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Cubic-interaction-induced deformations of higher-spin symmetries

Euihun Joung, Massimo Taronna

TL;DR

The paper systematically analyzes how cubic interactions among symmetric massless higher-spin fields deform gauge transformations and gauge algebras within a metric-like framework, clarifying distinctions between AdS and flat-space backgrounds. It introduces H-couplings, which do not deform gauge transformations, and distinguishes G-couplings that can induce deformations, connecting these to currents and HS curvatures. A four-class classification (I–IV) organizes couplings by their deformation behavior, with Class III/IV capturing gravitational HS interactions in AdS and the Aragone–Deser/Fradkin–Vasiliev mechanism. The work also connects local deformations to global HS algebras via Killing tensors and invariant bilinear forms, and provides explicit examples (e.g., 3–3–2 and 4–4–2 vertices) to illustrate the interplay between derivative counting, AdS dependence, and Jacobi-consistency constraints.

Abstract

The deformations of higher-spin symmetries induced by cubic interactions of symmetric massless bosonic fields are analyzed within the metric-like formalism. Our analysis amends the existing classification according to gauge-algebra deformations taking into account also gauge-transformation deformations. In particular, we identify a class of couplings which leave the gauge algebra Abelian but deform one (out of three) gauge transformation, and another class of couplings which deform all three gauge transformations in (A)dS but only two in the flat-space limit. The former class is related to higher-spin algebra multiplets (representations of the global algebra) together with the massless-massive-massive couplings, which we also briefly discuss. The latter class is what makes (A)dS a distinguished background for higher-spin interactions and includes in particular the gravitational interactions of higher-spin fields, retrospectively accounting for the Fradkin-Vasiliev solution to the Aragon-Deser problem. We also study the restriction of gauge symmetries to global symmetries (higher-spin algebra) discussing the invariant bilinear form and the cyclicity of the structure constants. A possible generalization of the analysis to partially-massless fields is also commented.

Cubic-interaction-induced deformations of higher-spin symmetries

TL;DR

The paper systematically analyzes how cubic interactions among symmetric massless higher-spin fields deform gauge transformations and gauge algebras within a metric-like framework, clarifying distinctions between AdS and flat-space backgrounds. It introduces H-couplings, which do not deform gauge transformations, and distinguishes G-couplings that can induce deformations, connecting these to currents and HS curvatures. A four-class classification (I–IV) organizes couplings by their deformation behavior, with Class III/IV capturing gravitational HS interactions in AdS and the Aragone–Deser/Fradkin–Vasiliev mechanism. The work also connects local deformations to global HS algebras via Killing tensors and invariant bilinear forms, and provides explicit examples (e.g., 3–3–2 and 4–4–2 vertices) to illustrate the interplay between derivative counting, AdS dependence, and Jacobi-consistency constraints.

Abstract

The deformations of higher-spin symmetries induced by cubic interactions of symmetric massless bosonic fields are analyzed within the metric-like formalism. Our analysis amends the existing classification according to gauge-algebra deformations taking into account also gauge-transformation deformations. In particular, we identify a class of couplings which leave the gauge algebra Abelian but deform one (out of three) gauge transformation, and another class of couplings which deform all three gauge transformations in (A)dS but only two in the flat-space limit. The former class is related to higher-spin algebra multiplets (representations of the global algebra) together with the massless-massive-massive couplings, which we also briefly discuss. The latter class is what makes (A)dS a distinguished background for higher-spin interactions and includes in particular the gravitational interactions of higher-spin fields, retrospectively accounting for the Fradkin-Vasiliev solution to the Aragon-Deser problem. We also study the restriction of gauge symmetries to global symmetries (higher-spin algebra) discussing the invariant bilinear form and the cyclicity of the structure constants. A possible generalization of the analysis to partially-massless fields is also commented.

Paper Structure

This paper contains 28 sections, 171 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. General procedure
  3. Summary of our results
  4. Organization of paper
  5. Review: cubic interactions
  6. Deformations of gauge transformations and Non-deforming couplings
  7. Deformations of gauge transformations
  8. Non-deforming couplings
  9. Flat-space case
  10. (A)dS case
  11. Deformations of gauge algebras and abelian couplings
  12. Deformations of gauge algebras
  13. Abelian interactions
  14. Flat-space case
  15. (A)dS case
  16. ...and 13 more sections

Figures (1)

  • Figure 1: Schematic diagram of the relation between $H$- and $G$-couplings