Finite action principle for Chern-Simons AdS gravity

TL;DR

This paper develops a finite, background-independent action principle for Chern-Simons AdS gravity in all odd dimensions by imposing boundary conditions that fix the extrinsic curvature, leading to a local boundary term $B_{2n}$ that renders the action finite on asymptotically AdS spacetimes. The five-dimensional case is worked out explicitly, with the boundary term $B_4$ and a first-order formalism, then generalized to $d=2n+1$ dimensions via a CS-like Lagrangian $I_{2n+1}$ and a compact integral representation involving $R_t^{ab}$. The Euclidean action reproduces the canonical-ensemble black hole thermodynamics, yielding finite expressions for mass and entropy, while conserved charges associated with asymptotic symmetries are obtained directly as surface integrals through Noether’s theorem. The construction also reveals a topology-dependent zero-point energy that can be interpreted as the Casimir energy of the dual CFT, connected to Weyl anomalies, and it opens avenues for comparison with other charge definitions and extensions to other gravity theories. The approach emphasizes the transgression-form structure as a key to the favorable finiteness and variational properties observed.

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.