On classical solutions of 4d supersymmetric higher spin theory

  • Jun Bourdier
  • Nadav Drukker
Open Access
Regular Article - Theoretical Physics


We present a simple construction of solutions to the supersymmetric higher spin theory based on solutions to bosonic theories. We illustrate this for the case of the Didenko-Vasiliev solution in arXiv:0906.3898, for which we have found a striking simplification where the higher-spin connection takes the vacuum value. Studying these solutions further, we check under which conditions they preserve some supersymmetry in the bulk, and when they are compatible with the boundary conditions conjectured to be dual to certain 3d SUSY Chern-Simons-matter theories. We perform the analysis for a variety of theories with \( \mathcal{N} \) = 2, \( \mathcal{N} \) = 3, \( \mathcal{N} \) = 4 and \( \mathcal{N} \) = 6 and find a rich spectrum of 1/4, 1/3 and 1/2-BPS solutions.


Higher Spin Gravity AdS-CFT Correspondence 


Open Access

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  1. [1]
    M.A. Vasiliev, Consistent equations for interacting massless fields of all spins in the first order in curvatures, Annals Phys. 190 (1989) 59 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    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].
  3. [3]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    S. Giombi and X. Yin, Higher spin gauge theory and holography: the three-point functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    S.H. Shenker and X. Yin, Vector models in the singlet sector at finite temperature, arXiv:1109.3519 [INSPIRE].
  6. [6]
    M.A. Vasiliev, Holography, unfolding and higher-spin theory, J. Phys. A 46 (2013) 214013 [arXiv:1203.5554] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  7. [7]
    S. Giombi and X. Yin, The higher spin/vector model duality, J. Phys. A 46 (2013) 214003 [arXiv:1208.4036] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  8. [8]
    C.-M. Chang, S. Minwalla, T. Sharma and X. Yin, ABJ triality: from higher spin fields to strings, J. Phys. A 46 (2013) 214009 [arXiv:1207.4485] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  9. [9]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    E. Sezgin and P. Sundell, An exact solution of 4D higher-spin gauge theory, Nucl. Phys. B 762 (2007) 1 [hep-th/0508158] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    V.E. Didenko and M.A. Vasiliev, Static BPS black hole in 4d higher-spin gauge theory, Phys. Lett. B 682 (2009) 305 [Erratum ibid. B 722 (2013) 389] [arXiv:0906.3898] [INSPIRE].
  12. [12]
    C. Iazeolla and P. Sundell, Families of exact solutions to Vasilievs 4D equations with spherical, cylindrical and biaxial symmetry, JHEP 12 (2011) 084 [arXiv:1107.1217] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    S.S. Gubser and W. Song, An axial gauge ansatz for higher spin theories, JHEP 11 (2014) 036 [arXiv:1405.7045] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    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].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    M.A. Vasiliev, Higher spin gauge theories: star product and AdS space, in The many faces of the superworld, M.A. Shifman ed., World Scientific, Singapore (2000), pg. 533 [hep-th/9910096] [INSPIRE].
  16. [16]
    X. Bekaert, N. Boulanger and P. Sundell, How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples, Rev. Mod. Phys. 84 (2012) 987 [arXiv:1007.0435] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    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].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    J. Engquist, E. Sezgin and P. Sundell, On N = 1, N = 2, N = 4 higher spin gauge theories in four-dimensions, Class. Quant. Grav. 19 (2002) 6175 [hep-th/0207101] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    S.E. Konstein and M.A. Vasiliev, Extended higher spin superalgebras and their massless representations, Nucl. Phys. B 331 (1990) 475 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    C. Iazeolla, E. Sezgin and P. Sundell, Real forms of complex higher spin field equations and new exact solutions, Nucl. Phys. B 791 (2008) 231 [arXiv:0706.2983] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    M.A. Vasiliev, Higher-rank fields, currents and higher spin holography, talk presented at Strings 2014, Princeton U.S.A. June 2014.Google Scholar
  22. [22]
    P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  24. [24]
    J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju, Indices for superconformal field theories in 3, 5 and 6 dimensions, JHEP 02 (2008) 064 [arXiv:0801.1435] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    C. Fronsdal, Massless fields with integer spin, Phys. Rev. D 18 (1978) 3624 [INSPIRE].ADSGoogle Scholar
  27. [27]
    B. Carter, Hamilton-Jacobi and Schrödinger separable solutions of Einsteins equations, Commun. Math. Phys. 10 (1968) 280 [INSPIRE].MATHGoogle Scholar
  28. [28]
    G.W. Gibbons, H. Lü, D.N. Page and C.N. Pope, The general Kerr-de Sitter metrics in all dimensions, J. Geom. Phys. 53 (2005) 49 [hep-th/0404008] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of MathematicsKing’s College LondonLondonUnited Kingdom

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