Finite action principle for Chern-Simons AdS gravity

TL;DR

This paper develops a finite action principle for Chern-Simons AdS gravity, ensuring well-defined variational principles and conserved charges in odd dimensions, and correctly reproduces black hole thermodynamics.

Contribution

It introduces a boundary term that makes the action finite and well-defined for asymptotically AdS solutions in odd-dimensional Chern-Simons gravity, without background fields.

Findings

Boundary term renders the action finite for asymptotically AdS solutions.

Correctly reproduces black hole thermodynamics in Euclidean formulation.

Defines conserved charges as surface integrals via Noether's theorem.

Abstract

A finite action principle for Chern-Simons AdS gravity is presented. The construction is carried out in detail first in five dimensions, where the bulk action is given by a particular combination of the Einstein-Hilbert action with negative cosmological constant and a Gauss-Bonnet term; and is then generalized for arbitrary odd dimensions. The boundary term needed to render the action finite is singled out demanding the action to attain an extremum for an appropriate set of boundary conditions. The boundary term is a local function of the fields at the boundary and is sufficient to render the action finite for asymptotically AdS solutions, without requiring background fields. It is shown that the Euclidean continuation of the action correctly describes the black hole thermodynamics in the canonical ensemble. Additionally, background independent conserved charges associated with the asymptotic symmetries can be written as surface integrals by direct application of Noether's theorem.