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String cosmology, hierarchies and marginal time evolution

String cosmology, hierarchies and marginal time evolution. Marios Petropoulos CPHT - Ecole Polytechnique Based on works with K. Sfetsos. 1. Motivations: FRW-like hierarchies in strings. Assume a string target space Can this be promoted to FRW-like

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String cosmology, hierarchies and marginal time evolution

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

  2. 1. Motivations:FRW-like hierarchies in strings Assume a string target space Can this be promoted to FRW-like with the usual “matter” content of string theory, P.M. PETROPOULOS CPHT-X

  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 P.M. PETROPOULOS CPHT-X

  4. Example with positive curvature and 4-D de Sitter space-time foliated with 3-D spheres (equal-time sections) P.M. PETROPOULOS CPHT-X

  5. One must solve the full (to some order in ) string equations: 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) P.M. PETROPOULOS CPHT-X

  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 evolutionin contrast to the time as an RG flow 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 (not homogeneous) • 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 • equal- leaf: (radius ) “bulk” theory “boundary” theory P.M. PETROPOULOS CPHT-X

  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 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. Also valid for 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 • the radial asymptotics of their target-space 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 • 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) 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. 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] P.M. PETROPOULOS CPHT-X

  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 P.M. PETROPOULOS CPHT-X

  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 • Thurston’s 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) P.M. PETROPOULOS CPHT-X

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

  27. Appendix: Maximally symmetric 4-D space-times • with spatial sections • Einstein-de Sitter with spatial sections • with spatial sections P.M. PETROPOULOS CPHT-X

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

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

  30. 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

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