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

Abstract

We construct a map at the level of cycles from the equivariant twisted K-homology of a compact, connected, simply connected Lie group $G$ to the Verlinde ring, which is inverse to the Freed–Hopkins–Teleman isomorphism. As an application, we prove that two of the proposed definitions of the quantization of a Hamiltonian loop group space—one via twisted K-homology of $G$ and the other via index theory on non-compact manifolds—agree with each other