Higher spin black hole entropy in three dimensions

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 $\varphi_{+}^{{{1 \left/ {3} \right.}}}$, 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 ${\varphi_{+}}$ has to be bounded from above according to ${{\left| {{\varphi_{+}}} \right|}^{{{1 \left/ {3} \right.}}}}\leq {A \left/ {{\sqrt{3}}} \right.}$. 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.