On Volumes of Subregions in Holography and Complexity

TL;DR

This work defines and analyzes a covariant codim-one volume inside the Ryu-Takayanagi surface as a subregion generalization of holographic complexity. It derives a general strip-volume prescription for time-independent, asymptotically AdS geometries, and shows that the volume can be non-monotonic in AdS black hole backgrounds while still tracking entanglement structure through phase-transition behavior. The authors demonstrate that the volume exhibits finite jumps at bulk minimal-surface transitions across several geometries, including strips, global AdS_3, confining backgrounds, and AdS black holes, and they compute explicit jump magnitudes in key examples. They also explore the codim-zero volume and the bulk action of the entanglement wedge for a sphere in pure AdS, and discuss implications for the holographic dual of subregion complexity and future extensions to dynamics and higher-derivative theories.

Abstract

The volume of the region inside the bulk Ryu-Takayanagi surface is a codimension-one object, and a natural generalization of holographic complexity to the case of subregions in the boundary QFT. We focus on time-independent geometries, and study the properties of this volume in various circumstances. We derive a formula for computing the volume for a strip entangling surface and a general asymptotically AdS bulk geometry. For an AdS black hole geometry, the volume exhibits non-monotonic behaviour as a function of the size of the entangling region (unlike the behaviour of the entanglement entropy in this setup, which is monotonic). For setups in which the holographic entanglement entropy exhibits transitions in the bulk, such as global AdS black hole, geometries dual to confining theories and disjoint entangling surfaces, the corresponding volume exhibits a discontinuous finite jump at the transition point (and so do the volumes of the corresponding entanglement wedges). We compute this volume discontinuity in several examples. Lastly, we compute the codim-zero volume and the bulk action of the entanglement wedge for the case of a sphere entangling surface and pure AdS geometry.

Figures (13)

• Figure 1: Left: Illustration of the double sided AdS black hole Penrose diagram dual to the thermofield double state. The left and right boundaries are where $CFT_L$ and $CFT_R$ live. The singularity is shown in dashed red. A few maximal codim-one surfaces are shown, including the one at $t\to \infty$ in green (which does not reach the singularity.). Right: Illustration of embedding diagram of a wormhole.
• Figure 2: A strip entangling region with width $l$ and length $L\to \infty$ on the boundary, and its corresponding minimal surface in the bulk. The boundary is at $z\rightarrow 0$.
• Figure 3: Left: Plot of $\Delta V \equiv V-V_{CFT}$ as a function of $d$ for a temperature perturbation. Right: Plot of $V-V_{CFT}$ as a function of $d$ for perturbations of the form $f_2(z)= 1+m z^{q}$. Curves from top to bottom correspond to: $q=d$, $q=d+1$, $q=d+2$, and $q=d+3$.
• Figure 4: Plots of $\Delta V \equiv V-V_{CFT}$ as a function of $l$ (we change $l$ and leave $T$ fixed). At $l=0$ we have $\Delta V=0$. Left: The $AdS_5$ black hole. Right: The $AdS_4$ black hole.
• Figure 5: The entanglement entropy entropy $\Delta S = S -S_{CFT}$ as a function of $l$ (we change $l$ and leave $T$ fixed) for an $AdS_5$ black hole and a strip entangling surface. $\Delta S$ starts at the origin, and has linear behavior at large $lT$.