A modular sewing kit for entanglement wedges

We relate the Riemann curvature of a holographic spacetime to an entangle- ment 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 cor- responding entanglement wedges. At leading order in 1/N and for sufficiently smooth HRRT surfaces, the modular Berry connection simply sews together the orthonormal co- ordinate systems covering neighborhoods of HRRT surfaces. This geometric perspective on entanglement is a promising new tool for connecting the dynamics of entanglement and gravitation.