Making Precise the Nothing at the Beginning of the Universe

Yu Nakayama, hep-th/0606127(Collaboration with S.J. Rey, Y. Sugawara)

Introduction Universe begins from the singularity. Theorem: under some assumptions, the univers

e (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

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?

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.

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

Winding tachyon at the beginning of the universe

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??

Open String Tachyon Condensation Decay of D-brane

Closed String Tachyon Condensation Decay of Space-(time) ?

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.

~

Time-like Sine-Liouville Theory As a toy model of MS scenario, we consider time-like Sine-Liouvill

e 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.

~

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 underst

and the real time tachyon condensation problem.

Revival of old idea that Liouville direction might be time

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

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)

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?

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

3. Sine-Liouville Theory 3-parameter action

Vertex operator:

Conformal condition:

Symmetry: U(1) conserved current

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

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.

Time-like Sine-Liouville Theory As a toy model of MS scenario, we consider time-like Sine-Liouvill

e theory (MS)

Obtained by

Fermionize by , so we obtain 2 fermions

MS studied the model with non-conventional Wick rotation in the semiclassical approach.

~

2pt function for neutral sector We compute 2pt function (Bogoliubov coefficient) by th

e analytic continuation from 3-parameter sine-Liouville

Apart from the renormalized cosmological constant part, integral converges and gives a phase (as Q 0).

When not a phase? Renormalized cosmological const governs the qualitati

ve feature of Bogoliubov coefficient

Depending on the sign, particle production is drastically different.

Is the singularity resolved? Due to the tachyon condensation, the geometry is effecti

vely cut-off around We can freely take a weak coupling limit near the singula

rity.

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

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

Conclusion String Theory is a candidate for

Theory of Everything

But…

Conclusion String Theory is a candidate for

Theory of Everything

But…

Exact treatment of α’ is very important!

also provides a Theory of Nothing