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On the coadjoint Virasoro action

Anton Alekseev, Eckhard Meinrenken

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.

On the coadjoint Virasoro action

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 of the universal cover of . We strengthen this bijection to a Morita equivalence of quasi-symplectic groupoids, integrating the Poisson structure on and the Cartan-Dirac structure on , respectively.
Paper Structure (29 sections, 21 theorems, 189 equations)

This paper contains 29 sections, 21 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Virasoro Lie algebra
  3. Hill operators
  4. Virasoro Lie algebra
  5. Coordinate descriptions
  6. The space of developing maps
  7. Developing maps
  8. The quotient map $q$
  9. Action of diffeomorphisms
  10. Stabilizers
  11. Local sections
  12. Morita equivalence for the coadjoint Virasoro action
  13. Groupoids
  14. The 2-form $\varpi_\mathcal{D}$
  15. Hamiltonian spaces
  16. ...and 14 more sections

Key Result

Proposition 3.4

The image of the map $q$ is the subset

Theorems & Definitions (72)

  • Example 2.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • proof
  • ...and 62 more