Einstein-AdS action, renormalized volume/area and holographic Renyi entropies

TL;DR

This paper establishes a connection between the renormalized volume in asymptotically AdS Einstein manifolds and the renormalized Euclidean gravity action, extending the concept to higher dimensions and linking it to holographic Renyi entropies.

Contribution

It introduces an explicit form for the renormalized volume in higher even-dimensional AAdS Einstein manifolds and relates it to holographic Renyi entropies via a new extremization principle.

Findings

Renormalized volume equals the renormalized Euclidean gravity action in 4 and 6 dimensions.

Renormalized Renyi entropy can be derived from the area of a cosmic brane with finite tension.

The approach generalizes to higher even dimensions and includes backreaction effects.

Abstract

We exhibit the equivalence between the renormalized volume of asymptotically anti-de Sitter (AAdS) Einstein manifolds in four and six dimensions, and their renormalized Euclidean bulk gravity actions. The action is that of Einstein gravity, where the renormalization is achieved through the addition of a single topological term. We generalize this equivalence, proposing an explicit form for the renormalized volume of higher even-dimensional AAdS Einstein manifolds. We also show that evaluating the renormalized bulk gravity action on the conically singular manifold of the replica trick results in an action principle that corresponds to the renormalized volume of the regular part of the bulk, plus the renormalized area of a codimension-2 cosmic brane whose tension is related to the replica index. Renormalized Renyi entropy of odd-dimensional holographic CFTs can thus be obtained from the renormalized area of the brane with finite tension, including the effects of its backreaction on the bulk geometry. The area computation corresponds to an extremization problem for an enclosing surface that extends to the AdS boundary,where the newly defined renormalized volume is considered.