Aspects of the BMS/CFT correspondence

  • Glenn Barnich
  • Cédric TroessaertEmail author


After a review of symmetries and classical solutions involved in the AdS3/CFT2 correspondence, we apply a similar analysis to asymptotically flat spacetimes at null infinity in 3 and 4 dimensions. In the spirit of two dimensional conformal field theory, the symmetry algebra of asymptotically flat spacetimes at null infinity in 4 dimensions is taken to be the semi-direct sum of supertranslations with infinitesimal local conformal transformations and not, as usually done, with the Lorentz algebra. As a first application, we derive how the symmetry algebra is realized on solution space. In particular, we work out the behavior of Bondi’s news tensor, mass and angular momentum aspects under local conformal transformations.


AdS-CFT Correspondence Gauge Symmetry Classical Theories of Gravity Space-Time Symmetries 


  1. [1]
    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] [SPIRES].zbMATHMathSciNetADSGoogle Scholar
  2. [2]
    M. Henneaux and C. Teitelboim, Asymptotically anti-de Sitter spaces, Commun. Math. Phys. 98 (1985) 391 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. [3]
    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 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  4. [4]
    M. Henneaux, Asymptotically anti-de Sitter universes in d = 3, 4 and higher dimensions, in proceedings of the Fourth Marcel Grossmann Meeting on General Relativity, Rome (1985), R. Ruffini ed., Elsevier Science Publishers B.V. (1986) pg. 959.Google Scholar
  5. [5]
    H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [SPIRES].ADSGoogle Scholar
  6. [6]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [SPIRES].MathSciNetADSGoogle Scholar
  7. [7]
    R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  8. [8]
    G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, arXiv:0909.2617 [SPIRES].
  9. [9]
    G. Barnich and G. Compere, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [Erratum ibid. 24 (2007) 3139] [gr-qc/0610130] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. [10]
    A. Ashtekar, J. Bicak and B.G. Schmidt, Behavior of Einstein-Rosen waves at null infinity, Phys. Rev. D 55 (1997) 687 [gr-qc/9608041] [SPIRES].MathSciNetADSGoogle Scholar
  11. [11]
    A. Ashtekar, J. Bicak and B.G. Schmidt, Asymptotic structure of symmetry reduced general relativity, Phys. Rev. D 55 (1997) 669 [gr-qc/9608042] [SPIRES].MathSciNetADSGoogle Scholar
  12. [12]
    A. Strominger, Black hole entropy from near-horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT correspondence, Phys. Rev. D 80 (2009) 124008 [arXiv:0809.4266] [SPIRES].ADSGoogle Scholar
  14. [14]
    S. Carlip, The statistical mechanics of the (2+1)-dimensional black hole, Phys. Rev. D 51 (1995) 632 [gr-qc/9409052] [SPIRES].MathSciNetADSGoogle Scholar
  15. [15]
    S.N. Solodukhin, Conformal description of horizon’s states, Phys. Lett. B 454 (1999) 213 [hep-th/9812056] [SPIRES].MathSciNetADSGoogle Scholar
  16. [16]
    S. Carlip, Black hole entropy from conformal field theory in any dimension, Phys. Rev. Lett. 82 (1999) 2828 [hep-th/9812013] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. [17]
    S. Carlip, Entropy from conformal field theory at Killing horizons, Class. Quant. Grav. 16 (1999) 3327 [gr-qc/9906126] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. [18]
    M.-I. Park and J. Ho, Comments on ’Black hole entropy from conformal field theory in any dimension’, Phys. Rev. Lett. 83 (1999) 5595 [hep-th/9910158] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  19. [19]
    M.-I. Park, Hamiltonian dynamics of bounded spacetime and black hole entropy: canonical method, Nucl. Phys. B 634 (2002) 339 [hep-th/0111224] [SPIRES].CrossRefADSGoogle Scholar
  20. [20]
    I. Sachs and S.N. Solodukhin, Horizon holography, Phys. Rev. D 64 (2001) 124023 [hep-th/0107173] [SPIRES].MathSciNetADSGoogle Scholar
  21. [21]
    J.-i. Koga, Asymptotic symmetries on Killing horizons, Phys. Rev. D 64 (2001) 124012 [gr-qc/0107096] [SPIRES].MathSciNetADSGoogle Scholar
  22. [22]
    S. Carlip, Near-horizon conformal symmetry and black hole entropy, Phys. Rev. Lett. 88 (2002) 241301 [gr-qc/0203001] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  23. [23]
    S. Silva, Black hole entropy and thermodynamics from symmetries, Class. Quant. Grav. 19 (2002) 3947 [hep-th/0204179] [SPIRES].zbMATHCrossRefADSGoogle Scholar
  24. [24]
    G. Kang, J.-i. Koga and M.-I. Park, Near-horizon conformal symmetry and black hole entropy in any dimension, Phys. Rev. D 70 (2004) 024005 [hep-th/0402113] [SPIRES].MathSciNetADSGoogle Scholar
  25. [25]
    J.-i. Koga, Universal properties from local geometric structure of Killing horizon, Class. Quant. Grav. 24 (2007) 3067 [gr-qc/0604054] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  26. [26]
    J.-i. Koga, Asymptotic symmetries on Kerr-Newman horizon without anomaly of diffeomorphism invariance, Class. Quant. Grav. 25 (2008) 045009 [gr-qc/0609120] [SPIRES].CrossRefADSGoogle Scholar
  27. [27]
    A. Ashtekar, Asymptotic quantization of the gravitational field, Phys. Rev. Lett. 46 (1981) 573 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  28. [28]
    A. Ashtekar, Asymptotic quantization: based on 1984 Naples lectures, Naples, Italy, Bibliopolis (1987) pg. 107, Monographs and textbooks in physical science, 2.Google Scholar
  29. [29]
    S. Hollands and A. Ishibashi, Asymptotic flatness and Bondi energy in higher dimensional gravity, J. Math. Phys. 46 (2005) 022503 [gr-qc/0304054] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  30. [30]
    S. Hollands and A. Ishibashi, Asymptotic flatness at null infinity in higher dimensional gravity, hep-th/0311178 [SPIRES].
  31. [31]
    S. Hollands and R.M. Wald, Conformal null infinity does not exist for radiating solutions in odd spacetime dimensions, Class. Quant. Grav. 21 (2004) 5139 [gr-qc/0407014] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  32. [32]
    K. Tanabe, N. Tanahashi and T. Shiromizu, On asymptotic structure at null infinity in five dimensions, arXiv:0909.0426 [SPIRES].
  33. [33]
    L. Susskind, Holography in the flat space limit, hep-th/9901079 [SPIRES].
  34. [34]
    J. Polchinski, S-matrices from AdS spacetime, hep-th/9901076 [SPIRES].
  35. [35]
    J. de Boer and S.N. Solodukhin, A holographic reduction of Minkowski space-time, Nucl. Phys. B 665 (2003) 545 [hep-th/0303006] [SPIRES].ADSGoogle Scholar
  36. [36]
    G. Arcioni and C. Dappiaggi, Exploring the holographic principle in asymptotically flat spacetimes via the BMS group, Nucl. Phys. B 674 (2003) 553 [hep-th/0306142] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  37. [37]
    G. Arcioni and C. Dappiaggi, Holography in asymptotically flat space-times and the BMS group, Class. Quant. Grav. 21 (2004) 5655 [hep-th/0312186] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  38. [38]
    S.N. Solodukhin, Reconstructing Minkowski space-time, hep-th/0405252 [SPIRES].
  39. [39]
    M. Gary and S.B. Giddings, The flat space S-matrix from the AdS/CFT correspondence?, Phys. Rev. D 80 (2009) 046008 [arXiv:0904.3544] [SPIRES].MathSciNetADSGoogle Scholar
  40. [40]
    T. Banks, A critique of pure string theory: Heterodox opinions of diverse dimensions, hep-th/0306074 [SPIRES].
  41. [41]
    C. Fefferman and C. Graham, Elie cartan et les mathématiques d’aujourd’hui, Conformal invariants, Astérisque (1985), pg. 95.Google Scholar
  42. [42]
    C. Graham and J. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991) 186.zbMATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  44. [44]
    M. Bañados, Three-dimensional quantum geometry and black holes, in Trends in Theoretical Physics II, H. Falomir, R.E. Gamboa Saravi and F.A. Schaposnik eds., American Institute of Physics Conference Series, vol. 484 (1999) pg. 147.Google Scholar
  45. [45]
    K. Skenderis and S.N. Solodukhin, Quantum effective action from the AdS/CFT correspondence, Phys. Lett. B 472 (2000) 316 [hep-th/9910023] [SPIRES].MathSciNetADSGoogle Scholar
  46. [46]
    C.R. Graham, Volume and area renormalizations for conformally compact Einstein metrics, math/9909042.
  47. [47]
    C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, Diffeomorphisms and holographic anomalies, Class. Quant. Grav. 17 (2000) 1129 [hep-th/9910267] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  48. [48]
    M. Rooman and P. Spindel, Aspects of (2+1) dimensional gravity: AdS 3 asymptotic dynamics in the framework of Fefferman-Graham-Lee theorems, Annalen Phys. 9 (2000) 161 [hep-th/9911142] [SPIRES].Google Scholar
  49. [49]
    K. Bautier, F. Englert, M. Rooman and P. Spindel, The Fefferman-Graham ambiguity and AdS black holes, Phys. Lett. B 479 (2000) 291 [hep-th/0002156] [SPIRES].MathSciNetADSGoogle Scholar
  50. [50]
    I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  51. [51]
    R. Penrose, Asymptotic properties of fields and space-times, Phys. Rev. Lett. 10 (1963) 66 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  52. [52]
    L.A. Tamburino and J.H. Winicour, Gravitational fields in finite and conformal Bondi frames, Phys. Rev. 150 (1966) 1039 [SPIRES].CrossRefADSGoogle Scholar
  53. [53]
    J. Winicour, Logarithmic asymptotic flatness, Found. Phys. 15 (1985) 605.CrossRefMathSciNetADSGoogle Scholar
  54. [54]
    M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  55. [55]
    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] [SPIRES].ADSGoogle Scholar
  56. [56]
    G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  57. [57]
    G. Barnich and G. Compere, Surface charge algebra in gauge theories and thermodynamic integrability, J. Math. Phys. 49 (2008) 042901 [arXiv:0708.2378] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  58. [58]
    T. Regge and C. Teitelboim, Role of surface integrals in the hamiltonian formulation of general relativity, Ann. Phys. 88 (1974) 286 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  59. [59]
    J.D. Brown and M. Henneaux, On the poisson brackets of differentiable generators in classical field theory, J. Math. Phys. 27 (1986) 489 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  60. [60]
    E.T. Newman and R. Penrose, Note on the Bondi-Metzner-Sachs group, J. Math. Phys. 7 (1966) 863 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  61. [61]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  62. [62]
    R. Penrose, Zero rest mass fields including gravitation: asymptotic behavior, Proc. Roy. Soc. Lond. A 284 (1965) 159 [SPIRES].MathSciNetADSGoogle Scholar
  63. [63]
    P.T. Chrusciel, M.A.H. MacCallum and D.B. Singleton, Gravitational waves in general relativity: 14. Bondi expansions and the polyhomogeneity of Scri, Proc. Roy. Soc. Lond. A 436 (1992) 299 [gr-qc/9305021] [SPIRES].ADSGoogle Scholar
  64. [64]
    R. Geroch, Asymptotic structure of space-time, in Symposium on the asymptotic structure of space-time, P. Esposito and L. Witten eds., Plenum, New York, U.S.A. (1977) pg. 1.Google Scholar
  65. [65]
    R. Wald, General relativity, The University of Chicago Press, Chicago, U.S.A. (1984).zbMATHGoogle Scholar
  66. [66]
    E. Witten, Baryons and branes in anti de Sitter space, talk given at Strings ’98

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.Physique Théorique et MathématiqueUniversité Libre de Bruxelles and International Solvay InstitutesBruxellesBelgium

Personalised recommendations