Pulling the Boundary into the Bulk

TL;DR

The paper introduces the holographic slice, a covariantly defined bulk object that captures a continuum of progressively coarse-grained boundary states and generalizes the HRRT prescription to general convex boundaries. By defining a renormalized leaf flow with evolution vector $s = \frac{1}{2}(\theta_k l + \theta_l k)$ and leveraging entropy inequalities, the authors show monotonic shrinkage of leaf area and entanglement entropy, while constraining the flow within entanglement wedges and preventing penetration into black hole interiors. Through explicit spacetime examples (Conical AdS, Black Holes, FRW, and asymptotically AdS/flat spacetimes), they illustrate how the holographic slice traverses entanglement shadows and delineates directly reconstructable regions. They further interpret the slice as a continuous tensor network (a generalization of cMERA) and connect its radial evolution to renormalization concepts, providing a covariant gauge fixing of bulk spacetime and a framework for understanding bulk emergence via distillable correlations and modular evolution.

Abstract

Motivated by the ability to consistently apply the Ryu-Takayanagi prescription for general convex surfaces and the relationship between entanglement and geometry in tensor networks, we introduce a novel, covariant bulk object - the holographic slice. The holographic slice is found by considering the continual removal of short range information in a boundary state. It thus provides a natural interpretation as the bulk dual of a series of coarse-grained holographic states. The slice possesses many desirable properties that provide consistency checks for its boundary interpretation. These include monotonicity of both area and entanglement entropy, uniqueness, and the inability to probe beyond late-time black hole horizons. Additionally, the holographic slice illuminates physics behind entanglement shadows, as minimal area extremal surfaces anchored to a coarse-grained boundary may probe entanglement shadows. This lets the slice flow through shadows. To aid in developing intuition for these slices, many explicit examples of holographic slices are investigated. Finally, the relationship to tensor networks and renormalization (particularly in AdS/CFT) is discussed.

Figures (11)

• Figure 1: $R(B_{\delta})$ is the entanglement wedge associated with the new leaf $\sigma_C^1$, where we have taken $C(p) = B_\delta(p)$. It is formed by intersecting the entanglement wedges associated with the complements of spherical subregions of size $\delta$ on the original leaf $\sigma$.
• Figure 2: The radial evolution procedure when restricted to a subregion $A$ results in a new leaf $\sigma(\lambda) = A(\lambda) \cup \overline{A}$, where $A$ is mapped to a subregion $A(\lambda)$ contained within $\text{EW}(A)$ (blue). The figure illustrates this for two values of $\lambda$ with $\lambda_2 < \lambda_1 < 0$ (dashed lines).
• Figure 3: The case of conical $\text{AdS}_3$ with $n=3$. The points $B$, $B'$, and $B"$ are identified. There are 3 geodesics from $A$ to $B$, of which generically only one is minimal. Here, we have illustrated the subregion $AB$ with $\alpha = \pi/6$, where two of the geodesics are degenerate. This is the case in which the HRRT surface probes deepest into the bulk, leaving a shadow region in the center. Nevertheless, the holographic slice spans the entire spatial slice depicted.
• Figure 4: The exterior of a two-sided eternal AdS black hole can be foliated by static slices (black dotted lines). The holographic slice (red) connects the boundary time slices at $t = t_1$ on the right boundary and $t = t_2$ on the left boundary to the bifurcation surface along these static slices.
• Figure 5: Penrose diagram of an AdS Vaidya spacetime formed from the collapse of a null shell (blue), resulting in the formation of an event horizon (green). Individual portions of the spacetime, the future and past of the null shell, are static. Thus, the holographic slice (red) can be constructed by stitching together a static slice in each portion.

Theorems & Definitions (17)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• Definition