Deformations of quasi-Hamiltonian spaces

TL;DR

This paper introduces deformations from quasi-Hamiltonian to Hamiltonian G-spaces, illustrating how key examples like the double G×G and moduli spaces deform to cotangent bundles and coadjoint orbits.

Contribution

It provides a framework for deforming quasi-Hamiltonian G-spaces into Hamiltonian G-spaces with explicit examples and applications.

Findings

Double G×G deforms to T*G

Conjugacy classes near identity deform to coadjoint orbits

Moduli space of flat G-connections deforms to T*G^{r+g}

Abstract

We introduce a notion of deformations of quasi-Hamiltonian $G$-spaces to Hamiltonian $G$-spaces and provide several examples. In particular, we show that the double $G \times G$ of a Lie group, viewed as a quasi-Hamiltonian $G \times G$-space, deforms smoothly to the cotangent bundle $T^*G$. Likewise, any conjugacy class of $G$ sufficiently close to the identity deforms to a coadjoint orbit. We further show that the moduli space of flat $G$-connections on a compact oriented surface of genus $g$ with $r+1$ boundary components deforms to $T^*G^{r+g}$.