Γ(E) ͷ Lie ΛҰରҰͰରԠɼ Lie ͷΛ

ѻΔͱΈͳΔɽͷҙຯͰɼLie ѥ Lie ͷҰൠԽͰΔɽM ҰͰΓɼ

anchor ρ = 0 ͰΔͱ Lie ѥ Lie ͱͳΔɽ

Lie g ʹରɼରͳ Lie glowast ఆͰΑʹɼE ʹରͳϕΫτϧଋ Elowast Λ༻

ɼಉଟମʹରͳ Lie ѥ Elowast = (Elowast ρlowast [middot middot]Elowast) ΛఆͰΔɽ[middot middot]Elowast Γ(Elowast)timesΓ(Elowast) rarr Γ(Elowast) [middot middot]E ʹରରͳ Lie ͰΔɽɼanchorހ map ρlowast Elowast rarr TM

ͰΔɽE ͱ Elowast ͷʹɼવʹ ⟨middot middot⟩ఆͰΔɽ

Lie ͷɼLie Λހ Schouten-Nijenhuis ހ [middot middot]S ʹҰൠԽͰɽLie ͷހ

co-bracket [middot middot]S ΛΜͰΔ 1-cocycle (37) ΛຬͷͰΕ Lie

ߏͰɼରͳ Lie Λ༠ಋͰɽͷਪͱɼLie ѥΒ Lie ѥ

(bialgebroid) ಘΒΕΔͱΛΔɽͷΊʹɼLie ހ [middot middot]E ͷҰൠԽΛߦɽɼ

ඞཁͱͳΔԋΛઌʹఆɽ

Γ(TM) Γ(T lowastM) ΒΛߏஙΔͷͱಉΑʹɼΓ(E) Γ(Elowast) Β p-vector

Γ(andpE) p-form Γ(andpElowast) ΛఆΔɽɼΓ(andpE) Γ(andpElowast) ʹରɼ௨ৗͷඍ p ܗ

ͱಉʹඍԋࢠ dɼdlowast ΛఆͰΔɽͱɼd ɼξ isin Γ(andpElowast) ʹରͷΑ

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

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[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

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A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

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prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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Physique theorique 53 (1990) 35

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[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

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and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

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[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

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[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

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[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

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meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

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[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

19

Δ༺ʹ [44]ɽXi isin Γ(E) ͱɼ

dξ(X1 Xp+1) =p+1)

i=1

(minus)i+1ρ(Xi) middotξ(X1 Xi Xp+1)

(

+)

iltj

(minus)i+jξ([Xi Xj ]E X1 Xi Xj Xp+1) (315)

ɼXi i ൪ͷ X ΛফڈΔͱҙຯͰΔɽͷఆΒɼd ҎԼͷΑͳ

ΛຬɽXY isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼ

d(ξ and η) = dξ and η minus ξ and dη

df(X) = ρ(X) middot fdξ([XY ]E) = ρ(X) middot (ξ(Y ))minus ρ(Y ) middot (ξ(X))minus ξ([XY ]E)

(316)

ಉʹɼΓ(E) ʹରΔ෦ ιX Γ(andpElowast)rarr Γ(andpminus1Elowast) ͷΑʹఆͰΔɽ

ιX(ξ(Y1 Ypminus1)) = ξ(XY1 Ypminus1) (317)

ΕΒͷΈΘΒɼΓ(E) ʹରΔ Lie ඍ LX Γ(andpElowast)rarr Γ(andpElowast) ҎԼͷΑʹఆ

ͰΔɽ

LX(ξ)(Y1 Yp) = ρ(X) middot (ξ(Y1 Yp))minusp)

i=1

ξ (Y1 [XYi]E Yp) (318)

ΕΒͷԋࢠɼҎԼͷΑͳΛຬɽXY isin Γ(E) f isin Cinfin(M) ξ isin Γ(andpElowast) ͱ

ͱɼ

L[XY ]E = LX middot LY minus LY middot LX

ι[XY ]E = LX middot ιY minus ιY middot LX

LX = d middot ιX + ιX middot dLfX(ξ) = fLX(ξ) + df and ιX(ξ)

LX⟨ξ Y ⟩ = ⟨LXξ Y ⟩+ ⟨ξLXY ⟩

(319)

ಉʹɼΓ(Elowast) ʹରΔ෦ͱ Lie ඍఆͰΔɽΕͰɼΓ(andpE) Γ(andpElowast) Λऔ

ΓѻΔΑʹͳɽʹɼLie ΛҾހ Γ(andpE) ͷͱுΔɽ[middot middot]E ͷҰ

ൠԽͱɼSchouten-Nijenhuis ހ [middot middot]S ҎԼͷΛຬͷͱఆΕΔɽ

A isin Γ(andpE) B isin Γ(andqE) ͱͱɼ

(i) [AB]S = minus(minus)(pminus1)(qminus1)[BA]S

(ii) ಛʹɼp = 1 ͷͱ [A f ]S = ρ(A) middot f (iii) [X middot ]S ɼΓ(andqE) ʹରɼp Δɽ༺ͷඍͱ

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

20

Lie ѥ (EElowast) Lie ѥҎԼͷΑʹఆΕΔɽE Λ Lie ѥɼElowast Λ E ͱରͳ Lie ѥͱ

ΔɽE ͷހΒߏ Schouten-Nijenhuis ހ [middot middot]S ͱɼΓ(andpE) ʹରΔඍ

ԋࢠ dlowast ͷʹɼҎԼͷ derivation

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S (320)

ΔɼEElowast Lie ѥ (EElowast) Λͳ [44]ɽ Lie ѥɼLie ͷҰൠԽͰΔɽड़ͷఆͷΘΓʹɼElowast Λݩʹ Schouten-

Nijenhuis ހ [middot middot]lowastS ΛߏɼΓ(andpElowast) ʹରΔඍԋࢠ d ͱͷʹ derivation Λ

ɼಉ Lie ѥ (EElowast) ಘΒΕΔɽderivation ɼલઅͷ Lie ͷͷ

1-cocycle ʹରԠΔɽM ҰͰɼѥͷ anchor 0 ͷͱɼLie ѥ Lie

ͱͳΔɽLie ͷͱಉʹɼLie ѥʹର Drinfelrsquod double ΛߦΔɼ

ͷՌಘΒΕΔߏɼLie ѥͰͳɼઅͰड़Δ Courant ѥͰΔ [35]ɽ

33 Courantѥ

ʹɼڀݚରΑΒΕ೦ͷுҰൠԽΒಘΒΕΔͱଟɽ

ઌड़ͷ Lie Lie ѥͰͳɽඍزԿͷͰΕͷͻͱ

ɼʮϕΫτϧଋͷுʯͰΔɽϕΫτϧ V ͱ ରϕΫτϧ V lowast ͷ V oplus V lowast

ʹɼLie Λ༩ހ Lie ΛఆͰɽͰɼϕΫτϧଋ E ͱରϕΫτϧଋ Elowast ͷ

E oplus Elowast ʹରɼͲͷΑͳހ༩ΒΕɼͲͷΑͳߏಘΒΕΔΖɽಛ

ʹଟମ M ͷଋ TM ͱ༨ଋ T lowastM ͷ TM oplus T lowastM ΛʹߟΔͱɼCourant ѥ

ͱߏݱΕΔͱ 1990 ʹΕ [17]ɽจݙ [17] ͰఏΕ Courant ހ

(22) ɼͷʹҰൠԽଋجʹ TM oplus T lowastM ΛڀݚରͱΔɼҰൠԽزԿ (generalized

geometry) ΕΔͱͱͳΔ [11]ɽҰൠԽزԿͰɼఈ (ଟମ) ഒԽɼ

ଋഒԽΕΔɽҰൠԽزԿʹɼݪจ [11] ͷஶͰΔ Hitchin ʹΑΔߨ

ϊʔτΔ [45] ΄ɼจݙ [16 31 46] ͳͲʹɽΕΒͷจݙͰɼͷܗঢ়Λنఆ

ΔͰΔۂᎇʹΕΔɽɼҰൠԽزԿͷهड़ͱϙΞιϯ

(Poisson) ଟମͰͷهड़ߟΒΕΔ [47 48]ɽཧతʹݭཧΒಘΒΕΔॏ

ཧͷɼNS-NS sector ʹରͷزԿతͳඳΛ༩ΔɽCourant Α༩ΒΕʹހ

Δͷɼඍಉ૬ͱ B ʹΑΔήʔδมΛΈΘɼҰൠԽଋʹରΔҰൠԽඍ

ಉ૬ͰΔɽจݙ [49] Ͱɼड़ͷϙΞιϯଟମͷҰൠԽزԿΛ༻ɼβ ॏཧ

ʹΕΔɽ

ΑΓҰൠʹɼE oplus Elowast ͷɼLie ѥͷ Drinfelrsquod double ʹΑ Courant ѥΛ

ஙͰΔͱߏ 1997 ʹΕ [35]ɽจݙ [17] ͰఏΕ Courant ݙͱɼจހ [35] Ͱ

ఏΕ Courant ݙঢ়ҟͳΔɽจܗͷހ [35] ͷ Courant తʹܗɼހ DFT ͷ C

ހ (216) ͱಉܗɼJacobi ΛܭՌҟͳΔͷͰɼ C ߏఆΔنހ

Courant ѥͰͳɽ

ͷઅͰɼॳΊʹҰൠతͳ Courant ѥͷఆΛ༩ΔɽCourant ͷఆހ 2 ௨

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

21

ΓͷՁͳΔͱΛհΔɽɼจݙ [35] ͷϨϰϡʔΛߦɽͷจݙͰఏ

Ε Courant ΛհͷɼLieހ ѥͷ Drinfelrsquod double Λߦɽ

ɼCourant ѥͷఆΛɽ

Courant ѥ ҎԼͷ 4 ͷߏͷ (C [middot middot]c ρc (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ C πminusrarrM

bull C ͷஅ Γ(C) ΛҾͱΔରশͳ Courant ހ [middot middot]c Γ(C)times Γ(C)rarr Γ(C)bull C Βଋ TM ͷ anchor map ρc C rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (C [middot middot]c ρc (middot middot)) ɼҎԼʹ 5 ͷཧ Axiom C1-C5 ΛຬɼCourant

ѥΛ [35]ɽɼD Γ(C) ͰΔɽࢠΔඍԋ༺ʹ

Axiom C1 e1 e2 e3 isin Γ(C) ͱΔɽCourant ͷހ Jacobiator ҎԼͷΑʹͳΔɽ

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ɼT (e1 e2 e3) =13 (([e1 e2]c e3) + cp) ͰΔɽcp cyclic permutation Λҙຯ

ΔɽT (e1 e2 e3) e ͷೖΕସʹશରশͰΔɽ

Axiom C2 e1 e2 isin Γ(C) ʹରɼ

ρc([e1 e2]c) = [ρc(e1) ρc(e2)] (322)

ɼ[middot middot] TM ͷ Lie ࢦΛހ

Axiom C3 Courant ରɼमਖ਼Εʹހ Leibniz ଇΔɽͳΘɼe1 e2 isinΓ(C) f isin Cinfin(M) ʹରɼ

[e1 fe2]c = f [e1 e2]c + (ρc(e1) middot f)e2 minus (e1 e2)Df (323)

Axiom C4 ρc middot D = 0 ͰΔɽݴΔͱɼf g isin Cinfin(M) ʹରɼ

(DfDg) = 0 (324)

Axiom C5 (middot middot) ͱ ρc ΛຬɽͳΘɼe1 e2 e3 isin Γ(C) ͱͱ

ρc(e1) middot (e2 e3) = ([e1 e2]c +D(e1 e2) e3)+ (e2 [e1 e3]c +D(e1 e3)) (325) จݙ [50] ʹΑΔͱɼཧͷɼAxiom C5 Axiom C3 Βಋग़ͰɼAxiom C2 Β

Axiom C4 Λಋग़ͰΔɽશʹಠΔͷ Axiom C1 ͷΈͰΔɽҎԼͷͰɼ

Axiom C1-C5 શಠͳͷͱѻɽ

ͰɼCourant ѥͷͷఆʹίϝϯτΔɽड़ͷରশͳ Courant ހ [middot middot]cͷΘΓʹɼDorfman bracket operator ͱ Γ(C) ʹରΔೋԋ Λ༻ఆΔ

Δ [34 53]ɽͷɼʹ Axiom C1-C5 ҎԼͷΑʹ Axiom C1prime-C5prime ʹஔ

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

22

ΘΔɽΕΒͷఆՁͰΔͱɼจݙ [34] ͰΕΔɽAxiom C1prime ΒΘ

ΔΑʹɼ Jacobi ͷΘΓʹɼࠨ Leibniz Λຬɽରশͳހ [middot middot]c ʹΑΔఆ

ΒɼCourant ѥ strong homotopy Lie (Linfin ) ͱղऍΔͱͰΔ [51]ɽ

ҰͰɼೋԋ ʹΑΔఆΒɼCourant ѥ Leibniz ѥͱղऍΔͱͰ

Δɽ

e1 (e2 e3) = (e1 e2) e3 + e2 (e1 e3) (Axiom C1prime)

ρc(e1 e2) = [ρc(e1) ρc(e2)] (Axiom C2prime)

e1 (fe2) = f(e1 e2) + (ρc(e1)f)e2 (Axiom C3prime)

e1 e1 = D(e1 e1) (Axiom C4prime)

ρc(e3)(e1 e2) = (e3 e1 e2)+ (e1 e3 e2) (Axiom C5prime)

ຊจͰɼରশͳހ [middot middot]c ʹΑΔఆʹΑΛ2ߦɽ

ଓɼ[35] ʹͷͱΓ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ Courant ѥʹ

ड़ΔɽલͷઅͰఆ௨Γɼରͳͷ Lie ѥ EElowast ΛಋೖΔɽEElowast ͷ

அͰಘΒΕΔϕΫτϧΛɼΕΕ Γ(E) = Xi Γ(Elowast) = ξi ͱΔɽʹɼE ͱ Elowast ͷʹ

ΑΓͳϕΫτϧଋ C = E oplus Elowast ΛఆΔɽei isin Γ(C)ɼͳΘ ei = Xi + ξi ͱͱɼ

Γ(C) ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (328)

ͷΑʹఆΔɽ(e1 e2)+ ରশ (e1 e2)minus ରশͰΔɽɼρ ҎԼͷΑʹఆ

Δɽρc(ei) = ρ(Xi) + ρlowast(ξi) (329)

Γ(C) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(C)

(Df ei) =1

2ρc(ei)f (330)

ͷΑʹఆΔɽड़ͷ ρ ͷఆͱΘΔͱɼD = d + dlowast ͱදͱͰΔɽɼ

Courant ހ [middot middot]c ΛମతʹҎԼͷΑʹఆΊΔɽ

[e1 e2]c = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (331)

2 Courant ɼDorfmanހ bracket operator ΛରশʹΉͱͰఆͰΔɽ

[e1 e2]c =1

2((e1 e2)minus (e2 e1)) (326)

ҰൠԽزԿͷจͰɼCourant ͱͷରൺͱހ Dorfman bracket ΛରশʹΉͱͰ ldquoDorfman

bracketrdquo

[e1 e2]d =1

2((e1 e2) + (e2 e1)) (327)

ΛఆΔΔɽDFT ͷจͰɼD ଋΛͱಘΒΕΔͷɼldquoDorfman߆ڧʹހ bracketrdquo Ͱͳ Dorfman bracket operator ͰΔɽ

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

23

ͷΑʹఆͱɼ(E oplusElowast [middot middot]C (middot middot)+ ρc) Courant ѥͱͳΔͱจݙ [35]

ͰΕɽ[e1 e2]c DFT ͷ C ͰΔ3ɽܗతʹಉܗͱހ

จݙ [35] ͰɼDrinfelrsquod double ͷٯΕΔɽΔ Courant ѥΛఆΔϕΫ

τϧଋ C ʹணΔɽC ɼઢܗܕ (middot middot)+ ʹతͳ෦ଋ (subbundle) EElowast ʹ

Ͱɼͳ Courant ހ [middot middot]c Γ(C) ͰดΔɼ(EElowast) વʹ Lie ѥ

ͷߏΛͱΕΔɽͰɼҎͷͰඞཁͱͳΔ Dirac ΛಋೖΔɽߏ

Definition 331 EElowast ҎԼͷͷΛຬͱɼC ͷ Dirac ͰΔͱɽߏ

bull େతͰΔɼͳΘ dimE = dimElowast = 12 dim C ͰΔɽ

bull E ʹดހ [middot middot] (Lie (ހ ΛఆͰΔɽͳΘɼҙͷ XY isin Γ(E) ʹର

ɼ[XY ] isin Γ(E) ͰΔΑͳ [middot middot] ଘࡏΔɽ

EElowast C ͷ Dirac lowastͰΔͱɼ(EEߏ E oplus Elowast) ͷѥߏʹΑΔ Manin triple

ͱݴΔɽΖΜɼશͷ Courant ѥ Lie ѥͷ Drinfelrsquod double ʹΑಘΒΕΔ

ΘͰͳɽٯʹɼCourant ѥඞ Dirac ΒͳɽݶʹΑͰΔͱߏ

34 DoubleʹΑΔ Vaismanѥͷߏ

Vaisman ѥͷΑʹఆΕΔɽ

Vaisman ѥ ҎԼͷ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)) ΛߟΔɽ

bull M ͷϕΫτϧଋ V πminusrarrM

bull V ͷஅ Γ(V) ΛҾͱΔ Vaisman ހ [middot middot]V Γ(V)times Γ(V)rarr Γ(V)bull V Βଋ TM ͷ anchor map ρV V rarr TM

bull ඇୀԽͰରশͳઢܗܕ (middot middot)

ͷ (V [middot middot]V ρV (middot middot)) ɼҎԼʹ 2 ͷఆ Axiom V1 V2 Λຬɼ

Vaisman ѥΛɽɼD Γ(V) ͰΔɽࢠΔඍԋ༺ʹ

Axiom V1 [e1 fe2]V = f [e1 e2]V + (ρV(e1) middot f)e2 minus (e1 e2)Df

Axiom V2 ρV(e1) middot (e2 e3) = ([e1 e2]V +D(e1 e2) e3)+ (e2 [e1 e3]V +D(e1 e3)) Courant ѥͷఆͱൺΔͱɼAxiom V1 Axiom C3 ͱɼAxiom V2 Axiom C5 ͱ

ରԠΔɽɼVaisman ѥ Courant ѥΛߋʹҰൠԽߏͰΔͱ

ΔɽɼAxiom C5 Β Axiom C3 ಋΕΔͱΔจݙ [50] ͷͱໃ६ͳɽ

จݙ [35] ͷ Courant ѥʹΔՌΒɼVaisman ѥԿΒͷߏͷ double Ͱಘ

ΒΕΔͷͱਪଌΕΔɽզʑɼAxiom V1 V2 Λຬ Vaisman ѥɼCourant ѥ

ͱ double ԽͷखଓʹΑಘΒΕΔͱΛ [13]ɽݤͱͳΔͷɼLie ѥ

3 ͱܥ [35] ͰతʹݟΕͷઌͰΔɽ

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

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[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

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[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

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[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

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[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

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[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

24

ʹΔ derivation (320) ͰΔɽҎԼͰɼderivation Λʹͷ Lie ѥ

ͷ double ΛͱΔͱͰɼVaisman ѥಘΒΕΔͱΛɽɼdouble ʹΑߏ

Vaisman ѥʹରɼޙΒ derivation Λͱจݙ [35] ͷ Courant algeroid

ಘΒΕΔɽ

ɼରͳͷ Lie ѥ E = (E [middot middot]E ρ) ͱ Elowast = (Elowast [middot middot]Elowast ρlowast) Λఆɼʹ

ΑϕΫτϧଋ V = E oplus Elowast Λ༩Δɽɼ(EElowast) Lie ѥΛͳͳͷͱԾ

ఆΔɽͳΘɼހͱඍͷʹ derivation (320) ͳͱΔɽΓ(V)ͷඇୀԽͳઢܗܕ (middot middot)plusmn Λ

(e1 e2)plusmn =1

2(⟨ξ1 X2⟩plusmn ⟨ξ2 X1⟩) (332)

ͷΑʹఆΔɽɼanchor ρV ҎԼͷΑʹఆΔɽ

ρV(ei) = ρ(Xi) + ρlowast(ξi) (333)

Γ(V) ʹରΔඍԋࢠ D Cinfin(M)rarr Γ(V)

(Df ei) =1

2ρV(ei)f (334)

ͷΑʹఆΔɽɼD = d + dlowast ͱදͱͰΔɽVaisman ݙจހ [52] Λߟʹ

ҎԼͷΑͳରশͳހͱఆΔɽ

[e1 e2]V = [X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minus

+ [ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minus (335)

ΕɼDFT ͷ C ͰΔɽɼ[35]ܗͱಉހ ͷ Courant ͰΔɽܗͱಉހ

ͷΑʹఆ 4 ͷߏͷ (V [middot middot]V ρV (middot middot)+) Courant ѥͰͳ Vaisman

ѥΛఆΔͱΛҎԼʹɽԾఆΒɼderivation (320) ͳͱʹ

ҙɼAxiom C1-C5 ΔͲΛΕΕΊΔɽຊจதʹɼܭͷཁ

ΛࡌΔɽࡉͳܭ Appendix A5-A11 ΛরͷͱɽAxiom C3 C5 ͷΈ

ɼଞͷ Axiom ΕΕ (V [middot middot]V ρV (middot middot)) Vaisman ѥͰΔͱΔɽ

ͷաఔͰඞཁͱͳΔॿతͳɼҎԼͷ 2 ͰΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]c e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus+ cp (336)

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus

minus ⟨[ξ1 ξ2]V X3⟩+ cp+ (337)

ΕΒͷͷಋग़ɼʹΘΔશܭ Appendix A5 ʹऩɽ

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

25

Axiom C1ͷ

ɼAxiom C1 ΛΔΊʹɼVaisman ހ [middot middot]V ͷ Jacobiator ΛܭΔɽ(321)

ͷࠨลΛలΔͱɼҎԼͷՌಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 (338)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (339)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (340)

ɼΓ(Elowast) ʹଐΔ I1 ʹɼΓ(E) ʹଐΔ I2 ʹͱΊɽҎͷܭɼI1 ʹ

ͷΈߦɽX ͱ ξ ΛೖΕସΔͱͰɼI2 ʹΔܭݱͰΔɽهͷ I1 ʹɼLie

ͷހ Jacobi ʹΑফΔΕΓɼΕΒΛཧΕҎԼͷܗͱͳΔɽ

I1 = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast + LLξ1X2minusLξ2X1ξ3 minus Ldlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (341)

ͰɼI1 ҎԼͷΛద༻Δɽͷͷಋग़ Appendix A7 ɽࡌʹ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (342)

Δͱɼऴతʹ I1 ҎԼͷΑʹͱΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (343)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (344)

ಉʹɼI2 ΔͱɼҎԼͷՌಘΒΕΔɽܭʹ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = minusLd(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E

(345)

ɼVaisman ހ [middot middot]V ͷ Jacobiator ͷܭՌͷΑʹͳΔɽ

[[e1 e2]V e3]V + cp = I1 + I2 = DT (e1 e2 e3)minus (J1 + J2 + cp) (346)

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

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[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

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mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

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Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

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no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

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Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

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[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

26

ɼJ1 J2 ҎԼͷΑͳͰΔɽ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (347)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

Axiom C2ͷ

ಉʹɼAxiom C2 Δɽɼ(322)ܭʹ ͷลΛ f ʹର༻ΔͱɼҎԼ

ͷಘΒΕΔɽρV([e1 e2]V) middot f = [ρV(e1) ρV(e2)]f (348)

(348) ͷࠨลͷΑʹలΕΔɽ

ρV([e1 e2]V) middot f

= [ρ(X1) ρ(X2)] middot f + ρ(Lξ1X2) middot f minus ρ(Lξ2X1) middot f minus1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f

+ [ρlowast(ξ1) ρlowast(ξ2)] middot f + ρlowast(LX1ξ2) middot f minus ρlowast(LX2ξ1) middot f +1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f(349)

ɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΓɼd0 Γ(T lowastM) ͷඍԋࢠͰΔɽ

(348) ͷӈลͷΑʹలΕΔɽ

[ρV(e1) ρV(e2)]f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρ(X1) ρlowast(ξ2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f (350)

ӈลͷ 2 ɼ3 ͷ X ξ ͷ cross term ɼҎԼͷ

ρlowast(ξ)ρ(X) middot f = minusρρlowastlowastd0⟨ξ X⟩ middot f + ⟨ξLdfX⟩+ ⟨dfLξX⟩ (351)

Λ༻ΔͱͰɼ

[ρ(X) ρlowast(ξ)]f = (ρ(X)ρlowast(ξ)minus ρlowast(ξ)ρ(X)) middot f= ρ(X)ρlowast(ξ) middot f + (ρρlowastlowastd0⟨ξ X⟩) middot f minus ⟨ξLdfX⟩ minus ρ(LξX) middot f (352)

ͱॻΔɽɼ(348) ͷࠨลͱӈลͷͷΑʹͳΔɽ

ρV([e1 e2]V) middot f minus [ρV(e1) ρV(e2)]f

= minus⟨ξ1LdfX2 minus [X2 dlowastf ]E

(⟩+ ⟨ξ2

LdfX1 minus [X1 dlowastf ]E

(⟩

+1

2

ρρlowastlowast + ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩) middot f (353)

Ұൠʹɼҙͷ Xi ξi ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

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54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

27

Axiom C3ͷ

(323) ͷࠨลɼei = Xi + ξi ΒɼҎԼͷΑʹॻΔɽ

[e1 fe2]V = [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (354)

ͰɼVaisman ͷఆΒɼ(354)ހ ͷӈลҎԼͷΑʹలΕΔɽ

[X1 fX2]V = [X1 fX2]E

[X1 fξ2]V = f [X1 ξ2]V + (ρ(X1) middot f)ξ2 minus1

2Df⟨ξ2 X1⟩

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1) middot f)X2 minus1

2Df⟨ξ1 X2⟩

[ξ1 fξ2]V = [ξ1 fξ2]Elowast (355)

(355) ͷ 4 ͷͷӈลΛશͱɼҎԼͷՌಘΒΕΔɽ

[e1 fe2]V = f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)+ (356)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

Axiom C4ͷ

(324) ͷࠨลҎԼͷΑʹలΕΔɽ

(DfDg)+ =1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (357)

Ұൠʹ anchor ρ ରশͰͳɽͳΘ ρρlowastlowast = minusρlowastρlowast ͳɽΑɼ(357) ͷӈ

ล 0 ʹͳΒɼAxiom C4 (V ρV [middot middot]V (middot middot)) ʹΕΔɽɼanchor ʹ

Δͷ lowastɼadjoint operator ͷҙຯͰΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (358)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽ

Derivation Δඞ anchor ରশʹͳΔͱ [44] ͷ Proposition

34 ͷ༰ΛݩʹΔɽҰൠʹɼLie ѥͷΒҎԼͷಘΒΕΔɽ

(LdfX + [dlowastfX]E) and Y

= minusfdlowast[XY ]E + LY dlowastX minus LXdlowastY

+

dlowast[X fY ]E minus LXdlowast(fY ) + LfY dlowastX

(359)

Proposition 34 Ͱɼ(359) ͷӈล derivation ΛͱͰ 0 ͱͳΔͱΛΔɽ

ͳΘɼderivation ຬΕɼਵҎԼͷΔͱͰ

Δɽ

LdfX + [dlowastfX]E = 0 (360)

(360) ຬΕͳݶΓɼρ ରশͱͳΒͳͱΕΔɽࡉ Appendix A10 র

Αɽɼderivation ΔͳΒඞ ρ ରশͱͳΓɼAxiom C4

Δɽٯʹɼderivation ΛͳͳΒɼanchor ରশͰͳΛഉআͰͳɽ

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

28

Axiom C5ͷ

ষͷͰ༩ T (e1 e2 e3) ͷ (336) ΒɼҎԼͷͷΔɽ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+ (361)

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (362)

ҎԼͷΑʹɼ (361)ɼ(362) ͷࠨลಉɼӈลಉΛอΕΔɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) +1

2ρV(e) middot (e1 e2)+ minus

1

2ρV(e1) middot (e e2)+

+ T (e e2 e1) +1

2ρV(e) middot (e2 e1)+ minus

1

2ρV(e2) middot (e e1)+ (363)

(363) ɼT ͷରশͳͲΒҎԼͷΑʹཧͰΔɽ

ρV(e) middot (e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ +1

2ρV(e1) middot (e e2)+ +

1

2ρV(e2) middot (e e1)+

([e e1]V +D(e e1)+ e2)+ + (e1 [e e2]V +D(e e2)+)+ (364)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5 ΔɽΕΒͷܭͷࡉ Appendix

A9 A11 ʹɽΕͰɼରͳ 2 ͷ Lie ѥͷ double ʹΑɼVaisman ѥ

ಘΒΕΔͱɽAxiom C5 ͱ Axiom C3 Δঢ়گɼจݙ [50] ͷͱໃ६

ͳɽ

Ͱɼٯʹ Vaisman ѥͷΛߟΈΑɽ[35] ͰߦΘΕΔ Courant ѥ

ͷ Dirac ෦ଋʹΑΔʹجΔΊʹɼ [35] ͷ෦ͷϨϰϡʔΛߦ

ɽCourant ѥ (C [middot middot]c ρc (middot middot)) (middot middot) େతͰΓɼͳʹ Courant

ހ [middot middot]c Γ(C) ͰดΔɼDirac ෦ଋ L L ʹΑͰΔɽɼ(L L)

વʹ Lie ѥͱͳΔɽΕΛΊʹඞཁͱͳͷɼҎԼʹఏΔ Proposition 23 ͱ

Lemma 52 ͰΔɽ

Proposition 23 in [35] L ɼCourant ѥ (C [middot middot]c ρc (middot middot)+) ͷઢܗܕ (middot middot)+ ʹ

తͳ෦ଋͰɼՄ4ͳͱ (L [middot middot]c ρc|L) Lie ѥͱͳΔɽ

ɼʮ(middot middot) ʹͳ෦ଋʯͱɼҙͷ XY isin Γ(L) ʹରɼ(XY ) = 0

ΔΑͳ෦ଋΛࢦɽρc|L ɼanchor ρc Β L Δ෦Λൈग़ͷͰ༺ʹ

ΔɽProposition 23 ɼAxiom C1 ͷ (321)

[[e1 e2]c e3]c + cp = DT (e1 e2 e3) (321)

ΒɼL ʹདྷΔ෦ (X ΛΉ) Λൈग़ͱͰʹͰΔɽ(321) ͷࠨล

ɼL ͷހ [middot middot]L ʹΑΔ Jacobiator ͱͳΔɽҰɼӈล T ͷఆ

T (e1 e2 e3) =1

3([e1 e2]c e3)+ cp (365)

4 ՄʹͷࡉͷষͰ༩ΔɽͻͱͱͰݴͱɼ෦ଋ L ͷ Lie ɽࢦดΔͱΛހ

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

29

ΒɼL (middot middot) ʹతͰΔͱΛԾఆΔͱඞ 0 ʹͳΔɽΑɼ[middot middot]L Jacobi

ΛຬͱΊΒΕɼL Lie ѥͱͳΔɽProposition 23 ΒɼCourant ѥͷ

Dirac ߏ L L ΕΕ Lie ѥͱͳΔɽL ͷ anchor ρc|L ͰɼL ͷ anchor ρc|L Ͱ

ఆΕΔɽ

Proposition 23 ΛཁͱɼL (middot middot) ʹతͳͱΛ༻ Axiom C5 ͷ

(325) ΒҎԼͷಋΔɽX isin Γ(L) ξ isin Γ(L) ͱͱɼ

[X ξ]c = minusLξX +1

2dlowast⟨ξ X⟩+ LXξ minus 1

2d⟨ξ X⟩ (366)

Lemma 52 in [35] L L ΛɼCourant ѥ C ʹରɼC = Loplus L ͱͳΔΑͳ Dirac ߏ

ͰΔͱΔɽͷͱɼҎԼͷΔɽɼ

Ldlowastfξ = minus[df ξ]L LdfX = minus[dlowastfX]L (367)

Ͱɼd dlowast ΕΕ Γ(andpL)Γ(andpL) ͰΔɽࢠΔඍԋ༺ʹ

Lemma 52 ҎԼͷΑʹΔɽɼAxiom C4 ΒɼҎԼͷΔɽ

ρlowast middot d = minusρ middot dlowast (368)

ΕΛɼAxiom C2 Ԡ༻ΔͱͰɼҎԼͷΘΔɽ

[ρlowast(ξ) ρ(X)] = ρc[ξ X]c

= ρc(LξX minus1

2dlowast⟨ξ X⟩ minus LXξ +

1

2d⟨ξ X⟩)

= ρ(LξX)minus ρlowast(LXξ) + ρlowast(d⟨ξ X⟩) (369)

ɼ (369) ͷ 1 ஈΒ 2 ஈͷมܗͰɼ (366) Λ༻ɽҰͰɼLie ѥͷ

Βɼ

ρlowast(d⟨ξ X⟩) middot f = [ρlowast(ξ) ρ(X)]f minus ρ(LξX) middot f + ρlowast(LXξ) middot f + ⟨Ldlowastfξ + [df ξ]L X⟩ (370)

(369) ͱ (370) ͷൺΒɼLdlowastfξ = minus[df ξ]L ͰΔɽ (369) ͱ (370) ͷ ξ ͱ X Λೖ

ΕସܭΕɼLdfX = minus[dlowastfX]L Δɽ

Lemma 52 ͷΛ೦ʹஔɼҰ Courant ͷހ Jacobiator ͷܭՌʹ

Δɽ

[[e1 e2]c e2]c + cp = DT (e1 e2 e3)minus (J1 + J2 + cp) (371)

ɼJ1 J2 (347) ͰͷͱಉͷͰΔ

J1 = ιX3

d[ξ1 ξ2]V minus Lξ1dξ2 + Lξ2dξ1

+ ιξ3

dlowast[X1 X2]V minus LX1dlowastX2 + LX2dlowastX1

J2 =Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]V

minus

Ld(e1e2)minusX3 + [dlowast(e1 e2)minus X3]V

(347)

ɼC = L oplus L Axiom C1 ΛຬͷͰɼඞવతʹ J1 + J2 + cp = 0 ͰΔɽՃɼ

Lemma 52 ΒɼJ2 = 0 ͰΔɽΑ J1 + cp = 0 ཁٻΕΔɽɼe1 = X1

e2 = X2 e3 = ξ3 ͱஔͱɼҎԼͷΛಘΔɽ

dlowast[X1 X2]L minus LX1dlowastX2 + LX2dlowastX1 = 0 (372)

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

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[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

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[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

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[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

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[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

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[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

30

ΕɼLie ѥͷ derivation (320) ͷͷͰΔɽΑɼ(Loplus L) Lie ѥ

ͱͳΔɽલઅͰड़௨ΓɼCourant ѥ C ͷ Dirac ߏ L L ɼશମͰ Manin triple

(C L L) ΛߏΔɽ

ͷΛͱʹɼVaisman ѥͷΛߟΔɽVaisman ѥͷ Dirac ɼCourantߏ

ѥͷͱಉʹɼΓ(V) ΛҾͱΔઢܗܕ (middot middot) ʹରΔɼେͳ෦ଋͰఆ

ͰΔɽL L V = Loplus L ͱͳΔΑͳ V ͷ Dirac ͰΔͱΔɽɼVߏ Vaisman

ѥͳͷͰɼAxiom C3 C5 Δɽɼ[35] ͷ Proposition 23 ɼಋग़Δࡍʹ

Axiom C1 Λ༻ΔͷͰɼVaisman ѥΛલఏͱΔͱͳɽΑɼL L ͷހ

Ұൠʹ Jacobi ΛຬͱݶΒɼL L Lie ѥʹͳΒͳߟΒΕΔɽՃɼ

Ծʹ L L Lie ѥͰͱ Axiom C2 C4 ΕΔΊɼLemma 52

ͳɽɼ(L L) ͷʹ derivation ɼ(L L) શମͰ Lie ѥΛͳ

ͳɽΕΛɼDFT ͷηοτΞοϓͰମతʹܭՌ Ͱɽରͳޙষͷ

Lie ѥͷ L ͱ Llowast Lie ѥʹͳΔͷɼ(LLlowast) શମͰ differential Gerstenhaber

ΛͳͱͷΈͰΔ [54]ɽΕɼඍԋࢠ d ( dlowast) Schouten-Nijenhuis

ͱΔͱͰΔɽހ

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

31

ୈ 4ষ

Vaisman algebroidݱΕΔزԿ

ύϥΤϧϛʔτ (para-Hermitian) ଟମʹɼDFT ͷഒԽવʹݱΕΔͱ [1415

26ndash28] ͳͲͰΕΔɽͷষͰɼύϥΤϧϛʔτଟମʹલઅͰ Vaisman ѥ

Λݱɼͷ Dirac ɼDFTجʹߏ ͷήʔδରশʹΔɽ

ͷষͷલͰɼѥͷهड़ʹඞཁͳతߏͷఆΛ༩ΔɽɼύϥΤϧϛʔτଟ

ମ M Λ༩ɼଋ TM ͷύϥෳૉ (para-complex) ΒવʹಋೖͰΔͱΛߏ

ΔɽΕʹਵΔ༿ߏ (foliation) ߏ [56] ΛಋೖΔɽՃɼDFT ͷύϥυ

ϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹɼѥߏͷݱʹඞཁͳ

ԋࢠΛཧΔɽষͷޙͰɼड़ͷఆΛ༻ࡍʹܭΛߦɼύϥΤϧϛʔτଟ

ମʹ Vaisman ѥΛߏஙΔɽɼલͷষͰछʑͷѥߏɼDFT Ͱ

ͲͷΑʹݱΕΔΛௐΔɽಛʹɼdeivation ͱ DFT ͷ߆ڧଋͷΛߟΔɽ

41 ύϥෳૉߏ

DFT ͷഒԽ M ͷඪܥ xM = (xmicro xmicro) ɼKK Ϟʔυʹ Fourier ඪͳڞ xmicro ͱ

winding Ϟʔυʹ Fourier ඪͳڞ xmicro ΛΈΘΔͱͰಋೖΕΔɽจݙ [1415] ʹ

ɼͷΑͳߏͷඪɼύϥΤϧϛʔτଟମʹવʹΈࠐΕΔͱࢦఠͳ

ΕɽύϥΤϧϛʔτଟମ M ΛఆΔʹɼύϥෳૉߏඞཁͱͳΔɽɼύϥෳ

ૉଟମҎԼͷΑʹఆΕΔɽ

Definition 411 (almost) ύϥෳૉଟମͱɼҎԼͷύϥෳૉߏ K ఆΕଟ

ମ M ͷͱͰΔɽK ɼK2 = 1 ΛຬΑͳ TM ͷݾಉܕͰΔɽ

௨ৗͷෳૉߏ J2 = minus1 Ͱ Nijenhuis ςϯιϧ 0 ʹͳΔͱ J ʹՄͱͳ

Δ [57]ɽΕͱಉʹɼҎԼͷ

NK(XY ) =1

4([K(X)K(Y )] + [XY ]minusK([K(X) Y ] + [XK(Y )]) (41)

0 ʹͳΔͱɼύϥෳૉߏ K ՄͰΔͱɽՄͳύϥෳૉߏύϥෳૉ

ͰΔɽKߏ Մͳͱ (KM) Λύϥෳૉଟମͱɽ

ύϥෳૉߏ K ͷݻ plusmn1 ʹԠɼM ͷଋ TM Λ K ͷݻ෦ଋ L L ʹͰ

ΔɽTM ͷɼҎԼͷӨԋࢠ P P ʹΑߦΘΕΔɽP P ɼTM Β L L Λબ

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

32

ͿΑͳͰΓɼҎԼͷΑʹఆΕΔɽ

P =1

2(1 +K) P =

1

2(1minusK) (42)

P P ʹΑಘΒΕΔ෦ଋ L L ɼdistribution ͰΔɽ

Definition 412 Distribution ɼҎԼͷΑʹఆΕΔɽҰൠʹɼM Λ m ͷݩ Cinfin ڃ

ଟମͱΔɽM ͷҙͷ x ͷ TxM ʹରɼx Βͳʹ n ݩ (n le m)

ͷ෦ ∆x ΛߟΔɽ∆ Λ M ͷશͷʹରΔ෦ ∆x ͷͱΔͱɼ∆ n

ͷݩ distribution ͱͳΔɽɼx Βͱɼ∆xʹ ʹଐΔϕΫτϧΒɼx

ͷશͷʹର n ΕΔɼͱҙຯͰΔɽͷϕΫτϧΛݩ

Distribution ͷ؆୯ͳɼn = 1 ͷͰΔɽM ʹରΔ 1 ͷ distribution ɼ

TM ͷஅʹΑΔϕΫτϧ Γ(TM) ͷيͱͳΔɽҙͷ XY isin Γ(L) ʹରɼLie ʹހ

ΑΔ [XY ]L ඞ L ʹଐΔͱɼdistribution L involutive ͰΔͱɽԾʹɼL L

involutive ͰΕɼҎԼͷ 0 ͱͳΔɽ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (43)

ΔͱɼҎԼͷ Frobenius ͷఆཧΒɼ(43) ʹΑ L L ͷՄධՁͰΔɽ

Theorem 413 Δ distribution L ՄͱͳΔͷ L involutive ͷͱͰΔɽ

ɼٯΔɽ

(43) ͰఏΕ NP ͱ NP Λͱɼ (41) ಘΒΕΔɽಋग़ Appendix A12 ʹ

ɽΓɼL ʹରͷՄͱ L ʹରͷՄಠΓɼL L ͷ

L ՄʹͳΔ (NP (XY ) = 0) ɼL ՄʹͳΔ (NP = 0) ΛߟΔ

ͱͰΔɽͷɼύϥෳૉߏͱ௨ৗͷෳૉߏͱͷେͳҧͰΔɽͷΒɼ

ҎԼͷΑͳଟମΛఆͰΔ [26 27]ɽ

Definition 414 (MK) ΛύϥෳૉଟମͱΔɽಛʹɼL K ʹՄͳͱ

ɼΕΛ L-ύϥෳૉ (L-para-complex) ଟମͱɽL ʹಉͰΔɽಛʹɼL ͱ

L ՄͰΕɼTM શମ K ʹՄ (NK = 0) ͱͳΓɼ(MK) ύϥෳ

ૉଟମʹͳΔɽ

42 ύϥΤϧϛʔτଟମ

ɼύϥΤϧϛʔτଟମΛఆΑɽʹ

Definition 421 (MK) ΛύϥෳૉଟମͱΔɽ(MK) neutral ͳܭ η TM timesTMrarr R Λͱɼ(M ηK) ΛύϥΤϧϛʔτଟମͱɽη ΛύϥΤϧϛʔτܭͱ

ɼη ͱ K ͷʹ η(KmiddotKmiddot) = minusη(middot middot) Δɽ

ఆΒɼXY isin Γ(L) ͱͱɼη(XY ) = 0 ͰΔɽՃɼη neutral ͳͷͰɼL L

ͷ rank 12 dimM ͱͳΔɽΑɼL L ύϥΤϧϛʔτܭ η େతͰΔɽʹ

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

33

dω = 0 dω = 0

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ (almost para-Kahler)

(γϯϓϨΫςΟοΫ)

NK = 0ύϥΤϧϛʔτ

(γϯϓϨΫςΟοΫ)

ύϥέʔϥʔ

(γϯϓϨΫςΟοΫ)

ද 41 Մͱ ω ͷ

ύϥΤϧϛʔτଟମʹɼK ͱ η ΒɼඇୀԽͳඍ 2 ܗ ω = ηK ΛఆͰΔɽω

ҰൠʹดܗͱݶΒͳɽω ΛɼҎԼʹఆΔΑͳɼγϯϓϨΫςΟοΫଟମͷγ

ϯϓϨΫςΟοΫ 2 ͱΈͳͱͰΔɽܗ

Definition 422 γϯϓϨΫςΟοΫ (almost symplectic) ଟମͱɼ 2n ଟମݩ

M ͱɼM ͷඍ 2 ܗ ω ͷͷͱΛɽω ΛγϯϓϨΫςΟοΫܗͱɽಛʹɼ

ω ดܗ (dω = 0) ͰΔɼγϯϓϨΫςΟοΫଟମͱͳΔɽͷͱɼdω = 0 Λ

ʮγϯϓϨΫςΟοΫܗʹରΔՄʯͱɽ

ΑɼύϥΤϧϛʔτଟମ (M ηK) γϯϓϨΫςΟοΫଟମͰΓɼK ʹ

ΔՄͱ ω ʹΔՄΛผݸʹѻΔɽΕΛ༻ɼද 41 ͷΑʹෳͷ

ଟମΛఆͰΔɽಛʹɼޙࠓͷͷதͱͳΔύϥΤϧϛʔτଟମͷఆΛվΊఏ

ɽ

Definition 423 M ͱɼύϥΤϧϛʔτܭ η TM times TM rarr Rɼύϥෳૉߏ K ͷΛ

ΔɽKߟ ՄͰɼω ՄͰͳͱɼ(MK η) ύϥΤϧϛʔτଟମͰΔɽ

ɼK ʹରΔՄ L L ʹରಠʹΔͱΒɼҎԼͷΑͳଟମఆ

ՄͰΔɽ

Definition 424 (MK) Λ L-ύϥෳૉଟମͱΔɽΕʹɼύϥΤϧϛʔτܭΛՃ

(M ηK) Λ L-ύϥΤϧϛʔτଟମͱɽL ʹಉͰɼL-ύϥΤϧϛʔτଟମΛ

ఆͰΔɽ

ʹͷΑه L L ͷΕՄͱͳΔΑʹଟମΛఆΔͱͰΔɼ

ΕഒԽΛهड़ΔʹͲͷΑʹӨڹΔͷΓΕͳɽɼ

L-ύϥΤϧϛʔτଟମͷηοτΞοϓΛ༻ɼϦʔϚϯزԿͰڞมඍΛಋೖΔͷͱ

ΕͰ M ͷඍΛਖ਼ɼDFT ͷ D ܭΛހ η ۂΔுͰΔͱ

ఠΓࢦ [26]ɼยଆՄͱঢ়ޙࠓگ༻ͱͳΔՄΓΔɽຊ

จͰҎޙɼL L ͷՄɼͳΘύϥΤϧϛʔτଟମΛ༻ഒԽͷهड़Λ

Δɽ

ύϥΤϧϛʔτଟମ M ͷ distribution L L η େతɼͳΘʹ dimL =

dim L = 12 dimM ͰΔɽɼK ՄͰΔͱΒ (41) 0 ͱͳΓɼ(43) Ͱ༩

NP NP 0 ͱͳΔɽɼL L ʹ involutiveɼΓΕΕʹดࢠΛఆ

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

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[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

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A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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J 73 (1994) 415

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[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

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[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

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[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

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Manifoldsrdquo Ann of Math 65 (1957) 391

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derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

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Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

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[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

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and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

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[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

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[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

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[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

34

ͰΔɽ3 ষͷ 331 અͷఆΛরΔͱɼL L TM ͷ Dirac ͰΔɽɼLߏ L

Frobenius ͷఆཧΒɼՄͰΔɽ

M ʹΑഒԽΛతʹهड़ΔʹΓɼL L ՄͰΔͱͷ༻ɼҎԼ

ʹ Frobenius ͷఆཧͷผͷදݱΛద༻ΔͱΘΔɽ

Theorem 425 Δ෦ଋ E sub TM ՄͱͳΔͷɼM ʹ༿ߏ F ఆͰΔ

ʹݶΒΕΔɽٯʹɼM ʹ༿ߏ F ఆͰΔͳΒɼՄͳ෦ଋ E ΛɼM ͷ

༿ߏ F ʹରΔଋͱඞ༩ΒΕΔɽ

ɼL L ՄͰΔɽFrobenius ͷఆཧΒɼL L ଋͰΔΑͳ༿ߏ F FΛఆͰΔɽ

L = TF and L = T F (44)

Mp F ͷ༿ͰΓɼM ͷ p Λ௨ΔΑͳɼF ͷ෦ͰΔɽF ༿ Mp ͷ

[p] M[p] ͰΔɽL ʹಉͰɼہॴඪ xmicro ɼMp ʹԊ༩ΒΕΔɽͷͱɼ

xmicro Mp ͰɼΔఆΛͱΔɽͷΕɼਤ 41 ͷΑͳֆΛඳͱɽɼ

M F (ઢͷ) ͱɼF جʹ (੨ઢͷ) ͷ 2 ௨ΓʹઍΓͰΔɽਤ 41

Ͱ ઢ F ͷ༿ɼ੨ઢ F ͷ༿ͰΔɽઢͱ੨ઢͷҐஔΛΊΔͷɼ

ඪ xmicro xmicro ͰΔɽxmicro ઢ (F ͷ༿) ʹԊɼxmicro ੨ઢ (F ͷ༿) ʹԊ༩ΒΕ

ΔɽɼF Β༿ΛҰຕબͿͱ xmicro ఆΛͱΔɽ

F F

p

M

ਤ 41 M ߏΕΔ༿ݱʹ F Fɽ༿ͷҰ෦ΛઢͰɽ

ͷઅͰΊΒΕͱΛ؆୯ʹͱΊΔɽύϥΤϧϛʔτଟମ M ͷେͷಛɼ

ଋ TM Λ L L ʹͱʹɼL ͷՄͱ L ͷՄશʹಠΔͱ

ɽͷಛΒɼFrobenius ͷఆཧΛ༻ M ʹ ೋछͷ༿ߏ F ͱ F ΛఆͰɽ

M ͷഒԽඪ (xmicro xmicro) Βɼͷ xmicro ඪΛൈग़ͱɼM ͷ༿ߏ F Β༿ΛҰ

ຕબͿͱͱઆͰɽΕɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ

ࢹΕΔཧͰΔɽ

ɼLʹޙ ͱ L ͷରଋͰΔ Llowast ͷʹίϝϯτɽύϥΤϧϛʔτܭ η

ɼTM = Loplus L Β T lowastM = Llowast oplus Llowast ͷͱΈͳΔɽΑɼη ҎԼͷͷಉܕ

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

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[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

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Trans Am Math Soc 63 (1948) 85

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A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

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Physique theorique 53 (1990) 35

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[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

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and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

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098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

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[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

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[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

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meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

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[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

35

ΛఆΔɽ

φ+ Lrarr Llowast and φminus Lrarr Llowast (45)

ΕΒͷɼL ͷϕΫτϧɼLlowast ͷܗͱಉҰࢹͰΔͱΛҙຯΔɽL ͷϕΫτ

ϧͱɼLlowast ͷܗʹಉͰΔɽΑɼҎԼͷ༩ΒΕΔɽ

Φ+ TMrarr Loplus Llowast and Φminus TMrarr Loplus Llowast (46)

ಛʹɼΦ+ ɼnatural isomorphism ͱɼഒԽزԿͱҰൠԽزԿΛͼΔͷʹར༻

ΕΔɽNatural isomorphism વʹ༩ΒΕΔͱɼύϥΤϧϛʔτଟମഒԽزԿͷ

తͳهड़ͱͳཧͰΔɽ

43 ύϥυϧϘʔίϗϞϩδʔ

લઅͰɼύϥΤϧϛʔτଟମ M ഒԽͷతͳهड़ͱ༻ͳͱͰɽ

ɼM Ͱ Vaisman ѥΛߏʹࡍஙΔʹɼඍ෦ͳͲͷԋࢠΛఆɼ

ܭͰΔΑʹॻԼඞཁΔɽͷઅͰ M ͷԋࢠʹड़ɼମܭ

ͷΊʹඞཁͳ४උΛߦɽ

ͷઅͰɼલઅͰఆ M ͷ distribution L L ʹରɼύϥυϧϘʔίϗϞϩδʔ

ΛఆΔɽΊʹɼM ͷ TM ʹରɼ௨ৗͷΛఆΔɽk શͷ

ରশςϯιϧͷஅ Γ(andkTM) ΛɼAk(M) ͱΔɽΔͱɼTM = Loplus L ͰΔͱΒɼ

Γ(L) ͱ Γ(L) ʹΛఆͰΔɽArs(M) Λ Γ((andrL) and (andsL)) ͱΔͱɼAk(M)

ҎԼͷΑʹͰΔɽ

Ak(M) =-

k=r+s

Ars(M) (47)

ɼͷΛىΑͳਖ਼४Өԋࢠͱ

πrs Ar+s(M)rarr Ars(M) (48)

ΛఆΔɽπrs ɼP P ʹΑ༠ಋΕΔɽମతͳදҎԼͷΑʹ༩ΒΕΔɽɼ

TM ͷΛΒӨԋࢠΛɼҎԼͷΑʹఆΊΔɽ

P =

0 00 1

$ P =

1 00 0

$ (49)

Βɼͱ r = 2 s = 0 ͷΛߟΔɽT isin A2(M) ɼදͰҎԼͷΑ

ʹॻΔɽ

TMN =

tmicroν t ν

micro

tmicroν tmicroν

$ (410)

ɼπ20 P ʹΑҎԼͷΑʹఆΕΔɽ

PMKPN

LTKL =

0 00 1

$tmicroν t ν

micro

tmicroν tmicroν

$0 00 1

$

=

0 00 tmicroν

$ (411)

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

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[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

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[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

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[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

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[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

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[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

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Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

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[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

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[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

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meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

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[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

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tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

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[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

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[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

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[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

36

tmicroν A20(M) ͷͰΔɽΕʹΑΓɼπ20(TMN ) = tmicroν ༠ಋΕΔɽಉʹɼπ11π02

ఆՄͰΔɽ

ɼLʹ ͱ L ΛҎԼͷΑʹ༩ΔɽࢠΔඍԋ༺ʹ

d Ars(M)rarr Ar+1s(M)

d Ars(M)rarr Ars+1(M) (412)

ΕΒͷඍ༻ૉɼύϥυϧϘʔ༻ૉͱݺΕɼ௨ৗͷෳૉଟମͷυϧϘʔ༻ૉ

ͱಉɼҎԼͷΛɽ

d2 = 0 dd + dd = 0 d2 = 0 (413)

ͷΑͳύϥυϧϘʔ༻ૉͷϕΩΒɼύϥυϧϘʔίϗϞϩδʔΛఆͰΔɽՃ

ɼA isin Γ(L) ͱ α isin Γ(L) ʹରΕΕ෦ΛఆͰΔɽ

ιA Ars(M)rarr Arminus1s(M)

ια Ars(M)rarr Arsminus1(M) (414)

ΑɼҎԼͷΑʹೋछͷ Lie ඍΛఆͰΔɽξ isin Ars(M) ʹରɼ

LAξ = (dιA + ιAd)ξ Lαξ = (dια + ιαd)ξ (415)

ΕͰɼM Ͱ Vaisman ѥΛݱΔʹඞཁͳԋࢠΛఆͰɽ

ܭΔɽύϥΤϧϛʔτߟඪදͰॻԼΛΛɼΕΒͷԋʹ η ʹΑɼ

A10(M) timesA01(M) ͷ Cinfin(MR)-ઢܕଘࡏΔɽ⟨⟨α A⟩⟩ΛରশରΛͱΔɼͱ

ͱఆΔɽϕΫτϧͱ 1 ͷܗ ⟨middot middot⟩ͱΔɼL ͱ L ରͰͳͱɼର

শରΛΔͱͰΔɽಛʹɼα isin A0s(M) A1 As isin A10(M) ͱΔͱɼ

⟨⟨α A1 and middot middot middot andAs⟩⟩ = α(A1 As) (416)

ಉʹɼA isin Ar0(M)α1 αr isin A01(M) ͱΔͱɼ

⟨⟨α1 and middot middot middot and αs A⟩⟩ = A(α1 αr) (417)

ɼα isin A0s(M) ͷͱɼιAα A0sminus1(M) ʹΕΔɽΕͰɼpairing ⟨⟨α A⟩⟩ʹ

ɼΓ(andrL)ج Β Γ(andrminus1L) ΛಘΔɼΓ(andsL) Β Γ(andsminus1L) ΛಘΔΛنఆͰ

ɽΕɼड़ͷ෦ͷఆͷͷͰΔɽ෦ɼpairing ΛମతʹҎԼͷΑ

ʹॻԼΔɽ

ιAα(A1 and middot middot middot andAsminus1) = α(AA1 Asminus1) (418)

ಉʹɼA isin Ar0(M) ͷͱɼιαA Arminus10(M) ʹΕΔɽΑɼ

ιαA(α1 and middot middot middot and αrminus1) = A(αα1 αsminus1) (419)

ɼα isin A0s(M)β isin A0bull(M) A isin Ar0(M) B isin Abull0(M) ͱͱɼιA ια ΕΕ

ҎԼͷΑʹ༻Δɽ

ιA(α and β) = (ιAα) and β + (minus1)sα and ιAβ (420)

ια(A andB) = (ιαA) andB + (minus1)rA and ιαB (421)

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

37

44 DFTʹΔ Vaismanѥ

ͷઅͰɼલઅͰͷ༰Λ༻ɼC ʹΔɽॳΊߏఆΔѥހ

ʹɼഒԽΛύϥΤϧϛʔτଟମ M ʹΑہॴඪܥΛ༻ܗͰମతʹنఆΔɽ

ɼલઅͰ L L ΛಘΔͷʹ༻Ө P P ͳͲΛߦͰදΔɽɼL L Lie ѥ

ͰΔͱΛͰɼ3 ষͰ༩ ඍԋࢠͳͲදΔɽɼ

L ͱ L ͷʹ derivation ͳͱΛɼ߆ڧଋͱ derivation

ͷΛௐΔɽ

DFT ΕΔഒԽͱɼ2Dݱʹ ͷฏୱͳύϥΤϧϛʔτଟମݩ M Λ༻Δ [26]ɽ

M ͷہॴඪܥΛ xM (M = 1 2D) ͱΔͱɼ TM partM ʹΑுΒΕΔɽલ

ͷઅͰड़௨ΓɼTM Өԋࢠ P P ʹΑ L L ʹΕΔɽΕʹԠೋछͷ

༿ߏ F ͱ F ΛఆͰͱΒɼM ͷඪ xM ɼF ͷ༿ʹԊඪ xmicro ͱ F ͷ༿

ʹԊඪ xmicro ʹΔͱͰɽΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞMpartM isin Γ(TM)

ҎԼͷΑʹɼL ଆͷͱ L ଆͷʹॻΔɽA isin Γ(L)α isin Γ(L) ͱΔͱɼ

ΞMpartM = Amicro(x x)partmicro + αmicro(x x)partmicro (422)

ɼύϥΤϧϛʔτଟମʹɼఆΒඞ neutral ܭ η ଘࡏΔɽ

ηMN =

0 11 0

$ (423)

Neutral ɽηࢦΛܭͷਖ਼ʹͳΔͱෛʹͳΔಉͳݻɼͷܭ ɼ

DFT ʹΔ O(DD) ෆมͳߏ (25) ʹରԠΔɽɼη ʹΑΓ L rarr L ༠ಋ

ΕɼഒԽϕΫτϧͷͷԼ ηMNAN = AM ΛنఆΔɽͷԼΛߟΔͱɼL ͱ L ͷର

ଋ Llowast ͱͷʹಉܕ (46) ଘࡏΔͱΘΔɽη ʹΑɼഒԽϕΫτϧͷͷ

η(middot middot) = (middot middot)+ ΛఆͰΔɽɼXY isin Γ(L) ʹରɼ⟨XY ⟩ = 0 ͰΔɽL ʹ

ಉͰΔɽɼL L ΕΕ TM ͷେͳ෦ଋͰΔɽ

ύϥΤϧϛʔτଟମ M ͰɼL L involutive ͳͷͰɼ ΕΕดࢠͱ Lie

ΛఆͰΔɽɼLހ ͷ Lie ΛҎԼͷΑʹఆΔɽABހ isin Γ(L) ͱͱɼ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro (424)

[middot middot]L Jacobi Λຬɼ Leibniz ଇΛຬɽΒʹɼanchor ͱͳଋ

ρL = idL Λ༻ΔͱɼL Lie ѥͱͳΔɽL ʹಉʹɼLie ѥΛఆͰΔɽΕ

ͰɼM ΕΔݱʹ Lie ѥ L L Λମతʹ༩ΒΕɽ

ɼ2ʹ ষͰ multi vector ΛಋೖɼLie Λހ Schouten-Nijenhuis ҰൠԽΑʹɼʹހ

[middot middot]L جʹ Schouten-Nijenhuis ΛදͰ༩Δɽɼkހ ͷ multi vector ͱ

ɼA isin Γ(andkL) ΛҎԼͷΑʹ༩Δɽ

A =1

kAmicro1middotmiddotmiddotmicrokpartmicro1 and middot middot middot and partmicrok (425)

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

38

ldquoodd rdquoඪ ͱɼζmicro = partmicro ΛಋೖΔɽζ άϥεϚϯͱѻͱͰɼpartpartζ ӈඍ

ͱͳΔɽͳΘɼ

part

partζmicron

(ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok) = (ζmicro1 middot middot middot ζmicron middot middot middot ζmicrok)

larrminuspart

partζmicron

= (minus1)kminusnζmicro1 middot middot middot ζmicron middot middot middot ζmicrok (426)

ɼζmicron ζmicron ΛফڈΔɼͱҙຯͰ༻ɽζ Λ༻Δͱɼk-vector A ҎԼͷΑ

ʹॻΔɽ

A =1

kAmicro1middotmiddotmiddotmicrokζmicro1 middot middot middot ζmicrok (427)

ΔͱɼSchouten-Nijenhuis ҎԼͷΑʹॻΔɽAހ isin Γ(andpL) B isin Γ(andqL) ͱͱɼ

[AB]S =

part

partζmicroA

$partmicroB minus (minus1)(pminus1)(qminus1)

part

partζmicroB

$partmicroA (428)

ಉͷखॱͰɼ[middot middot]L ΛுɼΓ(andkL) ΛҾͱΔ Schouten-Nijenhuis ހ [middot middot]lowastS ఆ

ͰΔɽҰൠʹɼଟମͷ multi vector ɼSchouten-Nijenhuis Αʹހ Gerstenhaber

Λͳ [58] ɼϕΫτϧଋ V ͷ Lie ѥ V rarrM ͱɼmulti vector Γ(andpV ) ʹΑΔ

Gerstenhaber ՁͰΔ [59]ɽ

k-vector Λ (k+1)-vector ʹͷɼલઅͰఆ ύϥυϧϘʔ༻ૉ d andkLrarr andk+1L

ͰΔɽd ҎԼͷΑʹఆΕΔɽX isin Γ(andkL)α isin Γ(L) ͱΔͱɼ

dX(α1 αk+1) =k+1)

i=1

(minus1)i+1ρL(αi)(X(α1 αi αk+1))

+)

iltj

(minus1)i+jX([αiαj ]lowastSα1 αi αj αk+1) (429)

αi i ൪ͷ α ΛফڈΔͱҙຯͰΔɽಛʹɼہॴඪΛ༻Δͱɼd k-vector

X ʹରҎԼͷΑʹ༻Δɽ

dX =1

kpartmicroXν1middotmiddotmiddotνk(x x)partmicro and partν1 and middot middot middot and partνk (430)

Ͱɼಛʹ k = 1 ͷΛܭΔͱͰɼd ͱ [middot middot]lowastS ΔͱΛΔɽd ͷఆ

(429) Βɼk = 1 ͷͱ

dA(α1α2) = (minus1)2ρL(α1)(A(α2)) + (minus1)3ρL(α2)(A(α1)) + (minus1)3A([α1α2]lowastS)

= ρL(α1)(A(α2))minus ρL(α2)(A(α1))minusA([α1α2]lowastS)

ɼA isin Γ(L)α1α2 isin Γ(L) ͰΔɽΑɼຊઅͷલͰ༩ύϥΤϧϛʔτଟମ

ͷہॴඪදΛ༻Δͱɼಛʹ ρlowast(α1) = α1micropartmicro ͱॻΔͱʹҙΕ

A([α1α2]lowastS) = ρL(α1)(A(α2))minus ρL(α2)(A(α1))minus dA(α1α2)

= α1micropartmicro(Aνα2ν)minus α2ν part

ν(Amicroα1micro)minus (partmicroAν minus partνAmicro)α1microα2ν

= Amicro(α1ν partνα2micro minus α2ν part

να1micro)

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

39

ɼd ͱ [middot middot]lowastS ΔͱΘΔɽಉͷखॱͰɼd ͱ [middot middot]S Δͱ

ΊΒΕΔɽ

ɼLieʹ ඍΛہॴඪදΔɽ෦ͷΑʹ༻ΔɽAB isin A10(M)αβ isinA01(M) ͱΔͱɼ

ιAβ = Amicroβmicro ιAdβ = (Aνpartνβmicro minusAνpartmicroβν)partmicro

ιαB = αmicroBmicro ιαdB = (αν part

νBmicro minus αν partmicroBν)partmicro (431)

Δͱɼ(415) Ͱఆ Lie ඍҎԼͷΑʹॻΔɽ

LAβ = (dιA + ιAd)β

= d(ιAβ) + ιA(dβ)

= d(Aνβν) + ιA(partmicroβν partmicro and partν)

= [(partmicroAν)βν +Aνpartmicroβν ]part

micro +Amicropartmicroβν partν minusAνpartmicroβν part

micro

= (Aνpartνβmicro + βνpartmicroAν)partmicro (432)

LαB = (dια + ιαd)B

= d(ανBν) + ια(part

microBνpartmicro and partν)

= [(partmicroαν)Bν + αν part

microBν ]partmicro + αmicropartmicroBνpartν minus αν part

microBνpartmicro

= (αν partνBmicro +Bν partmicroαν)partmicro (433)

45 DerivaitonͷΕ

ΕͰͷΒɼύϥΤϧϛʔτଟମ M ʹରͳͷ Lie ѥ (L [middot middot]L ρL d)ͱ (Llowast [middot middot]Llowast ρLlowast dlowast) ΛہॴඪදͰಘΔͱͰɽΕΒͷѥΛ༻ɼલͷষͰ

ఏ double ԽΔʹΑಘΒΕΔߏ (L L) ΛɽલͷষͰड़௨Γɼ

Lie ѥ Lie ѥ (L [middot middot]L ρL d) ͱɼL ͱରͳ Lie ѥ (Llowast [middot middot]lowastL ρL dlowast) ͷʹ

derivation ΔͱʹఆͰΔɽderivation ҎԼͷΑͳͰΔɽ

dlowast[XY ]S = [dlowastXY ]S + [X dlowastY ]S X Y isin Γ(andbullL) (434)

લઅͷ (428) ͰɼSchouten-Nijenhuis ඪʹΛදͷͰɼύϥΤϧϛʔτଟମހ

ΛΒΘʹ༩ Lie ѥ L L ʹରɼderivation ΔͲΊ

ΒΕΔΑʹͳɽࡍʹ (434) ͷࠨลΛܭΔͱɼͷՌಘΒΕΔɽ

d[AB]S = partmicro[AB]νSpartmicro and partν

= partmicro(AρpartρBν minusBρpartρA

ν)partmicro and partν

= (partmicroAρpartρBν +Aρpartρpart

microBν minus partmicroBρpartρAν minusBρpartρpart

microAν)partmicro and partν (435)

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

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calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

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prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

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Physique theorique 53 (1990) 35

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Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

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no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

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Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

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[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

40

ଓɼ(434) ͷӈลΛܭΔͱҎԼͷՌಘΒΕΔɽ

[dAB]S =

part

partζρdA

$partρB minus (minus1)0

part

partζρB

$partρdA

= (partmicroAρζmicro minus partρAmicroζmicro)partρBνζν minusBρpartρpart

microAνζmicroζν

= (partmicroAρpartρBν minus partρAmicropartρB

ν minusBρpartρpartmicroAν)partmicro and partν

[A dB]S = minus[dBA]S

= minus(partmicroBρpartρAν minus partρBmicropartρA

ν minusAρpartρpartmicroBν)partmicro and partν (436)

ɼderivation ͷΑʹΕΔɽ

d[AB]S = [dAB]S + [A dB]S + (partρAmicropartρBν + partρBνpartρA

micro)partmicro and partν (437)

ӈลͷଘࡏΒɼύϥΤϧϛʔτଟମʹݱΕΔͷ Lie ѥ L ͱ L(≃ Llowast) ͷ

ʹɼderivation ͳͱΘΔɽɼ(L L) Lie ѥΛͳͳ

ɽՃɼ 3 ষͰ௨ΓɼL ͱ L ͷ double Loplus L ʹΑಘΒΕΔߏ Vaisman ѥ

ͰΔɽͷ Vaisman ѥͷ anchor ρ = ρL + ρLɼඍԋࢠ D = d+ d ͰఆΕ

ΔɽɼඇୀԽͳઢܗܕɼରশରΛ༻ҎԼͷΑʹఆΕΔɽΞi isin Γ(TM) ͱ

ͱɼ

(Ξ1Ξ2) = (A+ α B + β) =1

2

⟨⟨α B⟩⟩+ ⟨⟨β A⟩⟩

(438)

ɼVaisman ఆΔɽΕɼDFTʹܗҎԼͷހ ͷ C ͰΔɽܗͱશಉހ

[Ξ1Ξ2]V = [A+ α B + β]V = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (439)

(Loplus L [middot middot]C ρV (middot middot)) ͷɼVaisman ѥͱͳΔɽ

Ͱɼ(437) ͷӈล 3 ʹணΔɽͷɼderivation ͷΕΛݱΔͰ

Γɼ

partρAmicropartρBν + partρBνpartρA

micro = ηKLpartKAmicropartLBν (440)

ͱॻͱͰΔɽΕɼʹ߆ڧଋ (221) ΛͱͰফΔܗͷͰΔɽ

Γɼ߆ڧଋΛͱͰɼL ͱ L ͷʹ derivation ΔΑʹͳΔɽΑ

ɼΕΒͷϖΞ (L L) Lie ѥΛఆΔɽɼ3 ষͰͷΒɼهͷ L oplus L

ʹର derivation Λͱจݙ [35] ͷ Courant ѥͱͳΔɽΑɼDFTʹΔ

ɼLݯىଋͷతͳ߆ڧ ͱ L ͷͷ derivation ͰΔͱݴΔɽΕɼจ

ݙ [36] ʹΔɼpre-DFT ѥʹର߆ڧଋΛͱ Courant ѥಘΒΕΔͱ

ఠͱகΔɽࢦ

ҰͰɼC ଋΛͱɼ(22)߆ڧରʹހ Ͱ [17] ͷ Courant ಘΒހ

ΕΔͷɼ2 ষͷޙͰड़௨ΓͰΔɽจݙ [35] ͷ Courant Βހ (22) ͷ Courant

ଋͱ߆ڧඞཁͰΔɽߟΛಘΔʹɼগހ derivation ͷͷਤ 42

ʹ΄ɼͷઅͰҾଓΛߦɽ

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

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[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

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[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

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mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

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[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

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[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

41

46 ҰൠزԿͱͷ

ͷઅͰɼલઅͰ L L ʹΑΔήʔδͱɼҰൠԽزԿͱͷΛΔɽ

DFT ͷήʔδରশΛنఆΔ C ԿʹΔزɼύϥΤϧϛʔτހ Vaisman ͱҰހ

கΔͱɼ[15 25] ͰࢦఠΕΔɽC ͷඍಉ૬ͱɼBඪมɼҰൠހ ͷ

U(1) ήʔδมΛಉʹѻɼO(DD) มͱͳΔΑʹΜͷͰΔɽVaismanڞ Λހ

ύϥΤϧϛʔτଟମͰݱΔʹɼ߆ڧଋΛཁٻΔඞཁͳɽΑɼ

Vaisman ଋΛͳঢ়ଶͰͷ߆ڧހ DFT ͷήʔδରশɼͳΘ ldquooff-shellrdquo Ͱ

ͷήʔδରশΛهड़ΔɽزԿతʹɼഒԽΛهड़ΔΊʹύϥෳૉߏΛಋೖ

ΔΊɼଋ TM L oplus L ͷܗʹͰɼL ͱ L ΕΕ Lie ѥͱͳΔɽL L

ଞʹΑΛɽL L distribution ͰΓɼDirac ͰΔɽLߏ L ͷހ

involutive ͰΓɼFrobenius ͷఆཧΒՄͰΔɽɼL L ΕΕ M ͷ

༿ߏ F F ͷଋͱͳΔɽxmicro ͰಛΒΕΔཧతɼ༿ߏΒ༿ΛҰຕબͿ

ͱͰఆΊΒΕΔ (༿ΛબͿͱͰɼxmicro ఆʹͳΔ)ɽɼύϥΤϧϛʔτܭ η ʹΑఆ

ΕΔಉܕΒɼL = TF ≃ Llowast = T lowastF ͳͷͰɼL ͷϕΫτϧɼLlowast ͷ 1 ͰࢹͱಉҰܗ

ΔɽɼL ଆͷ Lie ϕΫτϧଆͷήʔδύϥϝʔλހ ξmicroi ʹΑҰൠඪมΛ

ɼLࢧ ଆͷ Lie ހ 1 ଆͷήʔδύϥϝʔλܗ ξimicro ʹΑ B ͷήʔδมΛࢧ

ΔɽҰൠʹɼ1 ଆͷܗ Lie ހ 0 ʹͳΒͳͱΒɼO(DD) มͳڞ B ͷήʔδม

ɼඇՄͳ ldquooff-shellrdquo ରশͱுΕΔɽ

C-bracket

strong constraint part = 0

c-bracket

Vaisman bracket

derivation condition (14)

Courant bracket in (21)

[middot middot]lowast = 0

c-bracket

ਤ 42 Λɽͱͷހ

Ͱɼ߆ڧଋΛޙͷͱΛߟΈΑɽͷͱɼDFT ͷήʔδ C ހ

ͰดΔɽɼC ଋ߆ڧड़ΔʹɼهఆΕΔରশΛنΑʹހ

ඞຬΕΔඞཁΔɽ߆ڧଋΛղ؆୯ͳɼwinding ඪʹΔඍ

Λ 0 ʹΔ (partlowast = 0) ͱͰΔɽͷͱɼ1 ଆͷܗ Lie ހ 0 ͱͳΓɼC ހ (22)

ͷ c ހ [middot middot]c ʹมܗΔ (ਤ 42)ɽΕɼDFT ήʔδύϥϝʔλʹର ଋ߆ڧ

Λɼldquoon-shellrdquoɼͳΘཧతͳ෦ΛఆΔͱʹΑΓɼldquooff-shellrdquo ͰඇՄ

B ͷήʔδରশ ldquoon-shellrdquo ͰՄʹͳΔͱΛҙຯΔɽͷҙຯͰɼC ހ

ͷ ldquoon-shellrdquo ൛ͱରԠΔͷ c ͱΔɽతʹɼcހ ͷखॱͰಘΒΕΔɽހ

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

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[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

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[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

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[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

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[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

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[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

42

ɼVaisman ରɼderivationʹހ (320) ΛɽɼL ଆͷ Lie Λހ zero

bracket ([middot middot]L = 0) ͱΕΑɽܭʹΑతʹΑʹɼC ఆΔنހ

Vaisman ѥʹର derivation Λͱɼͷߏจݙ [35] ͷ Courant ѥͱͳ

Δɽɼ (L L) શମͰ Lie ѥΛɽΒʹɼ[middot middot]L = 0 ͱΕ C ހ (22)

ͷ c มԽΔɽͷʹހ c ΕΔݱʹԿزɼҰൠԽހ Courant ͳΒͳ΄ʹހ [11]ɽ

ύϥυϧϘʔίϗϞϩδʔΛنఆͱͰɼ߆ڧଋΛຬΑͳɼldquoon-shellrdquo ͷ

DFT ήʔδύϥϝʔλɼύϥυϧϘʔ༻ૉΛ༻Δͱ 0ɼͳΘύϥਖ਼ଇ

(para-holomorphic) ͳͱͳΔಛΔɽ

para-holomorphic dΦ = 0 (441)

Φ ɼഒԽͷҙͷήʔδύϥϝʔλͰΔɽΕɼΦ F ͷ༿ʹݶΕΔͷ

ʹɼύϥυϧϘʔ༻ૉ d d ͷೋछΔͱΒɼ߆ڧଋͷղͱҰҙͰ

ͳɽ߆ڧଋͷҰͷղɼΦ ύϥਖ਼ଇ (anti-para-holomorphic) ͳͱͳΔ

ͰΔɽ

anti-para-holomorphic dΦ = 0 (442)

ύϥਖ਼ଇͳɼF ͷ༿ʹݶΕΔɽΓɼxmicro ΛఆʹΔͱͰఆͰΔ winding

ʹଘࡏΔɽDFT ʹΔӡಈఔͷղͱɼwinding ʹґଘΔݱΕ

Δ [60ndash64]ɽ

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

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(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

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etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

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and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

43

ୈ 5ষ

ͱΊ

ຊमจͰɼݭཧʹΔ T ରΛനʹΜॏཧͰΔ Double Field Theory

ʹɼಛجʹ DFT ͷͷزԿతͳඳɼήʔδରশʹݱΕΔߏͳͲͷత

ͳଆ໘ʹɽ

ୈ 2 ষͰɼDFT ʹɼඞཁݶͳ෦Λ؆୯ʹͱΊɽഒԽඪΛ

ఏɼܭɼجຊͱͳΔ DFT ͳͲΛհɽɼҰൠԽ Lie ඍͷܭΒɼಛ

ʹ DFT ΛੜΔߏΕΔήʔδݱʹ C ड़ɼຊจͷओͰΔʹࡉʹހ

Vaisman ѥͱߏఏΕΔͱΛհɽɼཧత section ʹ

ड़ɼ߆ڧଋΛલޙͰͷ C ͷΔͷมԽɼDFTހ ͷزԿతͳΈͷ

ҧʹٴݴɽ

ୈ 3 ষͰɼࠓతͳΒɼجతͳߏͰΔ Lie Λग़ͱɼ

ঃʑʹߏΛҰൠԽͱͰ Vaisman ѥͷݫͳఆΛ༩ɽɼDrinfelrsquod

double ͱʹணɼVaisman ѥɼશମͰ Lie ѥΛͳͳΑͳ Lie ѥ

ͷϖΞ (EElowast) ͷ double ʹΑߏͰΔͱΛɽՃɼ[35] ͷ Courant ѥ

ʹΔʹج Vaisman ѥͷ Dirac ΕݱΈɽʹΑࢼʹΑΔΛߏ

Δͷ Dirac ߏ L L ≃ Llowast ͷʹ derivation ɼL L ≃ Llowast ͷશମͰ

Lie ѥΛͳͳͱΊΒΕɽ

ୈ 4 ষͰɼୈ 3 ষͰΒʹ Vaisman ѥͷ double Λ༻ɼୈߏ 2 ষͰհ

DFT ͷήʔδߏʹߟɽɼഒԽͷతͳඳͱɼ2D ͷݩ

ύϥΤϧϛʔτଟମ M ΛಋೖɽM ɼύϥෳૉߏ K ΛͱΒɼଋ TM Λ

େతͳ෦ଋ L L ͱવʹΔ (TM = Loplus L)ɽΕʹɼTM ͷஅ໘Ͱ༩Β

ΕΔഒԽϕΫτϧ Ξ = ΞMpartM ΛɼΞ = Amicropartmicro +αmicropartmicro ʹɽͰɼA isin Γ(L)α isin Γ(L)

ͰΔɽɼL L ΕΕ Lie ѥͰΔͱΛΊɽL L ʹରΔΛ

ΒɼLܭͳࡉஙɽߏఆɼύϥυϧϘʔίϗϞϩδʔΛن ଆʹ༻Δඍԋ

ࢠ d ɼLie ހ [middot middot]L ʹର derivation ΛຬͳͱΛɼL L Ұൠʹશ

ମͰ Lie ѥΛͳͳͱΛɽୈ 2 ষͰͷΒɼC ߏఆΔنހ

Vaisman ѥͰΔͱΛվΊɽɼDFTͷ߆ڧଋͷతͳݯىɼ

derivationͰΔͱΛɽ

ຊमจʹಘΒΕͳՌɼओʹతͳΒɼVaisman ѥ

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

44

ͷ Lie ѥͷ double ͰಘΒΕΔͱͱͱɼͷཧతͳԠ༻ͱɼ߆ڧଋͷ

తͳݯى derivation ͰΔͱΒʹͱͷͰΔɽͷՌɼ[13] ʹ

ग़൛ࡁΈͰΔɽ

ɹ

ͷలޙࠓ

ഒԽزԿΛ௨ DFT ͷཧߏΛཧղɼT ରʹ༩ΔӨڹΛΓͱ

ͱɼڀݚͷతͰΔɽޙʹɼDFT ഒԽزԿʹΔຊจʹऔΓҎ

ͷɼͷతʹԊޙࠓऔΓΈΛհɽͰʹड़௨

ΓɼDFT ʹ C ΕݴಛͷɼͷήʔδରশͰΓɼހ ແݶখͷ

(infinitesimal) ήʔδมͰΔɽΕʹରɼݶͷ (finite) ήʔδมʹߦΘ

ΕΔɽදతͳจ [19 65ndash71] ͰΔɽಛʹతͳΒݴͱɼLie Lie

ΒߏͰΔͱΒਪΔʹɼDFT ͷݶήʔδมɼVaisman ѥͷ ldquo

(integration)rdquo ΒΕΔɽগͳͱɼLieߟΑಘΒΕΔͱʹ ѥͷʹΑ

Poisson-Lie ಘΒΕΔ [72] ͱɼͷҰൠԽͰ Lie () ѥͷʹΑ (Poisson-)

Lie ѥ (groupoid) ಘΒΕΔͱ [4344] ͰʹΒΕΔɽɼΔछͷ Courant ѥ

ͷʹΑѥݱ1ΕΔΖɼͱ༧ଌΔ [7475]ɽͷΑͳɼʮʹΑ

ɼ Δ (ѥ) Β (ѥ) ΛಘΔʯͱɼldquocoquecigrue problemrdquo

2ͱݺΕΔɽɼVaisman ѥʹରΔ coquecigrue problem ղΕɼΕ

తʹڵຯՌͰΔͰͳɼDFT ͷήʔδରশͷزԿతͳݯىΛ༩ΔΖɽ

ɼ߆ڧଋΛͱɼDFT ͷดήʔδΛಘΔΊͷͰΓɼඞ

ཁͰͳɽΑΓҰൠతʹɼ߆ڧଋΛඞཁͱͳ DFT ηοτΞοϓߟΒΕ

Δ [76]ɽͷΑͳঢ়گͰɼຊमจͰѻ Vaisman ѥΑΓॏཁͳҙຯΛ

Ζɽ

ഒԽΛతʹهड़Δʹɼݱஈ֊ͰʑͳߟΒΕΔɽޙࠓΕ

Βͷͻͱͷਖ਼ͳهड़ʹߜΓࠐΕͷɼΔෳͷՁͳهड़ݟ

ΔͷఆͰͳɼΕʹΑঃʑʹΕΕͷͷཧΕ

ΖɽຊमจͰɼഒԽͷݱͱύϥΤϧϛʔτଟମΛ༻ɽΕͱಠ

Ξϓϩʔνͱ graded ԿΛ༻ΔΔز [21 77 78]ɽΕΒͷͷɼ

Courant ѥͱ QP ଟମՁͰΔͱ [79] ʹདྷಉͳͷͰΔɽC ߏͷހ

ͱ derived ހ [21 80] ͷΛௐΔͱɼgraded ԿͷزԿͱύϥΤϧϛʔτز

ཧΕΔͰΔɽɼύϥΤϧϛʔτزԿͷுͱɼϘϧϯزԿߟҊΕ

Δ [26 27]ɽϘϧϯزԿͰɼᎇۂͱͷΛهड़ΔزԿΛఆΔࢼ

ΈߦΘΕΓɼଞͷͱൺҰาਐΜͰΔɽϘϧϯزԿͰఆ DFT

ʹͲͷΑʹݱΕΔνΣοΫͰΕɼഒԽͷతͳهड़ͱϘϧϯزԿͷ৴ጪ

1 ѥɼతʹͱ୯ҐݩʹΔݩΛෳΑͳɼΛுߏͰΔɽࡉɼ[73] ͳͲʹɽѥ (groupoid) ʹɼจதͰΕҎࡉʹٴݴͳɽ

2 coquecigrue ɼ16 ͷϑϥϯεจͰΔʮΨϧΨϯνϡΞل (Gargantua) ޠʯʮύϯλάϦϡΤϧ(Pantagruel)ʯʹݱΕΔͷੜΛࢦɽ

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

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[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

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mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

45

Ζɽ

T ରͷுͰΔ Poisson-Lie T-duality [81 82] ͱͷΕΔɽtype II

ॏཧΛϕʔεʹଟମͰ DFT ΛߏஙΔͱɼPoisson-Lie T-duality ཧͷനͳର

শͱͳΔɽຊमจͰड़ double ʹΑΔ Vaisman ѥͷߏஙఈͷछΛΘ

ʹͲͷΑͳՌಘΒΕΔΊɽগͳͱݱଟମͰΔͷͰɼߦ

ɼ[35] ͷ Drinfelrsquod double ʹΑΔ Courant ѥݭཧʹ Poisson-Lie T-duality Λ

ཧղΔΊͷݤͱͳΔ [83ndash86]ɽ

ɼDFTʹޙ ମͷૉͳுͱɼରশΛ T ରΒ U ରʹΔͱ

ΒΕΔɽDFTߟ ΛϕʔεʹɼU ରΛനͳରশͱΔཧͱ Exceptional Field

Theory (EFT) ҊΕߟ [87]ɽEFT ͷήʔδରশ [88] ͳͲͰΕΔɼDFT

ҎʹزԿతͳඳෆͰΔɽૉʹߟΕɼEFT DFT ͷுͰΔΒɼEFT

ͷزԿతඳ ഒԽزԿΛแΔͰΔɽԾʹഒԽزԿମతʹهड़Ͱ

ͱɼΒʹΕுΕΔՄΔɽޙࠓڵຯ

ΈͰΔɽ

෦ΒҾଓஸೡʹࢦಋࢣߨͷࠤʑ৳ઌੜʹײਃɽࢲ

ςʔϚͱग़ձͷɼΔͱڀݚͷࠓ 3 લʹʮҰൠԽزԿͱ໘നͳࡐ

ΔͷͰɼΒଔڀݚۀͰΈͳʯͱհΛडͱͰɽ

ͷΓऔΓͳΕࠓͷࢲଘࡏΜɽڞಉڀݚͰΓɼଚܟઌഐͰΔത

ೋͷԘଠΜʹײਃɽӃʹਐɼमऴͰʹೋਓ

ͱڞಉڀݚͰจग़ΔͱເʹࢥͳͰɽΕͰͷਓੜͰҰ൪خ

ग़དྷͰɽվΊޚਃɽՃɼࠒʹͳΔཬେΑͼ

ଞେͷॾઌഐɼಉڃੜɼޙഐʹײɽޙʹɼओΛΊେ

ਃɽײʹतڭେͷංࢠ

ͰɼਐॳͳΑͳɼඇৗʹೱͳ म 2 Λա

ͱͰɽͱͰɽΘʹͱҰੜͷๅͰɽͷΑʹ

ॻͱɼΔͰେӃੜऴΘΔͷΑͳࡨΛɼԿಉॴͰ

ʹͳΓɽޙΒͰɼΕஅͱࢥΔΑʹؤுΓɽվΊΑΖ

ɽئ

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

46

A

ϊʔτܭ

A1 Cހͷ Jacobiatorͷಋग़

C ͷހ Jacobiator (223) ͷಋग़ΛߦɽC ͱހ D ༺ΔͱΛʹҎԼͷΑހ

Δɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

MC +

1

2ηMNpartN (ηKLΞ

K1 ΞL

2 ) (A1)

ɼC ހ D ΑҎԼͷΑʹఆͰΔɽʹހ

[Ξ1Ξ2]MC =

1

2([Ξ1Ξ2]

MD minus [Ξ2Ξ1]

MD ) (A2)

ҎͷΒɼ D ɼͷՌΛܭʹހ C Ԡ༻ΔɽDʹހ ҎԼͷހ

ΑʹॻΔɽ

[Ξ1Ξ2]MD = [Ξ1Ξ2]

M + ΞK2 partMΞ1K (A3)

ɼ[middot middot] ͷࢠͰΔɽҰൠʹɼLeibniz ҎԼͷΑʹ༩ΒΕΔͱʹج

ɼD ΔɽܭͷΛހ

[Ξ1 [Ξ2Ξ3]] = [[Ξ1Ξ2]Ξ3] + [Ξ2 [Ξ1Ξ3]] (A4)

(A3) ͷࠨลɼҎԼͷΑʹܭͰΔɽ

[Ξ1 [Ξ2Ξ3]D]MD

= [Ξ1 [Ξ2Ξ3]D]M + ηKL[Ξ2Ξ3]

KD partMΞL

1

= [Ξ1 [Ξ2Ξ3]]M + (ΞN

1 partN (ηKLΞK3 partMΞL

2 )minus (ηKLΞK3 partNΞL

2 )partNΞM1 )

+ ηKL[Ξ2Ξ3]KpartMΞL

1 + ηKL(ηIJΞI3part

KΞJ2 )part

MΞL1 (A5)

ҰͰɼ(A3) ͷҰҎԼͷΑʹܭͰΔɽ

[[Ξ1Ξ2]DΞ3]MD

= [[Ξ1Ξ2]DΞ3]M + ηKLΞ

K3 partM [Ξ1Ξ2]

LD

= [[Ξ1Ξ2]Ξ3]M + ((ηKLΞ

K2 partNΞL

1 )partNΞM3 minus ΞN

3 partN (ηKLΞK2 partMΞL

1 ))

+ ηKLΞK3 partM [Ξ1Ξ2]

L + ηKLΞK3 partM (ηIJΞ

I2part

LΞJ1 ) (A6)

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

47

ɼ(A3) ͷӈลશମͷΑʹॻΔɽ

[[Ξ1Ξ2]DΞ3]MD + [Ξ2 [Ξ1Ξ3]D]

MD

= [Ξ1 [Ξ2Ξ3]]M + ηKL([Ξ2Ξ3]

KpartMΞL1 + (ηIJΞ

I3part

KΞJ2 )part

MΞL1 + ΞN

1 partN (ΞK3 partMΞL

2 ))

+ ηKL(ΞK2 partNΞL

1 partNΞM3 minus ΞK

3 partNΞL1 partNΞM

2 ) (A7)

(A5) ΛҰ༻ཧΔͱɼD ಘΒΕΔɽҎԼͷʹހ

[Ξ1 [Ξ2Ξ3]D]MD = [[Ξ1Ξ2]DΞ3]

MD + [Ξ2 [Ξ1Ξ3]D]

MD + SCD(Ξ1Ξ2Ξ3)

M (A8)

SCD(Ξ1Ξ2Ξ3)M = ηKL(Ξ

K3 partNΞL

1 partNΞM2 minus ΞK

2 partNΞL1 partNΞM

3 minus ΞK3 partNΞL

2 partNΞM1 ) (A9)

SCD ɼ߆ڧଋΛͱফΔͰΔɽ

ͷՌΛ༻ɼC ͷހ Jacobiator ͷܭʹҠΔɽ(A2) ΒɼҎԼͷಘΒΕΔɽ

[[Ξ1Ξ2]CΞ3]C =1

2([[Ξ1Ξ2]CΞ3]D minus [Ξ3 [Ξ1Ξ2]C]D)

=1

4([[Ξ1Ξ2]DΞ3]D minus [[Ξ2Ξ1]DΞ3]D minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D)

(A10)

(A8) ͱ (A10) Βɼ

[[Ξ1Ξ2]CΞ3]C =1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3)

minus [Ξ2 [Ξ1Ξ3]D]D + [Ξ1 [Ξ2Ξ3]D]D + SCD(Ξ2Ξ1Ξ3)

minus [Ξ3 [Ξ1Ξ2]D]D + [Ξ3 [Ξ2Ξ1]D]D) (A11)

(A11) ͷ cyclic ෦ΛͱΊͳͱɼҎԼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp

=1

4([Ξ1 [Ξ2Ξ3]D]D minus [Ξ2 [Ξ1Ξ3]D]D minus SCD(Ξ1Ξ2Ξ3) + SCD(Ξ2Ξ1Ξ3) + cp) (A12)

ӈลΛɼD ͷހ Leibnitz (A8) ͷՌΛ༻ஔΔͱɼͷΑʹͳΔɽ

[[Ξ1Ξ2]CΞ3]C + cp =1

4([[Ξ1Ξ2]DΞ3]D minus SCD(Ξ2Ξ1Ξ3) + cp) (A13)

ͰɼC ͱހ D ͷހ (A1) Βɼ

[[Ξ1Ξ2]CΞ3]C = [[Ξ1Ξ2]CΞ3]D minus partbull([Ξ1Ξ2]CΞ3)+

= [[Ξ1Ξ2]DΞ3]D minus [partbull(Ξ1Ξ2)+Ξ3]D minus partbull([Ξ1Ξ2]CΞ3)+ (A14)

Ͱɼpartbull ɼηMN ͰඍԋͰΔɽɼ(Ξ1Ξ2)+ = (ηMNΞM1 ΞN

2 )2 Λҙ

ຯΔɽ(A14) ͷӈลೋҎԼͷΑʹܭͰΔɽ

[partbull(Ξ1Ξ2)+Ξ3]MD =

1

2

partN (ΞK

1 Ξ2K)partNΞM3 + (partMpartN (ΞK

1 Ξ2K)minus partNpartM (ΞK1 Ξ2K))ΞN

3

=1

2partN (ΞK

1 Ξ2K)partNΞM3 (A15)

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

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(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

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[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

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mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

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calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

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[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

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[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

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[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

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[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

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Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

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[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

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[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

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[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

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[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

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(2015) 002 [arXiv150405913 [hep-th]]

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meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

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[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

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structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

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[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

48

Ͱɼ(A14) ΛͱɼC ͰΔɽܭʹͷΑހ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A16)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A17)

ɼ

NC(Ξ1Ξ2Ξ3) =1

3

([Ξ1Ξ2]CΞ3)+ + cp

( (A18)

SCC(Ξ1Ξ2Ξ3) =1

3

[partbull(Ξ1Ξ2)+Ξ3]D minus SCD(Ξ2Ξ1Ξ3) + cp

( (A19)

ɼ

JC(Ξ1Ξ2Ξ3) = partbullNC(Ξ1Ξ2Ξ3) + SCC(Ξ1Ξ2Ξ3) (A20)

ΕͰɼC ͷހ Jacobiator ͰɽɼNCܭ Nijenhuis ςϯιϧͰΓɼSCC

ଋͰফΔͰΔɽ߆ڧ

A2 CހΒ Courantހͷ reduction

C ଋͷͱͰ߆ڧɼΕهΛͰදހ [17] ͰఏΕ Courant ހ

[X1 + ξ1 X2 + ξ2]c = [X1 X2] + (LX1ξ2 minus LX2ξ1) +1

2d0(ξ1(X2)minus ξ2(X1)) (A21)

มԽΔͱΛɽ Xi xmicro ଆɼξi xmicro ଆͷͰΔɽɼgauge ύϥϝʔλ

Ξ1Ξ2 ΛҎԼͷΑʹͰղΔɽ

ΞM1 =

αmicro

Amicro

$ ΞM

2 =

βmicro

Bmicro

$ (A22)

ͷදʹΑɼC ΒΕΔɽҎԼͷΑʹॻހ

[Ξ1Ξ2]MC = ΞK

1 partKΞM2 minus ΞK

2 partKΞM1 minus

1

2ηKL(Ξ

K1 partMΞL

2 minus ΞK2 partMΞL

1 )

= αν partνΞM

2 +AνpartνΞM2 minus βν part

νΞM1 minusBνpartνΞ

M1

minus 1

2(ανpart

MBν +AνpartMβν minus βνpartMAν minusBνpartMαν) (A23)

Ͱɼdoubled ఈͱج partM = (partmicro partmicro) ΛಋೖΔͱɼC ΒʹҎԼͷΑʹॻΔɽހ

[Ξ1Ξ2]C = [Ξ1Ξ2]MC ηMNpartN = αν part

νβmicropartmicro +Aνpartνβmicropart

micro minus βν partναmicropart

micro minusBνpartναmicropartmicro

+ αν partνBmicropartmicro +AνpartνB

micropartmicro minus βν partνAmicropartmicro minusBνpartνA

micropartmicro

minus 1

2(αν part

microBν +Aν partmicroβν minus βν partmicroAν minusBν partmicroαν)partmicro

minus 1

2(ανpartmicroB

ν +Aνpartmicroβν minus βνpartmicroAν minusBνpartmicroαν)part

micro (A24)

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

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[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

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[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

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[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

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[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

49

Ͱɼ෦ҎԼͷΑʹ Lie ͳͲͰͱΊΔɽހ

[AB]L = [AB]microLpartmicro = (AνpartνBmicro minusBνpartνA

micro)partmicro

[αβ]L =[αβ]L

(micropartmicro = (αν part

νβmicro minus βν partναmicro)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)part

micro = (βνpartmicroAν +Aνpartmicroβν)part

micro

dιAβ = d(Aνβν) = partmicro(Aνβν)partmicro = (βν partmicroAν +Aν partmicroβν)partmicro

LαB = (αν partνBmicro +Bν partmicroαν)partmicro

LAβ = (Aνpartνβmicro + βνpartmicroAν)partmicro (A25)

ͰɼL ௨ৗͷϕΫτϧʹΔ Lie ඍͰΔɽҰɼL winding ඪͷϕΫτϧ

ʹΔ Lie ඍͰΔɽಉʹɼ[middot middot]L ௨ৗͷ Lie middot]ɼހ middot]L winding ඪͷϕ

ΫτϧʹΔ Lie ͰΔɽɼCހ ʹҎԼͷΑހ partmicro ଆͱ partmicro ଆʹ

ͱΊΒΕΔɽ

[Ξ1Ξ2]MC partM = ([αβ]L)micropart

micro +Aνpartνβmicropartmicro minusBνpartναmicropart

micro + [AB]microLpartmicro + αν partνBmicropartmicro minus βν part

νAmicropartmicro

minus 1

2(2Aν partmicroβν minus (dιAβ)

micro + (dιBα)micro minus 2Bν partmicroαν)partmicro

minus 1

2((dιAβ)micro minus 2βνpartmicroA

ν minus (dιBα)micro + 2ανpartmicroBν)partmicro

=

[AB]microL + LαB

micro minus LβAmicro +

1

2(d(ιAβ minus ιBα))

micro

$partmicro

+

([αβ]L)micro + LAβmicro minus LBαmicro minus

1

2(d(ιAβ minus ιBα))micro

$partmicro (A26)

ͷܗɼCourant ΔݟʹΛΘΑހ [9]ɽ

[Ξ1Ξ2]C = [A+ α B + β]C = [AB]L + LAβ minus LBαminus1

2d(ιAβ minus ιBα)

+ [αβ]L + LαB minus LβA+1

2d(ιAβ minus ιBα) (A27)

(A27) ʹର߆ڧଋΛ (partlowast = 0) ͱɼӈลͷೋஈফ໓ɼC ހ Courant

ఏݩͷ Courant ܗͷހ [17] ʹมԽΔɽ

A3 T (e1 e2 e3)ͷ (336)ͷಋग़

ҎԼͷΛಋग़Δɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3 + ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus⟩ (336)

Εɼจݙ [35] ͷ Lemma 32 ʹΔͰΓɼT (e1 e2 e3) ΛఆʹԊܭΔ

ͱͰͱͰΔɽހͷܗಉͳͷͰɼຊจதͰఆ Vaisman ހ [middot middot]V Ͱܭ

ɼจݙ [35] ͷ Courant ހ [middot middot]c ͰܭಉಘΒΕΔɽ

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

50

ɼ([e1 e2]V e3)+ ʹΔͱɼ[middot middot]V (middot middot)+ ͷఆΑΓ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨ξ3Lξ1X2⟩ minus ⟨ξ3Lξ2X1⟩ minus ρlowast(ξ3)(e1 e2)minus

⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩ minus ⟨LX2ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A28)

ͰɼLie ඍͷʹணΔͱɼLie ඍͷଇΒҎԼͷΔɽ

⟨ξ3Lξ1X2⟩ = Lξ1⟨ξ3 X2⟩ minus ⟨[ξ1 ξ3]Elowast X2⟩⟨LX1ξ2 X3⟩ = LX1⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩

ɼจݙ [44] Ͱͷ d ͷఆΒɼ

LX1⟨ξ2 X3⟩ = ιX1d⟨ξ2 X3⟩ = ρ(X1)⟨ξ2 X3⟩ (A29)

ಉʹɼLξ1⟨ξ3 X2⟩ = ρlowast(ξ1)⟨ξ3 X2⟩ (A30)

Αɼ([e1 e2]V e3)+ ɼ(A28) ΒߋʹҎԼͷΑʹॻΒΕΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ cp

+1

2ρlowast(ξ1)⟨ξ3 X2⟩ minus ρlowast(ξ2)⟨ξ3 X1⟩ minus ρlowast(ξ3)(e1 e2)minus

+ ρ(X1)⟨ξ2 X3⟩ minus ρ(X2)⟨ξ1 X3⟩+ ρ(X3)(e1 e2)minus (A31)

Εʹɼρ(X1)(e2 e3)minus ρ(X2)(e3 e1)minus ͱ ρlowast(ξ1)(e2 e3) ρlowast(ξ2)(e3 e1)minus ΛΕΕҾ

ɼ८ճΔΛཧΔͱɼρ ͷఆΒ ([e1 e2]V e3)+ ऴతʹҎԼͷܗʹͱΔɽ

([e1 e2]V e3)+ =1

2⟨ξ3 [X1 X2]E⟩+ ⟨X3 [ξ1 ξ2]Elowast⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus

1

2ρ(e2)(e3 e1)+ (A32)

ͱɼ([e1 e2]V e3)+ ͷΛ८ճΛऔΕɼ(336) ͷӈลͱͳΔɽ

T (e1 e2 e3) equiv1

3(([e1 e2]V e3)+ + cp)

=1

2⟨[X1 X2]E ξ3⟩+ ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minuscp

+1

2ρ(e1)(e2 e3)+ minus ρ(e2)(e3 e1)+ + ρ(e2)(e3 e1)+

minus ρ(e3)(e1 e2)+ + ρ(e3)(e1 e2)+ minus ρ(e1)(e2 e3)+

=1

2⟨ξ3 [X1 X2]E⟩+ ⟨[ξ1 ξ2]Elowast X3⟩

+ ρ(X3)(e1 e2)minus minus ρlowast(ξ3)(e1 e2)minus + cp (A33)

Αɼ(336) ɽ

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

51

A4 T (e1 e2 e3)ͷ (337)ͷಋग़

ҎԼͷΛɽ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

Ε [35] ͷ Lemma 34 ʹΔͰΓɼ(middot middot)plusmn ͷఆΒΔɽ

(middot middot)plusmn ͷఆΒɼҎԼͷಘΒΕΔɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ⟨LX1ξ2 X3⟩minus ⟨LX2ξ1 X3⟩+ ⟨d(e1 e2)minus X3⟩ (A34)

Βʹɼ(A29) (A30) ΛӈลΛཧΔͱɼ

([e1 e2]V e3)minus + ([e1 e2]V e3)+ = ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus

(A35)

(A35) ͷΛ८ճΛͱΔͱɼҎԼͷΑʹͳΔɽதɼ(336) Λ༻ɽ

([e1 e2]V e3)minus + ([e1 e2]V e3)++ cp

= ([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ ρ(X1)⟨ξ2 X3⟩ minus ⟨ξ2 [X1 X3]E⟩minus ρ(X2)⟨ξ1 X3⟩+ ⟨ξ1 [X2 X3]E⟩+ ρ(X3)(e1 e2)minus+ cp (A36)

Ұ୴ɼcp ͷ෦શܭΔͱɼҎԼͷΑʹͱΔɽ

([e1 e2]V e3)minus + cp+ 3T (e1 e2 e3)

= ⟨[ξ1 ξ2]Elowast X3⟩+ 2⟨ξ3 [X1 X2]E⟩+ 3ρ(X3)(e1 e2)minus+ cp

= 4T (e1 e2 e3) + [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp] (A37)

ɼ(337)ܗͷมߦͷޙ ʹணΔΊʹແཧΓ 4T (e1 e2 e3) ΛͰΔɽ

(A37) ΛҠߦཧΕɼ(337) ͱͳΔɽ

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

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(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

52

A5 Axiom C1ͷ

Axiom C1 ͷࠨล [[e1 e2]V e3]V + cp ΛܭΔɽ[middot middot]V ͷఆΒɼ

[[e1 e2]V e3]V + cp = I1 + I2 (A38)

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + [d(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1

minus LX3d(e1 e2)minus + d([e1 e2]V e3)minus + cp (A39)

I2 = [[X1 X2]E X3]E + [Lξ1X2 minus Lξ2X1 X3]E minus [dlowast(e1 e2)minus X3]E

+ L[ξ1ξ2]Elowast+Lξ1X2minusLξ2X1+d(e1e2)minusX3

minus Lξ3 [X1 X2]E minus Lξ3Lξ1X2 + Lξ3Lξ2X1

+ Lξ3dlowast(e1 e2)minusminus dlowast([e1 e2]V e3)minus+ cp (A40)

Γ(Elowast) Λ I1ɼΓ(E) Λ I2 ͱɽI1 ͱ I2 ΄΅ಉܭͱͳΔͷͰɼI1 औΓग़

Δɽܭ

L[X1X2]E = [LX1 LX2 ]E (A41)

Λ༻Δͱɼ

L[X1X2]Eξ3 minus LX3LX1ξ2 + LX3LX2ξ1 + cp = 0 (A42)

ͳͷͰɼI1 ҎԼͷΑʹॻΔɽ

I1 = [[ξ1 ξ2]Elowast ξ3]Elowast + [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ L[X1X2]E+Lξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus LX3LX1ξ2 + LX3LX2ξ1 minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp

= [LX1ξ2 minus LX2ξ1 ξ3]Elowast minus [dlowast(e1 e2)minus ξ3]Elowast

+ LLξ1X2minusLξ2X1minusdlowast(e1e2)minusξ3

minus LX3 [ξ1 ξ2]Elowast minus Ld(e1e2)minus + d([e1 e2]V e3)minus+ cp (A43)

ͰɼLX3 [ξ1 ξ2]Elowast + cp ͷʹΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨X3 [ξ1 ξ2]Elowast⟩+ iX3d[ξ1 ξ2]Elowast

+ (ιX3Lξ1dξ2 minus ιX3Lξ1dξ2) + (ιX3Lξ2dξ1 minus ιX3Lξ2dξ1)

= d⟨X3 [ξ1 ξ2]Elowast⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A44)

ΒʹɼX isin Γ(E) ξ η isin Γ(Elowast) ͱͱɼҰൠʹ LX3 [ξ1 ξ2]Elowast ʹର

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

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54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

53

Δɽ(A45) ͷಋग़ͷઅͰߦɽΕΛ iX3Lξ1dξ2 ͱ iX3Lξ2dξ1 ద༻Δͱɼ

LX3 [ξ1 ξ2]Elowast + cp ҎԼͷܗʹͳΔɽ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3)(e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)+ cp (A46)

(A46) ͷಋग़ͷͷઅͰߦɽ

(A46) Λ (A43) ೖΔͱɼI1 ΒʹཧͰɼ

I1 = d([e1 e2]V e3)minus minus ρ(X3)(e1 e2)minus minus 2ρlowast(ξ3)(e1 e2)minus + ⟨[ξ1 ξ2]Elowast X3⟩ minusK1 minusK2+ cp

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A47)

ɼ(337) Βɼ

([e1 e2]V e3)minus + cp = T (e1 e2 e3)

+ [ρ(X3)(e1 e2)minus + 2ρlowast(ξ3)(e1 e2)minus minus ⟨[ξ1 ξ2]Elowast X3⟩+ cp](337)

ΛೖΔͱɼI1 ҎԼͷΑͳܗʹͳΔɽ

I1 = dT (e1 e2 e3)minus K1 +K2+ cp (A48)

ɼ

K1 = ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1)

K2 = Ldlowast(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast (A49)

I2 ʹಉͰΔɽ

I2 = dlowastT (e1 e2 e3)minus K3 +K4+ cp

K3 = ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

K4 = Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A50)

Αɼऴతʹ [[e1 e2]V e3]V + cp ͷܭՌҎԼͷΑʹಘΒΕΔɽ

[[e1 e2]V e3]V + cp = I1 + I2

= DT (e1 e2 e3)minus (J1 + J2 + cp) (A51)

ɼ

J1 = K1 +K3

= ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + ιξ3(dlowast[X1 X2]E minus LX1dlowastX2 + LX2dlowastX1)

J2 = K2 +K4

= Ld(e1e2)minusξ3 + [d(e1 e2)minus ξ3]Elowast + Ldlowast(e1e2)minusX3 + [dlowast(e1 e2)minus X3]E (A52)

Ұൠʹɼҙͷ Xi ξi f ʹର (J1 + J2 +cp) 0 ͰͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹ

ɼAxiom C1 ΕΔɽ

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

54

A6 (A45)ͷಋग़

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ ((A45))

Λಋग़ΔɽࠨลΛܭΔͷͷͰɼ⟨ιXLξdη Y ⟩ΛܭɼҎԼͷΑʹӈลͱ

Y ͰΛͱܗʹͳΔͱΛΊΔɽ

⟨ιXLξdη Y ⟩ = ⟨[ξLXη]Elowast Y ⟩ minus ⟨LLξXη Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩+ ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ (A53)

ɼLie ඍͷଇΒɼ

⟨ιXLξdη Y ⟩ = Lξ(dη(XY ))minus dη(LξXY )minus dη(XLξY ) (A54)

ӈลͷ 1 ɼ(A30) ΛͱͰ ρlowast(ξ)dη(XY ) ʹஔΔͱͰΔɽɼඍ

ૉͷఆΒɼ༺

dξ(XY ) = ρ(X)⟨ξ Y ⟩ minus ρ(Y )⟨ξ X⟩ minus ⟨ξ [XY ]E⟩ (A55)

ΔͷͰɼΕΛʹΊΔͱɼ(A54) ҎԼͷΑʹมܗͰΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)ρ(X)⟨ξ Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρlowast(ξ)⟨η [XY ]E⟩minus ρ(LξX)⟨η Y ⟩+ ρ(Y )⟨ηLξX⟩+ ⟨η [LξXY ]E⟩+ ρ(LξY )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩ minus ⟨η [LξYX]E⟩ (A56)

Βʹɼ(A29)(A30) ͱ (319) Λͱɼ(A56) ͷߦͷઌͷҎԼͷΑʹղͰΔɽ

ρlowast(ξ)ρ(X)⟨ξ Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩+ ρlowast(ξ)⟨η [XY ]E⟩ (A57)

ρ(LξX)⟨η Y ⟩ = ⟨LLξXη Y ⟩+ ⟨η [LξXY ]E⟩ (A58)

ρ(LξY )⟨η X⟩ = ⟨LLξY η X⟩+ ⟨η [LξYX]E⟩ (A59)

ΕΛೖཧΔͱɼͷଧফҎԼͷΑʹͱΔɽ

⟨ιXLξdη Y ⟩ = ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ⟨LLξXη Y ⟩+ ρ(Y )⟨ηLξX⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A60)

ͷՌΛɼanchor ΛΘʹ Y ͱͷͷܗͰදΑʹॻΔɽɼρ(Y )⟨ηLξX⟩ʹΔɽͷɼ(A29)(A30) Β

ρ(Y )⟨ηLξX⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩ minus ⟨d⟨[ξ η]V X⟩ Y ⟩ɽ (A61)

ͱදΔɽΕΛೖ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ρlowast(ξ)⟨LXη Y ⟩ minus ρlowast(ξ)ρ(Y )⟨η X⟩ minus ρ(X)⟨ηLξY ⟩+ ⟨LLξY η X⟩ (A62)

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

55

ಉʹɼ(A29)(A30) Λ (A62) ͷ 2 ஈΛཧΔͱɼҎԼͷΑʹͳΔɽ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩ minus ⟨ηLXLξY ⟩+ ⟨LLξY η X⟩

(A63)

Ͱɼ

minus⟨ηLXLξY ⟩+ ⟨LLξY η X⟩ = ⟨LLξY η X⟩+ ⟨LLξY η X⟩= ⟨d⟨η X⟩LξY ⟩ (A64)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]V X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]V Y ⟩ minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ ⟨d⟨η X⟩LξY ⟩ (A65)

Lie ඍͷଇΒɼ

⟨d⟨η X⟩LξY ⟩ = Lξ⟨d⟨η X⟩ Y ⟩ minus ⟨[ξ d⟨η X⟩]V Y ⟩ (A66)

ͳͷͰɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩minus Lξ⟨LY η X⟩ minus Lξ⟨ηLY X⟩+ Lξ⟨d⟨η X⟩ Y ⟩ (A67)

ͰɼҎԼͷܭΒஈଧফɽ

Lξ⟨d⟨η X⟩ Y ⟩ = LξLY ⟨η X⟩= Lξ⟨LY η X⟩+ Lξ⟨ηLY X⟩ (A68)

Αɼ

⟨ιXLξdη Y ⟩ = ⟨d(ρlowast(ξ)⟨η X⟩) Y ⟩+ ⟨d⟨[ξ η]Elowast X⟩ Y ⟩ minus ⟨LLξXη Y ⟩+ ⟨[ξLXη]Elowast Y ⟩ minus ⟨[ξ d⟨η X⟩]Elowast Y ⟩ (A69)

A7 (A46)ͷಋग़

લઅͷܭͰ༻ɼ

LX3 [ξ1 ξ2]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]V + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)minus d⟨[ξ1 ξ2]Elowast X3⟩+ iX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A46)

ͷಋग़Λߦɽɼࠨลͷ LX3 [ξ1 ξ2]Elowast ΛܭͱͰɼͷ८ճΛͱӈลʹͳΔ

ͱΛΔɽLX3 [ξ1 ξ2]Elowast ɼLie ඍͷఆʹҎԼͷΑʹలͰΔɽɼ

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

56

2 ͷҎԼɼιX3Lξ2dξ1 ͷͱɼLξ2dξ1 ͷΛΕΕΘͱΒҾΔɽ

ͷʹΑɼderivation ӨڹΔΛΔͱͰΔɽ

LX3 [ξ1 ξ2]Elowast = (dιX3 + ιX3d)[ξ1 ξ2]Elowast

= d⟨[ξ1 ξ2]Elowast X3⟩+ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1

+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A70)

Ҏͷܭɼ1 ஈͷ ιX3Lξ1dξ2 minus ιX3Lξ2dξ1 ͷʹΔͱ؆୯ʹͳΔɽલઅͷܭ

ΒɼҰൠʹ

ιXLξdη = [ξLXη]Elowast minus LLξXη + [d⟨η X⟩ ξ]Elowast + d(ρlowast(ξ)⟨η X⟩)minus d⟨[ξ η]Elowast X⟩ (A45)

ͱΔɽ(A45) Λ (A70) ʹೖɼଧফΛཧΔͱɼҎԼͷ

ಘΒΕΔɽܗ

LX3 [ξ1 ξ2]Elowast = minusd⟨[ξ1 ξ2]Elowast X3⟩+ [ξ1LX3ξ2]Elowast minus LLξ1X3ξ2 + [d⟨ξ2 X3⟩ ξ1]Elowast + d(ρlowast(ξ1)⟨ξ2 X3⟩)minus [ξ2LX3ξ1]Elowast + LLξ2X3ξ1 minus [d⟨ξ1 X3⟩ ξ2]Elowast minus d(ρlowast(ξ2)⟨ξ1 X3⟩)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) (A71)

ଓɼ(A71) ͷ८ճ LX3 [ξ1 ξ2]V + cp ΛܭΔɽ(A71) ͷ 2 ஈͱ 3 ஈʹΔ

ΕΕҎԼͷΑʹͱΔɽ

[ξ1LX3ξ2]Elowast minus [ξ2LX3ξ1]Elowast + cp = [LX1ξ2 minus LX2ξ1 ξ3]Elowast + cp (A72)

minus LLξ1X3ξ2 + LLξ2X3ξ1 + cp = LLξ1X2minusLξ2X1ξ3 + cp (A73)

[d⟨ξ2 X3⟩ ξ1]Elowast minus [d⟨ξ1 X3⟩ ξ2]Elowast + cp = +2[d(e1 e2)minus ξ3]Elowast + cp (A74)

d(ρlowast(ξ1)⟨ξ2 X3⟩)minus d(ρlowast(ξ2)⟨ξ1 X3⟩) + cp = 2d(ρlowast(ξ3) middot (e1 e2)minus) + cp (A75)

ɼ

LX3 [ξ1 ξ2]Elowast + cp = minusd⟨[ξ1 ξ2]Elowast X3 + [LX1ξ2 minus LX2ξ1 ξ3]Elowast + LLξ1X2minusLξ2X1ξ3

+ 2[d(e1 e2)minus ξ3]Elowast + 2d(ρlowast(ξ3) middot (e1 e2)minus)+ ιX3(d[ξ1 ξ2]Elowast minus Lξ1dξ2 + Lξ2dξ1) + cp (A76)

ͱՌಘΒΕΔɽΕɼ(A46) ͷӈลͷͷͰΔɽ

A8 Axiom C2ͷ

ܭͰͳͷͰɼ(322) ͷลΛΕΕҙͷ f isin Cinfin(M) ༻ɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]f (322rsquo)

ΔͲΛΔɽ

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

57

ลΛలͱɼρࠨ ͱ [middot middot]V ͷఆΒҎԼͷΑͳܗͱͳΔɽ

ρV([e1 e2]V)f = ρV([X1 + ξ1 X2 + ξ2]V)f

= ρ[X1 X2]E + Lξ1X2 minus Lξ2X1 minus dlowast(e1 e2)minusf+ ρlowast[ξ1 ξ2]Elowast + LX1ξ2 minus LX2ξ1 + d(e1 e2)minusf

= ρ([X1 X2]E)f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ ρlowast([ξ1 ξ2]Elowast)f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A77)

தɼdlowast = ρlowastlowastd0 d = ρlowastd0 ͰΔͱΛ༻ɽͰɼ[ρ(X1) ρ(X2)] ͱ [ρlowast(ξ1) ρlowast(ξ2)] ɼ

ρ ρlowast ͷఆΒ

[ρ(X1) ρ(X2)] = ρ([X1 X2]E)

[ρlowast(ξ1) ρlowast(ξ2)] = ρlowast([ξ1 ξ2]Elowast)

ΔͷͰɼ(A77) ҎԼͷΑʹॻͳΔɽɼ[middot middot] TM ͷ Lie ͰΔɽހ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)f minus ρ(Lξ2X1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + ρlowast(LX1ξ2)f minus ρlowast(LX2ξ1)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A78)

ɼ

[ρ(X) ρlowast(ξ)]f = minusρ(LξX)f + ρlowast(LXξ)f + (ρρlowastlowastd0⟨ξ X⟩)fminus ⟨ξLdfX⟩+ ρ(X)ρlowast(ξ)f minus ρlowast(LXξ)f (A79)

ͰΔͱΛ༻Δͱɼ(A78) ΒʹཧΔͱͰɼ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)f

minus 1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + ρ(Lξ1X2)minus ρlowast(LX2ξ1)minus ρρlowastlowastd0⟨ξ1 X2⟩f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f minus ρ(Lξ2X1)minus ρlowast(LX1ξ2)minus ρρlowastlowastd0⟨ξ2 X1⟩f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A80)

Βɼ[ρ(X) ρlowast(ξ)]f ΛͱΊΔɽෆΔҾɼா৲ΛΘΔͱҎԼͷΑ

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

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[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

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[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

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638 Springer-Verlag (2004) 107

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Trans Am Math Soc 63 (1948) 85

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Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

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[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

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[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

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no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

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[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

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[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

58

ʹͳΔɽ

ρV([e1 e2]V)f = [ρ(X1) ρ(X2)]f minus [ρ(X2) ρlowast(ξ1)]f

minus ⟨ξ1LdfX2⟩+ ρ(X2)ρlowast(ξ1)f minus ρlowast(LX2)ξ1f

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

+ [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

+ ⟨ξ2LdfX1⟩ minus ρ(X1)ρlowast(ξ2)f + ρlowast(LX1ξ2)f

+1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f

= [ρ(X1) ρ(X2)]f + [ρlowast(ξ1) ρ(X2)]f + [ρlowast(ξ1) ρlowast(ξ2)]f + [ρ(X1) ρlowast(ξ2)]f

minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩ (A81)

Βʹɼρ ͷఆΒ (A81) ஈͷ 4 ͱΊΔͱͰɼ

ρV([e1 e2]V)f = [ρV(e1) ρV(e2)]minus ⟨ξ1LdfX2 minus [X2 dlowastf ]E⟩+ ⟨ξ2LdfX1 minus [X1 dlowastf ]E⟩

+1

2ρρlowastlowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f +

1

2ρlowastρ

lowastd0(⟨ξ1 X2⟩ minus ⟨ξ2 X1⟩)f (A82)

Ұൠʹɼҙͷ Xi ξ1 ʹରӈล 0 ʹͳΒͳɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼ

Axiom C2 ΕΔɽ

A9 Axiom C3ͷ

ลࠨɼʹࡍ [e1 fe2]V ΛలɼӈลʹߦணͱΛɽρ ͷఆΒɼ

[e1 fe2]V = [X1 + fξ1 X2+fξ2]V

= [X1 fX2]V + [X1 fξ2]V + [ξ1 fX2]V + [ξ1 fξ2]V (A83)

Ͱɼ[middot middot]V ͷఆΒɼ

[X1 fξ2]V = minusLfξ2X1 +1

2dlowast(f⟨ξ2 X1⟩) + LX1(fξ2)minus

1

2d(f⟨ξ2 X1⟩)

= minusfLξ2X1 minus (dlowastf)⟨ξ2 X1⟩+1

2(dlowastf)⟨ξ2 X1⟩+

1

2fdlowast⟨ξ2 X1⟩

+ fd⟨ξ2 X1⟩+ ιXdfξ2 + f ιXdξ2 minus1

2(df)⟨ξ2 X1⟩ minus

1

2fd⟨ξ2 X1⟩

= f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Df⟨ξ2 X1⟩ (A84)

[ξ1 fX2]V ಉʹɼ[middot middot]V ͷఆΒɼ

[ξ1 fX2]V = f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩ (A85)

ɼL ͱ Llowast ΕΕ Lie ѥͰΔͱΒɼ

[X1 fX2]V = [X1 fX2]E = f [X1 X2]V + (ρ(X1)f)X2 (A86)

[ξ1 fξ2]V = [ξ1 fξ2]Elowast = f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2 (A87)

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

59

Αɼ

[e1 fe2]V = (A83)

= (A86) + (A84) + (A85) + (A87)

= f [X1 X2]V + (ρ(X1)f)X2

+ f [X1 ξ2]V + (ρ(X1)f)ξ2 minus1

2Dg⟨ξ2 X1⟩

+ f [ξ1 X2]V + (ρlowast(ξ1)f)X2 minus1

2Df⟨ξ1 X2⟩

+ f [ξ1 ξ2]V + (ρlowast(ξ1)f)ξ2

= f [e1 e2]V + (ρV(e1)f)e2 minusDf(e1 e2)minus (A88)

Αɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C3 Δɽ

A10 Axiom C4ͷ

D ͷఆΛͱɼ(324) ͷࠨลҎԼͷΑʹมܗͰΔɽɼΓ(T lowastM) ͷඍԋ

Λࢠ d0 ͱɽ

(DfDg)+ = (df + dlowastf dg + dlowastg)+

=1

2(⟨df dlowastg⟩+ ⟨dg dlowastf⟩)

=1

2(ρlowast(df)g + ρ(dlowastf)g)

=1

2(ρlowastρ

lowastd0f + ρρlowastlowastd0f)g (A89)

ɼ(A89) 0 ʹͳΓɼ(324) Δʹ ρlowastρlowast = minusρρlowastlowast ΓΕΑɽ

ॳΊʹɼderivation Λඞ anchor ρ ରশ ρρlowastlowast = minusρlowastρlowast ʹͳΔͱΛɽ

ɼຊจதͰड़௨Γ anchor ʹΔͷ lowast ɼadjoint operator ͷҙຯͰ

ΓɼΛ௨ݩͷԋࢠͷసஔͰఆΕΔɽ

ρ E rarr TM ρlowast T lowastM rarr Elowast

ρlowast Elowast rarr TM ρlowastlowast T lowastM rarr E (A90)

ɼρρlowastlowast T lowastM rarr TMɼρlowastρlowast T lowastM rarr TM ͰΔɽɽ

ҰൠʹɼʮΔԋࢠ OɿT lowastM rarr TM ରশͰΔʯͱɼҎԼͷΔɽશͷ

x isin Γ(T lowastM) ʹରɼ⟨Ox x⟩ = 0 (A91)

ԋࢠ O Λ ρρlowastlowastɼx Λ d0f isin Γ(TM) ʹஔΔͱͰҎԼͷಘΒΕΔɽɼ

f isin Γ(TM) d0 T lowastM ͷඍ༻ૉͰΔɽ

⟨ρρlowastlowast(d0f) d0f⟩ = ρρlowastlowast(d0f) middot f = 0 (A92)

Αɼ(A92) ΔͳΒ ρlowastρlowast = minusρρlowastlowast Δɽ

ԾʹɼҎԼͷΓͷͱΑɽ

LdfX + [dlowastfX]E = 0 (360)

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

60

ͷɼจݙ [44] ͷ Proposition 34 Βɼdarivation ΕਵΓ

ͰΔɽX Λ dlowastf ͰஔΔͱɼҎԼͷΔɽ

dlowastρρlowastlowast(d0f) middot f

= 0 (A93)

f Λ f2 ʹஔΕɼ

dlowastρρlowastlowast(d0f

2) middot f2= 0 (A94)

ҰͰɼ

ρρlowastlowast(d0f2) middot f2 = ⟨d0f2 ρρlowastlowastd0f

2⟩ = ⟨df2 dlowastf2⟩ = 4f2⟨df dlowastf⟩ (A95)

ͳͷͰɼρρlowastlowast(d0f) middot f

dlowastf

2 = dlowast

ρρlowastlowast(d0f) middot f(f2

minus dlowast

ρρlowastlowast(d0f) middot f

f2

=1

4dlowast

ρρlowastlowast(d0f

2) middot f2minus dlowast

ρρlowastlowast(d0f) middot f

f2 (A96)

Ͱɼӈลʹ (A93) (A94) ΒফΔΔͷͰɼρρlowastlowast(d0f) middot f

ρlowastlowastd0f

2 = 0 (A97)

ΒʹɼΕ d0f ͱͷͱࢥͱ

2fρρlowastlowast(d0f) middot f

2= 0 (A98)

ͱͳΔɽɼderivation ΕΕ 0 = ρρlowastlowast(d0f) middot f = ⟨ρρlowastlowast(d0f) ͱͳΔɽρρlowastlowast

ରশͰΔʹɼderivation ΛඞཁΔɽ

ҎԼͷܭهͷՌͷͰΔɽderivation ΛͳҰൠͷɼρlowastρlowast = minusρρlowastlowastͳΛഉআͰͳͱΛɽମతʹɼ(V [middot middot]V ρV (middot middot)+) ɼ

(324) ຬͳͱΛΔɽମతʹɼ(A92) ͷࠨลΛผͷͰಋग़ɼҰൠʹ 0 ʹ

ͳΒͳͱΛɽ

[middot middot]E ͷΒɼXY isin Γ(E) ͱΔͱ

dlowast[X fY ]E = dlowast(f [XY ]E + (ρ(X) middot f)Y )

= dlowast(f [XY ]E) + dlowast((ρ(X) middot f)Y )

= dlowastf and [XY ]E + f [XY ]E + dlowastρ(X) middot f and Y + ρ(X) middot fdlowastY (A99)

ͰΔͱΒɼҎԼͷΔɽ

[X dlowastf ]E and Y = LdfX and Y minus dlowast[X fY ]E + fdlowast[XY ]E

+ dlowastf and LXY + LXdlowastf and Y + (ρ(X) middot f)dlowastY + fLY dlowastX minus fLfY dlowastX(A100)

X Λ dlowastf ʹஔཧΔͱɼҎԼͷಘΒΕΔɽ

[dlowastf dlowastf ]E and Y = Ldfdlowastf and Y minus dlowast[dlowastf fY ]E + fdlowast[dlowastf Y ]E

+ dlowastf and LdlowastfY + (ρ(dlowastf) middot f)dlowastY= 0 (A101)

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

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[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

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[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

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[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

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[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

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[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

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[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

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[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

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[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

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[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

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[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

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[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

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[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

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[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

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[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

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[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

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mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

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[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

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[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

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[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

61

ΑɼҎԼͷΔɽ

Ldfdlowastf and Y = dlowast[dlowastf fY ]E minus fdlowast[dlowastf Y ]E minus dlowastf and LdlowastfY minus (ρ(dlowastf) middot f)dlowastY (A102)

ɼ(A102) ɼf Λ f2 ʹஔΔɽ

Ldf2dlowastf2andY = dlowast[dlowastf

2 f2Y ]Eminusf2dlowast[dlowastf2 Y ]Eminusdlowastf2andLdlowastf2Yminus(ρ(dlowastf2)middotf2)dlowastY (A102prime)

Βʹɼ(A92) Βలͱɼ

ρρlowastlowast(d0f2) middot f2 = ⟨ρρlowastlowast(d0f2) d0f

2⟩= ⟨dlowastf2 df2⟩= 4⟨fdlowastf fdf⟩= 4f2⟨dlowastf df⟩= 4(ρρlowastlowast(d0f) middot f)f2 (A103)

ͳͷͰɼΕΛล dlowast Ͱୟͱ

dlowast(ρρlowastlowast(d0f

2) middot f2) = 4dlowast((ρρlowastlowast(d0f) middot f)f2)

= 4f2dlowast(ρρlowastlowast(d0f) middot f) + 4(ρρlowastlowast(d0f) middot f)dlowastf2 (A104)

ͰΔͱΒɼ(A100) ͷΑʹॻΔͱͰΔɽ

4f2dlowast(ρρlowastlowast(d0f) middot f) and Y + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E +

minus(dlowastf2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus ρ(dlowastf

2)[f2]dlowastY( (A102rdquo)

ʹลࠨ (A102) ΛೖΔͱɼҎԼͷΑʹͳΔɽ

4f2dlowast[dlowastf fY ]E minus 4f2(dlowastf) and LdlowastfY minus 4f2LdlowastfdlowastY minus 4f2(ρ(dlowastf) middot f)dlowastY + 4(ρρlowastlowast(d0f) middot f)dlowastf2 and Y

= dlowast[dlowastf2 f2Y ]E minus (dlowastf

2) and Ldlowastf2Y minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY

ΕΛ ρρlowastlowast(d0f) middot fdlowastf2 ͷΔͰͱΊΔͱɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = minus4f2dlowast[dlowastf fY ]E + 4f2(dlowastf) and LdlowastfY

+ 4f2LdlowastfdlowastY + 4f2ρ(dlowastf) middot fdlowastY+ dlowast[dlowastf

2 f2Y ]E minus (dlowastf2) and Ldlowastf2Y

minus fLdlowastf2dlowastY minus (ρ(dlowastf2) middot f2)dlowastY (A105)

(A105) ͷลʹ ρ Λ༻Δͱɼࠨล

ρ((ρρlowastlowast(d0f) middot f)(dlowastf2)) = (ρρlowastlowast(d0f) middot f)ρ(dlowastf2)

= (ρρlowastlowast(d0f) middot f)ρρlowastlowast(d0f2)

= 2f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f)

ͰΔͱΒɼ

4ρρlowastlowast(d0f) middot fdlowastf2 and Y = 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and Y (A106)

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

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[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

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[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

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[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

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[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

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[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

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[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

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[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

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[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

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vol 319 (1990) 631

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theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

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Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

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and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

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[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

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Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

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calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

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78 (1963) 267

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prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

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J 73 (1994) 415

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[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

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[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

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brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

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[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

62

ͱͳΓɼ(A105) (ρ(X and Y ) = ρ(X) and ρ(Y ) ͰΔͳΒ) ҎԼͷΑʹॻͱͰΔɽ

8f(ρρlowastlowast(d0f) middot fρρlowastlowastd0f) and ρ(Y ) = minus4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A107)

ΒʹɼลͰ d0f ͱΛͱΔͱɼࠨล

8f ιd0f (((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y ))

= 8f ιd0f ((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ρ(Y )minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y )

= 8f(ρρlowastlowast(d0f) middot f)2 and Y minus 8f((ρρlowastlowast(d0f) middot f) ρρlowastlowastd0f) and ιd0fρ(Y ) (A108)

ͱͳΔͱΒɼ

8f(ρρlowastlowast(d0f) middot f)2 and Y

= 8f((ρρlowastlowast(d0f) middot f)ρρlowastlowastd0f) and id0fρ(Y )

minus 4f2ρ(dlowast[dlowastf fY ]E) + 4f2ρ(dlowastf) and ρ(LdlowastfY )

+ 4f2ρ(LdlowastfdlowastY ) + 4f2ρ(dlowastf) middot fρ(dlowastY )

+ ρ(dlowast[dlowastf2 f2Y ]E)minus ρ((dlowastf

2)) and ρ(Ldlowastf2Y )

minus fρ(Ldlowastf2dlowastY )minus (ρ(dlowastf2) middot f2)ρ(dlowastY ) (A109)

ลࠨఔͷζϨΔɼ (A92) ͷதԝͷݱΕΔɽҰൠʹҙͷ f Y ʹର

ӈลΒʹ 0 ͰͳͷͰɼρρlowastlowast(d0f) middot f 0 ʹͳΒͳՄΛഉআͰͳɽ

A11 Axiom C5ͷ

ρ(e)(e1 e2) = ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (325)

ΔͲΛΊΔɽΕɼ(A32) ΛͱͰʹΔɽ(A32) Βɼ

([e e1]V e2)+ = T (e e1 e2) +1

2ρV(e)(e1 e2)+ minus

1

2ρV(e1)(e e2)+

(e1 [e e2]V)+ = T (e e2 e1) +1

2ρV(e)(e2 e1)+ minus

1

2ρV(e2)(e e1)+

ͷ 2 ΛͱɼҎԼͷಘΒΕΔɽT (e e2 e1) શରশͳͷͰଧফɽ

([e e1]V e2)+ + (e1 [e e2]V)+

= T (e e1 e2) + T (e e2 e1) + ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+

= ρV(e)(e1 e2)+ minus1

2ρ(e1)(e e2)+ minus

1

2ρ(e2)(e e1)+ (A110)

(A110) Λ ρ(e)(e1 e2)+ ʹղͱɼҎԼͷΑʹͳΔɽ

ρ(e)(e1 e2)+ = ([e e1]V e2)++ (e1 [e e2]V)++1

2ρV(e1)(e e2)++

1

2ρV(e2)(e e1)+ (A111)

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

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(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

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[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

63

Ͱɼρ ͱ D ͷఆΒɼei = Xi + ξi ͳͷͰɼ

1

2ρV(e1)(e e2)+ =

1

2ρ(X1)(e e2)+ +

1

2ρlowast(ξ1)(e e2)+

=1

2(⟨X1 d(e e2)+⟩+ ⟨ξ1 dlowast(e e2)+⟩)

= (d(e e2)+ + dlowast(e e2)+ X1 + ξ1)+

= (D(e e2)+ e1)+ (A112)

ಉʹɼ1

2ρV(e2)(e e1)+ = (D(e e1)+ e2)+ (A113)

ΕΒΛ (A111) ʹೖɼ

ρV(e)(e1 e2)+ = ([e e1]V e2)+ + (e1 [e e2]V)+ + (D(e e2)+ e1)+ + (D(e e1)+ e2)+

= ([e e1]V +D(e e1) e2) + (e1 [e e2]V +D(e e2)) (A114)

(A114) ʹ (325) ͷͷͰΔɽΑɼ(V [middot middot]V ρV (middot middot)+) ʹɼAxiom C5

Δɽ

A12 NK = NP +NP ͰΔͱͷ

NK ɼ(41) Ͱ༩ɼύϥෳૉߏͷՄΛධՁΔΊͷͰΔɽNP (XY )

NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A115)

ͱͱɼNK(XY ) = NP (XY ) +NP (XY ) ͰΔͱΛΊΔɽ

Ͱɼ(42) ͷ P P ͷఆΒɼNP (XY ) NP (XY ) Λ K ΒΘʹͳΔΑʹॻԼ

ͱɼҎԼͷΑʹͳΔɽ

NP (XY ) = P [P (X) P (Y )]

=1

2(1minusK)

01

2(1 +K)X

1

2(1 +K)Y

1

=1

2(1minusK)

1

4[X +K(X) Y +K(Y )]

=1

2(1minusK)

1

4

XY +XK(Y ) +K(X)Y +K(X)K(Y )

minus Y X minus Y K(X)minusK(Y )X minusK(Y )K(X)(

=1

2(1minusK)

1

4([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

64

ಉʹɼ

NP (XY ) = P [P (X) P (Y )]

=1

2(1 +K)

01

2(1minusK)X

1

2(1minusK)Y

1

=1

2(1 +K)

1

4[X minusK(X) Y minusK(Y )]

=1

2(1 +K)

1

4

XY minusXK(Y )minusK(X)Y +K(X)K(Y )

minus Y X + Y K(X) +K(Y )X minusK(Y )K(X)(

=1

2(1 +K)

1

4([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

=1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

ɼ

NP (XY ) +NP (XY ) =1

8([XY ] + [XK(Y )] + [K(X) Y ] + [K(X)K(Y )])

+1

8(minusK[XY ]minusK[XK(Y )]minusK[K(X) Y ]minusK[K(X)K(Y )])

+1

8([XY ]minus [XK(Y )]minus [K(X) Y ] + [K(X)K(Y )])

+1

8(K[XY ]minusK[XK(Y )]minusK[K(X) Y ] +K[K(X)K(Y )])

=1

4([XY ]minusK[XK(Y )]minusK[K(X) Y ] + [K(X)K(Y )])

= NK(XY ) (A116)

ΑɼNP (XY ) NP (XY ) Λ

NP (XY ) = P [P (X) P (Y )] NP (XY ) = P [P (X) P (Y )] (A117)

ͱͱɼNK = NP +NP Δɽ

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

65

ݙจߟ

[1] T Kaluza ldquoOn the Problem of Unity in Physicrdquo Sitzungsber Preuss Akad Wiss Berlin

(Math Phys) (1921) 66

[2] O Klein ldquoQuantum Theory and Five-Dimensional Theory of Relativity (In German

and English)rdquo Z Phys 37 (1926) 895 High Energ Phys 5 (1986) 241

[3] C G Callan Jr E J Martinec M J Perry and D Friedan ldquoStrings in Background

Fieldsrdquo Nucl Phys B 262 (1985) 593

[4] J M Maldacena ldquoThe Large N limit of superconformal field theories and supergrav-

ityrdquo Int J Theor Phys 38 (1999) 1113 [Adv Theor Math Phys 2 (1998) 231] [hep-

th9711200]

[5] A Sen ldquoStrong - weak coupling duality in four-dimensional string theoryrdquo Int J Mod

Phys A 9 (1994) 3707 [hep-th9402002]

[6] N A Obers and B Pioline ldquoU duality and M theoryrdquo Phys Rept 318 (1999) 113

[hep-th9809039]

[7] A Giveon M Porrati and E Rabinovici ldquoTarget space duality in string theoryrdquo Phys

Rept 244 (1994) 77 [hep-th9401139]

[8] C Hull and B Zwiebach ldquoDouble Field Theoryrdquo JHEP 0909 (2009) 099

[arXiv09044664 [hep-th]]

[9] C Hull and B Zwiebach ldquoThe Gauge algebra of double field theory and Courant brack-

etsrdquo JHEP 0909 (2009) 090 [arXiv09081792 [hep-th]]

[10] A Einstein ldquoThe Foundation of the General Theory of Relativityrdquo Annalen Phys 49

(1916) no7 769 [Annalen Phys 14 (2005) 517] [Annalen Phys 354 (1916) no7 769]

[11] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquo Quart J Math 54 (2003) 281

[arXivmath0209099 [math-dg]]

[12] C M Hull ldquoDoubled Geometry and T-Foldsrdquo JHEP 0707 (2007) 080 [arXivhep-

th0605149]

[13] H Mori S Sasaki and K Shiozawa ldquoDoubled Aspects of Vaisman Algebroid and Gauge

Symmetry in Double Field Theoryrdquo J Math Phys 61 (2020) 013505 [arXiv190104777

[hep-th]]

[14] I Vaisman ldquoOn the geometry of double field theoryrdquo J Math Phys 53 (2012) 033509

[arXiv12030836 [mathDG]]

[15] I Vaisman ldquoTowards a double field theory on para-Hermitian manifoldsrdquo J Math Phys

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

66

54 (2013) 123507 [arXiv12090152 [mathDG]]

[16] M Gualtieri ldquoGeneralized complex geometryrdquo Oxford University PhD thesis

[arXivmath0401221 [mathDG]]

[17] T J Courant ldquoDirac Manifoldsrdquo Transactions of the American Mathematical Society

vol 319 (1990) 631

[18] O Hohm C Hull and B Zwiebach ldquoGeneralized metric formulation of double field

theoryrdquo JHEP 1008 (2010) 008 [arXiv10064823 [hep-th]]

[19] O Hohm and B Zwiebach ldquoTowards an invariant geometry of double field theoryrdquo J

Math Phys 54 (2013) 032303 [arXiv12121736 [hep-th]]

[20] A Deser and C Samann ldquoExtended Riemannian Geometry I Local Double Field The-

oryrdquo arXiv161102772 [hep-th]

[21] A Deser and C Samann ldquoDerived Brackets and Symmetries in Generalized Geometry

and Double Field Theoryrdquo PoS CORFU 2017 (2018) 141 [arXiv180301659 [hep-th]]

[22] K Lee ldquoTowards Weakly Constrained Double Field Theoryrdquo Nucl Phys B 909 (2016)

429 [arXiv150906973 [hep-th]]

[23] O Hohm C Hull and B Zwiebach ldquoBackground independent action for double field

theoryrdquo JHEP 1007 (2010) 016 [arXiv10035027 [hep-th]]

[24] C M Hull ldquoA Geometry for non-geometric string backgroundsrdquo JHEP 0510 (2005)

065 [hep-th0406102]

[25] D Svoboda ldquoAlgebroid Structures on Para-Hermitian Manifoldsrdquo J Math Phys 59

(2018) no12 122302 [arXiv180208180 [mathDG]]

[26] L Freidel F J Rudolph and D Svoboda ldquoGeneralised Kinematics for Double Field

Theoryrdquo JHEP 1711 (2017) 175 [arXiv170607089 [hep-th]]

[27] L Freidel F J Rudolph and D Svoboda ldquoA Unique Connection for Born Geometryrdquo

Commun Math Phys 372 (2019) no1 119 [arXiv180605992 [hep-th]]

[28] V E Marotta and R J Szabo ldquoPara ʖ Hermitian Geometry Dualities and Generalized

Flux Backgroundsrdquo Fortsch Phys 67 (2019) no3 1800093 [arXiv181003953 [hep-th]]

[29] W Siegel ldquoSuperspace duality in low-energy superstringsrdquo Phys Rev D 48 (1993) 2826

[hep-th9305073]

[30] W Siegel ldquoTwo vierbein formalism for string inspired axionic gravityrdquo Phys Rev D

47 (1993) 5453 [hep-th9302036]

[31] M Gualtieri ldquoGeneralized Complex Geometryrdquo Ann of Math 174 (2011) 75

[arXivmath0703298]

[32] M Grana R Minasian M Petrini and D Waldram ldquoT-duality Generalized Geometry

and Non-Geometric Backgroundsrdquo JHEP 0904 (2009) 075 [arXiv08074527 [hep-th]]

[33] I Y Dorfmann ldquoDirac Structures of Integrable Evolution Equationsrdquo Phys Lett

A125(5) (1987) 240

[34] D Roytenberg ldquoCourant algebroids derived brackets and even symplectic supermani-

foldsrdquo math9910078 [mathDG]

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

67

[35] Z-J Liu A Weinstein P Xu ldquoManin Triples for Lie Bialgebroidsrdquo J Differential

Geom 45 (1997) 547 [dg-ga9508013]

[36] A Chatzistavrakidis L Jonke F S Khoo and R J Szabo ldquoDouble Field Theory and

Membrane Sigma-Modelsrdquo JHEP 1807 (2018) 015 [arXiv180207003 [hep-th]]

[37] Y Kosmann-Schwarzbach ldquoLie bialgebras Poisson Lie groups and dressing transfor-

mationsrdquo Integrability of Nonlinear Systems Second edition Lecture Notes in Physics

638 Springer-Verlag (2004) 107

[38] C M Marle ldquoThe Schouten-Nijenhuis bracket and interior productsrdquo Journal of Ge-

ometry and Physics 23 (1997) 350

[39] C Chevalley and S Eilenberg ldquoCohomology Theory of Lie Groups and Lie Algebrasrdquo

Trans Am Math Soc 63 (1948) 85

[40] J A Schouten ldquoUber Differentialkonkomitanten zweier kontravarianten Grossenrdquo

Indag Math 2 (1940) 449 ldquoOn the differential operators of the first order in tensor

calculusrdquo In Cremonese Convegno Int Geom Diff Italia (1953) 1

A Nijenhuis ldquoJacobi-type identities for bilinear differential concomitants of certain ten-

sor fields Irdquo Indagationes Math 17 (1955) 390

[41] M Gerstenhaber ldquoThe Cohomology Structure of an Associative Ringrdquo Ann of Math

78 (1963) 267

[42] V G Drinfelrsquod ldquoQuantum groupsrdquo Proc Internat Congr Math (1986) 789

[43] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiables Relations entre pro-

prietes locales et globalesrdquo C R Acad Sci Paris Ser A-B 263 (1966) A907

[44] K C H Mackenzie and P Xu ldquoLie bialgebroids and Poisson groupoidsrdquo Duke Math

J 73 (1994) 415

[45] N Hitchin ldquoLectures on generalized geometryrdquo arXiv10080973[mathDG]

[46] P Koerber ldquoLectures on Generalized Complex Geometry for Physicistsrdquo Fortsch Phys

59 (2011) 169 [arXiv10061536 [hep-th]]

[47] T Asakawa H Muraki S Sasa and S Watamura ldquoPoisson-generalized geometry and

R-fluxrdquo Int J Mod Phys A 30 (2015) no17 1550097 [arXiv14082649 [hep-th]]

[48] T Asakawa H Muraki and S Watamura ldquoTopological T-duality via Lie algebroids and

Q-flux in Poisson-generalized geometryrdquo Int J Mod Phys A 30 (2015) no30 1550182

[arXiv150305720 [hep-th]]

[49] T Asakawa H Muraki and S Watamura ldquoGravity theory on Poisson manifold with

R-fluxrdquo Fortsch Phys 63 (2015) 683 [arXiv150805706 [hep-th]]

[50] K Uchino ldquoRemarks on the Definition of a Courant Algebroidrdquo Lett Math Phys 60

(2002) 171 [math0204010 [mathDG]]

[51] D Roytenberg and A Weinstein ldquoCourant algebroids and strongly homotopy Lie alge-

brasrdquo Lett Math Phys 46 (1998) 1 [math9802118[mathQA]]

[52] I Vaisman ldquoGeneralized Sasaki metrics on tangent bundlesrdquo arXiv13124279

[mathDG]

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

68

[53] P Severa ldquoLetters to Alan Weinstein about Courant algebroidsrdquo arXiv170700265

[mathDG]

[54] Y Kosmann-Schwarzbach ldquoExact Gerstenharber algebras and Lie bialgebroidsrdquo Acta

Appl Math 41 (1995) 153

[55] Y Kosmann-Schwarzbach F Magri ldquoPoisson-Nijenhuis structuresrdquo Ann de lrsquoIHP

Physique theorique 53 (1990) 35

[56] ଜ Ұ ldquo༿ͷτϙϩδʔrdquo ॻళؠ (1976)

[57] A Newlander and L Nirenberg ldquoComplex Analytic Coordinates in Almost Complex

Manifoldsrdquo Ann of Math 65 (1957) 391

[58] W M Tulczyjew ldquoThe graded Lie algebra of multivector fields and the generalized Lie

derivative of formsrdquo Bull Acad Pol Sci Ser Sci Math Astr Phys 22 (1974) 937

[59] A Vaintrob ldquoLie algebroids and homological vector fieldsrdquo Uspekhi Mat Nauk 52

no2(314) (1997) 161 translation in Russian Math Surveys 52 no2 (1997) 428

[60] T Kimura and S Sasaki ldquoGauged Linear Sigma Model for Exotic Five-branerdquo Nucl

Phys B 876 (2013) 493 [arXiv13044061 [hep-th]]

[61] J Berkeley D S Berman and F J Rudolph ldquoStrings and Branes are Wavesrdquo JHEP

1406 (2014) 006 [arXiv14037198 [hep-th]]

[62] D S Berman and F J Rudolph ldquoBranes are Waves and Monopolesrdquo JHEP 1505

(2015) 015 [arXiv14096314 [hep-th]]

[63] I Bakhmatov A Kleinschmidt and E T Musaev ldquoNon-geometric branes are DFT

monopolesrdquo JHEP 1610 (2016) 076 [arXiv160705450 [hep-th]]

[64] T Kimura S Sasaki and K Shiozawa ldquoWorldsheet Instanton Corrections to Five-branes

and Waves in Double Field Theoryrdquo JHEP 1807 (2018) 001 [arXiv180311087 [hep-th]]

[65] O Hohm and B Zwiebach ldquoLarge Gauge Transformations in Double Field Theoryrdquo

JHEP 1302 (2013) 075 [arXiv12074198 [hep-th]]

[66] J H Park ldquoComments on double field theory and diffeomorphismsrdquo JHEP 1306 (2013)

098 [arXiv13045946 [hep-th]]

[67] D S Berman M Cederwall and M J Perry ldquoGlobal aspects of double geometryrdquo

JHEP 1409 (2014) 066 [arXiv14011311 [hep-th]]

[68] M Cederwall ldquoThe geometry behind double geometryrdquo JHEP 1409 (2014) 070

[arXiv14022513 [hep-th]]

[69] C M Hull ldquoFinite Gauge Transformations and Geometry in Double Field Theoryrdquo

JHEP 1504 (2015) 109 [arXiv14067794 [hep-th]]

[70] U Naseer ldquoA note on large gauge transformations in double field theoryrdquo JHEP 1506

(2015) 002 [arXiv150405913 [hep-th]]

[71] S J Rey and Y Sakatani ldquoFinite Transformations in Doubled and Exceptional Spacerdquo

arXiv151006735 [hep-th]

[72] V G Drinfelrsquod ldquoHamiltonian structures on Lie groups Lie bialgebras and the geometric

meaning of classical Yang-Baxter equationsrdquo Dokl Akad Nauk SSSR 268(2) (1983) 285

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献

69

[73] R Loja Fernandes and M Crainic ldquoLectures on Integrability of Lie Bracketsrdquo

math0611259

[74] P Severa and M Siran ldquoIntegration of differential graded manifoldsrdquo [arXiv150604898

[mathDG]]

[75] D Li-Bland and P Severa ldquoIntegration of Exact Courant Algebroidsrdquo Elec-

tronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58

[arXiv11013996 [mathDG]]

[76] M Grana and D Marques ldquoGauged Double Field Theoryrdquo JHEP 1204 (2012) 020

[arXiv12012924 [hep-th]]

[77] A Deser and J Stasheff ldquoEven symplectic supermanifolds and double field theoryrdquo

Commun Math Phys 339 (2015) no3 1003 [arXiv14063601 [math-ph]]

[78] U Carow-Watamura N Ikeda T Kaneko and S Watamura ldquoDFT in supermanifold

formulation and group manifold as background geometryrdquo [arXiv181203464 [hep-th]]

[79] D Roytenberg ldquoOn the structure of graded symplectic supermanifolds and Courant

algebroidsrdquo [arXivmath0203110 [math-sg]]

[80] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Lett Math Phys 69 (2004) 61

[arXivmath0312524 [mathDG]]

[81] C Klimcik and P Severa ldquoDual nonAbelian duality and the Drinfeld doublerdquo Phys

Lett B 351 (1995) 455 [arXivhep-th9502122]

[82] R Von Unge ldquoPoisson Lie T pluralityrdquo JHEP 0207 (2002) 014 [arXivhep-th0205245]

[83] D Lust and D Osten ldquoGeneralised fluxes Yang-Baxter deformations and the O(dd)

structure of non-abelian T-dualityrdquo JHEP 1805 (2018) 165 [arXiv180303971 [hep-th]]

[84] V E Marotta F Pezzella and P Vitale ldquoDoubling T-Duality and Generalized Geom-

etry a Simple Modelrdquo JHEP 1808 (2018) 185 [arXiv180400744 [hep-th]]

[85] P Severa and F Valach ldquoCourant algebroids Poisson-Lie T-duality and type II super-

gravitiesrdquo [arXiv181007763 [mathDG]]

[86] S Demulder F Hassler and D C Thompson ldquoDoubled aspects of generalised dualities

and integrable deformationsrdquo arXiv181011446 [hep-th]

[87] O Hohm and H Samtleben ldquoExceptional Field Theory I E6(6) covariant Form of M-

Theory and Type IIBrdquo Phys Rev D 89 (2014) no6 066016 [arXiv13120614 [hep-th]]

[88] O Hohm and H Samtleben ldquoHigher Gauge Structures in Double and Exceptional Field

Theoryrdquo Fortsch Phys 67 (2019) no8-9 1910008 [arXiv190302821 [hep-th]]

• はじめに
• Double Field Theory
• • DFTに現れる幾何学
  • DFTの作用とC括弧
  • 物理的拘束条件
  • • 亜代数 (Algebroid)の導入
    • • Lie（双）代数と Drinfeld double
      • Lie亜代数
      • Courant亜代数
      • Doubleによる Vaisman亜代数の構成
      • • Vaisman algebroidが現れる幾何学
        • • パラ複素構造
          • パラエルミート多様体
          • パラドルボーコホモロジー
          • DFTにおける Vaisman亜代数
          • Derivaiton条件の破れ
          • 一般幾何学との関連
          • • まとめ
            • 計算ノート
            • • C括弧の Jacobiatorの導出
              • C括弧から Courant括弧への reduction
              • T(e1e2e3)の関係式 (336)の導出
              • T(e1e2e3)の関係式 (337)の導出
              • Axiom C1の確認
              • (A45)式の導出
              • (A46)式の導出
              • Axiom C2の確認
              • Axiom C3の確認
              • Axiom C4の確認
              • Axiom C5の確認
              • NK=NP+Nであることの確認
              • • 参考文献