- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Irregular higher-spin generating equations and chiral perturbation theory

V. E. Didenko*

  • *Contact author: didenko@lpi.ru
PDF

Phys. Rev. D 114, 066001 – Published 1 September, 2026

DOI: https://doi.org/10.1103/fwp4-l6rz

Abstract

We present a complementary approach to the standard Vasiliev framework for nonlinear higher-spin interactions in four dimensions, aimed at identifying their minimally nonlocal form. Our proposal introduces a generating system for higher-spin vertices at the level of classical equations, which we refer to as irregular, in contrast to the regular case described by Vasiliev. This system extends the recently proposed equations for (anti)holomorphic interactions by incorporating the mixed sector. Its perturbative series encompasses the entire (anti)holomorphic sector in the leading order, with vertices related to powers of the complex parity-breaking parameter η or η¯. The subsequent corrections facilitate the mixing of the two sectors, with vertices carrying mixed powers of η and η¯. Consistency relies on the nonlinear algebraic constraint, which is shown to be satisfied, at least, in the quadratic and cubic approximations. As a result, the previously discussed (anti)holomorphic interactions in the literature can be systematically extended to generate vertices of the form ηNη¯k and their conjugate, at least for k≤2 and any N. As a byproduct of our analysis, we also identify the new higher-spin structure dualities.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (75)

  1. M. A. Vasiliev, Higher spin gauge theories: Star product and AdS space, arXiv:hep-th/9910096.
  2. X. Bekaert, S. Cnockaert, C. Iazeolla, and M. A. Vasiliev, Nonlinear higher spin theories in various dimensions, arXiv:hep-th/0503128.
  3. V. E. Didenko and E. D. Skvortsov, Elements of Vasiliev theory, Lect. Notes Phys. 1028, 269 (2024).
  4. M. A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions, Phys. Lett. B 243, 378 (1990).
  5. M. A. Vasiliev, More on equations of motion for interacting massless fields of all spins in (3+1)-dimensions, Phys. Lett. B 285, 225 (1992).
  6. N. Boulanger and P. Sundell, An action principle for Vasiliev’s four-dimensional higher-spin gravity, J. Phys. A 44, 495402 (2011).
  7. M. Flato and C. Fronsdal, One massless particle equals two Dirac singletons: Elementary particles in a curved space. 6., Lett. Math. Phys. 2, 421 (1978).
  8. E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B644, 303 (2002); B660, 403(E) (2003).
  9. I. R. Klebanov and A. M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550, 213 (2002).
  10. R. G. Leigh and A. C. Petkou, Holography of the N=1 higher spin theory on AdS(4), J. High Energy Phys. 06 (2003) 011.
  11. E. Sezgin and P. Sundell, Holography in 4D (super) higher spin theories and a test via cubic scalar couplings, J. High Energy Phys. 07 (2005) 044.
  12. S. Giombi and X. Yin, Higher spin gauge theory and holography: The three-point functions, J. High Energy Phys. 09 (2010) 115.
  13. S. Giombi and X. Yin, Higher spins in AdS and twistorial holography, J. High Energy Phys. 04 (2011) 086.
  14. V. E. Didenko and M. A. Vasiliev, Test of the local form of higher-spin equations via AdS/CFT, Phys. Lett. B 775, 352 (2017).
  15. J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46, 214011 (2013).
  16. J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, Classical Quantum Gravity 30, 104003 (2013).
  17. N. Colombo and P. Sundell, Higher spin gravity amplitudes from zero-form charges, arXiv:1208.3880.
  18. V. E. Didenko and E. D. Skvortsov, Exact higher-spin symmetry in CFT: All correlators in unbroken Vasiliev theory, J. High Energy Phys. 04 (2013) 158.
  19. O. A. Gelfond and M. A. Vasiliev, Operator algebra of free conformal currents via twistors, Nucl. Phys. B876, 871 (2013).
  20. J. Engquist and P. Sundell, Brane partons and singleton strings, Nucl. Phys. B752, 206 (2006).
  21. E. Sezgin and P. Sundell, An exact solution of 4-D higher-spin gauge theory, Nucl. Phys. B762, 1 (2007).
  22. N. Colombo and P. Sundell, Twistor space observables and quasi-amplitudes in 4D higher spin gravity, J. High Energy Phys. 11 (2011) 042.
  23. E. Sezgin and P. Sundell, Geometry and observables in Vasiliev’s higher spin gravity, J. High Energy Phys. 07 (2012) 121.
  24. A. Scalea, On correlation functions as higher-spin invariants, Symmetry 15, 950 (2023).
  25. C. Sleight and M. Taronna, Higher spin interactions from conformal field theory: The complete cubic couplings, Phys. Rev. Lett. 116, 181602 (2016).
  26. S. F. Prokushkin and M. A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3-D AdS space-time, Nucl. Phys. B545, 385 (1999).
  27. N. Boulanger, P. Kessel, E. D. Skvortsov, and M. Taronna, Higher spin interactions in four-dimensions: Vasiliev versus Fronsdal, J. Phys. A 49, 095402 (2016).
  28. X. Bekaert, J. Erdmenger, D. Ponomarev, and C. Sleight, Quartic AdS interactions in higher-spin gravity from conformal field theory, J. High Energy Phys. 11 (2015) 149.
  29. C. Sleight and M. Taronna, Higher-spin gauge theories and bulk locality, Phys. Rev. Lett. 121, 171604 (2018).
  30. D. Ponomarev, A note on (non)-locality in holographic higher spin theories, Universe 4, 2 (2018).
  31. Y. Neiman, Quartic locality of higher-spin gravity in de Sitter and Euclidean anti-de Sitter space, Phys. Lett. B 843, 138048 (2023).
  32. M. A. Vasiliev, Holography, unfolding and higher-spin theory, J. Phys. A 46, 214013 (2013).
  33. F. Diaz, C. Iazeolla, and P. Sundell, Fractional spins, unfolding, and holography. Part I. Parent field equations for dual higher-spin gravity reductions, J. High Energy Phys. 09 (2024) 109.
  34. F. Diaz, C. Iazeolla, and P. Sundell, Fractional spins, unfolding, and holography. Part II. 4D higher spin gravity and 3D conformal dual, J. High Energy Phys. 10 (2024) 066.
  35. V. Lysov and Y. Neiman, Higher-spin gravity’s “string”: New gauge and proof of holographic duality for the linearized Didenko-Vasiliev solution, J. High Energy Phys. 10 (2022) 054.
  36. Y. Neiman, New diagrammatic framework for higher-spin gravity, Phys. Rev. Lett. 130, 171601 (2023).
  37. M. A. Vasiliev, Current interactions and holography from the 0-form sector of nonlinear higher-spin equations, J. High Energy Phys. 10 (2017) 111.
  38. M. A. Vasiliev, On the local frame in nonlinear higher-spin equations, J. High Energy Phys. 01 (2018) 062.
  39. O. A. Gelfond and M. A. Vasiliev, Homotopy operators and locality theorems in higher-spin equations, Phys. Lett. B 786, 180 (2018).
  40. 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
  41. V. E. Didenko, O. A. Gelfond, A. V. Korybut, and M. A. Vasiliev, Limiting shifted homotopy in higher-spin theory and spin-locality, J. High Energy Phys. 12 (2019) 086.
  42. O. A. Gelfond and M. A. Vasiliev, Spin-locality of higher-spin theories and star-product functional classes, J. High Energy Phys. 03 (2020) 002.
  43. V. E. Didenko, O. A. Gelfond, A. V. Korybut, and M. A. Vasiliev, Spin-locality of η2 and η¯2 quartic higher-spin vertices, J. High Energy Phys. 12 (2020) 184.
  44. O. A. Gelfond and A. V. Korybut, Manifest form of the spin-local higher-spin vertex ϒωCCCηη, Eur. Phys. J. C 81, 605 (2021).
  45. M. A. Vasiliev, Projectively-compact spinor vertices and space-time spin-locality in higher-spin theory, Phys. Lett. B 834, 137401 (2022).
  46. V. E. Didenko and A. V. Korybut, On z-dominance, shift symmetry and spin locality in higher-spin theory, J. High Energy Phys. 05 (2023) 133.
  47. M. A. Vasiliev, Differential contracting homotopy in higher-spin theory, J. High Energy Phys. 11 (2023) 048.
  48. P. T. Kirakosiants, D. A. Valerev, and M. A. Vasiliev, Quadratic corrections to the higher-spin equations by the differential homotopy approach, Nucl. Phys. B1023, 117290 (2026).
  49. V. E. Didenko, On holomorphic sector of higher-spin theory, J. High Energy Phys. 10 (2022) 191.
  50. A. V. Korybut, On consistency of the interacting (anti)holomorphic higher-spin sector, Eur. Phys. J. C 85, 885 (2025).
  51. V. E. Didenko and M. A. Povarnin, All vertices for unconstrained symmetric gauge fields, Phys. Rev. D 110, 126012 (2024); 111, 109901(E) (2025).
  52. M. A. Vasiliev, Consistent equations for interacting massless fields of all spins in the first order in curvatures, Ann. Phys. (N.Y.) 190, 59 (1989).
  53. N. Misuna, Unfolded dynamics approach and quantum field theory, J. High Energy Phys. 12 (2023) 119.
  54. N. Misuna, Scalar electrodynamics and Higgs mechanism in the unfolded dynamics approach, J. High Energy Phys. 12 (2024) 090.
  55. N. Misuna, Unfolded formulation of 4d Yang-Mills theory, Phys. Lett. B 870, 139882 (2025).
  56. C. Iazeolla, P. Sundell, and B. C. Vallilo, Unfolding the six-dimensional tensor multiplet, J. Phys. A 58, 365402 (2025).
  57. D. De Filippi, C. Iazeolla, and P. Sundell, Metaplectic representation and ordering (in)dependence in Vasiliev’s higher spin gravity, J. High Energy Phys. 07 (2022) 003.
  58. V. E. Didenko and A. V. Korybut, Interaction of symmetric higher-spin gauge fields, Phys. Rev. D 108, 086031 (2023); 109, 069901(E) (2024).
  59. V. E. Didenko and A. V. Korybut, Planar solutions of higher-spin theory. Nonlinear corrections, J. High Energy Phys. 01 (2022) 125.
  60. V. E. Didenko and I. S. Faliakhov, Symmetry breaking in the self-dual higher-spin theory, Phys. Rev. D 112, 106010 (2025).
  61. M. A. Vasiliev, Star-product functions in higher-spin theory and locality, J. High Energy Phys. 06 (2015) 031.
  62. R. R. Metsaev, Poincare invariant dynamics of massless higher spins: Fourth order analysis on mass shell, Mod. Phys. Lett. A 06, 359 (1991).
  63. R. R. Metsaev, Effective action in string theory, Ph.D. thesis, Lebedev Physical Institute, 1991.
  64. R. R. Metsaev, Light-cone gauge cubic interaction vertices for massless fields in AdS(4), Nucl. Phys. B936, 320 (2018).
  65. D. Ponomarev, Chiral higher spin theories and self-duality, J. High Energy Phys. 12 (2017) 141.
  66. M. Serrani, On classification of (self-dual) higher-spin gravities in flat space, J. High Energy Phys. 08 (2025) 032.
  67. T. Basile, Massless chiral fields in six dimensions, SciPost Phys. 19, 079 (2025).
  68. A. Sharapov, E. Skvortsov, A. Sukhanov, and R. Van Dongen, More on chiral higher spin gravity and convex geometry, Nucl. Phys. B990, 116152 (2023).
  69. A. Sharapov, E. Skvortsov, and R. Van Dongen, Strong homotopy algebras for chiral higher spin gravity via Stokes theorem, J. High Energy Phys. 06 (2024) 186.
  70. O. A. Gelfond, Moderately non-local ηη¯ vertices in the AdS4 higher-spin gauge theory, Eur. Phys. J. C 83, 1154 (2023).
  71. J. Engquist, E. Sezgin, and P. Sundell, On N=1, N=2, N=4 higher spin gauge theories in four-dimensions, Classical Quantum Gravity 19, 6175 (2002).
  72. J. Engquist, E. Sezgin, and P. Sundell, Superspace formulation of 4-D higher spin gauge theory, Nucl. Phys. B664, 439 (2003).
  73. M. A. Vasiliev, From Coxeter higher-spin theories to strings and tensor models, J. High Energy Phys. 08 (2018) 051.
  74. A. A. Tarusov, K. A. Ushakov, and M. A. Vasiliev, Linearized Coxeter higher-spin theories, J. High Energy Phys. 08 (2025) 052.
  75. https://rscf.ru/project/26-12-00240/.

Outline

Information

Phys. Rev. D 114, 066001– Published 1 September, 2026
  • Received 3 March 2026
  • Revised 9 June 2026
  • Accepted 6 August 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation