# A Modular Sewing Kit for Entanglement Wedges

## TL;DR

This paper introduces a geometric approach linking the Riemann curvature of holographic spacetimes to the Berry curvature of modular Hamiltonians in the dual CFT, providing new insights into entanglement and gravity.

## Contribution

It establishes a modular sewing framework connecting entanglement wedge geometry with CFT modular Hamiltonian properties, advancing the understanding of holographic entanglement.

## Key findings

- Modular Berry connection encodes relative bases of CFT subregions.
- Bulk dual relates edge-mode frames of entanglement wedges.
- At leading order, the connection stitches coordinate systems near HRRT surfaces.

## Abstract

We relate the Riemann curvature of a holographic spacetime to an entanglement property of the dual CFT state: the Berry curvature of its modular Hamiltonians. The modular Berry connection encodes the relative bases of nearby CFT subregions while its bulk dual, restricted to the code subspace, relates the edge-mode frames of the corresponding entanglement wedges. At leading order in 1/N and for sufficiently smooth HRRT surfaces, the modular Berry connection simply sews together the orthonormal coordinate systems covering neighborhoods of HRRT surfaces. This geometric perspective on entanglement is a promising new tool for connecting the dynamics of entanglement and gravitation.