The action for higher spin black holes in three dimensions

TL;DR

The paper shows that in three dimensions, higher-spin black holes described by $SL(N,\mathbb{R})\times SL(N,\mathbb{R})$ Chern-Simons theory admit a well-defined on-shell action that yields sensible thermodynamics. By decomposing the gauge fields into constant time and angular components and imposing holonomy (regularity) conditions, the authors derive a canonical structure with conjugate pairs $(Q_n,\sigma_n)$, establish integrability of the holonomy relations, and construct both the canonical action $I_1$ and the microcanonical Legendre transform $I_2$. The explicit $N=3$ case demonstrates these features with a concrete holonomy solution and verifies the expected derivatives of the on-shell action with respect to the chemical potentials, reinforcing the link between CS thermodynamics and the underlying $W_N$ symmetry. Finally, the work clarifies how the CS formulation connects to the usual metric description through horizon regularity and boundary terms, providing a robust topological route to the thermodynamics of higher-spin black holes.

Abstract

In the context of (2+1)--dimensional Chern-Simons SL(N,R)\times SL(N,R) gauge fields and spin N black holes we compute the on-shell action and show that it generates sensible and consistent thermodynamics. In particular, the Chern-Simons action solves the integrability conditions recently considered in the literature.