Local bulk physics from intersecting modular Hamiltonians

TL;DR

This work shows that local bulk observables in AdS/CFT can be constructed purely from CFT data by using intersecting extended modular Hamiltonians associated with boundary segments. The authors derive explicit smearing functions and bulk mode representations in the vacuum and BTZ backgrounds, including a complex coordinate (3.1) and derivative (3.1.1) form, momentum-space expansions (3.2), time dependence (3.3), and finite-temperature BTZ generalizations (3.4). The central idea is that bulk operators on RT surfaces commute with the corresponding modular Hamiltonians, and if two RT surfaces intersect, a bulk operator commuting with both must live at the intersection, yielding a metric-free route to bulk locality. These results connect modular flow with bulk geometry, recover known bulk observables without assuming a bulk metric, and provide a framework potentially extensible to higher dimensions and interacting or gauge sectors.

Abstract

We show that bulk quantities localized on a minimal surface homologous to a boundary region correspond in the CFT to operators that commute with the modular Hamiltonian associated with the boundary region. If two such minimal surfaces intersect at a point in the bulk then CFT operators which commute with both extended modular Hamiltonians must be localized at the intersection point. We use this to construct local bulk operators purely from CFT considerations, without knowing the bulk metric, using intersecting modular Hamiltonians. For conformal field theories at zero and finite temperature the appropriate modular Hamiltonians are known explicitly and we recover known expressions for local bulk observables.