Geometric K-homology and the Freed-Hopkins-Teleman theorem

TL;DR

This paper constructs a geometric map linking equivariant twisted K-homology of certain Lie groups to the Verlinde ring, providing an inverse to the Freed-Hopkins-Teleman isomorphism and demonstrating compatibility of different quantization definitions.

Contribution

It introduces a geometric map at the level of D-cycles that inverts the Freed-Hopkins-Teleman isomorphism for compact Lie groups.

Findings

The map is explicitly described at the D-cycle level.

The map is inverse to the Freed-Hopkins-Teleman isomorphism.

Compatibility of different quantization approaches is established.

Abstract

We describe a map from the equivariant twisted K-homology of a compact, connected, simply connected Lie group $G$ to the Verlinde ring. Our map is described at the level of `D-cycles' for the geometric twisted K-homology of $G$, and is inverse to the Freed-Hopkins-Teleman isomorphism. As an application, we show that two possible definitions of the `quantization' of a Hamiltonian loop group space are compatible with each other.