Making Precise the Nothing at the Beginning of the Universe

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

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

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

9. The tachyon at the end of the universe (MS) • Consider expanding universe (with S1 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 Revival of old idea that Liouville direction might be time • 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.

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

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 Adopting GS conjecture, where does non-phase come from? • 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…)

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