Topological aspects of generalized gravitational entropy

TL;DR

This paper explores the topological conditions under which the holographic entanglement entropy's homology constraint is satisfied, analyzing the role of branched covers in the gravitational entropy framework.

Contribution

It proves that the homology constraint holds if and only if the branched cover exists for all positive integers q, clarifying the topological requirements of the gravitational entropy method.

Findings

Homology can be violated if the branched cover exists only for some q.

The homology constraint is equivalent to the existence of branched covers for all q.

Examples illustrate potential violations of the homology constraint.

Abstract

The holographic prescription for computing entanglement entropy requires that the bulk extremal surface, whose area encodes the amount of entanglement, satisfies a homology constraint. Usually this is stated as the requirement of a (spacelike) interpolating surface that connects the region of interest and the extremal surface. We investigate to what extent this constraint is upheld by the generalized gravitational entropy argument, which relies on constructing replica symmetric q-fold covering spaces of the bulk, branched at the extremal surface. We prove (at the level of topology) that the putative extremal surface satisfies the homology constraint if and only if the corresponding branched cover can be constructed for every positive integer q. We give simple examples to show that homology can be violated if the cover exists for some values of q but not others, along with some other issues.