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Journal of High Energy Physics

, 2010:116 | Cite as

Higher spin interactions with scalar matter on constant curvature spacetimes: conserved current and cubic coupling generating functions

  • Xavier Bekaert
  • Elisa MeunierEmail author
Open Access
Article

Abstract

Cubic couplings between a complex scalar field and a tower of symmetric tensor gauge fields of all ranks are investigated on any constant curvature spacetime of dimension d ≥ 3. Following Noether’s method, the gauge fields interact with the scalar field via minimal coupling to the conserved currents. A symmetric conserved current, bilinear in the scalar field and containing up to r derivatives, is obtained for any rank r ≥ 1 from its flat spacetime counterpart in dimension d + 1, via a radial dimensional reduction valid precisely for the mass-square domain of unitarity in (anti) de Sitter spacetime of dimension d. The infinite collection of conserved currents and cubic vertices are summarized in a compact form by making use of generating functions and of the Weyl/Wigner quantization on constant curvature spaces.

Keywords

Gauge Symmetry AdS-CFT Correspondence 

References

  1. [1]
    M.A. Vasiliev, Higher spin gauge theories in various dimensions, Fortsch. Phys. 52 (2004) 702 [hep-th/0401177] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. [2]
    M.A. Vasiliev, Higher spin gauge theories in any dimension, Comptes Rendus Physique 5 (2004) 1101 [hep-th/0409260] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  3. [3]
    X. Bekaert, S. Cnockaert, C. Iazeolla and M.A. Vasiliev, Nonlinear higher spin theories in various dimensions, hep-th/0503128 [SPIRES].
  4. [4]
    F.A. Berends, G.J.H. Burgers and H. van Dam, Explicit construction of conserved currents for massless fields of arbitrary spin, Nucl. Phys. B 271 (1986) 42 [SPIRES].Google Scholar
  5. [5]
    X. Bekaert, Higher spin algebras as higher symmetries, arXiv:0704.0898 [SPIRES].
  6. [6]
    X. Bekaert, E. Joung and J. Mourad, On higher spin interactions with matter, JHEP 05 (2009) 126 [arXiv:0903.3338] [SPIRES].CrossRefADSGoogle Scholar
  7. [7]
    A. Fotopoulos, N. Irges, A.C. Petkou and M. Tsulaia, Higher-Spin Gauge Fields Interacting with Scalars: The Lagrangian Cubic Vertex, JHEP 10 (2007) 021 [arXiv:0708.1399] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  8. [8]
    R. Manvelyan and W. Rühl, Conformal coupling of higher spin gauge fields to a scalar field in AdS 4 and generalized Weyl invariance, Phys. Lett. B 593 (2004) 253 [hep-th/0403241] [SPIRES].ADSGoogle Scholar
  9. [9]
    R. Manvelyan and K. Mkrtchyan, Conformal invariant interaction of a scalar field with the higher spin field in AdS d, Mod. Phys. Lett. A 25 (2010) 1333 [arXiv:0903.0058] [SPIRES].MathSciNetADSGoogle Scholar
  10. [10]
    A. Fotopoulos and M. Tsulaia, Current Exchanges for Reducible Higher Spin Modes on AdS, arXiv:1007.0747 [SPIRES].
  11. [11]
    Y.M. Zinoviev, Spin 3 cubic vertices in a frame-like formalism, JHEP 08 (2010) 084 [arXiv:1007.0158] [SPIRES].CrossRefADSGoogle Scholar
  12. [12]
    T. Biswas and W. Siegel, Radial dimensional reduction: (Anti) de Sitter theories from flat, JHEP 07 (2002) 005 [hep-th/0203115] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    P.A.M. Dirac, The Electron Wave Equation in De-Sitter Space, Annals Math. 36 (1935) 657 [SPIRES].CrossRefMathSciNetGoogle Scholar
  14. [14]
    C. Fronsdal, Singletons and Massless, Integral Spin Fields on de Sitter Space (Elementary Particles in a Curved Space. 7, Phys. Rev. D 20 (1979) 848 [SPIRES].MathSciNetADSGoogle Scholar
  15. [15]
    R.R. Metsaev, Massless mixed symmetry bosonic free fields in d-dimensional anti-de Sitter space-time, Phys. Lett. B 354 (1995) 78 [SPIRES].MathSciNetADSGoogle Scholar
  16. [16]
    R.R. Metsaev, Arbitrary spin massless bosonic fields in d-dimensional anti-de Sitter space, hep-th/9810231 [SPIRES].
  17. [17]
    X. Bekaert, I.L. Buchbinder, A. Pashnev and M. Tsulaia, On higher spin theory: Strings, BRST, dimensional reductions, Class. Quant. Grav. 21 (2004) S1457 [hep-th/0312252] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  18. [18]
    K. Hallowell and A. Waldron, Constant curvature algebras and higher spin action generating functions, Nucl. Phys. B 724 (2005) 453 [hep-th/0505255] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  19. [19]
    G. Barnich and M. Grigoriev, Parent form for higher spin fields on anti-de Sitter space, JHEP 08 (2006) 013 [hep-th/0602166] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  20. [20]
    N. Boulanger, C. Iazeolla and P. Sundell, Unfolding Mixed-Symmetry Fields in AdS and the BMV Conjecture: I. General Formalism, JHEP 07 (2009) 013 [arXiv:0812.3615] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  21. [21]
    N. Boulanger, C. Iazeolla and P. Sundell, Unfolding Mixed-Symmetry Fields in AdS and the BMV Conjecture: II. Oscillator Realization, JHEP 07 (2009) 014 [arXiv:0812.4438] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  22. [22]
    K.B. Alkalaev and M. Grigoriev, Unified BRST description of AdS gauge fields, Nucl. Phys. B 835 (2010) 197 [arXiv:0910.2690] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  23. [23]
    A. Fotopoulos, K.L. Panigrahi and M. Tsulaia, Lagrangian formulation of higher spin theories on AdS space, Phys. Rev. D 74 (2006) 085029 [hep-th/0607248] [SPIRES].MathSciNetADSGoogle Scholar
  24. [24]
    D. Francia, J. Mourad and A. Sagnotti, (A)dS exchanges and partially-massless higher spins, Nucl. Phys. B 804 (2008) 383 [arXiv:0803.3832] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  25. [25]
    X. Bekaert and N. Boulanger, Gauge invariants and Killing tensors in higher-spin gauge theories, Nucl. Phys. B 722 (2005) 225 [hep-th/0505068] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  26. [26]
    G. Barnich, N. Bouatta and M. Grigoriev, Surface charges and dynamical Killing tensors for higher spin gauge fields in constant curvature spaces, JHEP 10 (2005) 010 [hep-th/0507138] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  27. [27]
    J. Mickelsson and J. Niederle, Contractions of representations of de Sitter groups, Commun. Math. Phys. 27 (1972) 167 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  28. [28]
    B. de Wit and I. Herger, Anti-de Sitter supersymmetry, Lect. Notes Phys. 541 (2000) 79 [hep-th/9908005] [SPIRES].CrossRefADSGoogle Scholar
  29. [29]
    P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Ann. Phys. 144 (1982) 249 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  30. [30]
    X. Bekaert, Comments on higher-spin symmetries, Int. J. Geom. Meth. Mod. Phys. 6 (2009) 285 [arXiv:0807.4223] [SPIRES].zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    D. Anselmi, Higher-spin current multiplets in operator-product expansions, Class. Quant. Grav. 17 (2000) 1383 [hep-th/9906167] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  32. [32]
    M.A. Vasiliev, Higher spin gauge theories: Star-product and AdS space, hep-th/9910096 [SPIRES].
  33. [33]
    S.E. Konstein, M.A. Vasiliev and V.N. Zaikin, Conformal higher spin currents in any dimension and AdS/CFT correspondence, JHEP 12 (2000) 018 [hep-th/0010239] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  34. [34]
    O.A. Gelfond, E.D. Skvortsov and M.A. Vasiliev, Higher spin conformal currents in Minkowski space, Theor. Math. Phys. 154 (2008) 294 [hep-th/0601106] [SPIRES].zbMATHCrossRefGoogle Scholar
  35. [35]
    S.F. Prokushkin and M.A. Vasiliev, Currents of arbitrary spin in A dS 3, Phys. Lett. B 464 (1999) 53 [hep-th/9906149] [SPIRES].MathSciNetADSGoogle Scholar
  36. [36]
    S.F. Prokushkin and M.A. Vasiliev, Cohomology of arbitrary spin currents in AdS 3, Theor. Math. Phys. 123 (2000) 415 [hep-th/9907020] [SPIRES].zbMATHCrossRefGoogle Scholar
  37. [37]
    F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz and D. Sternheimer, Deformation Theory and Quantization. 1. Deformations of Symplectic Structures, Ann. Phys. 111 (1978) 61 [SPIRES] [SPIRES].zbMATHCrossRefADSGoogle Scholar
  38. [38]
    F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz and D. Sternheimer, Deformation Theory and Quantization. 2. Physical A pplications, Ann. Phys. 111 (1978) 111 [SPIRES].zbMATHCrossRefADSGoogle Scholar
  39. [39]
    M.A. Vasiliev, Actions, charges and off-shell fields in the unfolded dynamics approach, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 37 [hep-th/0504090] [SPIRES].CrossRefMathSciNetGoogle Scholar
  40. [40]
    M. Grigoriev, Off-shell gauge fields from BRST quantization, hep-th/0605089 [SPIRES].
  41. [41]
    M. Taronna, Higher Spins and String Interactions, MSc Thesis, University of Pisa, Pisa Italy (2009) arXiv:1005.3061 [SPIRES].
  42. [42]
    A. Sagnotti and M. Taronna, String Lessons for Higher-Spin Interactions, Nucl. Phys. B 842 (2011) 299 [arXiv:1006.5242] [SPIRES].CrossRefADSGoogle Scholar
  43. [43]
    M. Flato and C. Frønsdal, One Massless Particle Equals Two Dirac Singletons: Elementary Particles in a Curved Space. 6, Lett. Math. Phys. 2 (1978) 421 [SPIRES].CrossRefADSGoogle Scholar
  44. [44]
    A. Sagnotti, E. Sezgin and P. Sundell, On higher spins with a strong S p(2, R) condition, in the proceedings of the First Solvay Workshop on Higher-Spin Gauge Theories, Brussels Belgium (2004) hep-th/0501156 [SPIRES].
  45. [45]
    D.P. Sorokin and M.A. Vasiliev, Reducible higher-spin multiplets in flat and AdS spaces and their geometric frame-like formulation, Nucl. Phys. B 809 (2009) 110 [arXiv:0807.0206] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  46. [46]
    D. Francia and A. Sagnotti, On the geometry of higher-spin gauge fields, Class. Quant. Grav. 20 (2003) S473 [hep-th/0212185] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  47. [47]
    D. Francia and A. Sagnotti, Higher-spin geometry and string theory, J. Phys. Conf. Ser. 33 (2006) 57 [hep-th/0601199] [SPIRES].CrossRefGoogle Scholar
  48. [48]
    I.L. Buchbinder, A.V. Galajinsky and V.A. Krykhtin, Quartet unconstrained formulation for massless higher spin fields, Nucl. Phys. B 779 (2007) 155 [hep-th/0702161] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  49. [49]
    A. Sagnotti and M. Tsulaia, On higher spins and the tensionless limit of string theory, Nucl. Phys. B 682 (2004) 83 [hep-th/0311257] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  50. [50]
    I.L. Buchbinder, A. Fotopoulos, A.C. Petkou and M. Tsulaia, Constructing the cubic interaction vertex of higher spin gauge fields, Phys. Rev. D 74 (2006) 105018 [hep-th/0609082] [SPIRES].MathSciNetADSGoogle Scholar
  51. [51]
    A. Fotopoulos and M. Tsulaia, Interacting Higher Spins and the High Energy Limit of the Bosonic String, Phys. Rev. D 76 (2007) 025014 [arXiv:0705.2939] [SPIRES].MathSciNetADSGoogle Scholar
  52. [52]
    A. Fotopoulos and M. Tsulaia, Current Exchanges for Reducible Higher Spin Multiplets and Gauge Fixing, JHEP 10 (2009) 050 [arXiv:0907.4061] [SPIRES].CrossRefADSGoogle Scholar
  53. [53]
    D. Francia, String theory triplets and higher-spin curvatures, Phys. Lett. B 690 (2010) 90 [arXiv:1001.5003] [SPIRES].ADSGoogle Scholar
  54. [54]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  55. [55]
    I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [SPIRES].MathSciNetADSGoogle Scholar
  56. [56]
    T. Leonhardt, A. Meziane and W. Rühl, On the proposed AdS dual of the critical O(N) σ-model for any dimension 2 < d< 4, Phys. Lett. B 555 (2003) 271 [hep-th/0211092] [SPIRES].ADSGoogle Scholar
  57. [57]
    A.C. Petkou, Evaluating the AdS dual of the critical O(N) vector model, JHEP 03 (2003) 049 [hep-th/0302063] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  58. [58]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  59. [59]
    S. Giombi and X. Yin, Higher Spin Gauge Theory and Holography: The Three-Point Functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [SPIRES].CrossRefADSGoogle Scholar
  60. [60]
    S. Giombi and X. Yin, Higher Spins in AdS and Twistorial Holography, arXiv:1004.3736 [SPIRES].

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© The Author(s) 2010

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

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Physique Théorique, Unité Mixte de Recherche 6083 du CNRSFédération de Recherche 2964 Denis Poisson, Université François RabelaisToursFrance

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