1 / 30

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

jaegar
Download Presentation

String cosmology, hierarchies and marginal time evolution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related