Modular Berry Connection for Entangled Subregions in AdS/CFT

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 conformal field theory, global conformal symmetry singles out a unique modular Berry connection, which we compute directly and in the dual three-dimensional anti-de Sitter (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.