Higher-dimensional Willmore energy as holographic entanglement entropy

TL;DR

This paper establishes a connection between the universal entanglement entropy term in 5D conformal field theories and a four-dimensional Willmore energy, providing a new geometric interpretation for holographic entanglement.

Contribution

It introduces a generalized Willmore energy functional for 5D holographic CFTs, linking entanglement entropy to conformal invariants and extending previous 3D results.

Findings

F(A) equals a 4D Willmore energy of the RT surface

The functional is UV finite and involves quartic extrinsic curvature terms

The round ball is not the global minimizer of F(A) in 5D CFTs

Abstract

The vacuum entanglement entropy of a general conformal field theory (CFT) in $d=5$ spacetime dimensions contains a universal term, $F(A)$, which has a complicated and non-local dependence on the geometric details of the region $A$ and the theory. Analogously to the previously known $d=3$ case, we prove that for CFTs in $d=5$ which are holographically dual to Einstein gravity, $F(A)$ is equal to a four-dimensional version of the ``Willmore energy'' associated to a doubled and closed version of the Ryu-Takayanagi (RT) surface of $A$ embedded in $\mathbb{R}^5$. This generalized Willmore energy is shown to arise from a conformal-invariant codimension-two functional obtained by evaluating six-dimensional Conformal Gravity on the conically-singular orbifold of the replica trick. The new functional involves an integral over the doubled RT surface of a linear combination of three quartic terms in extrinsic curvatures and is free from ultraviolet divergences by construction. We verify explicitly the validity of our new formula for various entangling regions and argue that, as opposed to the $d=3$ case, $F(A)$ is not globally minimized by a round ball $A=\mathbb{B}^4$. Rather, $F(A)$ can take arbitrarily positive and negative values as a function of $A$. Hence, we conclude that the round ball is not a global minimizer of $F(A)$ for general five-dimensional CFTs.