Summary: We consider the coupling of a symmetric spin-3 gauge field \(\varphi_{\mu \nu \rho}\) to three-dimensional gravity in a second-order metric-like formulation. The action that corresponds to an \(\mathrm{SL}(3,\mathbb R) \times \mathrm{SL}(3,\mathbb R)\) Chern-Simons theory in the frame-like formulation is identified to quadratic order in the spin-3 field. We apply our result to compute corrections to the area law for higher spin black holes using Wald’s entropy formula.
This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.