# Bulk locality from modular flow

## TL;DR

This paper explores how bulk operators in holography can be reconstructed from boundary data using modular flow, providing explicit formulas and generalizations for operators on the Ryu-Takayanagi surface.

## Contribution

It introduces a method to reconstruct bulk operators via boundary modular flow and generalizes the geodesic operator/OPE block dictionary to arbitrary states and regions.

## Key findings

- Explicit formulas for bulk operators using modular flow.
- Reconstruction of bulk operators on the Ryu-Takayanagi surface.
- Generalization of the geodesic operator/OPE block dictionary.

## Abstract

We study the reconstruction of bulk operators in the entanglement wedge in terms of low energy operators localized in the respective boundary region. To leading order in $N$, the dual boundary operators are constructed from the modular flow of single trace operators in the boundary subregion. The appearance of modular evolved boundary operators can be understood due to the equality between bulk and boundary modular flows and explicit formulas for bulk operators can be found with a complete understanding of the action of bulk modular flow, a difficult but in principle solvable task. We also obtain an expression when the bulk operator is located on the Ryu-Takayanagi surface which only depends on the bulk to boundary correlator and does not require the explicit use of bulk modular flow. This expression generalizes the geodesic operator/OPE block dictionary to general states and boundary regions.