Holographic Reconstruction of General Bulk Surfaces

TL;DR

The paper extends the differential-entropy program to reconstruct general bulk surfaces in any dimension by introducing a generalized area density and a foliation of the surface with tangent extremal surfaces. It derives a boundary formula for differential entropy, $S_{\rm diff}$, that equals the bulk area $A$ under suitable homology and regularity conditions, and explores when the tangent extremals may be nonminimal. Through a torus in AdS$_4$ and a sphere degeneration analysis, it demonstrates the method’s validity and highlights phase structure and boundary-term subtleties, while discussing the boundary interpretations, including entwinement, and the need for a covariant formulation. Overall, the work advances understanding of how bulk geometry can be encoded in boundary entanglement data and points toward a covariant, sub-AdS reconstruction framework.

Abstract

We propose a reconstruction of general bulk surfaces in any dimension in terms of the differential entropy in the boundary field theory. In particular, we extend the proof of Headrick et al. to calculate the area of a general class of surfaces, which have a 1-parameter foliation over a closed manifold. The area can be written in terms of extremal surfaces whose boundaries lie on ring-like regions in the field theory. We discuss when this construction has a description in terms of spatial entanglement entropy and suggest lessons for a more complete and covariant approach.

Figures (10)

• Figure 1: A minimal surface $M$ tangent to the original surface (not drawn) at a loop $K$. The choice of which side is left/right is fixed by demanding that the coordinate $\lambda$ increase from left to right. In other words, the intersection of the minimal surfaces $M(\lambda)$ and $M(\lambda+d\lambda)$ happens inside $M_R(\lambda)$ and $M_L(\lambda+d\lambda)$.
• Figure 2: Nomenclature used in the proof. Unprimed symbols denote quantities at $\lambda$ while primes mark quantities at $\lambda+d\lambda$. The other $d-2$ dimensions are denoted by $\vec{\eta}$.
• Figure 3: Left: reconstruction of a torus on a constant time slice of $AdS_4$. Middle: the infinitesimal version where the loop $K(\lambda)$ sweeps out a small area. Right: we have exploited the symmetry of AdS and moved the loop $K(\lambda)$ to the center of AdS.
• Figure 4: Left: foliation of a sphere by circles that degenerate at two points. Middle and right: the extremal surfaces $M(\lambda)$ constructed from circles $K(\lambda)$ as they approach a degeneration point.
• Figure 5: Three possible extremal surfaces that asymptote to the same ring-shaped region on the asymptotic boundary: thick, thin, and disconnected.