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 AdS3 dressed with boundary gravitons, the Berry phases associated with Brown-Henneaux diffeomorphisms provide a gravitational extension of Thomas precession.
Highlights
Eigenstate along a closed curve in parameter space is proportional to the area enclosed by that path on the sphere
A second purpose is to relate these phases to gravity: since Virasoro describes the asymptotic symmetries of gravitation on three-dimensional anti-de Sitter space (AdS3) [16], one may ask if the Berry phases obtained here have a bulk interpretation when thinking of Virasoro representations as particles dressed with boundary gravitons
In the previous pages we have seen how Virasoro symmetry can be used to evaluate the Berry phase picked up by a primary state as it undergoes a family of conformal transformations
Summary
We briefly review general aspects of Berry phases and apply them to unitary group representations, where the Berry connection is closely related to the Maurer-Cartan form. We refer e.g. to [28, chapter 10] and [29, chapter 17] or the reviews [30, 31] for an introduction to Berry phases; see [32, 33] for recent works on Berry phases in field theory. The second part of this section will rely on various tools in group theory and differential geometry; for more on this, see e.g. [34, chapter 5] or [35, chapter 5]. Note that our presentation will not be mathematically rigorous. All manifolds, bundles, functions and sections below are assumed to be smooth (except if explicitly stated otherwise)
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