A Modular Sewing Kit for Entanglement Wedges

TL;DR

<3-5 sentence high-level summary> We relate the curvature of a holographic spacetime to an entanglement property of the dual CFT state via the Berry curvature of modular Hamiltonians. The modular Berry connection encodes relative zero-mode frames across nearby subregions and, in the bulk, reduces to the relative embedding of entanglement-wedge coordinates with a spin-connection structure; its curvature reproduces the bulk Riemann tensor in a controlled regime. By leveraging JLMS within a code subspace and treating edge modes as gauge degrees of freedom, the paper establishes a concrete boundary-to-bulk correspondence between entanglement structure and spacetime geometry, and derives explicit parallel transport and holonomy formulas in tractable examples such as CFT$_2$ vacua and pure AdS$_3$. The framework opens routes to entanglement-based measures of multipartite correlations, and suggests how soft modes and bulk gauge dynamics may be encoded in modular Berry holonomies, with potential implications for bulk locality and holographic dynamics.

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.

Figures (4)

• Figure 1: A holographic representation of eqs. (\ref{['qubitstate']}-\ref{['Wtransform']}): the global state $W_{ij}$ of a bipartite holographic CFT is prepared by a tensor network that fills a spatial slice of the bulk spacetime (orange). The division of the CFT is illustrated with a red line that cuts through the bulk. The panels show two general examples of 'gauge transformations' of 'Wilson line' $W$. The focus of this paper will be on those gauge transformations, which localize on HRRT surfaces.
• Figure 2: The fiber bundle studied in this paper. The base comprises different modular Hamiltonians and the fibers are modular zero mode frames.
• Figure 3: A closed trajectory in the space of CFT regions. To avoid clutter and to clarify the holographic application, here we display the family of corresponding RT surfaces in the bulk of AdS.
• Figure 4: Modular Berry curvature in the bulk. The modular zero mode frames are marked with pairs of arrows that stand for the normal vectors $n_\alpha^M(\lambda, y)$ (which transform under orthogonal boosts); the distances between neighboring pairs reflect the extremal surface diffeomorphism frame. We parallel transport a zero mode frame from the bottom surface to the top surface along two different paths (red and blue); the mismatch between the resulting frames is the modular curvature. The mismatch between the locations of the red and blue arrows on the top is the surface diffeomorphism component of the curvature (\ref{['diffcurvature']}) while the mismatch between their directions is the boost component of the curvature (\ref{['boostcurvature']}).

Theorems & Definitions (1)

• Definition 1