Modular Berry Connection

TL;DR

This paper explores the modular Berry connection in 2d CFTs, linking it to geometric quantities in AdS$_3$ and proposing ways to measure it through bulk observer motions.

Contribution

It introduces a unique modular Berry connection in 2d CFT vacuum states and relates it to geometric lengths in AdS$_3$, connecting quantum information and holography.

Findings

The modular Berry connection is uniquely determined by conformal symmetry.

Wilson loops of the connection compute lengths of curves in AdS$_3$.

Modular Berry transformations can be observed by bulk observers with varying accelerations.

Abstract

The Berry connection describes transformations induced by adiabatically varying Hamiltonians. We study how zero modes of the modular Hamiltonian are affected by varying the region that supplies the modular Hamiltonian. In the vacuum of a 2d CFT, global conformal symmetry singles out a unique modular Berry connection, which we compute directly and in the dual AdS$_3$ picture. In certain cases, Wilson loops of the modular Berry connection compute lengths of curves in AdS$_3$, reproducing the differential entropy formula. Modular Berry transformations can be measured by bulk observers moving with varying accelerations.