Quantum Extremal Modular Curvature: Modular Transport with Islands

TL;DR

This paper introduces the Quantum Extremal Modular Curvature (QEMC), a novel concept extending modular Berry transport to include quantum extremal surfaces, providing new insights into bulk geometry and horizon physics in AdS/CFT.

Contribution

It develops the QEMC framework for analyzing bulk curvature in the presence of islands, revealing non-locality and locality properties in different limits, and offers a new method to probe behind horizons.

Findings

QEMC is generally non-local.

In the OPE limit, QEMC becomes local.

QEMC probes bulk Riemann curvature with islands.

Abstract

Modular Berry transport is a useful way to understand how geometric bulk information is encoded in the boundary CFT: The modular curvature is directly related to the bulk Riemann curvature. We extend this approach by studying modular transport in the presence of a non-trivial quantum extremal surface. Focusing on JT gravity on an AdS background coupled to a non-gravitating bath, we compute the modular curvature of an interval in the bath in the presence of an island: the Quantum Extremal Modular Curvature (QEMC). We highlight some important properties of the QEMC, most importantly that it is non-local in general. In an OPE limit, the QEMC becomes local and probes the bulk Riemann curvature in regions with an island. Our work gives a new approach to probe physics behind horizons.