Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields

  • A. CampoleoniEmail author
  • S. Fredenhagen
  • S. Pfenninger
  • S. Theisen


We discuss the emergence of \( \mathcal{W} \mbox{-algebras}\) as asymptotic symmetries of higher-spin gauge theories coupled to three-dimensional Einstein gravity with a negative cosmological constant. We focus on models involving a finite number of bosonic higher-spin fields, and especially on the example provided by the coupling of a spin-3 field to gravity. It is described by a SL(3) × SL(3) Chern-Simons theory and its asymptotic symmetry algebra is given by two copies of the classical \( {\mathcal{W}_3}\mbox{-algebra} \) with central charge the one computed by Brown and Henneaux in pure gravity with negative cosmological constant.


Field Theories in Lower Dimensions Chern-Simons Theories Conformal and W Symmetry 


  1. [1]
    C. Fronsdal, Massless Fields with Integer Spin, Phys. Rev. D 18 (1978) 3624 [SPIRES].ADSGoogle Scholar
  2. [2]
    X. Bekaert, N. Boulanger and P. Sundell, How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples, arXiv:1007.0435 [SPIRES].
  3. [3]
    M.A. Vasiliev, Higher-spin gauge theories in four, three and two dimensions, Int. J. Mod. Phys. D 5 (1996) 763 [hep-th/9611024] [SPIRES].MathSciNetADSGoogle Scholar
  4. [4]
    X. Bekaert, S. Cnockaert, C. Iazeolla and M.A. Vasiliev, Nonlinear higher spin theories in various dimensions, hep-th/0503128 [SPIRES].
  5. [5]
    C. Iazeolla, On the Algebraic Structure of Higher-Spin Field Equations and New Exact Solutions, arXiv:0807.0406 [SPIRES].
  6. [6]
    D. Francia and A. Sagnotti, On the geometry of higher-spin gauge fields, Class. Quant. Grav. 20 (2003) S473 [hep-th/0212185] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  7. [7]
    D. Sorokin, Introduction to the classical theory of higher spins, AIP Conf. Proc. 767 (2005) 172 [hep-th/0405069] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  8. [8]
    N. Bouatta, G. Compère and A. Sagnotti, An introduction to free higher-spin fields, hep-th/0409068 [SPIRES].
  9. [9]
    D. Francia and A. Sagnotti, Higher-spin geometry and string theory, J. Phys. Conf. Ser. 33 (2006) 57 [hep-th/0601199] [SPIRES]. CrossRefGoogle Scholar
  10. [10]
    A. Campoleoni, Metric-like Lagrangian Formulations for Higher-Spin Fields of Mixed Symmetry, Riv. Nuovo Cim. 033 (2010) 123 [arXiv:0910.3155] [SPIRES].Google Scholar
  11. [11]
    A. Sagnotti, E. Sezgin and P. Sundell, On higher spins with a strong Sp(2,R) condition, hep-th/0501156 [SPIRES].
  12. [12]
    A. Fotopoulos and M. Tsulaia, Gauge Invariant Lagrangians for Free and Interacting Higher Spin Fields. A Review of the BRST formulation, Int. J. Mod. Phys. A 24 (2009) 1 [arXiv:0805.1346] [SPIRES].MathSciNetADSGoogle Scholar
  13. [13]
    C. Aragone and S. Deser, Consistency Problems of Hypergravity, Phys. Lett. B 86 (1979) 161 [SPIRES].MathSciNetADSGoogle Scholar
  14. [14]
    C. Aragone and S. Deser, Hypersymmetry In D = 3 Of Coupled Gravity Massless Spin 5/2 System, Class. Quant. Grav. 1 (1984) L9 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  15. [15]
    M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3+ 1)-dimensions, Phys. Lett. B 243 (1990) 378 [SPIRES].MathSciNetADSGoogle Scholar
  16. [16]
    M.A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS(d), Phys. Lett. B 567 (2003) 139 [hep-th/0304049] [SPIRES].MathSciNetADSGoogle Scholar
  17. [17]
    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
  18. [18]
    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
  19. [19]
    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
  20. [20]
    A. Strominger, Black hole entropy from near-horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  21. [21]
    P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183 [hep-th/9210010] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  22. [22]
    A. Achúcarro and P.K. Townsend, A Chern-Simons Action for Three-Dimensional anti-de Sitter Supergravity Theories, Phys. Lett. B 180 (1986) 89 [SPIRES].ADSGoogle Scholar
  23. [23]
    E. Witten, (2 + 1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  24. [24]
    M.P. Blencowe, A Consistent Interacting Massless Higher Spin Field Theory In D = (2 + 1), Class. Quant. Grav. 6 (1989) 443 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  25. [25]
    E. Bergshoeff, M.P. Blencowe and K.S. Stelle, Area Preserving Diffeomorphisms And Higher Spin Algebra, Commun. Math. Phys. 128 (1990) 213 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  26. [26]
    M.A. Vasiliev, Higher Spin Algebras And Quantization On The Sphere And Hyperboloid, Int. J. Mod. Phys. A 6 (1991) 1115 [SPIRES].MathSciNetADSGoogle Scholar
  27. [27]
    S.F. Prokushkin and M.A. Vasiliev, Higher-spin gauge interactions for massive matter fields in 3D AdS space-time, Nucl. Phys. B 545 (1999) 385 [hep-th/9806236] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  28. [28]
    A. Bilal, V.V. Fock and I.I. Kogan, On the origin of W algebras, Nucl. Phys. B 359 (1991) 635 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  29. [29]
    J. de Boer and J. Goeree, W gravity from Chern-Simons theory, Nucl. Phys. B 381 (1992) 329 [hep-th/9112060] [SPIRES].CrossRefADSGoogle Scholar
  30. [30]
    J. Balog, L. Fehér, L. O’Raifeartaigh, P. Forgács and A. Wipf, Toda Theory And W Algebra From A Gauged WZNW Point Of View, Ann. Phys. 203 (1990) 76 [SPIRES].zbMATHCrossRefADSGoogle Scholar
  31. [31]
    M. Henneaux and S.-J. Rey, Nonlinear W(infinity) Algebra as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity, arXiv:1008.4579 [SPIRES].
  32. [32]
    T. Curtright, Massless Field Supermultiplets With Arbitrary Spin, Phys. Lett. B 85 (1979) 219 [SPIRES].ADSGoogle Scholar
  33. [33]
    C. Fronsdal, Singletons and Massless, Integral Spin Fields on de Sitter Space (Elementary Particles in a Curved Space VII), Phys. Rev. D 20 (1979) 848 [SPIRES].MathSciNetADSGoogle Scholar
  34. [34]
    B. Binegar, Relativistic Field Theories In Three-Dimensions, J. Math. Phys. 23 (1982) 1511 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  35. [35]
    M.A. Vasiliev, ’Gauge’ Form Of Description Of Massless Fields With Arbitrary Spin. (in Russian), Yad. Fiz. 32 (1980) 855 [Sov. J. Nucl. Phys. 32 (1980) 439] [SPIRES]. Google Scholar
  36. [36]
    M.A. Vasiliev, Free Massless Fields Of Arbitrary Spin In The De Sitter Space And Initial Data For A Higher Spin Superalgebra, Fortsch. Phys. 35 (1987) 741 [Yad. Fiz. 45 (1987) 1784] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  37. [37]
    V.E. Lopatin and M.A. Vasiliev, Free Massless Bosonic Fields Of Arbitrary Spin In d-dimensional de Sitter Space, Mod. Phys. Lett. A 3 (1988) 257 [SPIRES]. MathSciNetADSGoogle Scholar
  38. [38]
    M. Hamermesh, Group theory and its applications to physical problems, Dover Publications, New York U.S.A. (1969).Google Scholar
  39. [39]
    E.S. Fradkin and M.A. Vasiliev, Candidate to the Role of Higher Spin Symmetry, Ann. Phys. 177 (1987) 63 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  40. [40]
    J. Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph.D. Thesis, Massachusetts Institute of Technology, Massachusetts U.S.A. (1982).Google Scholar
  41. [41]
    A. Achúcarro and P.K. Townsend, Extended Supergravities In d = (2+ 1) As Chern-Simons Theories, Phys. Lett. B 229 (1989) 383 [SPIRES].ADSGoogle Scholar
  42. [42]
    M. Bañados, Global charges in Chern-Simons field theory and the (2+ 1) black hole, Phys. Rev. D 52 (1996) 5816 [hep-th/9405171] [SPIRES].Google Scholar
  43. [43]
    M. Bañados, T. Brotz and M.E. Ortiz, Boundary dynamics and the statistical mechanics of the 2+ 1 dimensional black hole, Nucl. Phys. B 545 (1999) 340 [hep-th/9802076] [SPIRES].CrossRefADSGoogle Scholar
  44. [44]
    M. Bañados, Three-dimensional quantum geometry and black holes, hep-th/9901148 [SPIRES].
  45. [45]
    S. Carlip, Conformal field theory, (2+ 1)-dimensional gravity and the BTZ black hole, Class. Quant. Grav. 22 (2005) R85 [gr-qc/0503022] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  46. [46]
    T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Ann. Phys. 88 (1974) 286 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  47. [47]
    R. Benguria, P. Cordero and C. Teitelboim, Aspects of the Hamiltonian Dynamics of Interacting Gravitational Gauge and Higgs Fields with Applications to Spherical Symmetry, Nucl. Phys. B 122 (1977) 61 [SPIRES].CrossRefADSGoogle Scholar
  48. [48]
    V.E. Didenko, A.S. Matveev and M.A. Vasiliev, BTZ black hole as solution of 3d higher spin gauge theory, Theor. Math. Phys. 153 (2007) 1487 [Teor. Mat. Fiz. 153 (2007) 158] [hep-th/0612161] [SPIRES].zbMATHCrossRefGoogle Scholar
  49. [49]
    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
  50. [50]
    C. Fefferman and C. R. Graham, Conformal Invariants in Elie Cartan et les Mathématiques d’aujourd’hui, Astèrisque (1985).Google Scholar
  51. [51]
    O. Coussaert, M. Henneaux and P. van Driel, The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961 [gr-qc/9506019] [SPIRES].zbMATHCrossRefADSGoogle Scholar
  52. [52]
    M. Bañados, K. Bautier, O. Coussaert, M. Henneaux and M. Ortiz, Anti-de Sitter/CFT correspondence in three-dimensional supergravity, Phys. Rev. D 58 (1998) 085020 [hep-th/9805165] [SPIRES].ADSGoogle Scholar
  53. [53]
    M. Henneaux, L. Maoz and A. Schwimmer, Asymptotic dynamics and asymptotic symmetries of three-dimensional extended AdS supergravity, Annals Phys. 282 (2000) 31 [hep-th/9910013] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  54. [54]
    M. Henneaux and C. Teitelboim, Asymptotically anti-de Sitter Spaces, Commun. Math. Phys. 98 (1985) 391 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  55. [55]
    P. Mathieu, Extended Classical Conformal Algebras and the Second Hamiltonian Structure of Lax Equations, Phys. Lett. B 208 (1988) 101 [SPIRES].MathSciNetADSGoogle Scholar
  56. [56]
    S. Okubo and J. Patera, General Indices Of Representations And Casimir Invariants, J. Math. Phys. 25 (1984) 219 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  57. [57]
    C.N. Pope and P.K. Townsend, Conformal Higher Spin In (2+ 1)-Dimensions, Phys. Lett. B 225 (1989) 245 [SPIRES].MathSciNetADSGoogle Scholar
  58. [58]
    A.V. Razumov and M.V. Saveliev, Lie Algebras, Geometry, and Toda-type Systems, Cambridge Lecture Notes in Physics, Cambridge University Press, Cambridge U.K. (1997).zbMATHCrossRefGoogle Scholar
  59. [59]
    F.A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, Extensions of the Virasoro Algebra Constructed from Kac-Moody Algebras Using Higher Order Casimir Invariants, Nucl. Phys. B 304 (1988) 348 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  60. [60]
    F.A. Bais, T. Tjin and P. van Driel, Covariantly coupled chiral algebras, Nucl. Phys. B 357 (1991) 632 [SPIRES].CrossRefADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  • A. Campoleoni
    • 1
    Email author
  • S. Fredenhagen
    • 1
  • S. Pfenninger
    • 1
  • S. Theisen
    • 1
  1. 1.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutGolmGermany

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