Topological String Theory and Black Holes Eurostrings 2006, Cambridge, UK - review - w/ C. Vafa, E.Verlinde, hep-th/ work in progress Robbert Dijkgraaf University of Amsterdam

Topological Strings Toy model (cf topology versus geometry) Exact BPS sector of superstrings Mathematical experiments to test physical intuition

Geometry of Calabi-Yau manifolds 3 complex dimensions Euclidean solutions of X ( 3, 0 ) - f orm ­ = d z 1 ^ d z 2 ^ d z 3 c 1 ( X ) = 0 K ae hl er f orm k = p ¡ 1 g i ¹ | d z i ^ d ¹ z ¹ | ; dk = 0

Exact Effective Actions F-terms for Weyl muliplet in 4 dim supergravity action CY top string partition function

A-Model (IIA): Exact worldsheet instantons f holomorphic map degree d Kähler moduli Localizes on

Gromov-Witten Invariants Exact instanton sum

B-Model (IIB string) Localizes on Complex moduli f almost constant maps

Mirror Symmetry classical quantum A-model B-Model

Fiberwise T-Duality [ Strominger, Yau, Zaslov ] Dual Torus Fibrations base

D-branes & Black holes

D-Branes Coherent sheaves IIA Special Lagrangians IIB homological mirror symmetry)

Charge Lattice (B-model) symplectic vector space H 3 ( X ; Z )

Period Map & Quantization moduli space of CY Lagrangian cone L=graph (dF 0 )  semi-classical state ψ ~ exp F 0 symplectic vector space hol 3-form dz 1  dz 2  dz 3

Special Geometry

Top String Partition Function = Wave Function Transforms as a wave function under Sp(2n,Z) change of canonical basis (A,B)

A-Model Kähler cone symplectic vector space complexified Kähler volume H ev ( X ; C ) h ® ; ¯ i = i n d ex D ® ­ ¯ ¤ 1 ¸ e k + i B plus world-sheet instanton corrections F 0 = t 3 6 ¸ 2

Charged objects: D-branes charged particles electric-magnetic charges Large volume: electric D0-D2 magnetic D4-D6

4d Black Holes D-brane Black Hole large charges Attractor CY

Attractor Mechanism near-horizon moduli t I

Quantization of Moduli Space “attractive” CY’s R e ­ 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 electric/magnetic charges

Mixed Ensemble OSV Conjecture

M-theory

Gopakumar-Vafa At strong coupling can integrate out (light) electric charges D0-D2 to obtain the effective action g s ! 1 M-theory limit g s virtual loops of M2 branes

5d Black Holes in M-theory Transversal rotations SO ( 4 ) ´ SU ( 2 ) L £ SU ( 2 ) R M2-branes with charge Q 2 H 2 ( X ; Z ) M2 Internal spin quantum numbers ( m L ; m R )

BPS degeneracies M2 Index of susy ground states (GV-inv) N m R Q = X m L ( ¡ 1 ) m L N m L ; m R Q 4d Quantum Hall system: wave functions lowest Landau level ª ( z 1 ; z 2 ) = X n 1 ; n 2 a n 1 ; n 2 z n 1 1 z n 2 2 Orbital angular momentum ( n 1 ; n 2 ) self-dual flux rotation space

GV Partition function Gas of 5d charged & spinning black holes Z ( ¸ ; t ) = Y n 1 ; n 2 Q ; m ³ 1 ¡ e ¸ ( n 1 + n 2 + m ) + t Q ´ ¡ N m Q 5d entropy N m Q » p Q 3 ¡ m 2

6+1 dim SUSY Gauge Theory Witten index counts D-brane bound states Z = T r £ ( ¡ 1 ) F e ¡ ¯ H ¤ Induced charges: non-trivial gauge bundle ( P ; Q ) ¼ c h ¤ ( E ) Reduction to moduli space of vacua Z » E u l er ( M E )

Donaldson-Thomas Invariants Single D6: U(1) gauge theory + singularities q = D 2 = c h 2 » T r F 2 i ns t an t ons t r i ngs k = D 0 = c h 3 » T r F 3 Z ( ¸ ; t ) = X k ; q DT ( k ; q ) e k¸ + q t

Lift to M-theory D6 → Taub-NUT geometry SO ( 4 ) angular momentum d s 2 TN = R 2 · 1 V ( d  + ~ A ¢ d ~ x ) 2 + V d ~ x 2 ¸ U ( 1 ) £ SO ( 3 ) Kaluza-Klein momentum [Gaiotto, Strominger,Yin]

Bound states with D0-D2 spinning M2-branes R q = X i Q i k = X i ( n i + m i ) Gauge theory quantum numbers

4 dim limit: Bound state of D6-D2 R ! 0 Z ( ¸ ; t ) = X k ; q DT ( k ; q ) e k¸ + q t Donaldson-Thomas Invariants

5 dim limit: R ! 1 Free gas of M2-branes Gopakumar-Vafa Invariants Z ( ¸ ; t ) = Y n 1 ; n 2 Q ; m ³ 1 ¡ e ¸ ( n 1 + n 2 + m ) + t Q ´ ¡ N m Q

Topological String Triality peturbative IIA strings Gromov-Witten M2-branes Gopakumar-Vafa D2-branes Donaldson-Thomas strong-weak 9-11 flip Taub-NUT

Simplest Calabi-Yau

g constant maps

Stat-Mech: 3d Partitions

Melting Crystals Reshetikhin,Okounkov,Vafa, Nekrasov,...

Universal Wave Function

Wave Function of String Theory Compactify on a 9-space X £ t i me ª 2 H X Flux/charge/brane sectors H X = M Q H Q X

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 ! S ym ¤ H

Hawking-Hartle Wave Function Sum over bounding geometries X B Include singularities (branes, black holes) ª = X B j B i

“Entropic Principle” Natural probability density on moduli space of string compactifications e S = j ª j 2 Depends on massless & massive d.o.f. peaked around moduli space

string theory on the near horizon geometry of the black hole AdS/CFT duality supersymmetric gauge theory on the brane superconformal quantum mechanics

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

cf. open/closed worldsheet duality string D - b rane E j E i 2 H c l ose d

Index theorem EF i n d ex D E ­ F ¤ = Z X c h ( E ) c h ( F ¤ ) b A

time T r H open ( ¡ 1 ) F = i n d ex D E ­ F ¤

h E ; F i = Z X c h ( E ) c h ( F ¤ ) b A EF

Supersymmetry breaking Non-susy boundary conditions Z ( ¯ ) = T r £ e ¡ ¯ H ¤ ¯ Positivity of H ¯ < ¯ 0 ) Z ( ¯ ) > Z ( ¯ 0 ) Ground states Prefers symmetric CY’s Z ( 1 ) = d i m H 0 = #h armon i c f orms ¸ E u l er

Space of All Calabi-Yau’s

Topology of Calabi-Yau spaces = X 0 X § g X = X 0 # § g b 3 = 0 b 2 = 0 Core

Non-Kahler CY are unique § g § g = # g ¡ S 3 £ S 3 ¢ Moduli space of complex structures d i m M g = g ¡ 1

Miles Reid’s Fantasy: “ There is only one CY space ” M g b 2 = 0 All CY connected through conifold transitions S 3 → S 2 b 2 = 1 Kähler CYs

SYZ: fibrations by Slag T 3 network of singularities S 1 shrinks

Two Vertices +- Mirror Symmetry topological vertex local Riemann surface

3d Topological Gauge Theory Wilson loops & graphs

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 on the moduli space of vacua Universal Calabi-Yau??? Many more surprises...

Happy 60 th, Michael!