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

TL;DR

This work demonstrates that the conventional holographic renormalization of AdS gravity can be realized through topological regularization by including the Gauss–Bonnet invariant with a fixed coupling, leading to a MacDowell–Mansouri–type action equivalent to a Weyl-squared theory and reproducing standard boundary counterterms. It further shows that adding a Pontryagin term and enforcing asymptotic self-duality fixes its coupling and yields a holographic stress tensor/Cotton tensor duality, $T_j^{\,i}=\pm (\ell^2/(8\pi G)) C_j^{\,i}$, with a dual self-dual boundary action in terms of the Weyl 2-form. The analysis connects topological invariants to holographic regularization, clarifies the boundary stress tensor structure, and outlines a general method to generate counterterm series in any even dimension from Euler-type terms, highlighting a deep topological underpinning of AdS/CFT renormalization. Together, these results provide a cohesive perspective on how global topological features control finite boundary data and dual boundary dynamics in 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 $T_{j}^{i}=\pm (\ell ^{2}/8πG)C_{j}^{i}$, 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.