Twisted K-homology and group-valued moment maps

TL;DR

This paper develops a quantization functor for pre-quantized quasi-Hamiltonian G-spaces at level k, mapping them to the fusion ring using twisted equivariant K-homology, with explicit fixed point formula computations.

Contribution

It introduces a new quantization framework connecting quasi-Hamiltonian spaces to the fusion ring via twisted K-homology, including explicit computational methods.

Findings

Derived a fixed point formula for quantization Q(M)

Computed Q(M) explicitly in several examples

Established a new link between geometric quantization and the Verlinde algebra

Abstract

Let G be a compact, simply connected Lie group. We develop a `quantization functor' from pre-quantized quasi-Hamiltonian G-spaces at level k to the fusion ring (Verlinde algebra) R_k(G). The quantization Q(M) is defined as a push-forward in twisted equivariant K-homology. It may be computed by a fixed point formula, similar to the equivariant index theorem for Spin_c-Dirac operators. Using the formula, we calculate Q(M) in several examples.