Dirac operators on quasi-Hamiltonian GG-spaces

We construct twisted spinor bundles as well as twisted pre-quantum bundles on quasi-Hamiltonian GG-spaces, using the spin representation of loop group and the Hilbert space of Wess–Zumino–Witten model. We then define a Hilbert space together with a Dirac operator acting on it. The main result of this paper is that we show the Dirac operator has a well-defined index given by positive energy representation of the loop group. This generalizes the geometric quantization of Hamiltonian GG-spaces to quasi-Hamiltonian GG-spaces.