Dirac operators on quasi-Hamiltonian G-spaces

TL;DR

This paper extends the concept of Dirac operators and quantization from Hamiltonian G-spaces to quasi-Hamiltonian G-spaces, linking geometric structures with positive energy representations of loop groups.

Contribution

It introduces twisted spinor and pre-quantum bundles on quasi-Hamiltonian G-spaces and constructs a Dirac operator whose index relates to loop group representations.

Findings

Constructed a Dirac operator on quasi-Hamiltonian G-spaces.

Established the index of the Dirac operator corresponds to positive energy loop group representations.

Generalized Hamiltonian G-space quantization to the quasi-Hamiltonian setting.

Abstract

We develop notions of twisted spinor bundle and twisted pre-quantum bundle on quasi-Hamiltonian G-spaces. The main result of this paper is that we construct a Dirac operator with index given by positive energy representation of loop group. This generalizes the quantization of Hamiltonian $G$-spaces to quasi-Hamiltonian G-spaces.