1. Topological String Theory and Black Holes Eurostrings 2006, Cambridge, UK - review - w/ C. Vafa, E.Verlinde, hep-th/0602087 - work in progress Robbert Dijkgraaf University of Amsterdam

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

4. 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

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

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

7. Gromov-Witten Invariants Exact instanton sum

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

9. Mirror Symmetry classical quantum A-model B-Model

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

11. D-branes & Black holes

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

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

14. 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

15. Special Geometry

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

17. 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

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

19. 4d Black Holes D-brane Black Hole large charges Attractor CY

20. Attractor Mechanism near-horizon moduli t I

21. Quantization of Moduli Space “attractive” CY’s R e ­ 2 H 3 ( X ; Z )

22. Black Hole Partition Function Witten index of susy gauge theory

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

24. Mixed Ensemble electric/magnetic charges

25. Mixed Ensemble OSV Conjecture

26. M-theory

27. 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

28. 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 )

29. 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

30. 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

31. 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 )

32. 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

33. 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]

34. 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

35. 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

36. 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

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

38. Simplest Calabi-Yau

39. g constant maps

40. Stat-Mech: 3d Partitions

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

43. Universal Wave Function

44. 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

45. Topology Change Finite energy transitions X ! X 0 ª 2 H Universal wave function, components on all geometries

46. Baby Universes Disconnected spaces X ! X 1 + X 2 Second quantization H ! S ym ¤ H

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

48. “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

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

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

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

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

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

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

55. 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

56. Space of All Calabi-Yau’s

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

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

59. 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

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

61. Two Vertices +- Mirror Symmetry topological vertex local Riemann surface

62. 3d Topological Gauge Theory Wilson loops & graphs

63. Universal Moduli Space of CYs?

64. Gell-Mann on renormalization: “Just because it’s infinite, it doesn’t mean it’s zero”

65. 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...

66. Happy 60 th, Michael!