Regularization of odd-dimensional AdS gravity: Kounterterms

TL;DR

This work presents Kounterterms, boundary terms depending on the extrinsic curvature, as a universal regularization mechanism for AdS gravity in all odd dimensions. By fixing ALAdS-compatible boundary conditions, the authors obtain a well-posed action principle and finite Noether charges, which split into a mass/anglular-momentum part $q(\xi)$ and a vacuum-energy part $q_0(\xi)$, the latter encoding the AdS Casimir-like energy. They demonstrate explicit finite results in 5D and 7D, including Schwarzschild–AdS, Kerr–AdS, and Clarkson–Mann solitons, with the Euclidean action yielding the standard entropy $S=\text{Area}/(4G)$. The general odd-dimensional construction extends to all $D=2n+1$, with a fixed coupling $c_{2n}$ and a Noether-charge structure $Q(\xi)=q(\xi)+q_0(\xi)$, where $q(\xi)$ vanishes for global AdS and $q_0(\xi)$ provides the vacuum energy, supporting a background-independent, topologically-informed regularization framework that aligns with, and potentially informs, holographic renormalization.

Abstract

As an alternative to the Dirichlet counterterms prescription, I introduce the concept of Kounterterms as the boundary terms with explicit dependence on the extrinsic curvature K_{ij} that regularize the AdS gravity action. Instead of a Dirichlet boundary condition on the metric, a suitable choice of the boundary conditions --compatible with any asymptotically AdS (AAdS) spacetime-- ensures a finite action principle for all odd dimensions. Background-independent conserved quantities are obtained as Noether charges associated to asymptotic symmetries and their general expression appears naturally split in two parts. The first one gives the correct mass and angular momentum for AAdS black holes and vanishes identically for globally AdS spacetimes. Thus, the second part is a covariant formula for the vacuum energy in AAdS spacetimes and reproduces the results obtained by the Dirichlet counterterms method in a number of cases. It is also shown that this Kounterterms series regularizes the Euclidean action and recovers the correct black hole thermodynamics in odd dimensions.