Topological regularization and self-duality in four-dimensional anti-de Sitter gravity

TL;DR

This paper demonstrates how adding topological invariants to four-dimensional AdS gravity regularizes the theory and relates the holographic stress tensor to the boundary Cotton tensor through self-duality conditions.

Contribution

It introduces a method to incorporate topological invariants in AdS gravity to achieve regularization and establishes a duality between the stress tensor and Cotton tensor via self-duality.

Findings

Regularization of AdS gravity via topological invariants

Relation between holographic stress tensor and boundary Cotton tensor

Procedure to generate counterterms in even-dimensional AdS gravity

Abstract

It is shown that the addition of a topological invariant (Gauss-Bonnet term) to the anti-de Sitter (AdS) gravity action in four dimensions recovers the standard regularization given by holographic renormalization procedure. This crucial step makes possible the inclusion of an odd parity invariant (Pontryagin term) whose coupling is fixed by demanding an asymptotic (anti) self-dual condition on the Weyl tensor. This argument allows to find the dual point of the theory where the holographic stress tensor is related to the boundary Cotton tensor as $T_{j}^{i}=\pm (\ell ^{2}/8\pi G)C_{j}^{i}$, which has been observed in recent literature in solitonic solutions and hydrodynamic models. A general procedure to generate the counterterm series for AdS gravity in any even dimension from the corresponding Euler term is also briefly discussed.