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The Topological String Partition Function as a Wave Function (of the Universe)

The Topological String Partition Function as a Wave Function (of the Universe). Erik Verlinde. Institute for Theoretical Physics University of Amsterdam. = BPS Type II Strings. Topological Strings. with 8 supercharges (N=2 in 4d). Introduction to Topological Strings

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The Topological String Partition Function as a Wave Function (of the Universe)

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

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

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

  5. Mirror symmetry: A-model B-model Hodge diamond 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 “shape”

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

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

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

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

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

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

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

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

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

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

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

  17. 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 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!

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

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

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

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

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

  23. Physics 2005 Conference Warwick, April 12, 2005 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

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

  25. Outline • 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

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

  27. obey Exact Hartle-Hawking wave function Wave functions

  28. Conclusion • 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.

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

  30. 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

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

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

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

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

  35. Dirac bracket Non-commutative moduli Holomorphic wave functions with inner product Reduced BPS phase space BPS condition = Constraint

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

  37. Summary and Conclusion • 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.

  38. Partition function: Partition Function

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

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

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