String cosmology, hierarchies and marginal time evolution

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String cosmology, hierarchies and marginal time
evolution

• Marios Petropoulos
• CPHT - Ecole Polytechnique
• Based on works with K. Sfetsos

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1. MotivationsFRW-like hierarchies in strings

• Assume a string
  target space
• Can this be promoted to FRW-like
• with the usual matter content of string
  theory,

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Straightforward in GR FRW space-times

• Assume homogeneous and isotropic
• Einstein equations lead to Friedmann-Lemaître
  equations for
• exact solutions
  maximally symmetric space-times

Hierarchical structure maximally symmetric 4-D
space-times foliated with 3-D maximally symmetric
spaces
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Example with positive curvature and
4-D de Sitter space-time
foliated with 3-D spheres (equal-time sections)
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More involved in string theory

• matter is not chosen arbitrarily dilaton,
  axion,
• there is an internal manifold
• there are two perturbation parameters
  expansions must be
  kept under control (small curvatures, small
  dilaton)

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Here hierarchies in exact string backgrounds

• Hierarchy of exact string backgrounds and
  precise relation
• is not foliated with
• appears as the boundary of
• World-sheet CFT structure parafermion-induced
  marginal deformations similar to those that
  deform a continuous NS5-brane distribution on a
  circle to an ellipsis
• Cosmological applications time as a marginal
  evolution in contrast to the time as an RG flow

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2. Geometric versus conformal cosets
Ordinary geometric cosets are not exact
string backgrounds

• Solve at most the lowest order (in )
  equations
• Have no dilaton because they have constant
  curvature
• Need antisymmetric tensors to get stabilized
• Have large isometry

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Conformal cosets
Gauged WZW models are exact string
backgrounds they are not ordinary geometric
cosets

• is the WZW on the group manifold of
• isometry of target space
• current algebras in the ws CFT, at
  level
• gauging spoils the symmetry (not
  homogeneous)
• Other background fields and dilaton

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Example

• plus corrections (known)
• central charge

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3. The three-dimensional case

• up to (known) corrections
• range
  
• choosing and flipping
  gives

Bars, Sfetsos 92
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Geometrical property of the background
bulk theory boundary theory

• Comparison with geometric coset
  
• at radius
• equal- leaf (radius )

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Proof check the background fields

• Metric in the asymptotic region at large
• Dilaton
• Conclusion
• decouples and supports a background charge
• the 2-D boundary is identified with
  
• using

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Also beyond the large- limit all-order in

• Check the corrections in metric and
  dilaton of
• and
• Check the central charges of the two ws CFTs

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4. In higher dimensions a hierarchy of gauged
WZW
bulk
large radial coordinate
boundary decoupled radial direction
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Also valid for Lorentzian spaces

• Lorentzian-signature gauged WZW
• Various similar hierarchies
• large radial coordinate ? time-like boundary
• remote time ? space-like boundary

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5. The world-sheet CFT viewpoint

• Observation
• and are two exact 2-D
  sigma-models
• the radial asymptotics of their target-space
  coincide
• Expectation
• A continuous one-parameter family such
  that

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The world-sheet CFT viewpoint

• Why?
• Both satisfy with the same
  asymptotics
• Consequence
• There must exist a marginal operator in
  s.t.

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The marginal operator

• In practice
• The marginal operator is read off in the
  asymptotic expansion of beyond leading
  order
• What is ?
• By analyzing the beta-function equations one
  observes that the would-be continuous parameter
  can be reabsorbed by a rescaling of the
  sigma-model fields
• up to a constant dilaton shift (known
  phenomenon)

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The asymptotics of beyond leading
order in the radial coordinate

• The metric (at large ) in the large- region
  beyond l.o.
• The marginal operator

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Conformal operators in

• A marginal operator has dimension
• In there is no isometry
  neither currents
• Parafermions (non-Abelian in higher dimensions)
• holomorphic
• anti-holomorphic
• Free boson with background charge ? vertex
  operators

The displayed expressions are semi-classical
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Back to the marginal operator

• The operator of reads
• Conformal weights match the operator is marginal

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The marginal operator for

• Generalization to
• Exact matching the operator is marginal

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6. Summary and final comments

• Novelty use of parafermions for building
  marginal operators
• Proving that
  is integrable from
• pure ws CFT techniques would be a tour de force
  
• Another instance circular NS5-brane distribution
• Continuous family of exact backgrounds circle ?
  ellipsis
• Marginal operator dressed bilinear of compact
  parafermions

Petropoulos, Sfetsos 06
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Back to the original motivation FRW

• Gauged WZW cosets of orthogonal groups instead of
  ordinary cosets
• exact string backgrounds
• not homogeneous
• Hierarchical structure
• not foliations (unlike ordinary cosets) but
• exact bulk and exact boundary string theories
• in Lorentzian geometries can be a set of
  initial data

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Time in string theory?

• In some regimes of string theory
• target time 2-D scale Liouville field
  (dilaton interplay between target space-time and
  world sheet)
• time evolution RG flow Ricci flow
• Thurstons geometrization conjecture target
  space converges universally with time towards a
  collection of homogeneous spaces and isotropic
  (if available)
• We are not in such a regime
• time evolution marginal
• no convergence towards homogeneous spaces (gauged
  WZW are not homogeneous)

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Appendix Maximally symmetric 3-D spaces
Cosets of (pseudo)orthogonal groups
constant scalar curvature
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Appendix Maximally symmetric 4-D space-times

• with
  spatial sections
• Einstein-de Sitter with spatial sections
• with
  spatial sections

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Appendix Lorentzian cosets space-like boundary
bulk
large radial coordinate
time-like boundary decoupled radial
direction
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Appendix Lorentzian cosets time-like boundary
bulk
remote time
space-like boundary decoupled asymptotic
time
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Appendix 3-D Lorentzian cosets and their central
charges

• The Lorentzian-signature three-dimensional gauged
  WZW models
• Their central charges