Surface Terms as Counterterms in the AdS/CFT Correspondence

TL;DR

The paper develops and applies a covariant boundary counterterm framework for regulating the gravitational action in asymptotically AdS spacetimes, extending the construction to d≤7. It demonstrates how this holographic renormalization approach yields finite actions, energies, and entropies for a variety of solutions, including AdS black holes and Taub–NUT/Bolt–AdS, without relying on background subtraction. It reveals a rich interplay between bulk geometry and dual CFT data, including Casimir energies and conformal anomalies, and discusses limitations arising from logarithmic divergences in even dimensions. The results illuminate how different boundary geometries and topologies map to distinct field-theory backgrounds and phases, and they connect gravitational thermodynamics to dual QFT insights, with implications for precursor states and strongly coupled dynamics on curved spaces.

Abstract

We examine the recently proposed technique of adding boundary counterterms to the gravitational action for spacetimes which are locally asymptotic to anti-de Sitter. In particular, we explicitly identify higher order counterterms, which allow us to consider spacetimes of dimensions d<=7. As the counterterms eliminate the need of ``background subtraction'' in calculating the action, we apply this technique to study examples where the appropriate background was ambiguous or unknown: topological black holes, Taub-NUT-AdS and Taub-Bolt-AdS. We also identify certain cases where the covariant counterterms fail to render the action finite, and we comment on the dual field theory interpretation of this result. In some examples, the case of vanishing cosmological constant may be recovered in a limit, which allows us to check results and resolve ambiguities in certain asymptotically flat spacetime computations in the literature.