¨uller) subject kshop m1. ators †spaces spinors † ies 2. sections †introduction ics 3. olds...
TRANSCRIPT
Glo
bally
hype
rbol
icLo
rent
zian
man
ifold
sw
ithsp
ecia
lhol
onom
y
Hel
gaB
aum
,Hum
bold
tUni
vers
ityof
Ber
lin
Lect
ure
atth
e20
06IM
AS
umm
erP
rogr
am:
Sym
met
ries
and
Ove
rdet
erm
ined
Sys
tem
sof
Par
tialD
iffer
entia
lEqu
atio
ns,
Min
ness
ota,
July
17-A
ugus
t4,2
006
Bef
ore
Ista
rtw
ithth
eto
pic
inth
etit
le,I
will
expl
ain
how
this
topi
cis
rela
ted
toth
esu
bjec
t
ofth
eS
umm
erP
rogr
aman
dto
som
eot
her
talk
sdu
ring
the
wor
ksho
p
(Coo
pera
tion
with
I.K
ath,
F.Le
itner
,Th.
Leis
tner
,Th.
Neu
kirc
hner
,A.G
alae
v,O
.Mul
ler)
1
1.R
elat
ion
toov
erde
term
ined
syst
ems
ofP
DE
and
conf
orm
ally
inva
riant
oper
ator
s
•S
pino
rson
curv
edsp
aces
•C
onfo
rmal
lyin
varia
ntop
erat
ors
onsp
inor
s•
Con
form
alK
illin
gsp
inor
san
dsp
ecia
lgeo
met
ries
2.H
olon
omy
ofco
nnec
tions
and
para
llels
ectio
ns
•G
ener
alin
trod
uctio
n•
Hol
onom
ygr
oups
ofsp
inco
nnec
tions
and
met
rics
3.H
olon
omy
grou
psof
Rie
man
nian
and
Lore
ntzi
anm
anifo
lds
•R
iem
anni
anm
anifo
lds
•Lo
rent
zian
man
ifold
s
4.G
loba
llyhy
perb
olic
Lore
ntzi
anm
anifo
lds
with
spec
ialh
olon
omy
and
para
llels
pino
rs
•gl
obal
lyhy
perb
olic
man
ifold
s•
Aco
nstr
uctio
n
2
Spi
nors
oncu
rved
spac
es
Let(
Mp,q
,g)b
ea
pseu
do-R
iem
anni
ansp
inm
anifo
ld(w
2(M
)=
0).
The
non
(M,g
)th
ere
isa
spec
ialc
ompl
exve
ctor
bund
leS
:=Q× S
pin
(p,q
)∆
(spi
nor
bund
le)
with
aco
varia
ntde
rivat
ive∇S
:Γ(S
)−→
Γ(T∗ M
⊗S
)(s
pin
conn
ectio
n)an
da
herm
itian
inne
rpr
oduc
t<·,·
>.
(n=
p+
q≥
3)
One
can
mul
tiply
vect
ors
and
spin
ors
X∈
TM
,ϕ∈
S7−→
X·ϕ
∈S
Clif
ford
prod
uct
such
that
the
follo
win
gru
les
hold
•(X
·Y+
Y·X
)·ϕ
=−2
g(X
,Y)ϕ
•<
X·ϕ
,ψ>
=(−
1)p−
1<
ϕ,X
·ψ>
•∇
S X(Y
·ϕ)
=(∇
g XY
)·ϕ
+Y·∇
S Xϕ
•X
(<ϕ
,ψ>
)=
<∇
S Xϕ
,ψ>
+<
ϕ,∇
S Xψ
>
3
Con
form
ally
inva
riant
oper
ator
son
spin
ors
The
Clif
ford
prod
uctµ
:T∗ M
⊗S−→
Sgi
ves
asp
littin
gof
the
bund
leof
1-fo
rms
onM
with
valu
esin
the
spin
orbu
ndle
T∗ M
⊗S
=Im
µ⊕
Ker
µ=
S⊕
Tw
Hen
ce,t
here
are
two
diffe
rent
ialo
pera
tors
of1-
orde
ron
spin
orfie
lds
D:=
pr S◦∇
S:
Γ(S
)→
Γ(T∗ M
⊗S
)→
Γ(S
)D
irac
oper
ator
P:=
pr T
w◦∇
S:
Γ(S
)→
Γ(T∗ M
⊗S
)→
Γ(T
w)
Twis
tor
oper
ator
Con
form
alco
varia
nce
: D(e
2σg)
=e−
n+
12
σD
(g)e
n−
12
σ
P(e
2σg)
=e−
1 2σ
P(g
)e−
1 2σ
Twis
tor
equa
tion
:P
ϕ=
0(ϕ
conf
orm
alK
illin
gsp
inor
)
Pϕ
=0⇔
∇S Xϕ
+1 nX·D
ϕ=
0O
verd
et.
PD
E⇒
talk
ofA
.Cap
1.O
p.in
aB
GG
talk
ofR
.Gov
er4
Con
form
alK
illin
gsp
inor
san
dsp
ecia
lLor
entz
ian
geom
etrie
s
Que
stio
n:F
orw
hich
Lore
ntzi
ansp
inm
anifo
lds
ther
eex
ist
solu
tions
ofth
eco
nfor
mal
Kill
ing
spin
oreq
uatio
n:∇S X
ϕ+
1 nX·D
ϕ=
0fo
ral
lvec
tor
field
sX
Rem
ark:
The
nam
eco
mes
from
the
follo
win
gfa
ct:
ϕ∈
Γ(S
)⇒
Vϕ∈X
(M):
g(V
ϕ,X
):=−
<X·ϕ
,ϕ>
Ifϕ
isa
conf
orm
alK
illin
gsp
inor
,th
anV
ϕis
atim
e-or
light
like
conf
orm
alve
ctor
field
with
the
sam
eze
ros
asϕ
.
Max
imal
num
ber
ofso
lutio
nsIf(M
,g)i
sco
nfor
mal
lyfla
tand
1-co
nnec
ted,
then
the
conf
orm
alK
illin
gsp
inor
equa
tion
has
the
max
imal
poss
ible
num
ber
ofin
depe
nden
tsol
utio
ns,n
amly
2[n 2
]+1.
Any
met
ricw
ithth
isnu
mbe
rof
inde
pend
ents
olut
ions
isco
nfor
mal
lyfla
t.(c
f.ta
lkof
A.C
ap).
Wha
tels
ein
the
non-
conf
orm
ally
flatc
ase
??
5
The
orem
:(F
.Lei
tner
2004
)
Let(
M,g
)be
aLo
rent
zian
man
ifold
with
”gen
eric
”con
form
alK
illin
gsp
inor
,the
n(M
,g)
islo
cally
conf
orm
aleq
uiva
lent
toon
eof
the
follo
win
gsp
aces
•P
rodu
ctof
(R,−
dt2
)w
itha
Ric
ci-fl
atR
iem
anni
anm
anifo
ldw
ithpa
ralle
lspi
nors
.
•Lo
rent
zian
Ein
stei
n-S
asak
iman
ifold
•Lo
rent
zian
Ein
stei
n-S
asak
iman
ifold×(
N,h
),w
here
(N,h
)is
aR
iem
anni
anE
inst
ein-
Sas
aki
man
ifold
,a
3-S
asak
i-man
ifold
,a
near
lyK
ahle
rm
anifo
ldor
aR
iem
anni
ansp
here
•F
effe
rman
spac
e(c
f.ta
lkof
K.H
irach
i)
•B
rinkm
ansp
ace
with
para
llels
pino
r
ϕis
”gen
eric
”iff
∗ϕ
has
noze
ros
∗V
ϕdo
esno
tcha
nge
the
caus
alty
pe
∗V
[ ϕha
sco
nsta
ntra
nk,w
here
rank
σ=
max{k|σ∧
(dσ)k6=
0}
6
All
thes
esp
ecia
lgeo
met
ries
are
inte
rest
ing.
The
first
4ty
pes
are
quite
wel
lund
erst
ood.
Aim
:U
nder
stan
dth
ela
stcl
ass
ofge
omet
ries:
Brin
kman
nsp
aces
Defi
nitio
n:A
Brin
kman
nsp
ace
isa
Lore
ntzi
anm
anifo
ldw
itha
para
llell
ight
like
vect
orfie
ld.
Suc
ha
man
ifold
has
spec
ialh
olon
omy
!!!
Que
stio
n:W
hati
skn
own
abou
tLor
entz
ian
man
ifold
sw
ithsp
ecia
lhol
onom
y??
?
7
Hol
onom
ygr
oups
and
para
llels
ectio
ns
Eve
ctor
bund
leov
erM
with
cova
riant
deriv
ativ
e∇,
x∈
M
Hol
x(E
,∇):={P
∇ γ:E
x→
Ex
para
llelt
rans
port
alon
gγ|γ
loop
inx}
PG
-prin
cipa
lbun
dle
over
Mw
ithpr
inci
palb
undl
eco
nnec
tion
ω,p∈
Px
Hol
p(P
,ω):={g∈
G|∃
loop
γin
xsu
chth
atγ∗ p(1
)=
p·g}
Letρ
:G→
GL
(V)
bea
repr
esen
tatio
n,E
:=P× G
Van
d∇
=∇ω
.F
ixin
ga
p∈
Px
give
san
isom
orph
ism
Ex'
Vsu
chth
at
Hol
x(E
,∇ω)
=ρ(H
olp(P
,ω))
Hol
onom
ypr
inci
ple:
The
reis
a1-
1co
rres
pond
ence
betw
een
{ϕ∈
Γ(E
)|∇
ωϕ
=0}
and
{v∈
V|ρ
(Hol
p(P
,ω))
v=
v}
={v∈
V|ρ∗(
hol p
(P,ω
))v
=0}
ifπ1(M
)=
0
8
Hol
onom
ygr
oups
ofsp
inco
nnec
tions
and
met
rics
Let(
Mp,q
,g)
bea
spin
man
ifold
with
the
fram
ebu
ndle
Pan
dsp
inst
ruct
ure
(Q,f
)).
λ:S
pin
(p,q
)−→
SO
(p,q
)2-
fold
cove
ring.
TM
:=P× S
O(p
,q)R
p,q
p∈
Px
fram
ein
x
S:=
Q× S
pin
(p,q
)∆
q∈
Qx
spin
fram
ein
x,f
(q)
=p
Hol
x(T
M,∇
g)
=H
olp(P
,ωL
C)⊂
SO
(p,q
)
Hol
x(S
,∇S)
=ρ(H
olq(Q
,ωL
C))⊂
ρ(S
pin
(p,q
))
The
nλ(H
olq(Q
,ωL
C))
=H
olx(T
M,∇
g)
hol q
(Q,ω
LC
)=
(λ∗)−
1hol x
(TM
,∇g)
Ups
hot:
One
can
deci
deth
eex
iste
nce
ofpa
ralle
lspi
nors
ifon
ekn
ows
the
holo
-no
my
grou
pof
(M,g
).If
(M,g
)is
sim
ply
conn
ecte
d,th
en
{ϕ∈
Γ(S
)|∇
Sϕ
=0}≡{v∈
∆|ρ∗(
λ−
1∗
(hol(
TM
,∇g))
)v=
0}9
Hol
onom
ygr
oups
ofR
iem
anni
anm
anifo
lds
(Mn,g
)R
iem
anni
anm
anifo
ld,c
ompl
ete,
sim
ply-
conn
ecte
d.
DeR
ham
Spi
tting
The
orem
:(G
.DeR
ham
1952
)(M
,g)'R
k×
(M1,g
1)×···×
(Mk,g
k)
whe
re(M
i,g i
)is
irred
ucib
le
Ber
ger’s
List
:(M
.Ber
ger
1955
)Le
t(M
n,g
)be
anirr
educ
ible
non-
loca
llysy
mm
etric
Rie
man
nian
man
ifold
.T
hen
the
holo
nom
ygr
oup
Hol
(M,g
) 0is
(up
toco
njug
atio
n)on
eof
the
follo
win
gon
ce
SO
(n)
gene
ricty
pe0
U(n 2
)K
ahle
r0
SU
(n 2)
Ric
ci-fl
at,K
ahle
r2
Sp(n 4
)H
yper
kahl
ern 4
+1
Sp(n 4
)·S
p(1
)qu
ater
nion
icK
ahle
r0
G2
n=
7,sp
ecia
lpar
alle
l3-f
orm
1S
pin
(7)
n=
8,sp
ecia
lpar
alle
l4-f
orm
1
10
Hol
onom
ygr
oups
ofsy
mm
etric
spac
es
Let(
M,g
)be
a1-
conn
ecte
dsy
mm
etric
spac
e,M
=G
/K,w
here
G⊂
Iso
m(M
,g)
isth
etr
ansv
ectio
ngr
oup
ofM
and
K=
Gx
the
stab
ilize
rof
apo
intx∈
M.
The
n
1.H
olx(M
,g)'
K
2.T
heho
lono
my
repr
esen
tatio
nH
olx(M
,g)→
SO
(TxM
,gx)
isgi
ven
byth
e
isot
ropy
repr
esen
tatio
nof
K.
irred
ucib
lesy
mm
etric
spac
esar
ecl
assi
fied
⇒th
eir
holo
nom
ygr
oups
are
know
n
11
Hol
onom
ygr
oups
ofLo
rent
zian
man
ifold
s
(Mn,g
)Lo
rent
zian
man
ifold
,com
plet
e,si
mpl
y-co
nnec
ted.
Wu
Spi
tting
The
orem
:(H
.Wu
1967
)
(M,g
)'
(N,h
)×
(M1,g
1)×···×
(Mk,g
k),
whe
re(M
i,g i
)are
flato
rirr
educ
ible
Rie
man
nian
man
ifold
san
d(N
,h)i
sa
Lore
ntzi
an
man
ifold
that
isei
gthe
r
•fla
t
•irr
educ
ible
or
•w
eakl
yirr
educ
ible
and
non-
irred
ucib
le,t
.m.
the
holo
nom
yre
pres
enta
tion
ρ:H
ol(N
,h)→
SO
(TxM
,gx)
has
nono
n-de
gene
rate
inva
riant
subs
pace
,
buta
dege
nera
tein
varia
nton
e.
12
The
orem
1(B
erge
r’slis
t,O
lmos
/DiS
cala
’01,
Bou
bel/Z
eghi
b’03
,Ben
oist
/del
aHar
pe’0
4)
Ifth
eho
lono
my
grou
pH
ol(N
,h)
ofa
sim
ply
conn
ecte
dLo
rent
zian
man
ifold
acts
irre-
duci
ble
than
Hol
(N,h
)=
SO
0(1
,n−
1)
The
reis
nosp
ecia
lirr
educ
ible
Lore
ntzi
anho
lono
my
!!!
LetH
ol(N
,h)
actw
eakl
y-irr
educ
ible
and
non-
irred
ucib
le.
IfW
isa
dege
nera
tein
varia
ntsu
bspa
ce,
then
W∩
W⊥
=R
v 0fo
ra
light
like
vect
or
v 0.
Hen
ce Hol
(N,h
)⊂
SO
(1,n−
1)Rv
0=
(R∗×
SO
(n−
2))nR
n−
2
13
The
orem
2(B
erar
d-B
erge
ry/Ik
emak
hen
’93,
Gal
aev’
04)
Let
h⊂
so(1
,n−
1)Rv
0=
(R⊕
so(n−
2))nR
n−
2be
aw
eakl
y-irr
educ
ible
sub-
alge
bra
and
g:=
pro
j so(n−
2)(h
)=
z(g)⊕
[g,g
]⊂so
(n−
2).
The
nth
ere
are
4ca
ses
•h
=(R⊕
g)nR
n−
2
•h
=gnR
n−
2
•h
=(g
raph(ϕ
)⊕
[g,g
])nR
n−
2,
whe
reϕ
:z(g
)→R
islin
ear
and
surje
ctiv
e
•h
=([
g,g
]⊕gra
ph(ψ
))nR
r,
whe
reR
n−
2=R
r⊕R
s,
0<
r,s
<n−
2g⊂
so(R
r)
ψ:z
(g)→R
slin
ear
and
surje
cive
14
The
orem
3(T
h.Le
istn
er20
03)
Let(N
n,h
)be
asi
mpl
y-co
nnec
ted
Lore
ntzi
anm
anifo
ldw
itha
wea
kly
irred
ucib
lean
d
non-
irred
ucib
leac
ting
holo
nom
ygr
oup
Hol
(N,h
)an
dle
t
G:=
pro
j SO
(n−
2)H
ol(N
,h)⊂
SO
(n−
2).
The
n
•G
isth
epr
oduc
tofR
iem
anni
anho
lono
my
grou
ps.
•(N
,h)
has
para
llels
pino
rsif
and
only
if
Hol
(N,h
)=
GnR
n−
2,
whe
reG
istr
ivia
lor
apr
oduc
tofS
U(k
),S
p(l
),G
2or
Spin
(7).
15
The
orem
4(A
.Gal
aev
2005
)
Any
grou
pap
pear
ing
inT
heor
em2
and
The
orem
3is
infa
ctth
eho
lono
my
grou
pof
aLo
rent
zian
man
ifold
.
A.G
alae
vco
nstr
ucte
dlo
cala
naly
ticm
etric
sfo
ral
ltyp
es(t
heco
uple
dty
pes
whe
reun
know
nbe
fore
):
N=R×R×R
n−
2
h(t
,s,x
)=
2dtd
s+
f(t
,s,x
)ds2
+2
n0 ∑ j
=1
uj(s
,x)d
xjds
+n−
2∑ j=
1
(dx
j)2
•f(t
,s,x
)=
...s
peci
alfo
rmin
the
four
case
s,us
esth
eco
uplin
gfu
nctio
nsϕ
and
ψ
•u
j(s
,x)=
Aj α
ikx
i xksα−
1A
. ...
com
esfr
oma
basi
sof
g=
pro
j so(n−
2)hol(
N,h
).
=⇒
The
clas
sific
atio
nof
holo
nom
ygr
oups
ofsi
mpl
y-co
nnec
ted
Lore
ntzi
anm
anifo
lds
isfin
ishe
d
16
Task
:D
escr
ibe
glob
alm
odel
sfo
rLo
rent
zian
man
ifold
sw
ithsp
ecia
lhol
onom
y
•Lo
rent
zian
sym
met
ricsp
aces
are
know
n:S
pace
form
s(M
inko
wsk
i,A
dS,
dS)
and
Cah
en-W
alla
chsp
aces
(Cah
en/W
alla
ch19
70)
(App
roac
hto
clas
sific
atio
nof
wea
kly-
irred
ucib
lesy
mm
etric
spac
es(n
on-s
emis
impl
e
tran
svec
tion
grou
p)by
I.K
ath,
M.O
lbric
h(2
004)
)
•Lo
rent
zian
hom
ogen
eous
spac
es(o
pen
prob
lem
,Th.
Neu
kirc
hner
)
•G
loba
llyhy
perb
olic
Lore
ntzi
anm
anifo
lds
(H.B
aum
,O.M
ulle
r(2
005)
)
•C
ompl
ete
Lore
ntzi
anm
anifo
lds
???
•??
?
Que
stio
n:W
hich
ofth
esp
ecia
lLor
entz
ian
holo
nom
ygr
oups
can
bere
aliz
edby
glob
ally
hype
rbol
icLo
rent
zian
man
ifold
s?
17
Glo
bally
hype
rbol
icLo
rent
zian
man
ifold
sw
ithsp
ecia
lhol
onom
y
Defi
nitio
n:A
Lore
ntzi
anm
anifo
ld(M
,g)
isca
lled
glob
ally
hype
rbol
iciff
•(M
,g)
isst
ongl
yca
usal
(for
exam
ple
ifth
ere
exis
tsa
cont
inou
sfu
nctio
nf
onM
whi
chis
stric
tlyin
crea
sing
alon
gan
yfu
ture
dire
cted
caus
alcu
rve)
•J
+(p
)∩
J−
(q)⊂
Mis
com
pact
for
allp
,q∈
MJ±
(p):={x∈
M|∃
γ:p→
xca
usal,↑
+(↓−
)}
Som
esp
ecia
lpro
pert
ies
ofgl
obal
lyhy
perb
olic
man
ifold
s
•N
orm
ally
hype
rbol
icop
erat
ors
have
angl
obal
and
uniq
uefo
rwar
dan
dba
ckw
ard
fund
amen
tals
olut
ion
•E
xist
ence
ofC
auch
ysu
rfac
es
•M
axim
alca
usal
geod
esic
s:p,q
∈M
,p≤
q.T
hen
ther
eex
ists
aca
usal
geod
esic
from
pto
qof
max
imal
leng
th.
18
A(v
ery)
part
iala
nsw
er:
The
orem
5(B
aum
/Mul
ler
2005
)
Any
Lore
ntzi
anho
lono
my
grou
pof
the
form
GnR
n−
2⊂
SO
(1,n−
1)
whe
reG⊂
SO
(n−2
)is
triv
ialo
rthe
prod
ucto
fgro
ups
ofth
efo
rmS
U(k
),S
p(l
),G
2
orS
pin
(7)
can
bere
aliz
edby
agl
obal
lyhy
perb
olic
Lore
ntzi
anm
anifo
ld(M
n,g
).
The
idea
fort
heco
nstr
uctio
nof
such
met
ricw
asin
spire
dby
apa
pero
fCh.
Bar
,P.G
audu
-
chon
,A.M
oroi
anu
(200
4)
19
Asp
ecia
lcon
stru
ctio
n
Let(
M,g
0)
bea
Rie
man
nian
spin
man
ifold
with
aC
odaz
zite
nsor
A
Asy
mm
etric
(1,1
)-te
nsor
field
with
(∇g0
XA
)(Y
)=
(∇g0
YA
)(X
).
Asp
inor
field
ϕ∈
Γ(S
M)
isca
lled
A-
Cod
azzi
spin
orif
∇S Xϕ
=iA
(X)·ϕ
for
allv
ecto
rfie
lds
X
The
orem
:(B
ar/G
audu
chon
/Mor
oian
u’04
,Bau
m/M
ulle
r’05)
Let(M
,g0)
bea
com
plet
eR
iem
anni
ansp
inm
anifo
ldw
ithan
A-C
odaz
zisp
inor
,th
enth
eLo
rent
zian
cylin
der
C:=
I×
M,
g C:=−d
t2+
(1−
2tA
)∗g 0
isgl
obal
lyhy
perb
olic
with
spec
ialh
olon
omy
and
apa
ralle
lspi
nor.
Que
stio
n:A
reth
ere
A-C
odaz
zisp
inor
s??
?H
owsu
chm
anifo
lds
look
like
for
inve
rtab
leA
?
20
The
case
ofin
vert
able
Cod
azzi
tens
ors
A
The
orem
:(B
aum
/Mul
ler’0
5)
Let(M
,g0)
bea
com
plet
eR
iem
anni
anm
anifo
ldw
ithA
-Cod
azzi
spin
orfo
ran
in-
vert
able
Cod
azzi
tens
orA
,an
dle
tal
leig
enva
lues
ofA
are
unifo
rmal
lybo
unde
daw
ay
from
zero
.T
hen
(M,g
0)'
(R×
F,
(A−
1)∗
(ds2
+e−
4sg F
))
whe
re(F
,h)
isa
com
plet
eR
iem
anni
anm
anifo
ldw
ithpa
ralle
lsp
inor
san
dA−
1is
a
Cod
azzi
-ten
sor
onth
ew
arpe
dpr
oduc
t(R×
F,d
s2+
e−4sg F
).A
ndvi
ceve
rsa.
21
The
orem
:(B
aum
/Mul
ler’0
5)
Let(
F,g
F)b
ea
com
plet
eR
iem
anni
anm
anifo
ldw
ithpa
ralle
lspi
nors
,Ta
Cod
azzi
ten-
sor
on(F
,gF)
with
eige
nval
ues
boun
ded
from
belo
w.T
defin
ess-
para
met
erfa
mili
es
ofap
prop
riate
Cod
azzi
tens
orsB
onth
ew
arpe
dpr
oduc
t(R×
F,d
s2+
e−4sg F
).Le
t
C(F
,B)
:=I×R×
F,
g C:=−d
t2+
(B−
2t)∗
(ds2
+e−
4sg F
).
The
n •(C
,gC
)is
agl
obal
lyhy
perb
olic
Brin
kman
spac
e.
•If
(F,h
)ha
sa
flatf
acto
r,th
enC
(F,B
)is
deco
mpo
sabl
e.
•If
(F,h
)is
(loca
lly)
apr
oduc
tof
irred
ucib
lefa
ctor
s,th
enC
(F,B
)is
wea
kly
irre-
duci
ble
and
Hol
0 (0,0
,x)(C
,gC
)=
(B−
1◦H
ol0 x(F
,gF)◦B
)nR
dim
F
22