Higher spin black hole entropy in three dimensions

TL;DR

The paper derives a gauge-invariant, nonperturbative entropy formula for three-dimensional black holes carrying a spin-3 field within SL(3,R)×SL(3,R) Chern-Simons gravity, expressed in terms of the horizon area A and the spin-3 horizon analogue φ_+ via S = (A/(4G)) cos((1/3) arcsin(3^(3/2) φ_+ / A^3)). This approach uses the first law in the canonical ensemble to compute δE from a surface integral, avoiding explicit higher-spin charge computations. The authors demonstrate consistency with known results in specific solutions (GKMP and CM), show perturbative agreement with CFPT, and discuss bounds on φ_+ and the multivalued nature of the entropy, as well as potential generalizations to higher spins and boundary duals. The work clarifies how higher-spin hair modifies black hole thermodynamics and reinforces the role of horizon data in defining entropy beyond the Bekenstein-Hawking area law.

Abstract

A generic formula for the entropy of three-dimensional black holes endowed with a spin-3 field is found, which depends on the horizon area A and its spin-3 analogue, given by the reparametrization invariant integral of the induced spin-3 field at the spacelike section of the horizon. From this result it can be shown that the absolute value of the spin-3 analogue of the area has to be bounded from above by A/3^(1/2). The entropy formula is constructed by requiring the first law of thermodynamics to be fulfilled in terms of the global charges obtained through the canonical formalism. For the static case, in the weak spin-3 field limit, our expression for the entropy reduces to the result found by Campoleoni, Fredenhagen, Pfenninger and Theisen, which has been recently obtained through a different approach.