double field theory and the geometry of duality › research › hep › seminars › string... ·...

Double Field Theory and the Geometry of

DualityCMH & Barton Zwiebach

arXiv:0904.4664, 0908.1792

CMH & BZ & Olaf HohmarXiv:1003.5027, 1006.4664

Closed String Field Theory

• Captures exotic and complicated structure of interacting string

• Non-polynomial, algebraic structure not Lie algebra, cocycles.

• On torus: winding modes, T-duality.

• Seek subsector capturing exotic structure & duality, simple enough to analyse explicitly

Strings on d-Torus

• Supergravity limit: symmetry O(d,d)

• String: Perturbative T-duality O(d,d;Z)

• Kaluza-Klein theory: includes momentum modes on torus

• Include string winding or brane wrapping modes? Duality symmetry?

Strings on Td

X = XL(! + ") + XR(! ! "), X = XL ! XR

X Xconjugate to momentum, to winding no.

dX = !dX !aX = "ab!bX

Strings on Td

X = XL(! + ") + XR(! ! "), X = XL ! XR



Need “auxiliary” for interacting theoryi) Vertex operators ii) String field Kugo & Zwiebach

X

eikL·XL , e

ikR·XR

![x, x, a, a]

Strings on Td

X = XL(! + ") + XR(! ! "), X = XL ! XR



Need “auxiliary” for interacting theoryi) Vertex operators ii) String field Kugo & Zwiebach

X

eikL·XL , e

ikR·XR

![x, x, a, a]

Doubled Torus 2d coordinatesTransform linearly underDoubled sigma model CMH 0406102

O(d, d; Z) X !!

xi

xi

"

String Field Theoryon Minkowski Space

String field ![X(!), c(!)]

Expand to get infinite set of fields

Xi(!)! xi, oscillators

gij(x), bij(x), !(x), . . . , Cijk...l(x), . . .

Integrating out massive fields gives field theory for

gij(x), bij(x), !(x)

String Field Theoryon a torus

String field ![X(!), c(!)]

Expand to get infinite set of double fields

Seek double field theory for

Xi(!)! xi, xi, oscillators

gij(x, x), bij(x, x), !(x, x), . . . , Cijk...l(x, x), . . .

gij(x, x), bij(x, x), !(x, x)

Double Field Theory

• Construct from SFT, DFT of “massless” fields

• Double field theory on doubled torus

• Novel symmetry, reduces to diffeos + B-field trans. in any half-dimensional subtorus

• Backgrounds depending on seen by particles, on seen by winding modes. Backgrounds with both: unfamiliar.

Earlier versions: Siegel, Tseytlin

{xa}{xa}

gij(x, x), bij(x, x), !(x, x)

• Restriction to “massless” fields NOT a low-energy limit

• Lowest terms in level expansion

• T-duality a manifest symmetry

• General solution of SFT: double fields

• DFT needed for non-geometric backgrounds

• Real dependence on full doubled geometry, dual dimensions not auxiliary or gauge artifact. Double geom. physical and dynamical

Strings on a Torus• Coordinates

• Momentum

• Winding

• Fourier transform

• Doubled Torus

• String Field Theory gives infinite set of fields

Rn!1,1 ! T d

xa ! xa + 2!

pi = (kµ, pa)

wa (pa, wa) ! Z2d

(kµ, pa, wa)! (yµ, xa, xa)

xa ! xa + 2!Rn!1,1 ! T 2d

n + d = D = 26 or 10

!(yµ, xa, xa)

xi = (yµ, xa)

T-Duality

• Interchanges momentum and winding

• Equivalence of string theories on dual backgrounds with very different geometries

• String field theory symmetry, provided fields depend on both Kugo, Zwiebach

• For fields not Buscher

• Aim: generalise to fields

!(y) !(y, x, x)

!(y, x, x)

x, x

Dabholkar & CMHGeneralised T-duality

Free field equn, M mass in D dimensions

M2 ! "(k2 + p2 + w2) =2!! (N + N " 2)

L0 ! L0 = N ! N ! pawa = 0

hij ! {hµ! , hµa, hab}

Constraint

N = N = 1 M2 = 0 pawa = 0Massless states

hij(yµ, xa, xa), bij(yµ, xa, xa), d (yµ, xa, xa)

Constrained fields

!! = 0

!(y, x, x)

! ! !

!xa

!

!xa

Constrained fields

!! = 0

!(y, x, x)

! ! !

!xa

!

!xa

Momentum space !(kµ, pa, wa) ! = pawa

Momentum space: Dimension n+2dCone: dimension n+2d-1pawa = 0

Constrained fields

!! = 0

!(y, x, x)

! ! !

!xa

!

!xa



DFT: fields on this cone, with discrete p,wProblem: naive product of fields on cone do not lie on cone. Vertices need projectors

Constrained fields

!! = 0

!(y, x, x)

! ! !

!xa

!

!xa



DFT: fields on this cone, with discrete p,wProblem: naive product of fields on cone do not lie on cone. Vertices need projectors

Restricted fields: Fields that depend on d of 2d torus momenta, e.g. orSimple subsector, no projectors needed, easier

!(kµ, pa) !(kµ, wa)

Torus Backgrounds

Gij =!

!µ! 00 Gab

", Bij =

!0 00 Bab

"

xi = {yµ, xa} = {0, xa}xi = {yµ, xa}

Eij ! Gij + Bij

Left and Right Derivatives

Di =!

!xi! Eik

!

!xk, Di =

!

!xi+ Eki

!

!xk

!! = 1

! =12(D2 ! D2) = !2

!

!xi

!

!xi

! =12(D2 + D2) D2 = GijDiDj

Quadratic Action

S(2) =!

[dxdx]" 1

2eij!eij +

14(Djeij)2 +

14(Dieij)2

!2 d DiDjeij ! 4 d ! d#

!eij = Dj"i + Di"j ,

!d = !14D · "! 1

4D · "

Invariant under

using constraint !! = !! = 0

eij ! eji , Di ! Di , Di ! Di , d! dDiscrete Symmetry

Comparison with Conventional Actions

Di = !i ! !i , Di = !i + !i

Take Bij = 0 !i ! Gik!

!xk

! = !2 + !2 ! = !2 !i!i

eij = hij + bij

Usual quadratic action!

dx L[h, b, d; ! ]

L[h, b, d; ! ] =14

hij!2hij +

12(!jhij)2 ! 2 d !i!j hij

!4 d !2 d +14

bij!2bij +

12(!jbij)2

Action + dual action + strange mixing terms

Double Field Theory Action

!hij = "i#j + "j#i + "i#j + "j #i ,

!bij = !("i#j ! "j#i)! ("i#j ! "j #i) ,

!d = ! " · # + " · # .

Diffeos and B-field transformations mixed. Cubic action found for full DFT

S(2) =!

[dxdx]"

L[h, b, d; ! ] + L[!h,!b, d; ! ]

+ (!khik)(!jbij) + (!khik)(!jbij)! 4 d !i!jbij

#

T-Duality Transformations of Background

g =!

a bc d

"! O(d, d; Z)

X !!

xi

xi

"

E! = (aE + b)(cE + d)"1

X ! =!

x!

x!

"= gX =

!a bc d

" !xx

"

T-duality

transforms as a vector

T-Duality is a Symmetry of the Action

Fields eij(x, x), d(x, x)

Background Eij

E! = (aE + b)(cE + d)"1

X ! =!

x!

x!

"= gX =

!a bc d

" !xx

"

M ! dt " E ct

M ! dt + Etct

Action invariant if:

eij(X) = Mik Mj

l e!kl(X

!)

d(X) = d!(X !)

With general momentum and winding dependence!

Projectors and Cocycles

Naive product of constrained fields does not satisfy constraint

String product has explicit projection

Double field theory requires projections, novel formsLeads to a symmetry that is not a Lie algebra, but is a homotopy lie algebra

SFT has non-local cocycles in vertices, DFT should tooCocycles and projectors not needed in cubic action

L!0 !1 = 0, L!0 !2 = 0 but

!A = 0,!B = 0 !(AB) != 0but

L!0 (!1!2) != 0

General fields

!(x)Fields on Spacetime M

!(x, x)

Restricted Fields on N, T-dual to M !(x!)

Subsector with fields and parameters all restricted to M or N

• Constraint satisfied on all fields and products of fields• No projectors or cocycles• T-duality covariant: independent of choice of N• Can find full non-linear form of gauge transformations• Full gauge algebra, full non-linear action

M,N null wrt O(D,D) metric ds2 = 2dxidxi

Background Independence?

Action S(E, e, d)

!Eij = !"ij

!eij = "ij !12!"i

kekj ! "kjeik

"+ O(e2)

!!S = 0is background independent:

Constant shift


Action S(E, e, d)

!Eij = !"ij

!eij = "ij !12!"i

kekj ! "kjeik

"+ O(e2)

!!S = 0

Background independent field: !!Eij = 0

Eij ! Eij + eij +12ei

kekj +O(e3)

is background independent:

Constant shift


Action S(E, e, d)

!Eij = !"ij

!eij = "ij !12!"i

kekj ! "kjeik

"+ O(e2)

!!S = 0

Background independent field: !!Eij = 0

Eij ! Eij + eij +12ei

kekj +O(e3)

E ! E +!1" 1

2e"!1

e

is background independent:

Constant shift

• Write action and transformations in terms of

• and derivatives

• Find manifestly background independent forms that agree with action and transformations to lowest order

• Unique BI terms that agree with lowest order results. Complete non-linear structure!

Eij , dDi = !i ! Eik!k

Di = !i + Eki!k

S =!

dxdx e!2d"! 1

4gikgjlDpEkl DpEij

+14gkl

#DjEikDiEjl + DjEki DiElj

$

+#Did DjEij + Did DjEji

$+ 4DidDid

%

BI Action and Transformations

S =!

dxdx e!2d"! 1

4gikgjlDpEkl DpEij

+14gkl

#DjEikDiEjl + DjEki DiElj

$

+#Did DjEij + Did DjEji

$+ 4DidDid

%

!Eij = "M#MEij

+Di"j ! Dj "i +Di"kEkj + Dj"

kEik

!!d = !12"M#M + #M"M d

BI Action and Transformations

• Remarkable action, reduces to familiar ones

• uses stringy combination

• checked gauge invariant

• Invariant under O(D,D) if dimensions all non-compact. Broken to discrete subgroup by boundary conditions.

Eij = gij + bij

2-derivative action

S = S(0)(!, !) + S(1)(!, !) + S(2)(!, !)

Write in terms of usual fieldsS(0)

Eij = gij + bij e!2d =!"ge!2!

2-derivative action

S = S(0)(!, !) + S(1)(!, !) + S(2)(!, !)



!dx!"ge!2!

"R + 4(!")2 " 1

12H2

#Gives usual action (+ surface term)

S(0) = S(E , d, !)

2-derivative action

S = S(0)(!, !) + S(1)(!, !) + S(2)(!, !)



!dx!"ge!2!

"R + 4(!")2 " 1

12H2


S(0) = S(E , d, !) S(2) = S(E!1, d, !)T-dual!

2-derivative action

S = S(0)(!, !) + S(1)(!, !) + S(2)(!, !)



!dx!"ge!2!

"R + 4(!")2 " 1

12H2


S(0) = S(E , d, !) S(2) = S(E!1, d, !)T-dual! strange mixed termsS(1)

E !(X !) = (aE(X) + b)(cE(X) + d)"1

Generalised T-duality transformations:

d!(X !) = d(X)

h in O(d,d) acts on toroidal coordinates only

X !M !!

x!i

x!i

"= hXM =

#a bc d

$ #xi

xi

$

Buscher if fields independent of toroidal coordinatesGeneralisation to case without isometries

!M !!

!i

!i

"

!MN =!

0 II 0

"

O(D,D) Covariant Notation

(!, !)! !M

M = 1, ..., 2D

XM !!

xi

xi

"

Parameters

Gauge Algebra

C-Bracket:

[!!1 , !!2 ] = ![!1,!2]C

[!1,!2]C ! [!1,!2]"12

!MN!PQ !P[1 "N !Q

2]

Lie bracket + metric term

Parameters restricted to NDecompose into vector + 1-form on NC-bracket reduces to Courant bracket on N

!M (X)

Same covariant form of gauge algebra found in similar context by Siegel

Symmetry is Reducible

!M = !MN"N#Parameters of the formdo not act

cf 2-form gauge fieldParameters of the formdo not act

!B = d"! = d"

Gauge algebra determined up to such transformations

J(!1,!2,!3) ! [ [!1,!2] ,!3 ] + cyclic "= 0

Jacobi Identities not satisfied!

for both C-bracket and Courant-bracket

How can bracket be realised as a symmetry algebra?

[ [!!1 , !!2 ] , !!3 ] + cyclic = !J(!1,!2,!3)

J(!1,!2,!3) ! [ [!1,!2] ,!3 ] + cyclic "= 0

Jacobi Identities not satisfied!

for both C-bracket and Courant-bracket

How can bracket be realised as a symmetry algebra?

Resolution:

J(!1,!2,!3)M = !MN"N#

!J(!1,!2,!3) does not act on fields

[ [!!1 , !!2 ] , !!3 ] + cyclic = !J(!1,!2,!3)

Generalised Metric Formulation

HMN =!

gij !gikbkj

bikgkj gij ! bikgklblj

".

2 Metrics on double space HMN , !MN


HMN =!

gij !gikbkj


".

HMN ! !MPHPQ!QN

HMPHPN = !MN


Constrained metric


HMN =!

gij !gikbkj


".

hPMhQ

NH!PQ(X !) = HMN (X)

X ! = hX h ! O(D,D)

HMN ! !MPHPQ!QN

HMPHPN = !MN


Constrained metric

Covariant Transformation

L =18HMN!MHKL !NHKL !

12HMN!NHKL !LHMK

! 2 !Md !NHMN + 4HMN !Md !Nd

S =!

dxdx e!2d L

O(D,D) covariant action


12HMN!NHKL !LHMK


S =!

dxdx e!2d L

!!HMN = "P #PHMN

+ (#M"P ! #P "M )HPN + (#N"P ! #P "N )HMP

Gauge Transformation



12HMN!NHKL !LHMK


S =!

dxdx e!2d L

!!HMN = "P #PHMN

+ (#M"P ! #P "M )HPN + (#N"P ! #P "N )HMP

!!HMN = !L!HMN

Gauge Transformation

Rewrite as “Generalised Lie Derivative”


Generalised Lie Derivative

!L!AMN ! !P "P AM

N

+("M!P""P !M )APN + ("N!P " "P !N )AM

P

A M1...N1...


!L!AMN ! !P "P AM

N

+("M!P""P !M )APN + ("N!P " "P !N )AM

P

!L!AMN = L!AM

N ! !PQ!MR "Q#R APN

+ !PQ!NR "R#Q AMP

A M1...N1...


!L!AMN ! !P "P AM

N

+("M!P""P !M )APN + ("N!P " "P !N )AM

P

!L!AMN = L!AM

N ! !PQ!MR "Q#R APN

+ !PQ!NR "R#Q AMP

A M1...N1...

! "L!1 , "L!2

#= ! "L[!1,!2]C

Algebra given by C-bracket

D-Bracket!A, B

"D! #LAB

!A, B

"M

D=

!A, B

"M

C+

12!M

#BNAN

$

D-Bracket!A, B

"D! #LAB

!A, B

"M

D=

!A, B

"M

C+

12!M

#BNAN

$

Not skew, but satisfies Jacobi-like identity!A,

!B, C

"D

"D

=!!

A, B"D

", C

"D

+!B,

!A, C

"D

"D

D-Bracket!A, B

"D! #LAB

On restricting to null subspace NC-bracket ➞ Courant bracketD-bracket ➞ Dorfman bracketGen Lie Derivative ➞ GLD of Grana, Minasian, Petrini and Waldram

!A, B

"M

D=

!A, B

"M

C+

12!M

#BNAN

$

Not skew, but satisfies Jacobi-like identity!A,

!B, C

"D

"D

=!!

A, B"D

", C

"D

+!B,

!A, C

"D

"D

Generalized scalar curvature

R ! 4HMN!M!Nd" !M!NHMN

" 4HMN!Md !Nd + 4!MHMN !Nd

+18HMN!MHKL!NHKL "

12HMN!MHKL !KHNL




+18HMN!MHKL!NHKL "

12HMN!MHKL !KHNL

S =!

dx dx e!2dR




+18HMN!MHKL!NHKL "

12HMN!MHKL !KHNL

S =!

dx dx e!2dR

!!R = !L!R = "M#MR!! e!2d = "M (#Me!2d)

Gauge Symmetry




+18HMN!MHKL!NHKL "

12HMN!MHKL !KHNL

S =!

dx dx e!2dR

!!R = !L!R = "M#MR!! e!2d = "M (#Me!2d)

Gauge Symmetry

Field equations give gen. Ricci tensor

M-Theory

• Strings: Doubled torus e.g. T7→ T14

• M-Theory: M-torus e.g. T7→ T56

• U-duality E7 acts on T56

• 7 torus coordinates + 49 coords conjugate brane wrapping modes

Extended Geometry

• Extended geometry: metric and 3-form gauge field

• Generalised metric

• Rewrite of 11-d sugra action in terms of this

CMH; Pacheco & Waldram

Hillman; Berman & Perry

Double Field Theory• Captures some of the magic of string theory

• Constructed cubic action, quartic could have new stringy features

• T-duality symmetry, cocycles, homotopy Lie, constraints

• For fields restricted to null subspace, have full non-linear action and gauge transformations.

• Background independent, duality covariant

• Courant bracket gauge algebra

• Stringy issues in simpler setting than SFT

• Geometry? Meaning of curvature?

• Use for non-geometric backgrounds?

• Generalised Geometry doubles Tangent space, DFT doubles coordinates.

• Full theory (no restriction)? Does it close on a geometric action with just these fields?

• General spaces, not tori?

• Doubled geometry physical and dynamical

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