Topological terms, AdS_2n gravity and renormalized Entanglement Entropy of holographic CFTs

TL;DR

This paper introduces a topological renormalization scheme for holographic entanglement entropy in odd-dimensional CFTs, utilizing boundary Chern forms to cancel divergences and relate entropy to topological invariants.

Contribution

It extends the topological renormalization method to arbitrary odd dimensions in AdS/CFT, incorporating Chern forms as boundary counterterms for entanglement entropy.

Findings

The renormalized entropy can be expressed via Euler characteristic and AdS curvature.

The boundary Chern form cancels divergences in entanglement entropy.

The method applies to studying CFT parameters like central charges.

Abstract

We extend our topological renormalization scheme for Entanglement Entropy to holographic CFTs of arbitrary odd dimensions in the context of the AdS/CFT correspondence. The procedure consists in adding the Chern form as a boundary term to the area functional of the Ryu-Takayanagi minimal surface. The renormalized Entanglement Entropy thus obtained can be rewritten in terms of the Euler characteristic and the AdS curvature of the minimal surface. This prescription considers the use of the Replica Trick to express the renormalized Entanglement Entropy in terms of the renormalized gravitational action evaluated on the conically-singular replica manifold extended to the bulk. This renormalized action is obtained in turn by adding the Chern form as the counterterm at the boundary of the 2n-dimensional asymptotically AdS bulk manifold. We explicitly show that, up to next-to-leading order in the holographic radial coordinate, the addition of this boundary term cancels the divergent part of the Entanglement Entropy. We discuss possible applications of the method for studying CFT parameters like central charges.