Renormalized AdS gravity and holographic entanglement entropy of even-dimensional CFTs

TL;DR

This paper develops a holographic method to compute renormalized entanglement entropy for even-dimensional conformal field theories using Einstein gravity, incorporating Kounterterms and the replica trick, revealing universal relations to central charges.

Contribution

It introduces a novel renormalization approach for holographic entanglement entropy in even-dimensional CFTs using adapted Kounterterm techniques and conical singularity analysis.

Findings

Derived a general formula for renormalized entanglement entropy.

Showed the boundary counterterm decomposes at conical singularities.

Established the proportionality of entropy to the central charge a.

Abstract

We derive a general formula for renormalized entanglement entropy in even dimensional CFTs holographically dual to Einstein gravity in one dimension higher. In order to renormalize, we adapt the Kounterterm method to asymptotically locally AdS manifolds with conical singularities. On the gravity side, the computation considers extrinsic counterterms and the use of the replica trick a la Lewkowycz-Maldacena. The boundary counterterm B_d is shown to satisfy a key property, in direct analogy to the Euler density: when evaluated on a conically singular manifold, it decomposes into a regular part plus a codimension-2 version of itself located at the conical singularity. The renormalized entropy thus obtained is shown to correspond to the universal part of the holographic entanglement entropy, which for spherical entangling surfaces is proportional to the central charge a that is the subject of the a-theorem. We also review and elucidate various aspects of the Kounterterm approach, including in particular its full compatibility with the Dirichlet condition for the metric at the conformal boundary, that is of standard use in holography.