Exact string backgrounds from boundary data

1. Exact string backgrounds from boundary data Marios Petropoulos CPHT - Ecole Polytechnique Based on works with K. Sfetsos

2. Some motivations:FLRW-like hierarchy in strings Isotropy & homogeneityof space & cosmic fluid co-moving frame with Robertson-Walker metric Homogeneous, maximally symmetric space: P.M. PETROPOULOS CPHT-X

3. Maximally symmetric 3-D spaces Cosets of (pseudo)orthogonal groups constant scalar curvature: P.M. PETROPOULOS CPHT-X

4. FLRW space-times • Einstein equations lead to Friedmann-Lemaître equations for • exact solutions: maximally symmetric space-times Hierarchical structure:maximally symmetric space-times foliated with 3-D maximally symmetric spaces P.M. PETROPOULOS CPHT-X

5. Maximally symmetric space-times • with spatial sections • Einstein-de Sitter with spatial sections • with spatial sections P.M. PETROPOULOS CPHT-X

6. 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” P.M. PETROPOULOS CPHT-X

7. 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 P.M. PETROPOULOS CPHT-X

8. Conformal 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 Gauged WZW models are exact string backgrounds – they are not ordinary geometric cosets P.M. PETROPOULOS CPHT-X

9. Example • plus corrections (known) • central charge P.M. PETROPOULOS CPHT-X

10. 3. The three-dimensional case • up to (known) corrections: • range • choosing and flipping gives [Bars, Sfetsos 92] P.M. PETROPOULOS CPHT-X

11. Geometrical property of the background Comparison with geometric coset • at radius • fixed- leaf: (radius ) “bulk” theory “boundary” theory P.M. PETROPOULOS CPHT-X

12. 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 P.M. PETROPOULOS CPHT-X

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 CFT’s: P.M. PETROPOULOS CPHT-X

14. 4. In higher dimensions: a hierarchy of gauged WZW bulk large radial coordinate boundary decoupled radial direction P.M. PETROPOULOS CPHT-X

15. Lorentzian spaces • Lorentzian-signature gauged WZW • Various similar hierarchies: • large radial coordinate time-like boundary • remote time space-like boundary P.M. PETROPOULOS CPHT-X

16. 5. The world-sheet CFT viewpoint • Observation: • andare two exact 2-D sigma-models • some corners of their respective target spaces coincide • Expectation: A continuous one-parameter family such that P.M. PETROPOULOS CPHT-X

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

18. 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 P.M. PETROPOULOS CPHT-X

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 P.M. PETROPOULOS CPHT-X

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 P.M. PETROPOULOS CPHT-X

21. Back to the marginal operator The operator of reads • Conformal weights match: the operator is marginal P.M. PETROPOULOS CPHT-X

22. The marginal operator for Generalization to • Exact matching: the operator is marginal P.M. PETROPOULOS CPHT-X

23. 6. 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] P.M. PETROPOULOS CPHT-X

24. 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 P.M. PETROPOULOS CPHT-X

25. Appendix: Lorentzian cosets & time-like boundary bulk large radial coordinate time-like boundary decoupled radial direction P.M. PETROPOULOS CPHT-X

26. Appendix: Lorentzian cosets & space-like boundary bulk remote time space-like boundary decoupled asymptotic time P.M. PETROPOULOS CPHT-X

27. Appendix: 3-D Lorentzian cosets and their central charges • The Lorentzian-signature three-dimensional gauged WZW models • Their central charges: P.M. PETROPOULOS CPHT-X