On the coadjoint Virasoro action

TL;DR

This paper establishes a deep geometric correspondence between coadjoint orbits of the Virasoro algebra and conjugacy classes in a certain subset of the universal cover of SL(2,R), using Morita equivalence of quasi-symplectic groupoids.

Contribution

It extends the bijection between Virasoro coadjoint orbits and conjugacy classes to a Morita equivalence of quasi-symplectic groupoids, integrating relevant Poisson and Dirac structures.

Findings

Established Morita equivalence of quasi-symplectic groupoids.

Integrated Poisson structure on Virasoro dual with Cartan-Dirac structure.

Strengthened the geometric understanding of Virasoro coadjoint orbits.

Abstract

The set of coadjoint orbits of the Virasoro algebra at level 1 is in bijection with the set of conjugacy classes in a certain open subset $\widetilde{\rm SL}(2,\mathbb{R})_+$ of the universal cover of ${\rm SL}(2,\mathbb{R})$. We strengthen this bijection to a Morita equivalence of quasi-symplectic groupoids, integrating the Poisson structure on $\mathfrak{vir}^*_\mathsf{1}(S^1)$ and the Cartan-Dirac structure on $\widetilde{\rm SL}(2,\mathbb{R})_+$, respectively.