Naturality of FHT isomorphism

TL;DR

This paper constructs modified functors to achieve naturality of the FHT isomorphism between twisted equivariant K-theory and the Verlinde ring for certain group homomorphisms, extending previous results.

Contribution

It introduces quasi functors that make the FHT isomorphism natural for tori and extends these constructions to broader classes of Lie groups.

Findings

Constructed quasi functors t.e.K and RL for tori.

Proved natural isomorphism among three quasi functors.

Extended the framework to Lie groups with torsion-free fundamental groups.

Abstract

Freed, Hopkins and Teleman constructed an isomorphism between twisted equivariant K-theory of compact Lie group $G$ and the "Verlinde ring" of the loop group of $G$. We call this isomorphism FHT isomorphism. However, it does not hold naturality with respect to group homomorphisms. We construct two "quasi functors" $t.e.K$ (a modification of twisted equivariant K-theory) and $RL$ (a modification of representation group of loop groups) so that FHT isomorphism is natural transformation between two "quasi functors" for tori, that is, we construct two "induced homomorphisms" of the "quasi functors" $t.e.K$ and $RL$ for a group homomorphism whose tangent map is injective between two tori. In fact, we construct another quasi functor $char$ and verify that three quasi functors are naturally isomorphic. Moreover, we extend the quasi functor $t.e.K$ and $char$ to compact connected Lie group with torsion-free fundamental group and group homomorphism satisfying "the decomposable condition", and verify that they are isomorphic. This is a generalization of a result in [FHT1].