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Consistent couplings between spin-2 and spin-3 massless fields

Nicolas Boulanger, Serge Leclercq

TL;DR

This work uses BRST–BV cohomology to classify and construct first-order non-Abelian couplings between massless spin-2 and spin-3 fields in flat $n>3$ dimensions, without restricting the derivative order. It identifies two parity-invariant algebra-deforming candidates at leading order: a three-derivative (3-2-2) vertex and a four-derivative (2-3-3) vertex; second-order analysis shows the former is obstructed while the latter remains consistent under certain internal-symmetry conditions and can coexist with spin-3 self-coupling. Explicit cubic vertices are derived for both cases, with the 3-2-2 vertex reproducing known structures (Berends–Burgers–van Dam) and the 2-3-3 vertex presenting a new, consistent coupling. The results reinforce that flat-space higher-spin interactions are highly constrained and hint at the necessity of additional fields or curved backgrounds (e.g., AdS) to realize a complete nonlinear theory, potentially connected to the flat limit of Vasiliev-type higher-spin algebras.

Abstract

We solve the problem of constructing consistent first-order cross-interactions between spin-2 and spin-3 massless fields in flat spacetime of arbitrary dimension n > 3 and in such a way that the deformed gauge algebra is non-Abelian. No assumptions are made on the number of derivatives involved in the Lagrangian, except that it should be finite. Together with locality, we also impose manifest Poincare invariance, parity invariance and analyticity of the deformations in the coupling constants.

Consistent couplings between spin-2 and spin-3 massless fields

TL;DR

This work uses BRST–BV cohomology to classify and construct first-order non-Abelian couplings between massless spin-2 and spin-3 fields in flat dimensions, without restricting the derivative order. It identifies two parity-invariant algebra-deforming candidates at leading order: a three-derivative (3-2-2) vertex and a four-derivative (2-3-3) vertex; second-order analysis shows the former is obstructed while the latter remains consistent under certain internal-symmetry conditions and can coexist with spin-3 self-coupling. Explicit cubic vertices are derived for both cases, with the 3-2-2 vertex reproducing known structures (Berends–Burgers–van Dam) and the 2-3-3 vertex presenting a new, consistent coupling. The results reinforce that flat-space higher-spin interactions are highly constrained and hint at the necessity of additional fields or curved backgrounds (e.g., AdS) to realize a complete nonlinear theory, potentially connected to the flat limit of Vasiliev-type higher-spin algebras.

Abstract

We solve the problem of constructing consistent first-order cross-interactions between spin-2 and spin-3 massless fields in flat spacetime of arbitrary dimension n > 3 and in such a way that the deformed gauge algebra is non-Abelian. No assumptions are made on the number of derivatives involved in the Lagrangian, except that it should be finite. Together with locality, we also impose manifest Poincare invariance, parity invariance and analyticity of the deformations in the coupling constants.

Paper Structure

This paper contains 24 sections, 6 theorems, 68 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Free theory and BRST settings
  3. Free Theory
  4. BRST spectrum and differential
  5. BRST deformations
  6. Cohomological results
  7. Cohomology of $\gamma$
  8. Invariant Poincaré lemma
  9. Cohomology of $\delta$ modulo $d\,$: $H^n_k(\delta \vert\, d)$
  10. Definition of the $D$-degree
  11. Definition of the $D$-degree
  12. First-order consistent deformations
  13. Equations
  14. Classification of the gauge algebras
  15. Computation of the gauge transformations
  16. ...and 9 more sections

Key Result

Proposition 1

The cohomology of $\gamma$ is isomorphic to the space of functions depending on Thus, identifying with zero any $\gamma$-exact term in $H(\gamma)$, we have if and only if where $\{F^A_{\mu\nu\rho}\}$ stands for the completely symmetrized derivatives $\partial^{}_{(\sigma_1}\ldots\partial^{}_{\sigma_k}F^A_{\mu\nu\rho)}$ of the Fronsdal tensor, while $\widehat{T}^A_{\rho\mu\vert\nu}$ denotes the

Theorems & Definitions (6)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6