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