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Journal of High Energy Physics

, 2019:86 | Cite as

Limiting shifted homotopy in higher-spin theory and spin-locality

  • V.E. DidenkoEmail author
  • O.A. Gelfond
  • A.V. Korybut
  • M.A. Vasiliev
Open Access
Regular Article - Theoretical Physics
  • 22 Downloads

Abstract

Higher-spin vertices containing up to quintic interactions at the LagTangian level are explicitly calculated in the one-form sector of the non-linear unfolded higher-spin equations using a 𝛽 →-∞-shifted contracting homotopy introduced in the paper. The problem is solved in a background independent way and for any value of the complex parameter 𝜂 in the higher-spin equations. All obtained vertices are shown to be spin-local containing a finite number of derivatives in the spinor space for any given set of spins. The vertices proportional to 𝜂2 and \( {\overline{\eta}}^2 \) are in addition ultra-local, i.e., zero-forms that enter into the vertex in question are free from the dependence on at least one of the spinor variables y or \( \overline{y} \). Also the 𝜂2 and \( {\overline{\eta}}^2 \) vertices are shown to vanish on any purely gravitational background hence not contributing to the higher-spin current interactions on AdS4. This implies in particular that the gravitational constant in front of the stress tensor is positive being proportional to \( \eta \overline{\eta} \). It is shown that the 𝛽-shifted homotopy technique developed in this paper can be reinterpreted as the conventional one but in the 𝛽-dependent deformed star product.

Keywords

Higher Spin Gravity Higher Spin Symmetry Gauge-gravity correspondence 

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]
    M.A. Vasiliev, More on equations of motion for interacting massless fields of all spins in (3 +I)-dimensions, Phys. Lett.B 285 (1992) 225 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    O.A. Gelfond and M.A. Vasiliev, Homotopy operators and locality theorems in higher-spin equations, Phys. Lett.B 786 (2018) 180 [arXiv:1805.11941] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    V.E. Didenko, O.A. Gelfond, A.V. Korybut and M.A. Vasiliev, Homotopy properties and lower-order vertices in higher-spin equations, J. Phys.A 51 (2018) 465202 [arXiv:1807.00001] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    M.A. Vasiliev, Equations of motion of interacting massless fields of all spins as a free differential algebra, Phys. Lett.B 209 (1988) 491 [INSPIRE].
  5. [5]
    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
  6. [6]
    M.A. Vasiliev, Dynamics of massless higher spins in the second order in curvatures, Phys. Lett.B 238 (1990) 305 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    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] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    M.A. Vasiliev, Star-product functions in higher-spin theory and locality, JHEP06 (2015) 031 [arXiv:1502.02271] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    O.A. Gelfond and M.A. Vasiliev, Spin-locality of higher-spin theories and star-product functional classes, arXiv:1910.00487 [INSPIRE].
  10. [10]
    A.K.H. Bengtsson, I. Bengtsson and L. Brink, Cubic interaction terms for arbitrarily extended supermultiplets, Nucl. Phys.B 227 (1983) 41 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    F.A. Berends, G.J.H. Burgers and H. Van Dam, On spin three selfinteractions, Z. Phys.C 24 (1984) 247 [INSPIRE].ADSGoogle Scholar
  12. [12]
    F.A. Berends, G.J.H. Burgers and H. van Dam, On the theoretical problems in constructing interactions involving higher spin massless particles, Nucl. Phys.B 260 (1985) 295 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    R.R. Metsaev, Poincaré invariant dynamics of massless higher spins: fourth order analysis on mass shell, Mod. Phys. Lett.A 6 (1991) 359 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    R.R. Metsaev, S matrix approach to massless higher spins theory. 2: the case of internal symmetry, Mod. Phys. Lett.A 6 (1991) 2411 [INSPIRE].
  15. [15]
    E.S. Fradkin and M.A. Vasiliev, Candidate to the role of higher spin symmetry, Annals Phys.177 (1987) 63 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    M.A. Vasiliev, Extended higher spin superalgebras and their realizations in terms of quantum operators, Fortsch. Phys.36 (1988) 33 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    S.E. Konstein and M.A. Vasiliev, Extended higher spin superalgebras and their massless representations, Nucl. Phys.B 331 (1990) 475 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    R.R. Metsaev, Light-cone gauge cubic interaction vertices for massless fields in AdS 4, Nucl. Phys.B 936 (2018) 320 [arXiv:1807.07542] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    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].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    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].
  21. [21]
    X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, Quartic AdS interactions in higher-spin gravity from conformal field theory, JHEP11 (2015) 149 [arXiv:1508.04292] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    C. Sleight and M. Taronna, Higher-spin gauge theories and bulk locality, Phys. Rev. Lett.121 (2018) 171604 [arXiv:1704.07859] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    D. Ponomarev, A note on (non)-locality in holographic higher spin theories, Universe4 (2018) 2 [arXiv:1710.00403] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    O. Aharony, G. Gur-Ari and R. Yacoby, d = 3 bosonic vector models coupled to Chern-Simons gauge theories, JHEP03 (2012) 037 [arXiv:1110.4382] [INSPIRE].
  25. [25]
    S. Giombi, S. Minwalla, S. Prakash, S.P. Trivedi, S.R. Wadia and X. Yin, Chern-Simons theory with vector fermion matter, Eur. Phys. J.C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    M.A. Vasiliev, Current interactions and holography from the 0-form sector of nonlinear higher-spin equations, JHEP10 (2017) 111 [arXiv:1605.02662] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    M.A. Vasiliev, On the local frame in nonlinear higher-spin equations, JHEP01 (2018) 062 [arXiv:1707.03735] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    O.A. Gelfond and M.A. Vasiliev, Current interactions from the one-form sector of nonlinear higher-spin equations, Nucl. Phys.B 931 (2018) 383 [arXiv:1706.03718] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    S. Giombi and X. Yin, The higher spin/vector model duality, J. Phys.A 46 (2013) 214003 [arXiv:1208.4036] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  30. [30]
    N. Boulanger, P. Kessel, E.D. Skvortsov and M. Taronna, Higher spin interactions in four-dimensions: Vasiliev versus Fronsdal, J. Phys.A 49 (2016) 095402 [arXiv:1508.04139] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    M.A. Vasiliev, Higher spin gauge theories: star product and AdS space, hep-th/9910096 [INSPIRE].
  32. [32]
    M.A. Vasiliev, Triangle identity and free differential algebra of massless higher spins, Nucl. Phys.B 324 (1989) 503 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys.A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  34. [34]
    O.A. Gelfond and M.A. Vasiliev, Unfolded equations for current interactions of 4d massless fields as a free system in mixed dimensions, J. Exp. Theor. Phys.120 (2015) 484 [arXiv:1012.3143] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    N. Misuna, On current contribution to Fronsdal equations, Phys. Lett.B 778 (2018) 71 [arXiv:1706.04605] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    D. De Filippi, C. Iazeolla and P. Sundell, Fronsdal fields from gauge functions in Vasiliev’s higher spin gravity, JHEP10 (2019) 215 [arXiv:1905.06325] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    R. Bonezzi, N. Boulanger, D. De Filippi and P. Sundell, Noncommutative Wilson lines in higher-spin theory and correlation functions of conserved currents for free conformal fields, J. Phys.A 50 (2017) 475401 [arXiv:1705.03928] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    C. Iazeolla, E. Sezgin and P. Sundell, On exact solutions and perturbative schemes in higher spin theory, Universe4 (2018) 5 [arXiv:1711.03550] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    F.A. Berezin and M.A. Shubin, Schrödinger equation, Moscow University Press, Moscow, Russia (1983).zbMATHGoogle Scholar
  40. [40]
    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].
  41. [41]
    S.F. Prokushkin and M.A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3D AdS space-time, Nucl. Phys.B 545 (1999) 385 [hep-th/9806236] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    M.A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A) dS d, Phys. Lett.B 567 (2003) 139 [hep-th/0304049] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    M.A. Vasiliev, From Coxeter higher-spin theories to strings and tensor models, JHEP08 (2018) 051 [arXiv:1804.06520] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    C. Iazeolla and P. Sundell, Families of exact solutions to Vasiliev’s 4D equations with spherical, cylindrical and biaxial symmetry, JHEP12 (2011) 084 [arXiv:1107.1217] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    C. Iazeolla and P. Sundell, Biaxially symmetric solutions to 4D higher-spin gravity, J. Phys.A 46 (2013) 214004 [arXiv:1208.4077] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    V.E. Didenko and M.A. Vasiliev, unpublished.Google Scholar
  47. [47]
    V.E. Didenko, N.G. Misuna and M.A. Vasiliev, Charges in nonlinear higher-spin theory, JHEP03 (2017) 164 [arXiv:1512.07626] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • V.E. Didenko
    • 1
    Email author
  • O.A. Gelfond
    • 1
    • 2
  • A.V. Korybut
    • 1
  • M.A. Vasiliev
    • 1
  1. 1.TK Tamm Department of Theoretical PhysicsLebedev Physical InstituteMoscowRussia
  2. 2.Federal State Institution “Scientific Research Institute for System Analysis of the Russian Academy of Science”MoscowRussia

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