1. Making Precise the Nothing at the Beginning of the Universe Yu Nakayama, hep-th/0606127 (Collaboration with S.J. Rey, Y. Sugawara)

2. Introduction Universe begins from the singularity. Theorem: under some assumptions, the universe (cosmological solution of Einstein’s GR) has an initial singularity (Penrose, Hawking) Several ways out in string/higher dimensional theories  String cosmology (T-duality, dilaton)  Brane cosmology (cyclic universe)  Winding tachyon condensation

3. String Theory at the singularity (time-like) orbifold singularity?  YES (with SUSY) Black hole singularity?  Probably yes (dual D-brane) (space-like) singularity? Can string theory coexist with singularities?  time-dependent system. Based on exact construction (orbifold, coset), the theory is defined, but divergence in amplitudes?

4. Tachyon censorship Big-bang / Big-crunch singularities (MS) Naked Singularities (ASP…) Singularities inside the blackhole (Horowitz) Localized Tachyon condensation provides a new way to resolve singularities.

5. Plan of the Talk Introduction Winding tachyon at the beginning of the universe (McGreevy-Silverstein scenario) Time-like sine-Liouville theory and resolution of the singularity Summary

6. Winding tachyon at the beginning of the universe

7. Closed string tachyon condensation Open string tachyon condensation  Decay of unstable D-brane (Sen’s conjecture)  Checked in many ways Open string field theory Rolling Tachyon  Many applications (Brane) Inflational Cosmology Classification of D-branes (K-theory, Derived category…) Closed string tachyon condensation  Decay of unstable space-time?  Many applications? Resolving singularity? Cosmological applications? Classification of space-time??

8. Open String Tachyon Condensation Decay of D-brane Closed String Tachyon CondensationDecay of Space-(time) ?

9. The tachyon at the end of the universe (MS) Consider expanding universe (with S 1 circle) If we choose SS-like compactification, winding tachyon appears t  0. Classical singularity in GR is removed by winding tachyon condensation! Initial singularity of space-time would be resolved by the winding tachyon condensation. ~

10. Time-like Sine-Liouville Theory As a toy model of MS scenario, we consider time-like Sine- Liouville theory (analytic continuation of 3-sine-Liouville: Kim et al) Obtained by Fermionize by, so we obtain 2 fermions MS studied the model with non-conventional Wick rotation in the semiclassical approach. ~

11. Analytic continuation of Liouville theory Idea: noncritical string needs Liouville direction to compensate Weyl anomaly. Take Q  0 or b  i so that we have critical string  For Hermiticity of the action, we need to Wick rotate  Worldsheet cosmological constant becomes real time tachyon condensation The structure of Liouville theory has been well- understood in this ten years  Suitable analytic continuation will be useful to understand the real time tachyon condensation problem. Revival of old idea that Liouville direction might be time

12. Time-like Liouville Field Theory Action C=1 theory with time-dependent tachyon condensation Minisuperspace approximation: Euclidean continuation is given by the Liouville theory with negative cosmological const: Wick rotate the Liouville action

13. Interpretation of 2pt function Vertex operator V (Euclidean mode) is expanded by later (free) mode R is related to Bogoliubov coefficient In the minisuperspace approximation (not a phase!) Minisuperspace 2pt function governs vacuum particle production as Bogoliubov coefficient This should also hold in string theory (conjecture: GS)

14. Beyond minisuperspace Exact 2pt function Substitute Bogoliubov coefficient Carefully regularizing, renormalized cosmological const is negative Then we reproduce minisuperspace result (ST) Higher correlation functions are much subtler (ST, Schomerus…) Adopting GS conjecture, where does non-phase come from?

15. Time-like sine-Liouville theory and resolution of the singularity

16. 3. Sine-Liouville Theory 3-parameter action Vertex operator: Conformal condition: Symmetry: U(1) conserved current

17. 2-parameter model (BF) Suppose Infinitely many symmetry appears Due to the duality, 2-parameter sine-Liouville is much better-understood. For this value of q, model is rotation of usual sine-Liouville + free boson. So FZZ dual to coset

18. 2pt function for neutral sector (KLPR) Can be computed by Teschner’s trick (at least in the neutral sector) Vertex operator: Remarks  No dual relation. Answer is not unique.  Agreement with BF in 2-parameter limit.  Renormalized cosmological constant should be correct.

19. Time-like Sine-Liouville Theory As a toy model of MS scenario, we consider time-like Sine- Liouville theory (MS) Obtained by Fermionize by, so we obtain 2 fermions MS studied the model with non-conventional Wick rotation in the semiclassical approach. ~

20. 2pt function for neutral sector We compute 2pt function (Bogoliubov coefficient) by the analytic continuation from 3-parameter sine- Liouville Apart from the renormalized cosmological constant part, integral converges and gives a phase (as Q  0).

21. When not a phase? Renormalized cosmological const governs the qualitative feature of Bogoliubov coefficient Depending on the sign, particle production is drastically different.

22. Is the singularity resolved? Due to the tachyon condensation, the geometry is effectively cut-off around We can freely take a weak coupling limit near the singularity. Bogoliubov particle production is a function of A. If the transverse dimension is less than 4. The theory shows no diverging particle production (small back reaction). Torus partition function also shows an imaginary part when

23. Summary Closed string tachyon condensation is interesting  Resolution of singularity  New geometrical interpretation Time-like Liouville approach is promising  Exact in alpha’ corrections  Beyond the minisuperspace approximation End (beginning) of the universe  Time-like sine-Liouville approach  Exact 2pt function  Evaluation of particle production

24. Conclusion String Theory is a candidate for Theory of Everything But…

25. Conclusion String Theory is a candidate for Theory of Everything But… Exact treatment of α’ is very important! also provides a Theory of Nothing