Higher-spin Chern–Simons theories in odd dimensions

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 × 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 Frønsdal equations on AdS 4 .