Exact higher-spin symmetry in CFT: all correlators in unbroken Vasiliev theory

  • V.E. Didenko
  • E.D. Skvortsov


All correlation functions of conserved currents of the CFT that is dual to unbroken Vasiliev theory are found as invariants of higher-spin symmetry in the bulk of AdS. The conformal and higher-spin symmetry of the correlators as well as the conservation of currents are manifest, which also provides a direct link between the Maldacena-Zhiboedov result and higher-spin symmetries. Our method is in the spirit of AdS/CFT, though we never take any boundary limit or compute any bulk integrals. Boundary-to-bulk propagators are shown to exhibit an algebraic structure, living at the boundary of SpH(4), semidirect product of Sp(4) and the Heisenberg group. N-point correlation function is given by a product of N elements.


AdS-CFT Correspondence Conformal and W Symmetry Global Symmetries Gauge Symmetry 


  1. [1]
    A. Belavin, A.M. Polyakov and A. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    J. Maldacena and A. Zhiboedov, Constraining Conformal Field Theories with A Higher Spin Symmetry, arXiv:1112.1016 [INSPIRE].
  3. [3]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, arXiv:1204.3882 [INSPIRE].
  4. [4]
    N. Colombo and P. Sundell, Higher Spin Gravity Amplitudes From Zero-form Charges, arXiv:1208.3880 [INSPIRE].
  5. [5]
    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].MathSciNetADSMATHGoogle Scholar
  6. [6]
    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  8. [8]
    M.A. Vasiliev, Holography, Unfolding and Higher-Spin Theory, arXiv:1203.5554 [INSPIRE].
  9. [9]
    M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions, Phys. Lett. B 243 (1990) 378 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    M.A. Vasiliev, Properties of equations of motion of interacting gauge fields of all spins in (3+1)-dimensions, Class. Quant. Grav. 8 (1991) 1387 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    M.A. Vasiliev, More on equations of motion for interacting massless fields of all spins in (3+1)-dimensions, Phys. Lett. B 285 (1992) 225 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    M.A. Vasiliev, Higher spin gauge theories in four-dimensions, three-dimensions and two-dimensions, Int. J. Mod. Phys. D 5 (1996) 763 [hep-th/9611024] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    M.A. Vasiliev, Higher spin gauge theories: Star product and AdS space, hep-th/9910096 [INSPIRE].
  14. [14]
    M. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS(d), Phys. Lett. B 567 (2003) 139 [hep-th/0304049] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [Erratum ibid. B 660 (2003) 403] [hep-th/0205131] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    I. Klebanov and A. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    M. Flato and C. Fronsdal, One Massless Particle Equals Two Dirac Singletons: Elementary Particles in a Curved Space. 6., Lett. Math. Phys. 2 (1978) 421 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    S. Giombi and X. Yin, Higher Spin Gauge Theory and Holography: The Three-Point Functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    S. Giombi and X. Yin, Higher Spins in AdS and Twistorial Holography, JHEP 04 (2011) 086 [arXiv:1004.3736] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    E. Sezgin and P. Sundell, An Exact solution of 4 − D higher-spin gauge theory, Nucl. Phys. B 762 (2007) 1 [hep-th/0508158] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    M.A. Vasiliev, Extended higher spin superalgebras and their realizations in terms of quantum operators, Fortsch. Phys. 36 (1988) 33 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    E. Sezgin and P. Sundell, Geometry and Observables in Vasilievs Higher Spin Gravity, JHEP 07 (2012) 121 [arXiv:1103.2360] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    N. Colombo and P. Sundell, Twistor space observables and quasi-amplitudes in 4D higher spin gravity, JHEP 11 (2011) 042 [arXiv:1012.0813] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    E. Sezgin and P. Sundell, Supersymmetric Higher Spin Theories, arXiv:1208.6019 [INSPIRE].
  25. [25]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    S. Giombi, S. Prakash and X. Yin, A Note on CFT Correlators in Three Dimensions, arXiv:1104.4317 [INSPIRE].
  27. [27]
    E. Fradkin and M.A. Vasiliev, Candidate to the Role of Higher Spin Symmetry, Annals Phys. 177 (1987) 63 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    S. Konshtein and M.A. Vasiliev, Massless representations and admissibility condition for higher Spin superalgebras, Nucl. Phys. B 312 (1989) 402 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    S. Konstein and M.A. Vasiliev, Extended higher spin superalgebras and their massless representations, Nucl. Phys. B 331 (1990) 475 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    X. Bekaert, S. Cnockaert, C. Iazeolla and M. Vasiliev, Nonlinear higher spin theories in various dimensions, hep-th/0503128 [INSPIRE].
  31. [31]
    M.A. Vasiliev, Equations of motion of interacting massless fields of all spins as a free differential algebra, Phys. Lett. B 209 (1988) 491 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    M.A. Vasiliev, Consistent equations for interacting massless fields of all spins in the first order in curvatures, Annals Phys. 190 (1989) 59 [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  33. [33]
    S. Giombi and X. Yin, The Higher Spin/Vector Model Duality, arXiv:1208.4036 [INSPIRE].
  34. [34]
    E. Sezgin and P. Sundell, Holography in 4D (super) higher spin theories and a test via cubic scalar couplings, JHEP 07 (2005) 044 [hep-th/0305040] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    S. Giombi et al., Chern-Simons Theory with Vector Fermion Matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    C.-M. Chang, S. Minwalla, T. Sharma and X. Yin, ABJ Triality: from Higher Spin Fields to Strings, arXiv:1207.4485 [INSPIRE].
  37. [37]
    V. Didenko and E. Skvortsov, Towards higher-spin holography in ambient space of any dimension, arXiv:1207.6786 [INSPIRE].
  38. [38]
    P. Kraus and E. Perlmutter, Probing higher spin black holes, JHEP 02 (2013) 096 [arXiv:1209.4937] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    M. Soloviev, Generalized Weyl correspondence and Moyal multiplier algebras, Theor. Math. Phys. 173 (2012) 1359.CrossRefGoogle Scholar
  40. [40]
    O. Gelfond and M. Vasiliev, Sp(8) invariant higher spin theory, twistors and geometric BRST formulation of unfolded field equations, JHEP 12 (2009) 021 [arXiv:0901.2176] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    N. Boulanger and P. Sundell, An action principle for Vasilievs four-dimensional higher-spin gravity, J. Phys. A 44 (2011) 495402 [arXiv:1102.2219] [INSPIRE].MathSciNetGoogle Scholar
  42. [42]
    N. Boulanger, N. Colombo and P. Sundell, A minimal BV action for Vasilievs four-dimensional higher spin gravity, JHEP 10 (2012) 043 [arXiv:1205.3339] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    V. Didenko and M. Vasiliev, Static BPS black hole in 4d higher-spin gauge theory, Phys. Lett. B 682 (2009) 305 [arXiv:0906.3898] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    V. Didenko and M. Vasiliev, Free field dynamics in the generalized AdS (super)space, J. Math. Phys. 45 (2004) 197 [hep-th/0301054] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  45. [45]
    O. Gelfond and M. Vasiliev, Unfolding Versus BRST and Currents in Sp(2M) Invariant Higher-Spin Theory, arXiv:1001.2585 [INSPIRE].
  46. [46]
    X. Bekaert and M. Grigoriev, Manifestly conformal descriptions and higher symmetries of bosonic singletons, SIGMA 6 (2010) 038 [arXiv:0907.3195] [INSPIRE].MathSciNetGoogle Scholar
  47. [47]
    M. Vasiliev, Relativity, causality, locality, quantization and duality in the S(p)(2M) invariant generalized space-time, hep-th/0111119 [INSPIRE].
  48. [48]
    M. Vasiliev, Higher spin conserved currents in Sp(2M) symmetric space-time, Russ. Phys. J. 45 (2002) 670 [hep-th/0204167] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    O. Gelfond, E. Skvortsov and M. Vasiliev, Higher spin conformal currents in Minkowski space, Theor. Math. Phys. 154 (2008) 294 [hep-th/0601106] [INSPIRE].CrossRefMATHGoogle Scholar
  50. [50]
    O. Gelfond and M. Vasiliev, Higher Spin Fields in Siegel Space, Currents and Theta Functions, JHEP 03 (2009) 125 [arXiv:0801.2191] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    O. Gelfond and M. Vasiliev, Unfolded Equations for Current Interactions of 4d Massless Fields as a Free System in Mixed Dimensions, arXiv:1012.3143 [INSPIRE].
  52. [52]
    M. Vasiliev and V. Zaikin, On Sp(2M) invariant Green functions, Phys. Lett. B 587 (2004) 225 [hep-th/0312244] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    I. Bars, Map of Wittens * to Moyals *, Phys. Lett. B 517 (2001) 436 [hep-th/0106157] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    I. Bars, MSFT: Moyal star formulation of string field theory, hep-th/0211238 [INSPIRE].

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Lebedev Institute of PhysicsMoscowRussia
  2. 2.Albert Einstein InstitutePotsdamGermany

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