Virasoro Entanglement Berry Phases

TL;DR

This paper explores the geometric and symplectic structure of entanglement in 2d CFTs through Virasoro Berry phases, connecting boundary parallel transport to bulk symplectic forms in holography.

Contribution

It introduces a novel approach to understanding entanglement via Virasoro Berry phases and relates boundary transport to bulk symplectic geometry in holographic settings.

Findings

Berry curvature is the Kirillov-Kostant form on a new coadjoint orbit.

Boundary parallel transport computes a bulk symplectic form.

Proposes a definition for the symplectic form on an entanglement wedge.

Abstract

We study the parallel transport of modular Hamiltonians encoding entanglement properties of a state. In the case of 2d CFT, we consider a change of state through action with a suitable diffeomorphism on the circle: one that diagonalizes the adjoint action of the modular Hamiltonian. These vector fields exhibit kinks at the interval boundary, thus together with their central extension they differ from usual elements of the Virasoro algebra. The Berry curvature associated to state-changing parallel transport is the Kirillov-Kostant symplectic form on an associated coadjoint orbit, one which differs appreciably from known Virasoro orbits. We find that the boundary parallel transport process computes a bulk symplectic form for a Euclidean geometry obtained from the backreaction of a cosmic brane, with Dirichlet boundary conditions at the location of the brane. We propose that this gives a reasonable definition for the symplectic form on an entanglement wedge.