Higher-dimensional Willmore energy as holographic entanglement entropy

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.

Figures (5)

• Figure 1: Ball-shaped entangling region $A$ with radius $\rho=R$ and its cobordant ($\partial A=\partial \Sigma$) codimension-two surface $\Sigma$.
• Figure 2: Double-copied surfaces $\Sigma$ and $\Sigma'$ glued along the umbilical line defined by $\partial \Sigma=\partial \Sigma'$.
• Figure 3: Infinite strip entangling region $A$ of width $l$ in the $u$ direction. We introduced $L_i$ as IR regulators for the transversal directions $x_i$. The surface $\Sigma$ is cobordant ($\partial A=\partial \Sigma$), where $\partial \Sigma$ is composed by two parallel boundaries located at $u=\pm l/2$, i.e., $\partial \Sigma=\partial \Sigma^{l/2}\cup\partial \Sigma^{-l/2}$.
• Figure 4: Double-copied surfaces $\Sigma$ and $\Sigma'$ glued along the umbilical line defined by $\partial \Sigma_1=\partial \Sigma_1'$ located at $u=\pm l/2$. The introduction of the IR regulators $L_i$ defines a pair of symmetric boundaries $\partial \Sigma_2$ and $\partial \Sigma_2'$ pointing upwards and downwards along the transverse directions $x_i$.
• Figure 5: Linear-logarithmic plot of $\mathbf W_5\left(\Sigma_{a}^{(i)}\right)/ \mathbf W_5\left(\mathbb S^4\right)$ with $i=1,2$, representing the $(1,1,1,1,a)$ (blue line) and $(1,1,1,a,a)$ (red line) ellipsoids respectively for different values of $a$. The values represented for the former are analytical while the latter are numerical. The red region for small values of $a$ indicates that the $(1,1,1,a,a)$ ellipsoid oscillates wildly between $-\infty$ and $+\infty$ in this regime. We also include the point $a=1$ corresponding to a round sphere $\mathbb S^4$ and represented by an eight-pointed star. As we can see, while $\mathbb S^4$ --- corresponding to an $A=\mathbb B^4$ holographic entangling region --- is a local minimum of the generalized Willmore functional, this is neither bounded from below nor from above for general regions.