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

2. NAPLES 2006P.M. PETROPOULOS CPHT-X2 1.Some motivations: FLRW-like hierarchy in strings Isotropy & homogeneity of space & cosmic fluid co-moving frame with Robertson-Walker metric Homogeneous, maximally symmetric space:

3. NAPLES 2006P.M. PETROPOULOS CPHT-X3 Maximally symmetric 3-D spaces constant scalar curvature: Cosets of (pseudo)orthogonal groups

4. NAPLES 2006P.M. PETROPOULOS CPHT-X4 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

5. NAPLES 2006P.M. PETROPOULOS CPHT-X5 Maximally symmetric space-times  with spatial sections  Einstein-de Sitter with spatial sections  with spatial sections

6. NAPLES 2006P.M. PETROPOULOS CPHT-X6 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”

7. NAPLES 2006P.M. PETROPOULOS CPHT-X7 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

8. NAPLES 2006P.M. PETROPOULOS CPHT-X8 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  Other background fields: and dilaton

9. NAPLES 2006P.M. PETROPOULOS CPHT-X9 Example   plus corrections (known)  central charge

10. NAPLES 2006P.M. PETROPOULOS CPHT-X10 3.The three-dimensional case  up to (known) corrections:  range  choosing and flipping gives [Bars, Sfetsos 92]

11. NAPLES 2006P.M. PETROPOULOS CPHT-X11 Geometrical property of the background “bulk” theory “boundary” theory Comparison with geometric coset  at radius  fixed- leaf: (radius )

12. NAPLES 2006P.M. PETROPOULOS CPHT-X12 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. NAPLES 2006P.M. PETROPOULOS CPHT-X13 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:

14. NAPLES 2006P.M. PETROPOULOS CPHT-X14 4.In higher dimensions: a hierarchy of gauged WZW bulk boundary decoupled radial direction large radial coordinate

15. NAPLES 2006P.M. PETROPOULOS CPHT-X15 Lorentzian spaces  Lorentzian-signature gauged WZW  Various similar hierarchies: large radial coordinate  time-like boundary remote time  space-like boundary

16. NAPLES 2006P.M. PETROPOULOS CPHT-X16 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 

17. NAPLES 2006P.M. PETROPOULOS CPHT-X17 The world-sheet CFT viewpoint  Why? Both satisfy with the same asymptotics  Consequence: There must exist a marginal operator in s.t. 

18. NAPLES 2006P.M. PETROPOULOS CPHT-X18 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

19. NAPLES 2006P.M. PETROPOULOS CPHT-X19 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. NAPLES 2006P.M. PETROPOULOS CPHT-X20 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. NAPLES 2006P.M. PETROPOULOS CPHT-X21 Back to the marginal operator The operator of reads  Conformal weights match: the operator is marginal

22. NAPLES 2006P.M. PETROPOULOS CPHT-X22 The marginal operator for Generalization to  Exact matching: the operator is marginal

23. NAPLES 2006P.M. PETROPOULOS CPHT-X23 6.Final comments  Novelty: u se 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. NAPLES 2006P.M. PETROPOULOS CPHT-X24 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

25. NAPLES 2006P.M. PETROPOULOS CPHT-X25 Appendix: Lorentzian cosets & time-like boundary bulk time-like boundary decoupled radial direction large radial coordinate

26. NAPLES 2006P.M. PETROPOULOS CPHT-X26 Appendix: Lorentzian cosets & space-like boundary bulk space-like boundary decoupled asymptotic time remote time

27. NAPLES 2006P.M. PETROPOULOS CPHT-X27 Appendix: 3-D Lorentzian cosets and their central charges  The Lorentzian-signature three-dimensional gauged WZW models  Their central charges: