Properties of Causal Holographic Information

TL;DR

This work studies causal holographic information χ_Σ, defined by the area of the causal information surface Ξ_Σ, as a covariant bulk dual to boundary subregions. The authors derive the universal logarithmic term of χ_Σ in four-dimensional boundary theories, showing it is nonlocal and Weyl-invariant, and they provide explicit expressions in flat-boundary cases that depend on the local causal diamond height τ and extrinsic geometry. Extending to Gauss–Bonnet gravity, they demonstrate how the log-term decomposes into the central charges a and c via a Wald-type entropy generalization, illustrating the deep connection between χ_Σ and CFT data. They also discuss speculative CFT interpretations of χ, including density-matrix coarse-graining concepts and ambiguities in non-Abelian gauge theories, offering a framework for understanding how bulk causality maps to boundary information beyond entanglement entropy. Overall, the paper advances the understanding of subregion duality by characterizing a nonlocal, universal conformal invariant and exploring its possible CFT duals and implications for bulk locality.

Abstract

Causal holographic information [1] is a variant of the Ryu-Takayanagi proposal for the entanglement entropy of a spatial region in the context of AdS/CFT, but with the bulk surface defined by causality rather than extremality. We investigate the properties of causal holographic information, focusing in particular on the universal coefficient of the logarithmically divergent term. We find that this coefficient contains a novel conformal invariant that cannot be written as an integral of local quantities. By considering higher curvature corrections in the bulk, we identify the coefficient of the a and c central charges in 4 dimensions. Finally, we speculate about which CFT quantity could correspond to the causal holographic information.

Figures (4)

• Figure 1: Visualization of the causal information surface $\Xi_{\Sigma}$ (red) reaching into the bulk (z-direction), constructed from the causal diamond of $\Sigma$. The area $\Sigma$ is indicated by the blue line.
• Figure 2: Visualization of the region $\Sigma$ and the boundary $\partial \Sigma$ (left) on a (1+1)-dimensional boundary and the construction of the causal diamond $\Diamond_{\Sigma}$ (shaded area) consisting of the union of the future and past domains of dependence $D^{+}_{\partial M}\left(\Sigma \right)$ and $D^{-}_{\partial M}\left(\Sigma \right)$ (right).
• Figure 3: Visualization of the boundary causal diamond $\Diamond_{\Sigma}$ in case of a (2+1)-dimensional boundary. The set of future "caustics" $\mathcal{C}^+$ is indicated by the blue line.
• Figure 4: Visualization of the quantity $\tau$. Note that that for each point on $\partial \Sigma$ there is a unique point on $\mathcal{C}^+$ separated by a vector that is proportional to the future directed inward pointing null normal vector. For a surface $\partial \Sigma$ that does not lie on a constant time slice, a single function $\tau$ is not sufficient.