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 Einsteins 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

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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?
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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.
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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

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Winding tachyon at the beginning of the universe
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Closed string tachyon condensation

• Open string tachyon condensation ? Decay of
  unstable D-brane (Sens 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??

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

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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
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Time-like Liouville Field Theory
Wick rotate the Liouville action

• Action
• C1 theory with time-dependent tachyon
  condensation
• Minisuperspace approximation
• Euclidean continuation is given by the Liouville
  theory with negative cosmological const

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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)
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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)

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Time-like sine-Liouville theory and resolution of
the singularity
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3. Sine-Liouville Theory

• 3-parameter action
• Vertex operator
• Conformal condition
• Symmetry U(1) conserved current

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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
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2pt function for neutral sector (KLPR)

• Can be computed by Teschners 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.

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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.

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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).

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When not a phase?

• Renormalized cosmological const governs the
  qualitative feature of Bogoliubov coefficient
• Depending on the sign, particle production is
  drastically different.

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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

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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

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Conclusion

• String Theory is a candidate for Theory of
  Everything
• But

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Conclusion

• String Theory is a candidate for Theory of
  Everything
• But
• Exact treatment of a is very important!

also provides a Theory of Nothing