# Berry Phases on Virasoro Orbits

## TL;DR

This paper explores how unitary representations of the Virasoro algebra exhibit Berry phases when acted upon by conformal transformations tracing closed paths, with implications for gravitational physics in AdS_3.

## Contribution

It demonstrates the exact computation of Berry phases in Virasoro representations using the Maurer-Cartan form and relates these phases to gravitational effects like Thomas precession.

## Key findings

- Berry phases can be exactly computed for Virasoro orbits.
- These phases persist when combining left- and right-moving sectors.
- Virasoro Berry phases extend Thomas precession to gravitational contexts.

## Abstract

We point out that unitary representations of the Virasoro algebra contain Berry phases obtained by acting on a primary state with conformal transformations that trace a closed path on a Virasoro coadjoint orbit. These phases can be computed exactly thanks to the Maurer-Cartan form on the Virasoro group, and they persist after combining left- and right-moving sectors. Thinking of Virasoro representations as particles in AdS_3 dressed with boundary gravitons, the Berry phases associated with Brown-Henneaux diffeomorphisms provide a gravitational extension of Thomas precession.