Holography, Entanglement Entropy, and Conformal Field Theories with Boundaries or Defects

TL;DR

This paper extends the Casini–Huerta–Myers mapping that relates entanglement entropy to thermal entropy on hyperbolic space to boundary and defect conformal field theories, proving that for hemispherical entangling surfaces the reduced density matrix is equivalent to a thermal state on $ ext{R} imes ext{H}^{d-1}$. In holographic BCFTs/DCFTs with Einstein gravity duals, the hyperbolic horizon’s Bekenstein–Hawking entropy reproduces the RT minimal surface, establishing RT’s conjecture in these settings. The authors then exploit this framework to compute EE and Rényi entropies for DCFTs with probe-brane defects, showing that in the probe limit the order-$ u L^{2-n}$ corrections can be obtained from the probe-brane action in the undeformed background, without backreaction. They apply the method to both bottom-up models and top-down string/M-theory constructions, obtaining explicit results for D3/D7, D3/D5, ABJM with M5/M2$'$-branes, and highlighting cases where boundary terms are essential to correctly capture EE. Overall, the work provides a versatile, symmetry-based toolkit for EE in BCFTs/DCFTs, linking defect degrees of freedom to universal contributions and central charges, and enabling tractable calculations across a broad class of holographic theories.

Abstract

We study entanglement entropy (EE) in conformal field theories (CFTs) in Minkowski space with a planar boundary or with a planar defect of any codimension. In any such boundary CFT (BCFT) or defect CFT (DCFT), we consider the reduced density matrix and associated EE obtained by tracing over the degrees of freedom outside of a (hemi-)sphere centered on the boundary or defect. Following Casini, Huerta, and Myers, we map the reduced density matrix to a thermal density matrix of the same theory on hyperbolic space. The EE maps to the thermal entropy of the theory on hyperbolic space. For BCFTs and DCFTs dual holographically to Einstein gravity theories, the thermal entropy is equivalent to the Bekenstein-Hawking entropy of a hyperbolic black brane. We show that the horizon of the hyperbolic black brane coincides with the minimal area surface used in Ryu and Takayanagi's conjecture for the holographic calculation of EE. We thus prove their conjecture in these cases. We use our results to compute the Rényi entropies and EE in DCFTs in which the defect corresponds to a probe brane in a holographic dual.

Figures (2)

• Figure 1: Cartoons depicting two different "slicings" of $AdS_{d+1}$, the Poincaré and the $AdS_d$ slicings, which make manifest different subgroups of the $SO(2,d)$ isometry. (Figure adapted from ref. Aharony:2003qf.) (a.) Poincaré slicing, with the metric in eq. \ref{['adspoin']}. We suppress all directions except for two: the horizontal axis, $r_{\perp}$, and the vertical axis, $z$. The solid horizontal line is the $AdS_{d+1}$ boundary $z \rightarrow 0$, the heavy dashed horizontal line is the Poincaré horizon $z \to \infty$, and the thin dashed lines are surfaces of fixed $z$, the Poincaré slices, which make manifest the $SO(1,d-1) \in SO(2,d)$. (b.)$AdS_d$ slicing, where the metric is written as in eq. \ref{['adsslicing']} with $n=1$. The $Z$ and $x$ directions are depicted with arrows. The thin dashed lines are surfaces of constant $x$, the $AdS_d$ slices, which make manifest the $SO(2,d-1) \in SO(2,d)$, as approproate for a fictitious $n=1$ defect at $r_{\perp} = 0$.
• Figure 2: A schematic representation of the two classes of solutions appearing in our proof that for surfaces in $AdS_{d+1}$ approaching a sphere of radius $R$ at the boundary, eq. \ref{['zsol']} produces the global minimum of the area functional ${\mathcal{A}}$. The solutions are curves parameterized in terms of their radius $\zeta$ as a function of the angular coordinate $\varphi \in [0,\pi/2]$ in the quadrant spanned by $r$ and $z$. The first class of solutions, of which the heavy-dashed curve $\zeta_1(\varphi)$ is a representative, are valued on the domain $[0,\pi/2]$. In the second class, of which the dotted curve $\zeta_2(\varphi)$ is a representative, $\zeta(\varphi) \to \infty$ at some angle $\varphi_0 \in (0,\pi/2)$, denoted by the thin-dashed line. The solution producing the global minimum of ${\mathcal{A}}$ is in the first class, and is simply the constant function $\zeta(\varphi)=R$.