On the quantization of conjugacy classes

Abstract

Let $G$ be a compact, simple, simply connected Lie group. A theorem of Freed-Hopkins-Teleman identifies the level $k\ge 0$ fusion ring $R_k(G)$ of $G$ with the twisted equivariant $K$-homology at level $k+h^v$, where $h^v$ is the dual Coxeter number of $G$. In this paper, we will review this result using the language of Dixmier-Douady bundles. We show that the additive generators of the group $R_k(G)$ are obtained as $K$-homology push-forwards of the fundamental classes of pre-quantized conjugacy classes in $G$.