1. The Topological String Partition Function as a Wave Function (of the Universe) Erik Verlinde Institute for Theoretical Physics University of Amsterdam

2. Topological Strings = BPS Type II Strings with 8 supercharges (N=2 in 4d) Introduction to Topological Strings A-model Partition Function and BPS counting in 5D B-model Partition Function as a Wave-function 4D Black Hole Entropy and the OSV Conjecture A Hartle-Hawking wave function for Flux compactifications: “The Entropic Principle”

3. = twisted N=2 SCFT Nilpotent BRST-charge: BRST-exact stress energy: Topological CFT Physical operators Chiral ring

4. Topological Strings on a Calabi-Yau Topological Sigma model Operators become forms Physical operators closed forms on the Calabi-Yau BRST charge = exterior derivative Chiral ring = “quantum” cohomology ring of CY.

5. A- and B-model A-model: physical operators are (n,m)-forms with n=m (1,1)-forms => Kahler deformations of CY “size” B-model: physical operators are (n,m)-forms with n=3-m (2,1)-forms => Complex structure deformations of CY Hodge diamond Mirror symmetry: A-model B-model “shape”

6. Free Energy String amplitude: integrated correlation function Free energy: = generating function computes F-terms in space time effective action of the form

7. Partition Function Full Free energy: Partition function: Coupling constants: parametrize background A-model : complexified Kahler moduli B-model: complex structure moduli

8. A-model amplitudes 3-point function: intersection form + worldsheet instantons Genus 0 free energy: obtained by integrating Higher genus: counts the number of holomorphic curves in homology class n I

9. Counting 5D BPS States M-theory on CY => 5D SUGRA Wrapped membranes = Charged BPS States We like to know Bekenstein-Hawking entropy of extremal spinning Black Holes predicts

10. Schwinger calculation of single D2-D0 boundstate in graviphoton field Gopakumar, Vafa Take Euclidean time circle as 11 th dimension in M-theory. Spin couples to graviphoton Counting 5D BPS States suggests rewritting of free energy

11. Gopakumar, Vafa Total free energy can be rewritten in terms of integer invariants as Counting 5D BPS States For the partition function this gives the product formula

12. Counting 5D BPS States Conjecture The l.h.s. describes a “free” gas of “single” BPS states. If true the 5D black hole partition function equals

13. B-model amplitudes 3-point function: obtained by differentiation Genus 0 free energy: from periods of holomorphic 3-form Higher genus: from holomorphic anomaly

14. B-model partition sum as a wave function Holomorphic anomaly in terms of partition function Background independent wave functions expresses background dependence, exactly like a wavefunction obtained by quantizing the 3rd cohomology Witten Dijkgraaf, Vonk, EV

15. B-model partition sum as a wave function The 3rd cohomology The decomposition leads to background dependent wave functions has a natural symplectic form EV Background independent decomposition leads to real wavefunctions

16. 4D Black Hole Entropy from Topological Strings Cardoso, de Wit, Mohaupt Ooguri, Strominger, Vafa Entropy as Legendre transform Semiclassical entropy Mixed partition function factorizes as

17. Exact Counting of 4D Black Hole States? OSV-conjecture: # BPS states is Wigner function Is this exact? Can one use product formula to obtain integral numbers? No! Recent connection with 5D black holes using Taub-NUT Shih, Strominger, Xi For these our conjectured formula is Cheng,Dijkgraaf, Manschot, EV work in progress

18. Flux Compactifications Fluxes through cycles Type IIB string on CY Superpotential for moduli fieldsModuli stabilization

19. BPS Black holes as Flux Vacua Entropy Electric and magnetic charges Graviphoton charge Attractor Mechanism Attractor Equations Type IIB string on CY

20. Near Horizon Geometry as Cosmological Model Euclidean metric with gauge choice Attractor flow equation Black Hole Entropy Ferrara, Gibbons, Kallosh

21. Hartle-Hawking wave function The wave functions obey Ooguri, Vafa, EV

22. Flux vacua as wave functions on moduli space Relative probability determined by entropy Flux Wave Functions............ Moduli fixed by fluxes : discrete points. The Entropic Principle Flux Vacua Entropic Principle Nature is (most likely) described by state of maximal entropy Constructive way to select vacua (in contrast with “Anthropic Principle”)

24. The Entropic Principle: A Hartle-Hawking Wave Function for String Compactification* Erik Verlinde Institute for Theoretical Physics University of Amsterdam * based on work with H. Ooguri and C. Vafa Physics 2005 Conference Warwick, April 12, 2005

25. A-model partition sum: a product formula Resummation of free energy Gopakumar, Vafa In terms of integral invariants gives the product formula

26. Flux vacua and moduli stabilization Cosmological model: type IIB on Attractor flow and the Wheeler-de Witt equation `Exact´ Hartle-Hawking wave function and topological strings Outline

27. Wheeler-De Witt equation Quantizing the BPS flow equation gives the BPS WDW equation +c.c Probality density peaked near Attractror value Natural Normalization => Entropy

28. Wave functions obey Exact Hartle-Hawking wave function

29. Evidence has been given for the identification of the topological string partition function with the `exact’ euclidean Hartle-Hawking wave function in mini superspace for Type IIB theory on a CY x S2. Our description leads for each flux vacuum to a probability density on the moduli space. Relative probalities between different flux vacua is determined by an `entropic’ instead of `anthropic’ principle. The continuation to Minkowski signature is presumably possible if one allows supersymmetry to be broken, but needs further investigation. The implications for more general 4d flux compactifications are worth studying. Conclusion

31. Flux vacua as discrete points in the moduli space Each point has a priori equal probability Discrete Flux Vacua Flux vacua as wave functions on the moduli space Relative probability determined by entropy Flux Wave Functions............ Moduli determined by fluxes Flux vacua A Hartle-Hawking Wavefunction for Flux Vacua

32. Outline Flux vacua and BPS black holes Moduli stabilization and attractor mechanism Cosmological model: type IIB on Attractor flow and the Wheeler-de Witt equation Exact Entropy and topological strings Attractor equations as canonical transformation `Exact´ Hartle-Hawking wave function

33. Flux Vacua Type IIB string on CY Fluxes through 3-cyclesComplex structure moduli Kahler potential Superpotential for moduli fields Scalar potential Moduli stabilization

34. Moduli Stabilization BPS condition Attractor Equations gives Kahler metric on Moduli Space Gauge choice

35. Cosmological model Euclidean metric Type IIB string on CYxS2xS1 Gauge choice BPS flow equations Combined BPS flow equation

36. Wheeler-De Witt equation Quantizing the BPS flow equation Normalization => Entropy gives the BPS WDW equation +c.c Peaked near Attractor value

37. Reduced BPS phase space BPS condition = Constraint Dirac bracket Holomorphic wave functionswith inner product Non-commutative moduli

38. Attractor equations as canonical transformation represent canonical transformation Attractor equations Topological string partition function Quantization of 3rd cohomology

39. Topological Strings have “real” physical applications in 4D (and 5D) type II (and M-theory) on a Calabi-Yau space, in particular in describing the entropy of BPS black holes. A proof that the 5D BPS states counted by the topological string is sufficient to explain the 5D black hole entropy is still missing. An interesting connection between 4D and 5D black holes suggest Our description leads for each flux vacuum to a probability density on the moduli space. Relative probalities between different flux vacua is determined by an `entropic’ instead of `anthropic’ principle. The continuation to Minkowski signature is presumably possible if one allows supersymmetry to be broken, but needs further investigation. The implications for more general 4d flux compactifications are worth studying. Summary and Conclusion

40. Partition Function Partition function:

41. Partition Function String amplitude: integrated correlation function => generating function of string amplitudes Free energy: Coupling constants: A-model: Kahler moduli B-model: Complex structure moduli Partition Function

42. 4D Black Hole Entropy from Topological Strings Cardoso, de Wit, Mohaupt Ooguri, Strominger, Vafa Entropy as Legendre transform Semiclassical entropy # BPS states as Wigner function