Higher-spin Chern-Simons theories in odd dimensions

TL;DR

The paper develops bosonic higher-spin gauge theories in odd dimensions using Chern-Simons actions based on infinite-dimensional extensions ho(D-1,2) of the AdS algebra SO(D-1,2). It introduces a symmetric invariant tensor and shows how the CS action naturally contains Lovelock gravity with dynamical torsion, enabling nontrivial HS couplings on suitable backgrounds. Focusing on D=5, the authors linearize around AdS4×S1 and demonstrate that spin-3 dynamics reduce, in the HS torsion-free sector, to the compensator form of the Fronsdal equations on AdS4. This work establishes a concrete, gauge-invariant action principle for interacting HS fields in odd dimensions and opens avenues for exploring HS dynamics on non-maximally symmetric backgrounds and their potential connections to broader theories like M-theory.

Abstract

We construct consistent bosonic higher-spin gauge theories in odd dimensions D>3 based on Chern-Simons forms. The gauge groups are infinite-dimensional higher-spin extensions of the Anti-de Sitter groups SO(D-1,2). We propose an invariant tensor on these algebras, which is required for the definition of the Chern-Simons action. The latter contains the purely gravitational Chern-Simons theories constructed by Chamseddine, and so the entire theory describes a consistent coupling of higher-spin fields to a particular form of Lovelock gravity. It contains topological as well as non-topological phases. Focusing on D=5 we consider as an example for the latter an AdS_4 x S^1 Kaluza-Klein background. By solving the higher-spin torsion constraints in the case of a spin-3 field, we verify explicitly that the equations of motion reduce in the linearization to the compensator form of the Fronsdal equations on AdS_4.