Towards metric-like higher-spin gauge theories in three dimensions

TL;DR

This work establishes a concrete metric-like action for a symmetric spin-3 field coupled to 3D gravity as the quadratic truncation of the frame-like SL(3,R) × SL(3,R) Chern-Simons theory and verifies it by mapping from frame-like variables and by recovering Wald entropy for higher-spin black holes. It shows that in D=3 there exists a unique, two-derivative coupling between gravity and a spin-3 field at leading order, related to the CS frame-like theory, and provides explicit gauge transformations and their closure on-shell. The results offer a clear bridge between frame-like and metric-like formulations in 3D higher-spin gravity and illuminate entropy corrections for higher-spin black holes, while highlighting tensions with some earlier entropy proposals and outlining paths for extending the framework. The work thus advances the geometric understanding of higher-spin interactions in three dimensions and sets the stage for broader SL(N) generalizations and deeper explorations of metric-like nonlinearities.

Abstract

We consider the coupling of a symmetric spin-3 gauge field to three-dimensional gravity in a second order metric-like formulation. The action that corresponds to an SL(3,R) x SL(3,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.