Topological String Theory

and Black Holes

Eurostrings 2006, Cambridge, UK

- review- w/ C. Vafa, E.Verlinde,

hep-th/0602087 - work in progress

Robbert DijkgraafUniversity of Amsterdam

Topological Strings

• Toy model (cf topology versus geometry)

• Exact BPS sector of superstrings

• Mathematical experiments to testphysical intuition

Geometry of Calabi-Yau manifolds

3 complex dimensions Euclidean solutions of 0R

X

(3,0)-form = dz1 ^dz2 ^dz3

c1(X ) = 0

Kaehler form k =p

¡ 1gi ¹|dzi ^d¹z¹| ;

dk = 0

Exact Effective Actions

4CY

4 4 2 ( ) gg gravd x d F t W

F-terms for Weyl muliplet in4 dim supergravity action

CY

top string partition function

2 2

0

exp ( )gtop g

g

Z F t

s gravg F

A-Model (IIA): Exact worldsheet instantons

f

holomorphic mapdegree d

3 CY X

Kähler moduli 1,1[ ] ( )t k H X

Localizes on 0f

Gromov-Witten Invariants

,( ) g ddt

gd

F t GW e

# maps

Exact instanton sum 2 ( , )d H X

genus g

B-Model (IIB string)

Localizes on 0df Complex moduli 2,1 ( )t H X

f

almostconstant maps

Mirror Symmetry

X

X

0 ( )F t

classical

/0 0,

0

( ) tdd

d

F t N e

quantum

A-model B-Model

Fiberwise T-Duality[Strominger, Yau, Zaslov]

X X

Dual Torus Fibrations

2,1h

3T

X

base

1,1h

3 *( )T

X

base

D-branes &Black holes

D-Branes

X0( ) ( )evenY H X K X

Coherent sheaves

IIA

X

13( ) ( )Y H X K X

Special Lagrangians

IIB

homological mirror

symmetry)

Charge Lattice (B-model)

3( , )

X

H X

a b

symplectic vector space

H 3(X ;Z)

Period Map & Quantization

3( , )

X

H X

a b

moduli space of CY

Lagrangian cone L=graph (dF0) semi-classical state ψ ~ exp F0

L

symplectic vector space

hol 3-form dz1 dz2 dz3

Special Geometry

3, ( , )jiA B H X

i

j

i

A

j B

q

p

0i i

Fp

q

3 CY X

Top String Partition Function = Wave Function

2 2exp ( ), gtop g

g

Z F t

Transforms as a wave function underSp(2n,Z) change of canonical basis (A,B)

A-Model

Kähler cone

symplectic vector space

complexifiedKähler volume

H ev(X ;C)

h®;¯i = index D® ¯ ¤

1¸

ek+iB

plus world-sheet instanton corrections

F0 =t3

6 2

Charged objects: D-branes

3CY time

charged particles

( , ) ( , )evQ P H X

electric-magnetic charges

Large volume:•electric D0-D2•magnetic D4-D6

4d Black Holes

3CY time

D-brane Black Hole

large charges

Attractor CY

charges ( , )IIP Q

Attractor Mechanism

near-horizonmoduli tI

Quantization of Moduli Space3( , )H X

“attractive”CY’s

Re 2 H 3(X ;Z)

Black Hole Partition Function

Witten index of susy gauge

theory

Exact Black Hole Entropy[G. Lopes Cardoso, T. Mohaupt, B. de Wit]

[ Ooguri, Strominger, Vafa]

Mixed Ensemble

( , ) ( , )BHQ

IIQZ P e P Q

( , ) exp #BHP Q S states

electric/magnetic charges

Mixed Ensemble

0 0

( , )

exp ( ) ( )

PBHZ P

F P i F P i

( , ) ( ) ( )BH top topZ P P i P i

OSV Conjecture

M-theory

Gopakumar-VafaAt strong coupling can integrate out(light) electric charges D0-D2 to obtain theeffective action

gs ! 1

charges

( , ) log det QQ

F t

3 1CY S time

M-theory limit

gs

virtual loopsof M2 branes

5d Black Holes in M-theory4CY time

Transversal rotationsSO(4) ´ SU(2)L £ SU(2)R

M2-branes with charge

Q 2 H2(X ;Z)M2M2

Internal spin quantum numbers

(mL ;mR )

BPS degeneracies

M2

Index of susy ground states (GV-inv)

N mRQ =

X

mL

(¡ 1)mL N mL ;mRQ

4d Quantum Hall system:wave functions lowest Landau level

ª (z1;z2) =X

n1 ;n2

an1 ;n2zn1

1 zn22

CY

4 2 Orbital angular momentum

(n1;n2)self-dual flux

rotation

space

GV Partition function

Gas of 5d charged & spinning black holes

Z(¸;t) =Y

n1;n2Q;m

³1¡ e (n1+n2+m)+tQ

´ ¡ N mQ

5d entropy

N mQ »

pQ3 ¡ m2

6+1 dim SUSY Gauge Theory

Witten index counts D-brane bound states

Z = Tr£(¡ 1)F e¡ ¯ H

¤

Induced charges: non-trivial gauge bundle

(P;Q) ¼ch¤(E)

Reduction to moduli space of vacua

Z » Euler(M E)

Donaldson-Thomas Invariants

Single D6: U(1) gauge theory + singularities

q= D2= ch2 » TrF 2

instanton strings

k = D0= ch3 » TrF 3

Z(¸;t) =X

k;q

DT(k;q)ek¸ +qt

Lift to M-theory

D6 → Taub-NUT geometry

4SO(4)

angular momentum

ds2T N = R2

·1V

(dÂ+ ~A ¢d~x)2 +Vd~x2

¸

3 1SU(1) £ SO(3)

Kaluza-Klein momentum

[Gaiotto, Strominger,Yin]

Bound states with D0-D2

spinning M2-branes

R

q=X

i

Qi

k =X

i

(ni + mi )

Gauge theory quantum numbers

4 dim limit:

Bound state of D6-D2

R ! 0

3

Z(¸;t) =X

k;q

DT(k;q)ek¸ +qt

Donaldson-Thomas Invariants

5 dim limit:

4

R ! 1

Free gas of M2-branes

Gopakumar-Vafa Invariants

Z(¸;t) =Y

n1 ;n2Q;m

³1¡ e (n1+n2+m)+tQ

´ ¡ N mQ

Topological String Triality

peturbative IIA stringsGromov-Witten

M2-branesGopakumar-

Vafa

D2-branesDonaldson-

Thomas

stro

ng-w

eak 9-11 flip

Taub-NUT

3Simplest Calabi-Yau

31 2 3, ,z z z

3

2 2 23,0 1

2 (2 2)(2 2)!( )

g

g gg g

M g g g

B BGW c H

g

constant maps

1 2

2

2,0

0

1( )

, 0

3d partitions of

exp

1

g gtop g

g

n n

n n

N

N

Z GW

e

e

Stat-Mech: 3d Partitions

2 3

0

1 1 3 6 ...nn

topn

Z q q q q

Melting Crystals

Reshetikhin,Okounkov,Vafa, Nekrasov,...

UniversalWave Function

Wave Function of String Theory

Compactify on a 9-space

X £ time

ª 2 HX

Flux/charge/brane sectors

HX =M

Q

HQX

Topology Change

Finite energy transitions

X ! X 0

ª 2 H

Universal wave function, components on all geometries

Baby Universes

Disconnected spaces

X ! X 1 + X 2

Second quantization

H ! Sym¤H

Hawking-Hartle Wave Function

Sum over bounding geometries

X = @B

Include singularities (branes, black holes)

ª =X

B

jB i

“Entropic Principle”

Natural probability density on moduli space of string compactifications

eS = jª j2

Depends on massless & massive d.o.f.

peaked aroundmoduli space

string theory on the near horizon geometry of the

black hole

AdS/CFT duality

22

3AdS S CY

supersymmetricgauge theory on

the brane

superconformalquantum mechanics

Hawking-Hartle Wave Function [Ooguri,E.Verlinde,Vafa]

Euclideantime

22

3AdS S CY

top top

cf. open/closed worldsheet duality

string

D-brane E

jEi 2 Hclosed

Index theorem

E F

indexDE F ¤ =

Z

Xch(E)ch(F ¤) bA

time

TrH open(¡ 1)F = indexDE F ¤

time

hE;F i =

Z

Xch(E)ch(F ¤) bA

E F

Supersymmetry breaking

Non-susy boundary conditions

Z(¯) = Tr£e¡ ¯ H

¤ ¯

Positivity of H¯ < ¯0 ) Z(¯) > Z(¯0)

Ground states

Prefers symmetric CY’s

Z(1 ) = dimH0 = #harmonic forms

¸ Euler

Space of AllCalabi-Yau’s

Topology of Calabi-Yau spaces

=

X 0X § g

X = X 0#§ g

b3 = 0 b2 = 0

Core

Non-Kahler CY are unique

§ g

§ g = #g¡S3 £ S3

¢

Moduli space of complex structures

dimM g = g¡ 1

Miles Reid’s Fantasy:“There is only one CY space”

M g

b2 = 0

All CY connected through conifoldtransitions S3 → S2

b2 = 1Kähler CYs

CY

3S

3T

SYZ: fibrations by Slag T3

network ofsingularities

S1 shrinks

3

Two Vertices

+ -

MirrorSymmetry

topological vertex local Riemann surface

3

3d Topological Gauge Theory

Wilson loops & graphs

3S

Universal Moduli Space of CYs?

Gell-Mann on renormalization:

“Just because it’s infinite, it doesn’t mean it’s zero”

Topological Strings

• Compute BPS black hole degeneracies(gauge-gravity dualities)

• Interesting probability distribution onthe moduli space of vacua

• Universal Calabi-Yau???

• Many more surprises...

Happy 60th, Michael!