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Tensor gauge fields in arbitrary representations of GL(D,R): II. Quadratic actions

Xavier Bekaert, Nicolas Boulanger

TL;DR

The paper provides explicit quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary $GL(D,\mathbb{R})$ irreps, written compactly with Levi-Civita tensors. It proves that the resulting field equations propagate the correct massless degrees of freedom and are equivalent to previously proposed local non-Lagrangian equations, unifying multiple higher-spin formalisms. A frame-like, MacDowell–Mansouri–style reformulation is established that does not require trace constraints in the tangent indices and reduces to the flat-space limit of a curved-background construction. For mixed-symmetry fields, the work completes the Bargmann–Wigner and Fierz–Pauli programmes in arbitrary dimensions, providing a universal covariant non-local action and demonstrating on-shell equivalence to known local formulations. Overall, the results connect metric-like and frame-like approaches, extend non-local higher-spin dynamics to arbitrary representations, and suggest avenues for curved-background generalizations and non-local non-Abelian extensions.

Abstract

Quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreducible representations of the general linear group in D-dimensional Minkowski space are explicitly written in a compact form by making use of Levi-Civita tensors. The field equations derived from these actions ensure the propagation of the correct massless physical degrees of freedom and are shown to be equivalent to non-Lagrangian local field equations proposed previously. Moreover, these actions allow a frame-like reformulation a la MacDowell-Mansouri, without any trace constraint in the tangent indices.

Tensor gauge fields in arbitrary representations of GL(D,R): II. Quadratic actions

TL;DR

The paper provides explicit quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreps, written compactly with Levi-Civita tensors. It proves that the resulting field equations propagate the correct massless degrees of freedom and are equivalent to previously proposed local non-Lagrangian equations, unifying multiple higher-spin formalisms. A frame-like, MacDowell–Mansouri–style reformulation is established that does not require trace constraints in the tangent indices and reduces to the flat-space limit of a curved-background construction. For mixed-symmetry fields, the work completes the Bargmann–Wigner and Fierz–Pauli programmes in arbitrary dimensions, providing a universal covariant non-local action and demonstrating on-shell equivalence to known local formulations. Overall, the results connect metric-like and frame-like approaches, extend non-local higher-spin dynamics to arbitrary representations, and suggest avenues for curved-background generalizations and non-local non-Abelian extensions.

Abstract

Quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreducible representations of the general linear group in D-dimensional Minkowski space are explicitly written in a compact form by making use of Levi-Civita tensors. The field equations derived from these actions ensure the propagation of the correct massless physical degrees of freedom and are shown to be equivalent to non-Lagrangian local field equations proposed previously. Moreover, these actions allow a frame-like reformulation a la MacDowell-Mansouri, without any trace constraint in the tangent indices.

Paper Structure

This paper contains 32 sections, 7 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Completely symmetric tensor gauge fields
  3. Bargmann--Wigner programme
  4. Local, constrained approach of Fronsdal
  5. Curvature tensors of de Wit, Freedman and Weinberg
  6. Non-local, unconstrained approach of Francia and Sagnotti
  7. Higher-derivative, unconstrained approach
  8. Fierz--Pauli programme
  9. Non-local actions of Francia and Sagnotti
  10. Non-local actions in terms of differential forms
  11. Non-local actions à la MacDowell and Mansouri
  12. Mixed-symmetry tensor gauge fields
  13. Bargmann--Wigner programme
  14. Local, constrained approach of Labastida
  15. Higher derivative, unconstrained approach
  16. ...and 17 more sections

Key Result

Lemma 1

Let $\cal P$ be a differential hyperform of $\Omega_{(s)}({\mathbb R}^D)\,$. Then,

Theorems & Definitions (7)

  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Proposition 2
  • Lemma 3