A note on covariant action integrals in three dimensions

TL;DR

The paper analyzes covariant action principles for 2+1 dimensional gravity with negative cosmological constant, focusing on the BTZ black hole. It shows that a covariant, finite regularized action yielding the canonical ensemble can be obtained in three dimensions, and that this action is precisely the three-dimensional Chern-Simons form without additional boundary terms. The authors connect the boundary contributions to the horizon entropy and the AdS3/CFT framework, deriving the correct thermodynamic quantities from the CS formulation. These results highlight a special simplicity in 3D, where the right action for black hole thermodynamics aligns with CS theory and has clear implications for holography and boundary dynamics.

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.