A note on covariant action integrals in three dimensions

TL;DR

This paper calculates the partition function for the 2+1 black hole using a covariant action approach, clarifies thermodynamic ensemble definitions, and discusses the relation to conformal field theory.

Contribution

It demonstrates that the covariant action equals the Chern-Simons action without boundary terms and correctly reproduces thermodynamic properties.

Findings

The covariant action matches the Chern-Simons action.

The action is finite and yields correct free energy.

Extremum occurs at fixed temperature and angular velocity.

Abstract

We compute -in the saddle point approximation- the partition function for the 2+1 black hole using the Gibbons-Hawking approach. Some issues concerning the definition of thermodynamical ensembles are clarified. It is pointed out that the right action in covariant form is exactly equal to the Chern-Simons action with no added boundary terms. This action is finite, yields the right canonical free energy and has an extremum when the temperature and angular velocity are fixed. The correspondence with a 1+1 $CFT$ is indicated.