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String cosmology, hierarchies and marginal time evolution

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Title: String cosmology, hierarchies and marginal time evolution


1
String cosmology, hierarchies and marginal time
evolution
  • Marios Petropoulos
  • CPHT - Ecole Polytechnique
  • Based on works with K. Sfetsos

2
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,

3
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
4
Example with positive curvature and
4-D de Sitter space-time
foliated with 3-D spheres (equal-time sections)
5
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)

6
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

7
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

8
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

9
Example
  • plus corrections (known)
  • central charge

10
3. The three-dimensional case
  • up to (known) corrections
  • range
  • choosing and flipping
    gives

Bars, Sfetsos 92
11
Geometrical property of the background
bulk theory boundary theory
  • Comparison with geometric coset
  • at radius
  • equal- leaf (radius )

12
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

13
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

14
4. In higher dimensions a hierarchy of gauged
WZW
bulk
large radial coordinate
boundary decoupled radial direction
15
Also valid for Lorentzian spaces
  • Lorentzian-signature gauged WZW
  • Various similar hierarchies
  • large radial coordinate ? time-like boundary
  • remote time ? space-like boundary

16
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

17
The world-sheet CFT viewpoint
  • Why?
  • Both satisfy with the same
    asymptotics
  • Consequence
  • There must exist a marginal operator in
    s.t.

18
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)

19
The asymptotics of beyond leading
order in the radial coordinate
  • The metric (at large ) in the large- region
    beyond l.o.
  • The marginal operator

20
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
21
Back to the marginal operator
  • The operator of reads
  • Conformal weights match the operator is marginal

22
The marginal operator for
  • Generalization to
  • Exact matching the operator is marginal

23
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
24
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

25
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)

26
Appendix Maximally symmetric 3-D spaces
Cosets of (pseudo)orthogonal groups
constant scalar curvature
27
Appendix Maximally symmetric 4-D space-times
  • with
    spatial sections
  • Einstein-de Sitter with spatial sections
  • with
    spatial sections

28
Appendix Lorentzian cosets space-like boundary
bulk
large radial coordinate
time-like boundary decoupled radial
direction
29
Appendix Lorentzian cosets time-like boundary
bulk
remote time
space-like boundary decoupled asymptotic
time
30
Appendix 3-D Lorentzian cosets and their central
charges
  • The Lorentzian-signature three-dimensional gauged
    WZW models
  • Their central charges
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