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: