On the quantization of conjugacy classes

TL;DR

This paper reviews the Freed-Hopkins-Teleman theorem linking the fusion ring of a Lie group to twisted equivariant K-homology, emphasizing the role of conjugacy classes and Dixmier-Douady bundles.

Contribution

It provides a K-homology perspective on the theorem, illustrating how conjugacy classes generate the fusion ring via push-forward maps.

Findings

Fusion ring R_k(G) is generated by conjugacy classes.

K-homology push-forwards produce the additive generators.

Dixmier-Douady bundles clarify the twisted K-homology structure.

Abstract

Let G be a compact, simple, simply connected Lie group. A theorem of Freed-Hopkins-Teleman identifies the level k fusion ring R_k(G) of G with the twisted equivariant K-homology at level k+h, where h is the dual Coxeter number. In this paper, we 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 conjugacy classes in G.