Coarse-grained entropy and causal holographic information in AdS/CFT

TL;DR

The paper investigates how certain coarse-grained entropies in holography relate to bulk geometric quantities. It argues that the boundary one-point entropy ${\\mathscr S}^{(1)}$ is dual to the causal holographic information $\\chi$, and the future one-point entropy ${\\mathfrak S}^{(1)}$ is dual to the future causal information $\\phi$, with the latter obeying a nontrivial second law that mirrors Hawking's area theorem. Through definitions, properties, and concrete examples (e.g., geon spacetimes and thermal states), the authors build a case for these dualities in the large $N$ limit and without boundary sources, and discuss potential tests and generalizations to broader bulk regions. The work provides a framework linking field-theoretic coarse graining to bulk locality and thermodynamic-like evolution, offering a potential path to understanding the bulk emergence of spacetime and the holographic encoding of information under coarse graining.

Abstract

We propose bulk duals for certain coarse-grained entropies of boundary regions. The `one-point entropy' is defined in the conformal field theory by maximizing the entropy in a domain of dependence while fixing the one-point functions. We conjecture that this is dual to the area of the edge of the region causally accessible to the domain of dependence (i.e. the `causal holographic information' of Hubeny and Rangamani). The `future one-point entropy' is defined by generalizing this conjecture to future domains of dependence and their corresponding bulk regions. We show that the future one-point entropy obeys a nontrivial second law. If our conjecture is true, this answers the question "What is the field theory dual of Hawking's area theorem?"

Figures (5)

• Figure 1: A sketch of the causal wedge construction of Hubeny:2012wa. $D[{\cal A}]$ is the boundary domain of dependence of $\cal A$ and $\Xi_{\cal A}$ extends into the bulk (see text).
• Figure 2: When $\cal C$ is a Cauchy surface $\chi_{\cal C}$ is calculated from the area of $\Xi^T_{\cal C}$. ${\cal B}_{\pm T}$ are slices of a foliation of boundary Cauchy surfaces and $\Xi^T_{\cal C}$ is the intersection of their respective past and future horizons. This construction addresses non-perturbative late time quantum effects such to Poincaré recurrences and black hole evaporation.
• Figure 7: A sketch of the construction of $\Phi _{\cal A}$ described in the text. $D[{\cal A}]$ is the boundary domain of dependence of $\cal A$ and $\Phi _{\cal A}$ extends into the bulk (see text).
• Figure 8: A sketch of the construction of $\Psi_{{\cal A}_-,{\cal A}_+}$ described in the text. $D[{\cal A}_-] = D[{\cal A}_+]$ is the boundary domain of dependence of ${\cal A}_\pm$ and $\Psi_{{\cal A}_-,{\cal A}_+}$ extends into the bulk (see text).
• Figure 9: Various insertions of sources on the vacuum AdS boundary. In each figure the solid line to the right represents the AdS boundary and $\cal A$ is a spherical region. (a) By causality ${\mathfrak E}_{\cal A}$ is unperturbed by the sources however $\Xi_{\cal A}$ is moved due to focusing of light rays (shown schematically by the dashed lines). However this focusing does not change $\chi$ since the past horizon has vanishing expansion. (b) An ingoing and outgoing source which gives $\chi_{\cal A} > S_{\cal A} = {\mathscr S}^{(1)} _{\cal A}$.