exact string backgrounds from boundary data marios petropoulos cpht - ecole polytechnique based on...
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Exact string backgrounds from boundary data
Marios PetropoulosCPHT - Ecole Polytechnique
Based on works with K. Sfetsos
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1. Some motivations:FLRW-like hierarchy in strings
Isotropy & homogeneity of space & cosmic fluid co-moving frame with Robertson-Walker metric
Homogeneous, maximallysymmetric space:
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Maximally symmetric 3-D spaces
constant scalar curvature:
Cosets of (pseudo)orthogonal groups
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FLRW space-times
Einstein equations lead to Friedmann-Lemaître equations for
exact solutions: maximally symmetric space-times
Hierarchical structure: maximally symmetricspace-times foliated with 3-D maximallysymmetric spaces
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Maximally symmetric space-times
with spatial sections
Einstein-de Sitter with spatial sections
with spatial sections
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Situation 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
Potential cosmological applications for space-like “boundaries”
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2. Geometric versus conformal cosets
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:
Ordinary geometric cosets are not exact string backgrounds
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Conformal cosets
Gauged WZW models areexact string backgrounds – theyare 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
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
fixed- leaf: (radius )
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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 CFT’s:
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4. In higher dimensions: a hierarchy of gauged WZW
bulk
boundary decoupled radial direction
large radial coordinate
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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
some corners of their respective target spaces 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
The idea
the larger is the deeper is the coincidence of the target spaces of and
the sigma-models and must have coinciding target spaces beyond the asymptotic corners
In practice
The marginal operator is read off in the asymptotic expansion of beyond leading order
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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. Final comments
Novelty: use of parafermions for building marginal operators
Proving that is integrable frompure 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 FLRW
Gauged WZW cosets of orthogonal groups instead of ordinary cosets exact string backgrounds not maximally symmetric
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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Appendix: Lorentzian cosets & time-like boundary
bulk
time-like boundary decoupled radial direction
large radial coordinate
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Appendix: Lorentzian cosets & space-like boundary
bulk
space-like boundary decoupled asymptotic time
remote 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: