Advertisement

Journal of High Energy Physics

, 2010:106 | Cite as

Gauge fields in (A)dS d within the unfolded approach: algebraic aspects

  • E. D. Skvortsov
Article

Abstract

It has recently been shown that generalized connections of the (A)dS d symmetry algebra provide an effective geometric and algebraic framework for all types of gauge fields in (A)dS d , both for massless and partially-massless. The equations of motion are equipped with a nilpotent operator called σ whose cohomology groups correspond to the dynamically relevant quantities like differential gauge parameters, dynamical fields, gauge invariant field equations, Bianchi identities etc. In the paper the σ -cohomology is computed for all gauge theories of this type and the field-theoretical interpretation is discussed. In the simplest cases the σ -cohomology is equivalent to the ordinary Lie algebra cohomology.

Keywords

Gauge Symmetry Field Theories in Higher Dimensions 

References

  1. [1]
    T. Curtright, Generalized gauge fields, Phys. Lett. B 165 (1985) 304 [SPIRES].MathSciNetADSGoogle Scholar
  2. [2]
    J.M.F. Labastida, Massless particles in arbitrary representations of the Lorentz group, Nucl. Phys. B 322 (1989) 185 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  3. [3]
    Y.M. Zinoviev, On massive mixed symmetry tensor fields in Minkowski space and (A)dS, hep-th/0211233 [SPIRES].
  4. [4]
    X. Bekaert and N. Boulanger, Tensor gauge fields in arbitrary representations of GL(D,R): duality and Poincaré lemma, Commun. Math. Phys. 245 (2004) 27 [hep-th/0208058] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  5. [5]
    Y.M. Zinoviev, First order formalism for massive mixed symmetry tensor fields in Minkowski and (A)dS spaces, hep-th/0306292 [SPIRES].
  6. [6]
    Y.M. Zinoviev, First order formalism for mixed symmetry tensor fields, hep-th/0304067 [SPIRES].
  7. [7]
    K.B. Alkalaev, Two-column higher spin massless fields in AdS d, Theor. Math. Phys. 140 (2004) 1253 [Teor. Mat. Fiz. 140 (2004) 424] [hep-th/0311212] [SPIRES].MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    K.B. Alkalaev, Mixed-symmetry massless gauge fields in AdS 5, Theor. Math. Phys. 149 (2006) 1338 [Teor. Mat. Fiz. 149 (2006) 47] [hep-th/0501105] [SPIRES].MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    X. Bekaert and N. Boulanger, Tensor gauge fields in arbitrary representations of GL(D,R). II: quadratic actions, Commun. Math. Phys. 271 (2007) 723 [hep-th/0606198] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  10. [10]
    P.Y. Moshin and A.A. Reshetnyak, BRST approach to Lagrangian formulation for mixed-symmetry fermionic higher-spin fields, JHEP 10 (2007) 040 [arXiv:0707.0386] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  11. [11]
    I.L. Buchbinder, V.A. Krykhtin and H. Takata, Gauge invariant Lagrangian construction for massive bosonic mixed symmetry higher spin fields, Phys. Lett. B 656 (2007) 253 [arXiv:0707.2181] [SPIRES].MathSciNetADSGoogle Scholar
  12. [12]
    E.D. Skvortsov, Frame-like actions for massless mixed-symmetry fields in Minkowski space, Nucl. Phys. B 808 (2009) 569 [arXiv:0807.0903] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    A.A. Reshetnyak, On Lagrangian formulations for mixed-symmetry HS fields on AdS spaces within BFV-BRST approach, arXiv:0809.4815 [SPIRES].
  14. [14]
    E.D. Skvortsov, Mixed-symmetry massless fields in Minkowski space unfolded, JHEP 07 (2008) 004 [arXiv:0801.2268] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  15. [15]
    A. Campoleoni, D. Francia, J. Mourad and A. Sagnotti, Unconstrained higher spins of mixed symmetry. I. Bose fields, Nucl. Phys. B 815 (2009) 289 [arXiv:0810.4350] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  16. [16]
    Y.M. Zinoviev, Toward frame-like gauge invariant formulation for massive mixed symmetry bosonic fields, Nucl. Phys. B 812 (2009) 46 [arXiv:0809.3287] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  17. [17]
    K.B. Alkalaev, M. Grigoriev and I.Y. Tipunin, Massless Poincaré modules and gauge invariant equations, Nucl. Phys. B 823 (2009) 509 [arXiv:0811.3999] [SPIRES].CrossRefGoogle Scholar
  18. [18]
    Y.M. Zinoviev, Frame-like gauge invariant formulation for mixed symmetry fermionic fields, Nucl. Phys. B 821 (2009) 21 [arXiv:0904.0549] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  19. [19]
    E.D. Skvortsov, Gauge fields in (anti)-de Sitter space and connections of its symmetry algebra, J. Phys. A 42 (2009) 385401 [arXiv:0904.2919] [SPIRES].Google Scholar
  20. [20]
    Y.M. Zinoviev, Towards frame-like gauge invariant formulation for massive mixed symmetry bosonic fields. II. General Young tableau with two rows, Nucl. Phys. B 826 (2010) 490 [arXiv:0907.2140] [SPIRES].CrossRefGoogle Scholar
  21. [21]
    K.B. Alkalaev and M. Grigoriev, Unified BRST description of AdS gauge fields, arXiv:0910.2690 [SPIRES].
  22. [22]
    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
  23. [23]
    R.R. Metsaev, Arbitrary spin massless bosonic fields in d-dimensional anti-de Sitter space, hep-th/9810231 [SPIRES].
  24. [24]
    L. Brink, R.R. Metsaev and M.A. Vasiliev, How massless are massless fields in AdS d, Nucl. Phys. B 586 (2000) 183 [hep-th/0005136] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  25. [25]
    S. Deser and R.I. Nepomechie, Gauge invariance versus masslessness in de Sitter space, Ann. Phys. 154 (1984) 396 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  26. [26]
    S. Deser and R.I. Nepomechie, Anomalous propagation of gauge fields in conformally flat spaces, Phys. Lett. B 132 (1983) 321 [SPIRES].MathSciNetADSGoogle Scholar
  27. [27]
    A. Higuchi, Symmetric tensor spherical harmonics on the N sphere and their application to the de Sitter group SO(N, 1), J. Math. Phys. 28 (1987) 1553 [Erratum ibid. 43 (2002) 6385] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  28. [28]
    S. Deser and A. Waldron, Gauge invariances and phases of massive higher spins in (A)dS, Phys. Rev. Lett. 87 (2001) 031601 [hep-th/0102166] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  29. [29]
    S. Deser and A. Waldron, Null propagation of partially massless higher spins in (A)dS and cosmological constant speculations, Phys. Lett. B 513 (2001) 137 [hep-th/0105181] [SPIRES].MathSciNetADSGoogle Scholar
  30. [30]
    S. Deser and A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B 607 (2001) 577 [hep-th/0103198] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  31. [31]
    S. Deser and A. Waldron, Arbitrary spin representations in de Sitter from dS/CFT with applications to dS supergravity, Nucl. Phys. B 662 (2003) 379 [hep-th/0301068] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  32. [32]
    Y.M. Zinoviev, On massive high spin particles in (A)dS, hep-th/0108192 [SPIRES].
  33. [33]
    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].CrossRefADSGoogle Scholar
  34. [34]
    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].CrossRefADSGoogle Scholar
  35. [35]
    S.W. MacDowell and F. Mansouri, Unified geometric theory of gravity and supergravity, Phys. Rev. Lett. 38 (1977) 739 [Erratum ibid. 38 (1977) 1376] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  36. [36]
    K.S. Stelle and P.C. West, Spontaneously broken de Sitter symmetry and the gravitational holonomy group, Phys. Rev. D 21 (1980) 1466 [SPIRES].MathSciNetADSGoogle Scholar
  37. [37]
    M.A. Vasiliev, Cubic interactions of bosonic higher spin gauge fields in AdS 5, Nucl. Phys. B 616 (2001) 106 [Erratum ibid. B 652 (2003) 407] [hep-th/0106200] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  38. [38]
    K.B. Alkalaev, O.V. Shaynkman and M.A. Vasiliev, On the frame-like formulation of mixed-symmetry massless fields in (A)dS d, Nucl. Phys. B 692 (2004) 363 [hep-th/0311164] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  39. [39]
    E.D. Skvortsov and M.A. Vasiliev, Geometric formulation for partially massless fields, Nucl. Phys. B 756 (2006) 117 [hep-th/0601095] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  40. [40]
    M.A. Vasiliev, Equations of motion of interacting massless fields of all spins as a free differential algebra, Phys. Lett. B 209 (1988) 491 [SPIRES].MathSciNetADSGoogle Scholar
  41. [41]
    M.A. Vasiliev, Consistent equations for interacting massless fields of all spins in the first order in curvatures, Ann. Phys. 190 (1989) 59 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  42. [42]
    D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Etudes Sci. Publ. Math. 47 (1977) 269.MATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    R. D’Auria and P. Fré, Geometric supergravity in D = 11 and its hidden supergroup, Nucl. Phys. B 201 (1982) 101 [Erratum ibid. B 206 (1982) 496] [SPIRES].CrossRefADSGoogle Scholar
  44. [44]
    R. D’Auria, P. Fré, P.K. Townsend and P. van Nieuwenhuizen, Invariance of actions, rheonomy and the new minimal N = 1 supergravity in the group manifold approach, Ann. Phys. 155 (1984) 423 [SPIRES].CrossRefADSGoogle Scholar
  45. [45]
    P. van Nieuwenhuizen, Free graded differential superalgebras, invited talk given at 11th Int. Colloq. on Group Theoretical Methods in Physics, Istanbul Turkey August 23–28 1982.Google Scholar
  46. [46]
    M.A. Vasiliev, Dynamics of massless higher spins in the second order in curvatures, Phys. Lett. B 238 (1990) 305 [SPIRES].MathSciNetADSGoogle Scholar
  47. [47]
    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
  48. [48]
    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
  49. [49]
    V.E. Didenko, A.S. Matveev and M.A. Vasiliev, Unfolded description of AdS 4 Kerr black hole, Phys. Lett. B 665 (2008) 284 [arXiv:0801.2213] [SPIRES].MathSciNetADSGoogle Scholar
  50. [50]
    V.E. Didenko, A.S. Matveev and M.A. Vasiliev, Unfolded dynamics and parameter flow of generic AdS 4 black hole, arXiv:0901.2172 [SPIRES].
  51. [51]
    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
  52. [52]
    O.V. Shaynkman and M.A. Vasiliev, Scalar field in any dimension from the higher spin gauge theory perspective, Theor. Math. Phys. 123 (2000) 683 [Teor. Mat. Fiz. 123 (2000) 323] [hep-th/0003123] [SPIRES].MATHCrossRefGoogle Scholar
  53. [53]
    M.A. Vasiliev, On conformal, SL(4, R) and Sp(8, R) symmetries of 4d massless fields, Nucl. Phys. B 793 (2008) 469 [arXiv:0707.1085] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  54. [54]
    O.V. Shaynkman, I.Y. Tipunin and M.A. Vasiliev, Unfolded form of conformal equations in M dimensions and o(M + 2)-modules, Rev. Math. Phys. 18 (2006) 823 [hep-th/0401086] [SPIRES].MATHCrossRefMathSciNetGoogle Scholar
  55. [55]
    M.A. Vasiliev, Unfolded representation for relativistic equations in (2 + 1) anti-de Sitter space, Class. Quant. Grav. 11 (1994) 649 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  56. [56]
    X. Bekaert, S. Cnockaert, C. Iazeolla and M.A. Vasiliev, Nonlinear higher spin theories in various dimensions, hep-th/0503128 [SPIRES].
  57. [57]
    C. Fronsdal, Massless particles, orthosymplectic symmetry and another type of Kaluza-Klein theory, UCLA/85/TEP/10 [SPIRES].
  58. [58]
    M.A. Vasiliev, Relativity, causality, locality, quantization and duality in the Sp(2M) invariant generalized space-time, hep-th/0111119 [SPIRES].
  59. [59]
    G. Barnich, M. Grigoriev, A. Semikhatov and I. Tipunin, Parent field theory and unfolding in BRST first-quantized terms, Commun. Math. Phys. 260 (2005) 147 [hep-th/0406192] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  60. [60]
    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
  61. [61]
    A.O. Barut and R. Raczka, Theory of group representations and applications, World Scientific, Singapore (1986).MATHGoogle Scholar
  62. [62]
    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
  63. [63]
    S. Kumar, Kac-Moody groups, their flag varieties and representation theory, Springer Verlag, U.S.A. (2001).Google Scholar
  64. [64]
    M.A. Vasiliev, Bosonic conformal higher-spin fields of any symmetry, arXiv:0909.5226 [SPIRES].
  65. [65]
    D. Francia and A. Sagnotti, Free geometric equations for higher spins, Phys. Lett. B 543 (2002) 303 [hep-th/0207002] [SPIRES].MathSciNetADSGoogle Scholar
  66. [66]
    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
  67. [67]
    X. Bekaert and N. Boulanger, On geometric equations and duality for free higher spins, Phys. Lett. B 561 (2003) 183 [hep-th/0301243] [SPIRES].MathSciNetADSGoogle Scholar
  68. [68]
    D. Francia and A. Sagnotti, Minimal local Lagrangians for higher-spin geometry, Phys. Lett. B 624 (2005) 93 [hep-th/0507144] [SPIRES].MathSciNetADSGoogle Scholar
  69. [69]
    A.S. Matveev and M.A. Vasiliev, On dual formulation for higher spin gauge fields in (A)dS d, Phys. Lett. B 609 (2005) 157 [hep-th/0410249] [SPIRES].MathSciNetADSGoogle Scholar
  70. [70]
    J. Hong and S.-J. Kang, Introduction to quantum groups and crystal bases, American Mathematical Society, U.S.A. (2002).MATHGoogle Scholar
  71. [71]
    O.V. Shaynkman and M.A. Vasiliev, Higher spin conformal symmetry for matter fields in 2 + 1 dimensions, Theor. Math. Phys. 128 (2001) 1155 [Teor. Mat. Fiz. 128 (2001) 378] [hep-th/0103208] [SPIRES].MATHCrossRefGoogle Scholar
  72. [72]
    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
  73. [73]
    E.S. Fradkin and M.A. Vasiliev, On the gravitational interaction of massless higher spin fields, Phys. Lett. B 189 (1987) 89 [SPIRES].ADSGoogle Scholar
  74. [74]
    K.B. Alkalaev, O.V. Shaynkman and M.A. Vasiliev, Lagrangian formulation for free mixed-symmetry bosonic gauge fields in (A)dS d, JHEP 08 (2005) 069 [hep-th/0501108] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  75. [75]
    K.B. Alkalaev, O.V. Shaynkman and M.A. Vasiliev, Frame-like formulation for free mixed-symmetry bosonic massless higher-spin fields in AdS d, hep-th/0601225 [SPIRES].
  76. [76]
    M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A 173 (1939) 211 [SPIRES].MathSciNetADSGoogle Scholar
  77. [77]
    L.P.S. Singh and C.R. Hagen, Lagrangian formulation for arbitrary spin. 1. The boson case, Phys. Rev. D 9 (1974) 898 [SPIRES].ADSGoogle Scholar
  78. [78]
    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
  79. [79]
    S.E. Konshtein and M.A. Vasiliev, Massless representations and admissibility condition for higher spin superalgebras, Nucl. Phys. B 312 (1989) 402 [SPIRES].CrossRefADSGoogle Scholar
  80. [80]
    M.A. Vasiliev, Higher spin superalgebras in any dimension and their representations, JHEP 12 (2004) 046 [hep-th/0404124] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  81. [81]
    R.R. Metsaev, Cubic interaction vertices of totally symmetric and mixed symmetry massless representations of the Poincaré group in D = 6 space-time, Phys. Lett. B 309 (1993) 39 [SPIRES].MathSciNetADSGoogle Scholar
  82. [82]
    R.R. Metsaev, Generating function for cubic interaction vertices of higher spin fields in any dimension, Mod. Phys. Lett. A 8 (1993) 2413 [SPIRES].MathSciNetADSGoogle Scholar
  83. [83]
    R.R. Metsaev, Cubic interaction vertices for massive and massless higher spin fields, Nucl. Phys. B 759 (2006) 147 [hep-th/0512342] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  84. [84]
    N. Boulanger and S. Cnockaert, Consistent deformations of (p, p)-type gauge field theories, JHEP 03 (2004) 031 [hep-th/0402180] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  85. [85]
    P. de Medeiros and C. Hull, Exotic tensor gauge theory and duality, Commun. Math. Phys. 235 (2003) 255 [hep-th/0208155] [SPIRES].MATHCrossRefADSGoogle Scholar
  86. [86]
    P. de Medeiros and C. Hull, Geometric second order field equations for general tensor gauge fields, JHEP 05 (2003) 019 [hep-th/0303036] [SPIRES].CrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.I.E. Tamm Department of Theoretical PhysicsP.N.Lebedev Physical InstituteMoscowRussia

Personalised recommendations