Proof of the Holographic Formula for Entanglement Entropy

TL;DR

The paper provides a general, semiclassical proof of the holographic entanglement entropy formula for CFTs with AdS duals, showing that the entropy is computed by the area of a bulk minimal surface via $S = \tilde{\mathcal{A}}/(4G_N^{(d+1)})$ obtained from a replica-trick path integral with conical singularities. It extends the framework to Randall-Sundrum brane worlds, yielding a decomposition of the brane entropy into thermal and vacuum pieces and clarifying the relation between brane and bulk gravitational couplings. It also analyzes higher-order corrections, including the conformal anomaly in four dimensions and extrinsic-curvature terms, and examines the impact of bulk higher-curvature terms like Gauss-Bonnet on the holographic formula. The work solidifies the geometric encoding of entanglement in holographic theories and broadens applicability to non-static setups, brane-world scenarios, and theories with higher-curvature corrections.

Abstract

Entanglement entropy for a spatial partition of a quantum system is studied in theories which admit a dual description in terms of the anti-de Sitter (AdS) gravity one dimension higher. A general proof of the holographic formula which relates the entropy to the area of a codimension 2 minimal hypersurface embedded in the bulk AdS space is given. The entanglement entropy is determined by a partition function which is defined as a path integral over Riemannian AdS geometries with non-trivial boundary conditions. The topology of the Riemannian spaces puts restrictions on the choice of the minimal hypersurface for a given boundary conditions. The entanglement entropy is also considered in Randall-Sundrum braneworld models where its asymptotic expansion is derived when the curvature radius of the brane is much larger than the AdS radius. Special attention is payed to the geometrical structure of anomalous terms in the entropy in four dimensions. Modification of the holographic formula by the higher curvature terms in the bulk is briefly discussed.

Figures (3)

• Figure 1: Spaces ${\cal M}$ and ${\cal M}_n$ (for $n=3$) are shown schematically on upper and lower pictures. The theory at temperature $T$ is defined on a space $\Sigma_1\cup\Sigma_2$ shown as an interval.
• Figure 2: A Euclidean BTZ black hole $\tilde{\cal M}$ is shown as a solid torus. The shaded region is the hypersurface $\tilde{\Sigma}_1$ with the boundary $\Sigma_1$. It lies inside the torus. The manifold $\tilde{\cal M}_n$ corresponding to the BTZ black hole is obtained by cutting and gluing $n$ copies of $\tilde{\cal M}$ along $\tilde{\Sigma}_1$.
• Figure 3: A slice by a plane of the torus shown on figure \ref{['f2']}. The Euclidean horizon $\rho=0$ lies on the slice. The shaded regions are the hypersurfaces $\tilde{\Sigma}_1$ and $\tilde{\Sigma}_2$. Their boundaries inside the slices, $\tilde{\cal B}_1$ and $\tilde{\cal B}_2$, are the geodesic lines.