The covariant action of higher spin black holes in three dimensions

TL;DR

This paper develops boundary terms for higher spin theories in AdS3, enabling a covariant, on-shell action approach to compute black hole thermodynamics and explore boundary $ ext{W}$-gravity.

Contribution

It introduces boundary terms compatible with Dirichlet conditions in higher spin AdS3 theories, facilitating thermodynamic calculations of higher spin black holes.

Findings

Successfully reproduces the thermodynamics of a known higher spin black hole.

Provides a covariant Euclidean action framework for higher spin black hole analysis.

Comments on the boundary induced $ ext{W}$-gravity theory.

Abstract

We propose a set of boundary terms for higher spin theories in AdS$_3$ that lead to a well-defined variational principle compatible with Dirichlet boundary conditions for the metric and higher spin fields. These boundary terms are valid for higher spin theories in the Fefferman-Graham gauge and they allow us to compute the canonical free energy of higher spin black holes directly from the Euclidean, covariant, on-shell action. Using these results we reproduce the thermodynamics of the higher spin black hole of Ammon, Gutperle, Kraus, and Perlmutter and comment on the corresponding theory of induced $\cal{W}$-gravity at the boundary.