Skip to main content
SpringerLink
Log in

Non-Abelian cubic vertices for higher-spin fields in AdS d

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We use the Fradkin-Vasiliev procedure to construct the full set of non-Abelian cubic vertices for totally symmetric higher spin gauge fields in AdS d space. The number of such vertices is given by a certain tensor-product multiplicity. We discuss the one-to-one relation between our result and the list of non-Abelian gauge deformations in flat space obtained elsewhere via the cohomological approach. We comment about the uniqueness of Vasiliev’s simplest higher-spin algebra in relation with the (non)associativity properties of the gauge algebras that we classified. The gravitational interactions for (partially)-massless (mixed)-symmetry fields are also discussed. We also argue that those mixed-symmetry and/or partially-massless fields that are described by one-form connections within the frame-like approach can have non-Abelian interactions among themselves and again the number of non-Abelian vertices should be given by tensor product multiplicities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Fradkin and M.A. Vasiliev, Cubic interaction in extended theories of massless higher spin fields, Nucl. Phys. B 291 (1987) 141 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. E. Fradkin and M.A. Vasiliev, On the gravitational interaction of massless higher spin fields, Phys. Lett. B 189 (1987) 89 [INSPIRE].

    ADS  Google Scholar 

  3. S. MacDowell and F. Mansouri, Unified geometric theory of gravity and supergravity, Phys. Rev. Lett. 38 (1977) 739 [Erratum ibid. 38 (1977) 1376] [INSPIRE].

  4. F. Mansouri, Superunified theories based on the geometry of local (super)gauge invariance, Phys. Rev. D 16 (1977) 2456 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  5. K. Stelle and P.C. West, Spontaneously broken de Sitter symmetry and the gravitational holonomy group, Phys. Rev. D 21 (1980) 1466 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  6. M. 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] [INSPIRE].

  7. K. Alkalaev, FV-type action for AdS 5 mixed-symmetry fields, JHEP 03 (2011) 031 [arXiv:1011.6109] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. Y. Zinoviev, On electromagnetic interactions for massive mixed symmetry field, JHEP 03 (2011) 082 [arXiv:1012.2706] [INSPIRE].

    Article  ADS  Google Scholar 

  9. N. Boulanger, E. Skvortsov and Y. Zinoviev, Gravitational cubic interactions for a simple mixed-symmetry gauge field in AdS and flat backgrounds, J. Phys. A 44 (2011) 415403 [arXiv:1107.1872] [INSPIRE].

    Google Scholar 

  10. Y. Zinoviev, Gravitational cubic interactions for a massive mixed symmetry gauge field, Class. Quant. Grav. 29 (2012) 015013 [arXiv:1107.3222] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. N. Boulanger and E. Skvortsov, Higher-spin algebras and cubic interactions for simple mixed-symmetry fields in AdS spacetime, JHEP 09 (2011) 063 [arXiv:1107.5028] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  12. M. Vasiliev, Cubic vertices for symmetric higher-spin gauge fields in (A)dS d , Nucl. Phys. B 862 (2012) 341 [arXiv:1108.5921] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  13. X. Bekaert, N. Boulanger and S. Leclercq, Strong obstruction of the Berends-Burgers-van Dam spin-3 vertex, J. Phys. A 43 (2010) 185401 [arXiv:1002.0289] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  15. G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rept. 338 (2000) 439 [hep-th/0002245] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. N. Boulanger, S. Leclercq and P. Sundell, On the uniqueness of minimal coupling in higher-spin gauge theory, JHEP 08 (2008) 056 [arXiv:0805.2764] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. 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].

    Article  ADS  Google Scholar 

  18. M. Vasiliev, Higher spin superalgebras in any dimension and their representations, JHEP 12 (2004) 046 [hep-th/0404124] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. K. Alkalaev, On manifestly sp(2) invariant formulation of quadratic higher spin Lagrangians, JHEP 06 (2008) 081 [arXiv:0711.3639] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. G. Barnich and M. Henneaux, Consistent couplings between fields with a gauge freedom and deformations of the master equation, Phys. Lett. B 311 (1993) 123 [hep-th/9304057] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  21. M. Henneaux, Consistent interactions between gauge fields: the cohomological approach, Contemp. Math. 219 (1998) 93 [hep-th/9712226] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  22. A. Sagnotti and M. Taronna, String lessons for higher-spin interactions, Nucl. Phys. B 842 (2011) 299 [arXiv:1006.5242] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  23. A. Fotopoulos and M. Tsulaia, On the tensionless limit of string theory, off - Shell higher spin interaction vertices and BCFW recursion relations, JHEP 11 (2010) 086 [arXiv:1009.0727] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. R. Manvelyan, K. Mkrtchyan and W. Ruehl, A generating function for the cubic interactions of higher spin fields, Phys. Lett. B 696 (2011) 410 [arXiv:1009.1054] [INSPIRE].

    ADS  Google Scholar 

  25. R. Metsaev, BRST-BV approach to cubic interaction vertices for massive and massless higher-spin fields, Phys. Lett. B 720 (2013) 237 [arXiv:1205.3131] [INSPIRE].

    ADS  Google Scholar 

  26. R. Metsaev, Generating function for cubic interaction vertices of higher spin fields in any dimension, Mod. Phys. Lett. A 8 (1993) 2413 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  27. 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 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  28. R. Metsaev, Cubic interaction vertices of massive and massless higher spin fields, Nucl. Phys. B 759 (2006) 147 [hep-th/0512342] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. A. Sagnotti, Notes on strings and higher spins, arXiv:1112.4285 [INSPIRE].

  30. X. Bekaert and E. Meunier, Higher spin interactions with scalar matter on constant curvature spacetimes: conserved current and cubic coupling generating functions, JHEP 11 (2010) 116 [arXiv:1007.4384] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  31. E. Joung, L. Lopez and M. Taronna, Solving the Noether procedure for cubic interactions of higher spins in (A)dS, arXiv:1207.5520 [INSPIRE].

  32. R. Manvelyan, R. Mkrtchyan and W. Ruehl, Radial reduction and cubic interaction for higher spins in (A)dS space, Nucl. Phys. B 872 (2013) 265 [arXiv:1210.7227] [INSPIRE].

    ADS  Google Scholar 

  33. M.G. Eastwood, Higher symmetries of the laplacian, Annals Math. 161 (2005) 1645 [hep-th/0206233] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  34. V. 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 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  35. C. Iazeolla and P. Sundell, A fiber approach to harmonic analysis of unfolded higher-spin field equations, JHEP 10 (2008) 022 [arXiv:0806.1942] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. G. Barnich, M. Henneaux and R. Tatar, Consistent interactions between gauge fields and the local BRST cohomology: the example of Yang-Mills models, Int. J. Mod. Phys. D 3 (1994) 139 [hep-th/9307155] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  37. X. Bekaert, N. Boulanger and M. Henneaux, Consistent deformations of dual formulations of linearized gravity: a no go result, Phys. Rev. D 67 (2003) 044010 [hep-th/0210278] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  38. N. Boulanger and S. Cnockaert, Consistent deformations of [p,p] type gauge field theories, JHEP 03 (2004) 031 [hep-th/0402180] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. X. Bekaert, N. Boulanger and S. Cnockaert, No self-interaction for two-column massless fields, J. Math. Phys. 46 (2005) 012303 [hep-th/0407102] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. R. Metsaev, Cubic interaction vertices for fermionic and bosonic arbitrary spin fields, Nucl. Phys. B 859 (2012) 13 [arXiv:0712.3526] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  41. S. Deser, E. Joung and A. Waldron, Partial masslessness and conformal gravity, J. Phys. A (2012) [arXiv:1208.1307] [INSPIRE].

  42. S. Deser and A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B 607 (2001) 577 [hep-th/0103198] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  43. E. Skvortsov and M. Vasiliev, Geometric formulation for partially massless fields, Nucl. Phys. B 756 (2006) 117 [hep-th/0601095] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. K. Alkalaev, O. Shaynkman and M. Vasiliev, On the frame - Like formulation of mixed symmetry massless fields in (A)dS(d), Nucl. Phys. B 692 (2004) 363 [hep-th/0311164] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  45. K. Alkalaev, Two column higher spin massless fields in AdS d , Theor. Math. Phys. 140 (2004) 1253 [hep-th/0311212] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  46. 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].

    Article  MathSciNet  ADS  Google Scholar 

  47. E. Skvortsov, Gauge fields in (A)dS(d) within the unfolded approach: algebraic aspects, JHEP 01 (2010) 106 [arXiv:0910.3334] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  48. 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].

    Article  MathSciNet  ADS  Google Scholar 

  49. E. Skvortsov, Gauge fields in (A)dS(d) and connections of its symmetry algebra, J. Phys. A 42 (2009) 385401 [arXiv:0904.2919] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  50. Y. Zinoviev, Toward frame-like gauge invariant formulation for massive mixed symmetry bosonic fields, Nucl. Phys. B 812 (2009) 46 [arXiv:0809.3287] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  51. Y. Zinoviev, Frame-like gauge invariant formulation for mixed symmetry fermionic fields, Nucl. Phys. B 821 (2009) 21 [arXiv:0904.0549] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  52. Y. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  53. K. Alkalaev, O. Shaynkman and M. Vasiliev, Lagrangian formulation for free mixed-symmetry bosonic gauge fields in (A)dS(d), JHEP 08 (2005) 069 [hep-th/0501108] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  54. K. Alkalaev, O. Shaynkman and M. Vasiliev, Frame-like formulation for free mixed-symmetry bosonic massless higher-spin fields in AdS d , hep-th/0601225 [INSPIRE].

  55. X. Bekaert, N. Boulanger, S. Cnockaert and S. Leclercq, On Killing tensors and cubic vertices in higher-spin gauge theories, Fortsch. Phys. 54 (2006) 282 [hep-th/0602092] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3 + 1)-dimensions, Phys. Lett. B 243 (1990) 378 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  57. 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].

    MathSciNet  ADS  Google Scholar 

  58. E. Skvortsov and Y. Zinoviev, Frame-like actions for massless mixed-symmetry fields in Minkowski space. Fermions, Nucl. Phys. B 843 (2011) 559 [arXiv:1007.4944] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. I. Buchbinder, V. Krykhtin and L. Ryskina, Lagrangian formulation of massive fermionic totally antisymmetric tensor field theory in AdS d space, Nucl. Phys. B 819 (2009) 453 [arXiv:0902.1471] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  60. Y. Zinoviev, Note on antisymmetric spin-tensors, JHEP 04 (2009) 035 [arXiv:0903.0262] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  61. X. Bekaert, B. Knaepen and C. Schomblond, Couplings of gravity to antisymmetric gauge fields, Phys. Lett. B 481 (2000) 89 [hep-th/0003170] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  62. G. Chapline and N. Manton, Unification of Yang-Mills theory and supergravity in ten-dimensions, Phys. Lett. B 120 (1983) 105 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  63. D.Z. Freedman and P. Townsend, Antisymmetric tensor gauge theories and nonlinear σ-models, Nucl. Phys. B 177 (1981) 282 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  64. R. Metsaev, Arbitrary spin massless bosonic fields in d-dimensional Anti-de Sitter space, Lect. Notes Phys. 524 (1997) 331 [hep-th/9810231] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  65. L. Brink, R. Metsaev and M.A. Vasiliev, How massless are massless fields in AdS d , Nucl. Phys. B 586 (2000) 183 [hep-th/0005136] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  66. D. Kaparulin, S. Lyakhovich and A. Sharapov, Consistent interactions and involution, JHEP 01 (2013) 097 [arXiv:1210.6821] [INSPIRE].

    Article  ADS  Google Scholar 

  67. E. Joung, L. Lopez and M. Taronna, Generating functions of (partially-)massless higher-spin cubic interactions, JHEP 01 (2013) 168 [arXiv:1211.5912] [INSPIRE].

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Mons — UMONS, 20 Place du Parc, 700, Mons, Belgium

    Nicolas Boulanger & Dmitry Ponomarev

  2. Albert Einstein Institute, Am Mühlenberg 1, D-14476, Golm, Germany

    E. D. Skvortsov

  3. Lebedev Institute of Physics, Leninsky pr-t, 53, 119991, Moscow, Russia

    E. D. Skvortsov

Authors
  1. Nicolas Boulanger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry Ponomarev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. D. Skvortsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitry Ponomarev.

Additional information

ArXiv ePrint: 1211.6979

Research Associate of the Fund for Scientific Research-FNRS (Belgium). (Nicolas Boulanger)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boulanger, N., Ponomarev, D. & Skvortsov, E.D. Non-Abelian cubic vertices for higher-spin fields in AdS d . J. High Energ. Phys. 2013, 8 (2013). https://doi.org/10.1007/JHEP05(2013)008

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP05(2013)008

Keywords

Search

Navigation