Berezin quantization, conformal welding and the Bott-Virasoro group

TL;DR

This paper explores the quantization of conformal welding and its relation to the Bott-Virasoro group, using Berezin formalism to lift symplectic transformations to operators on Fock space, revealing new insights into the energy functional.

Contribution

It introduces a novel Berezin quantization approach to conformal welding, providing new representations of the Bott-Virasoro cocycle and identities for the Takhtajan-Teo energy functional.

Findings

Quantization of conformal welding via Berezin formalism.

New representation of the Bott-Virasoro cocycle.

Identity for the Takhtajan-Teo energy functional.

Abstract

Following Nag-Sullivan, we study the representation of the group ${\rm Diff}^+(S^1)$ of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides a triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift provides quantization of conformal welding, gives a new representative of the Bott-Virasoso cocylce class, and leads to a surprising identity for the Takhtajan-Teo energy functional on ${\rm Diff}^+(S^1)$.