On shape dependence of holographic entanglement entropy in AdS4/CFT3

TL;DR

This paper investigates how the shape of regions affects the finite part of holographic entanglement entropy in AdS4/CFT3, providing analytic formulas and numerical checks for various geometries and backgrounds.

Contribution

It derives explicit analytic expressions for the finite entanglement entropy term depending on shape and normal vectors, including for stationary and time-dependent spacetimes in AdS4.

Findings

Finite term equals the Willmore energy for AdS4.

Analytic formulas for minimal surfaces in various backgrounds.

Numerical validation for elliptical and non-convex regions.

Abstract

We study the finite term of the holographic entanglement entropy of finite domains with smooth shapes and for four dimensional gravitational backgrounds. Analytic expressions depending on the unit vectors normal to the minimal area surface are obtained for both stationary and time dependent spacetimes. The special cases of AdS4, asymptotically AdS4 black holes, domain wall geometries and Vaidya-AdS backgrounds have been analysed explicitly. When the bulk spacetime is AdS4, the finite term is the Willmore energy of the minimal area surface viewed as a submanifold of the three dimensional flat Euclidean space. For the static spacetimes, some numerical checks involving spatial regions delimited by ellipses and non convex domains have been performed. In the case of AdS4, the infinite wedge has been also considered, recovering the known analytic formula for the coefficient of the logarithmic divergence.