Advertisement

Classification of boundary gravitons in AdS3 gravity

  • Alan Garbarz
  • Mauricio Leston
Open Access
Article

Abstract

We revisit the description of the space of asymptotically AdS3 solutions of pure gravity in three dimensions with a negative cosmological constant as a collection of coadjoint orbits of the Virasoro group. Each orbit corresponds to a set of metrics related by diffeomorphisms which do not approach the identity fast enough at the boundary. Orbits contain more than a single element and this fact manifests the global degrees of freedom of AdS3 gravity, being each element of an orbit what we call boundary graviton. We show how this setup allows to learn features about the classical phase space that otherwise would be quite difficult. Most important are the proof of energy bounds and the characterization of boundary gravitons unrelated to BTZs and AdS3. In addition, it makes manifest the underlying mathematical structure of the space of solutions close to infinity. Notably, because of the existence of a symplectic form in each orbit, being this related with the usual Dirac bracket of the asymptotic charges, this approach is a natural starting point for the quantization of different sectors of AdS3 gravity. We finally discuss previous attempts to quantize coadjoint orbits of the Virasoro group and how this is relevant for the formulation of AdS3 quantum gravity.

Keywords

Space-Time Symmetries Global Symmetries Models of Quantum Gravity Black Holes 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    E. Witten, Coadjoint Orbits of the Virasoro Group, Commun. Math. Phys. 114 (1988) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A.A. Kirillov, Lectures on the orbit method. Graduate studies in mathematics Volume 64, American Mathematical Society (2004).Google Scholar
  5. [5]
    A. Castro, T. Hartman and A. Maloney, The Gravitational Exclusion Principle and Null States in Anti-de Sitter Space, Class. Quant. Grav. 28 (2011) 195012 [arXiv:1107.5098] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [gr-qc/9302012] [INSPIRE].ADSGoogle Scholar
  8. [8]
    T. Nakatsu, H. Umetsu and N. Yokoi, Three-dimensional black holes and Liouville field theory, Prog. Theor. Phys. 102 (1999) 867 [hep-th/9903259] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    J. Navarro-Salas and P. Navarro, Virasoro orbits, AdS 3 quantum gravity and entropy, JHEP 05 (1999) 009 [hep-th/9903248] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    N.M.J. Woodhouse, Geometric quantization, Second edition, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press (1992).Google Scholar
  11. [11]
    L. Guieu, C. Roger, Lalgèbre et le groupe de Virasoro, aspects géométriques et algébriques, généralisations, Les Publications CRM Montréal (2006).Google Scholar
  12. [12]
    J. Balog, L. Feher and L. Palla, Coadjoint orbits of the Virasoro algebra and the global Liouville equation, Int. J. Mod. Phys. A 13 (1998) 315 [hep-th/9703045] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    C. Li and J. Lucietti, Three-dimensional black holes and descendants, arXiv:1312.2626 [INSPIRE].
  14. [14]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. Navarro-Salas and P. Navarro, A Note on Einstein gravity on AdS 3 and boundary conformal field theory, Phys. Lett. B 439 (1998) 262 [hep-th/9807019] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    M. Bañados, Three-dimensional quantum geometry and black holes, hep-th/9901148 [INSPIRE].
  17. [17]
    J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    M. Schottenloher, A mathematical introduction to conformal field theory, Second edition. Lect. Notes Phys. 759, Springer, Heidelberg Berlin, Germany (2008).Google Scholar
  20. [20]
    C. Scarinci and K. Krasnov, The universal phase space of AdS 3 gravity, Commun. Math. Phys. 322 (2013) 167 [arXiv:1111.6507] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  21. [21]
    G. Barnich and B. Oblak, Holographic positive energy theorems in three-dimensional gravity, arXiv:1403.3835 [INSPIRE].
  22. [22]
    O. Mišković and J. Zanelli, On the negative spectrum of the 2 + 1 black hole, Phys. Rev. D 79 (2009) 105011 [arXiv:0904.0475] [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    H.K. Kunduri and J. Lucietti, Classification of near-horizon geometries of extremal black holes, Living Rev. Rel. 16 (2013) 8 [arXiv:1306.2517] [INSPIRE].CrossRefzbMATHGoogle Scholar
  24. [24]
    A. Echeverria-Enriquez, M.C. Muñoz-Lecanda, N. Roman-Roy and C. Victoria-Monge, Mathematical foundations of geometric quantization, Extracta Math. 13 (1998) 135 [math-ph/9904008] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  25. [25]
    A.A. Kirillov, D.V. Yur’ev, Kähler geometry of the infinite-dimensional homogeneous space M = Diff +(S1)/Rot(S1), Func. Anal. Appl. 21 (1987) 284.CrossRefzbMATHGoogle Scholar
  26. [26]
    R. Goodman and N.R. Wallach, Projective unitary positive-energy representations of Diff(S 1), J. Funct. Anal. 63 (1985) 299MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    H. Salmasian and K.-H. Neeb, Classification of positive energy representations of the Virasoro group, arXiv:1402.6572 [INSPIRE].

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Instituto de Física de La Plata (IFLP), CONICET & Departamento de FísicaUniversidad Nacional de la PlataLa PlataArgentina
  2. 2.Instituto de Astronomıa y Física del Espacio, Pabellón IAFE-CONICET, Ciudad UniversitariaBuenos AiresArgentina

Personalised recommendations