Causal shadow and non-local modular flow: from degeneracy to perturbative genesis by correlation

TL;DR

This paper investigates the structure of causal shadows in AdS/CFT, analyzing their degeneracy and perturbative corrections, and explores how quantum effects and mutual information influence their properties and reconstruction capabilities.

Contribution

It introduces a perturbation theory for causal shadows, linking their degeneracy to modular flow properties and quantum corrections from bulk mutual information.

Findings

Degenerate causal shadows originate from local modular flow.

Quantum corrections induce non-degeneracy at order G_N.

Perturbative analysis reveals positivity conditions for causal shadows.

Abstract

Causal shadows are bulk space-time regions between the entanglement wedges and the causal wedges, their existence encodes deep aspects of the entanglement wedge reconstruction in the context of subregion duality in AdS/CFT. In this paper, we study the perturbation theory of the causal shadows and their relation to the properties of the associated modular flows. We first revisit the cases of degenerate causal shadows based on known examples, and discuss the origin for their degeneracy via the local nature of the modular flow. We then focus on the perturbative case in which the CFT subregion consists of two spheres separated by a large distance $L\gg R_{1,2}$. The RT surfaces still agree with the causal horizons, giving a degenerate causal shadow classically. We compute the corrections to the quantum extremal surfaces (Q.E.S) from the bulk mutual information, which then give rise to a non-degenerate causal shadow at order $G_N$. We end by discussing the causal shadow perturbation theory more generally, in particular we explore the possibility of extracting the positivity conditions characterizing perturbative causal shadows in the boundary CFTs.