Corner contributions to holographic entanglement entropy in AdS4/BCFT3

TL;DR

This work computes holographic corner contributions to entanglement entropy in AdS$_4$/BCFT$_3$ with a flat boundary, focusing on two-dimensional regions that feature corners on or near the BCFT boundary. It derives analytic expressions for the boundary corner function $F_ ext{α}(γ)$ and the tip-boundary corner function $ extsf{F}_ ext{α}(ω,γ)$ via minimal-surface analyses in AdS$_4$ with a brane at angle α, and confirms these with Surface Evolver numerics. Key results include the identification of a critical slope α_c, detailed limiting regimes (γ→0, γ→π/2), and universal relations tying corner data to boundary stress-tensor coefficients $A_T$, with special cases recovering known AdS$_4$/CFT$_3$ results. The findings illuminate how BCFT boundary conditions and geometry shape entanglement structure and provide a foundation for extensions to higher dimensions and non-flat boundaries.

Abstract

We study the holographic entanglement entropy of spatial regions with corners in the AdS4/BCFT3 correspondence by considering three dimensional boundary conformal field theories whose boundary is a timelike plane. We compute analytically the corner function corresponding to an infinite wedge having one edge on the boundary. A relation between this corner function and the holographic one point function of the stress tensor is observed. An analytic expression for the corner function of an infinite wedge having only its tip on the boundary is also provided. This formula requires to find the global minimum among two extrema of the area functional. The corresponding critical configurations of corners are studied. The results have been checked against a numerical analysis performed by computing the area of the minimal surfaces anchored to some finite domains containing corners.

Figures (29)

• Figure 1: Examples of finite two dimensional regions $A$ (yellow domains) containing the kinds of corners considered in this manuscript. Left: $A$ is a domain in the plane with three corners and two different kinds of vertices. Right: $A$ is a domain in the half plane with three corners whose boundary $\partial A$ intersects the boundary of the BCFT$_3$ (solid black line). The three vertices in $\partial A$ are also on the boundary of the BCFT$_3$ and they belong to two different classes of vertices. In both panels, the red curve corresponds to the entangling curve $\partial A \cap \partial B$, whose length provides the area law term in (\ref{['ee cft3 corner intro']}) and in (\ref{['ee bcft3 corner intro']}).
• Figure 2: Configurations of adjacent domains containing corners (yellow regions) in the half plane $x\geqslant 0$ (grey region) which have been used in Sec. \ref{['sec constraints']} to constrain the corner functions through the strong subadditivity.
• Figure 3: Triangulated surface in $\mathbb{H}_3$ which approximates the minimal area surface $\hat{\gamma}_A$ corresponding to a single drop region $A$ in the $z=0$ plane, as discussed in Sec. \ref{['sec single drop']}. The boundary $\partial A$ (red curve) lies in the $z=0$ plane and it is characterised by $L=1$ and $\theta = \pi/3$. The UV cutoff is $\varepsilon = 0.03$. The triangulation has been obtained with Surface Evolver by setting $\partial A$ at $z=\varepsilon$.
• Figure 4: Corner function for a vertex with two edges in AdS$_4$/CFT$_3$. The blue curve corresponds to the analytic expression given by (\ref{['cusp exact result']}) and (\ref{['theta_V expression']}) found in Drukker:1999zq. The points labeled by the red triangles have been found through the numerical analysis based on Surface Evolver (see Sec. \ref{['sec single drop']} and the appendix \ref{['app numerics']}). The inset highlights the domain corresponding to opening angles close to $\pi$.
• Figure 5: Triangulated surfaces in $\mathbb{H}_3$ approximating the minimal area surfaces $\hat{\gamma}_A$ which correspond to two different double drop regions $A$ described in Sec. \ref{['sec double drop ads4']}. For these domains $\phi_1 = \phi_2 \equiv \phi$ and $\varphi_1 = \varphi_2 = \pi - \phi$. The boundary $\partial A$ (red curve) belongs to the $z=0$ plane and the UV cutoff is $\varepsilon = 0.03$. Top: $L=2$ and $\phi=1.4$ (below $\phi_c=\pi/2$). Bottom: $L=1$ and $\phi=2.2$ (above $\phi_c = \pi/2$).