Advertisement

Journal of High Energy Physics

, 2018:167 | Cite as

Conformal higher-spin gravity: linearized spectrum = symmetry algebra

  • Thomas Basile
  • Xavier Bekaert
  • Euihun JoungEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

The linearized spectrum and the algebra of global symmetries of conformal higher-spin gravity decompose into infinitely many representations of the conformal algebra. Their characters involve divergent sums over spins. We propose a suitable regularization adapted to their evaluation and observe that their characters are actually equal. This result holds in the case of type-A and type-B (and their higher-depth generalizations) theories and confirms previous observations on a remarkable rearrangement of dynamical degrees of freedom in conformal higher-spin gravity after regularization.

Keywords

Conformal Field Theory Higher Spin Gravity Higher Spin Symmetry 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    H. Weyl, A new extension of relativity theory, Annalen Phys. 59 (1919) 101 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    M.A. Vasiliev, Extended higher spin superalgebras and their realizations in terms of quantum operators, Fortsch. Phys. 36 (1988) 33 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    M.A. Vasiliev, Quantization on sphere and high spin superalgebras, JETP Lett. 50 (1989) 374 [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    M.A. Vasiliev, Higher spin algebras and quantization on the sphere and hyperboloid, Int. J. Mod. Phys. A 6 (1991) 1115 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    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] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  7. [7]
    M. Grigoriev, Off-shell gauge fields from BRST quantization, hep-th/0605089 [INSPIRE].
  8. [8]
    A.A. Sharapov and E.D. Skvortsov, Formal higher-spin theories and Kontsevich-Shoikhet-Tsygan formality, Nucl. Phys. B 921 (2017) 538 [arXiv:1702.08218] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A.A. Sharapov and E.D. Skvortsov, Hochschild cohomology of the Weyl algebra and Vasilievs equations, arXiv:1705.02958 [INSPIRE].
  10. [10]
    E.S. Fradkin and A.A. Tseytlin, Conformal supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    E.S. Fradkin and V.Ya. Linetsky, Cubic interaction in conformal theory of integer higher spin fields in four-dimensional space-time, Phys. Lett. B 231 (1989) 97 [INSPIRE].
  12. [12]
    E.S. Fradkin and V.Ya. Linetsky, Superconformal higher spin theory in the cubic approximation, Nucl. Phys. B 350 (1991) 274 [INSPIRE].
  13. [13]
    E.S. Fradkin and V.Ya. Linetsky, Conformal superalgebras of higher spins, Mod. Phys. Lett. A 04 (1989) 2363.Google Scholar
  14. [14]
    E.S. Fradkin and V. Ya. Linetsky, Conformal superalgebras of higher spins, Annals Phys. 198 (1990) 252 [INSPIRE].
  15. [15]
    E.S. Fradkin and M.A. Vasiliev, Cubic interaction in extended theories of massless higher spin fields, Nucl. Phys. B 291 (1987) 141 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    E.S. Fradkin and M.A. Vasiliev, On the gravitational interaction of massless higher spin fields, Phys. Lett. B 189 (1987) 89 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    E.S. Fradkin and V.Ya. Linetsky, A superconformal theory of massless higher spin fields in D = (2 + 1), Mod. Phys. Lett. A 4 (1989) 731 [Annals Phys. 198 (1990) 293] [INSPIRE].
  18. [18]
    J.H. Horne and E. Witten, Conformal gravity in three-dimensions as a gauge theory, Phys. Rev. Lett. 62 (1989) 501 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    C.N. Pope and P.K. Townsend, Conformal higher spin in (2 + 1)-dimensions, Phys. Lett. B 225 (1989) 245 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    O.V. Shaynkman and M.A. Vasiliev, Higher spin conformal symmetry for matter fields in (2 + 1)-dimensions, Theor. Math. Phys. 128 (2001) 1155 [hep-th/0103208] [INSPIRE].CrossRefzbMATHGoogle Scholar
  21. [21]
    M.A. Vasiliev, Bosonic conformal higher-spin fields of any symmetry, Nucl. Phys. B 829 (2010) 176 [arXiv:0909.5226] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    B.E.W. Nilsson, Towards an exact frame formulation of conformal higher spins in three dimensions, JHEP 09 (2015) 078 [arXiv:1312.5883] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  23. [23]
    O.V. Shaynkman, Bosonic Fradkin-Tseytlin equations unfolded, JHEP 12 (2016) 118 [arXiv:1412.7743] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    B.E.W. Nilsson, On the conformal higher spin unfolded equation for a three-dimensional self-interacting scalar field, JHEP 08 (2016) 142 [arXiv:1506.03328] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    T. Basile, R. Bonezzi and N. Boulanger, The Schouten tensor as a connection in the unfolding of 3D conformal higher-spin fields, JHEP 04 (2017) 054 [arXiv:1701.08645] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    O.V. Shaynkman, Bosonic Fradkin-Tseytlin equations unfolded. Irreducible case, arXiv:1807.00142 [INSPIRE].
  27. [27]
    A.A. Tseytlin, On limits of superstring in AdS 5 × S 5, Theor. Math. Phys. 133 (2002) 1376 [hep-th/0201112] [INSPIRE].CrossRefGoogle Scholar
  28. [28]
    A.Y. Segal, Conformal higher spin theory, Nucl. Phys. B 664 (2003) 59 [hep-th/0207212] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    X. Bekaert, E. Joung and J. Mourad, Effective action in a higher-spin background, JHEP 02 (2011) 048 [arXiv:1012.2103] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    X. Bekaert and M. Grigoriev, Notes on the ambient approach to boundary values of AdS gauge fields, J. Phys. A 46 (2013) 214008 [arXiv:1207.3439] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  31. [31]
    X. Bekaert and M. Grigoriev, Higher order singletons, partially massless fields and their boundary values in the ambient approach, Nucl. Phys. B 876 (2013) 667 [arXiv:1305.0162] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    L. Bonora et al., One-loop effective actions and higher spins, JHEP 12 (2016) 084 [arXiv:1609.02088] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    L. Bonora et al., One-loop effective actions and higher spins. Part II, JHEP 01 (2018) 080 [arXiv:1709.01738] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    R. Bonezzi, Induced action for conformal higher spins from worldline path integrals, Universe 3 (2017) 64 [arXiv:1709.00850] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    L. Bonora et al., Worldline quantization of field theory, effective actions and L structure, JHEP 04 (2018) 095 [arXiv:1802.02968] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    J. Maldacena, Einstein gravity from conformal gravity, arXiv:1105.5632 [INSPIRE].
  37. [37]
    G. Anastasiou and R. Olea, From conformal to Einstein gravity, Phys. Rev. D 94 (2016) 086008 [arXiv:1608.07826] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    A.A. Tseytlin, On partition function and Weyl anomaly of conformal higher spin fields, Nucl. Phys. B 877 (2013) 598 [arXiv:1309.0785] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    R.R. Metsaev, Arbitrary spin conformal fields in (A)dS, Nucl. Phys. B 885 (2014) 734 [arXiv:1404.3712] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    T. Nutma and M. Taronna, On conformal higher spin wave operators, JHEP 06 (2014) 066 [arXiv:1404.7452] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    M. Grigoriev and A. Hancharuk, On the structure of the conformal higher-spin wave operators, arXiv:1808.04320 [INSPIRE].
  42. [42]
    E.S. Fradkin and A.A. Tseytlin, Instanton zero modes and beta functions in supergravities. 2. Conformal supergravity, Phys. Lett. B 134 (1984) 307.Google Scholar
  43. [43]
    A.A. Tseytlin, Effective action in de Sitter space and conformal supergravity (in Russian), Yad. Fiz. 39 (1984) 1606 [Sov. J. Nucl. Phys. 39 (1984) 1018] [INSPIRE].
  44. [44]
    S. Deser and R.I. Nepomechie, Gauge invariance versus masslessness in de Sitter space, Annals Phys. 154 (1984) 396 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    E. Joung and K. Mkrtchyan, A note on higher-derivative actions for free higher-spin fields, JHEP 11 (2012) 153 [arXiv:1209.4864] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    S. Giombi et al., AdS description of induced higher-spin gauge theory, JHEP 10 (2013) 016 [arXiv:1306.5242] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    M. Grigoriev and A.A. Tseytlin, On conformal higher spins in curved background, J. Phys. A 50 (2017) 125401 [arXiv:1609.09381] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  48. [48]
    M. Beccaria and A.A. Tseytlin, On induced action for conformal higher spins in curved background, Nucl. Phys. B 919 (2017) 359 [arXiv:1702.00222] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    M. Beccaria and A.A. Tseytlin, C T for conformal higher spin fields from partition function on conically deformed sphere, JHEP 09 (2017) 123 [arXiv:1707.02456] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  50. [50]
    S. Acevedo, R. Aros, F. Bugini and D.E. Diaz, On the Weyl anomaly of 4D conformal higher spins: a holographic approach, JHEP 11 (2017) 082 [arXiv:1710.03779] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    E. Joung, S. Nakach and A.A. Tseytlin, Scalar scattering via conformal higher spin exchange, JHEP 02 (2016) 125 [arXiv:1512.08896] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    M. Beccaria, S. Nakach and A.A. Tseytlin, On triviality of S-matrix in conformal higher spin theory, JHEP 09 (2016) 034 [arXiv:1607.06379] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    P. Hähnel and T. McLoughlin, Conformal higher spin theory and twistor space actions, J. Phys. A 50 (2017) 485401 [arXiv:1604.08209] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  54. [54]
    T. Adamo, P. Hähnel and T. McLoughlin, Conformal higher spin scattering amplitudes from twistor space, JHEP 04 (2017) 021 [arXiv:1611.06200] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    T. Adamo, S. Nakach and A.A. Tseytlin, Scattering of conformal higher spin fields, JHEP 07 (2018) 016 [arXiv:1805.00394] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  56. [56]
    T. Adamo and L. Mason, Conformal and Einstein gravity from twistor actions, Class. Quant. Grav. 31 (2014) 045014 [arXiv:1307.5043] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    O.V. Shaynkman, I.Yu. 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] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    R.R. Metsaev, Light cone form of field dynamics in Anti-de Sitter space-time and AdS/CFT correspondence, Nucl. Phys. B 563 (1999) 295 [hep-th/9906217] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    R.R. Metsaev, Ordinary-derivative formulation of conformal totally symmetric arbitrary spin bosonic fields, JHEP 06 (2012) 062 [arXiv:0709.4392] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    R.R. Metsaev, Gauge invariant two-point vertices of shadow fields, AdS/CFT and conformal fields, Phys. Rev. D 81 (2010) 106002 [arXiv:0907.4678] [INSPIRE].ADSMathSciNetGoogle Scholar
  61. [61]
    R.R. Metsaev, Conformal totally symmetric arbitrary spin fermionic fields, arXiv:1211.4498 [INSPIRE].
  62. [62]
    R.R. Metsaev, Mixed-symmetry fields in AdS 5 , conformal fields and AdS/CFT, JHEP 01 (2015) 077 [arXiv:1410.7314] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    A. Chekmenev and M. Grigoriev, Boundary values of mixed-symmetry massless fields in AdS space, Nucl. Phys. B 913 (2016) 769 [arXiv:1512.06443].ADSCrossRefzbMATHGoogle Scholar
  64. [64]
    M. Beccaria, X. Bekaert and A.A. Tseytlin, Partition function of free conformal higher spin theory, JHEP 08 (2014) 113 [arXiv:1406.3542] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    M. Beccaria and A.A. Tseytlin, Iterating free-field AdS/CFT: higher spin partition function relations, J. Phys. A 49 (2016) 295401 [arXiv:1602.00948] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  66. [66]
    M. Beccaria and A.A. Tseytlin, On higher spin partition functions, J. Phys. A 48 (2015) 275401 [arXiv:1503.08143] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  67. [67]
    A.Y. Segal, Point particle-symmetric tensors interaction and generalized gauge principle, Int. J. Mod. Phys. A 18 (2003) 5021 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    A. Mikhailov, Notes on higher spin symmetries, hep-th/0201019 [INSPIRE].
  69. [69]
    R.R. Metsaev, CFT adapted gauge invariant formulation of arbitrary spin fields in AdS and modified de Donder gauge, Phys. Lett. B 671 (2009) 128 [arXiv:0808.3945] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  70. [70]
    S. Giombi and X. Yin, Higher spin gauge theory and holography: the three-point functions, JHEP 09 (2010) 115 [arXiv:0912.3462].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    E. Joung and J. Mourad, Boundary action of free AdS higher-spin gauge fields and the holographic correspondence, JHEP 06 (2012) 161 [arXiv:1112.5620] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    G.W. Gibbons, M.J. Perry and C.N. Pope, Partition functions, the Bekenstein bound and temperature inversion in Anti-de Sitter space and its conformal boundary, Phys. Rev. D 74 (2006) 084009 [hep-th/0606186] [INSPIRE].ADSMathSciNetGoogle Scholar
  73. [73]
    F.A. Dolan, Character formulae and partition functions in higher dimensional conformal field theory, J. Math. Phys. 47 (2006) 062303 [hep-th/0508031] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    M.G. Eastwood and J.W. Rice, Conformally invariant differential operators on Minkowski space and their curved analogues, Commun. Math. Phys. 109 (1987) 207 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, The shadow operator formalism for conformal algebra. vacuum expectation values and operator products, Lett. Nuovo Cim. 4S2 (1972) 115 [INSPIRE].
  76. [76]
    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].
  77. [77]
    E. Angelopoulos and M. Laoues, Singletons on AdS n, talk given at the Conference Moshe Flato, September 5–8, Dijon, France (1999).Google Scholar
  78. [78]
    M.A. Vasiliev, Higher spin superalgebras in any dimension and their representations, JHEP 12 (2004) 046 [hep-th/0404124] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  79. [79]
    T. Basile, X. Bekaert and E. Joung, Twisted flato-fronsdal theorem for higher-spin algebras, JHEP 07 (2018) 009 [arXiv:1802.03232].ADSCrossRefzbMATHGoogle Scholar
  80. [80]
    S. Giombi, I.R. Klebanov and A.A. Tseytlin, Partition functions and Casimir energies in higher spin AdS d+1 /CFT d, Phys. Rev. D 90 (2014) 024048 [arXiv:1402.5396].ADSGoogle Scholar
  81. [81]
    J.-B. Bae, E. Joung and S. Lal, One-loop test of free SU(N) adjoint model holography, JHEP 04 (2016) 061 [arXiv:1603.05387] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  82. [82]
    T. Basile, E. Joung, S. Lal and W. Li, Character integral representation of Zeta function in AdS d+1 : I. derivation of the general formula, JHEP 10 (2018) 091 [arXiv:1805.05646] [INSPIRE].
  83. [83]
    T. Basile, E. Joung, S. Lal and W. Li, Character integral representation of zeta function in AdS d+1 . Part II. Application to partially-massless higher-spin gravities, JHEP 07 (2018) 132 [arXiv:1805.10092] [INSPIRE].
  84. [84]
    E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [hep-th/0205131] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  85. [85]
    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] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  86. [86]
    I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  87. [87]
    E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
  88. [88]
    A. Sever and A. Shomer, A note on multitrace deformations and AdS/CFT, JHEP 07 (2002) 027 [hep-th/0203168] [INSPIRE].ADSCrossRefGoogle Scholar
  89. [89]
    R.G. Leigh and A.C. Petkou, SL(2, ℤ) action on three-dimensional CFTs and holography, JHEP 12 (2003) 020 [hep-th/0309177] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  90. [90]
    M.A. Vasiliev, Holography, unfolding and higher-spin theory, J. Phys. A 46 (2013) 214013 [arXiv:1203.5554] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  91. [91]
    G. Compere and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav. 25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  92. [92]
    X. Bekaert, E. Joung and J. Mourad, Comments on higher-spin holography, Fortsch. Phys. 60 (2012) 882 [arXiv:1202.0543] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  93. [93]
    C. Brust and K. Hinterbichler, Free k scalar conformal field theory, JHEP 02 (2017) 066 [arXiv:1607.07439].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  94. [94]
    F. Gliozzi, A.L. Guerrieri, A.C. Petkou and C. Wen, The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points, JHEP 04 (2017) 056 [arXiv:1702.03938] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  95. [95]
    R.R. Metsaev, Long, partial-short and special conformal fields, JHEP 05 (2016) 096 [arXiv:1604.02091] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  96. [96]
    C. Brust and K. Hinterbichler, Partially massless higher-spin theory, JHEP 02 (2017) 086 [arXiv:1610.08510] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  97. [97]
    A.G. Nikitin, Generalized killing tensors of arbitrary rank and order, Ukrainian Math. J. 43 (1991) 734.MathSciNetCrossRefzbMATHGoogle Scholar
  98. [98]
    A.G. Nikitin and O.I. Prylypko, Generalized Killing tensors and symmetry of Klein-Gordon-Fock equations, math-ph/0506002.
  99. [99]
    T. Basile, X. Bekaert and N. Boulanger, Flato-Fronsdal theorem for higher-order singletons, JHEP 11 (2014) 131 [arXiv:1410.7668] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  100. [100]
    M. Grigoriev and E.D. Skvortsov, Type-B formal higher spin gravity, JHEP 05 (2018) 138 [arXiv:1804.03196].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  101. [101]
    S. Giombi, I.R. Klebanov and Z.M. Tan, The ABC of higher-spin AdS/CFT, Universe 4 (2018) 18 [arXiv:1608.07611] [INSPIRE].ADSCrossRefGoogle Scholar
  102. [102]
    J.B. Bae, E. Joung and S. Lal, A note on vectorial AdS 5 /CFT 4 duality for spin-j boundary theory, JHEP 12 (2016) 077 [arXiv:1611.00112].ADSCrossRefGoogle Scholar
  103. [103]
    E. Joung and K. Mkrtchyan, Partially-massless higher-spin algebras and their finite-dimensional truncations, JHEP 01 (2016) 003 [arXiv:1508.07332] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  104. [104]
    J. Lepowsky, A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Alg. 49 (1977) 496.MathSciNetCrossRefzbMATHGoogle Scholar
  105. [105]
    A. Bourget and J. Troost, The conformal characters. JHEP 04 (2018) 055 [arXiv:1712.05415].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  106. [106]
    R.R. Metsaev, Massless mixed symmetry bosonic free fields in d-dimensional Anti-de Sitter space-time, Phys. Lett. B 354 (1995) 78 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  107. [107]
    R.R. Metsaev, Arbitrary spin massless bosonic fields in d-dimensional Anti-de Sitter space, Lect. Notes Phys. 524 (1999) 331 [hep-th/9810231] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  108. [108]
    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] [INSPIRE].
  109. [109]
    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] [INSPIRE].
  110. [110]
    E.D. Skvortsov, Gauge fields in (A)dS(d) and connections of its symmetry algebra, J. Phys. A 42 (2009) 385401 [arXiv:0904.2919] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  111. [111]
    E.D. Skvortsov, Gauge fields in (A)dS(d) within the unfolded approach: algebraic aspects, JHEP 01 (2010) 106 [arXiv:0910.3334] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  112. [112]
    S. Gwak, E. Joung, K. Mkrtchyan and S.-J. Rey, Rainbow vacua of colored higher-spin (A)dS 3 gravity, JHEP 05 (2016) 150 [arXiv:1511.05975] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  113. [113]
    A.A. Tseytlin, Weyl anomaly of conformal higher spins on six-sphere, Nucl. Phys. B 877 (2013) 632 [arXiv:1310.1795] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  114. [114]
    A.Yu. Artsukevich and M.A. Vasiliev, On dimensional degression in AdS(d), Phys. Rev. D 79 (2009) 045007 [arXiv:0810.2065] [INSPIRE].ADSGoogle Scholar
  115. [115]
    T. Kobayashi, B. Ørsted, P. Somberg and V. Souček, Branching laws for verma modules and applications in parabolic geometry. I, Adv. Math. 285 (2015) 1796 [arXiv:1305.6040].
  116. [116]
    G. Barnich, X. Bekaert and M. Grigoriev, Notes on conformal invariance of gauge fields, J. Phys. A 48 (2015) 505402 [arXiv:1506.00595].MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of Physics and Research Institute of Basic ScienceKyung Hee UniversitySeoulKorea
  2. 2.Institut Denis Poisson, Université de Tours, Université d’Orléans, CNRSToursFrance

Personalised recommendations