Verlinde modules and quantization

TL;DR

This paper constructs categories related to Lie groups and twists to define Verlinde modules and demonstrates their quantization, linking geometric K-homology with the Verlinde algebra through an equivalence relation.

Contribution

It introduces the categories C(G, h) and D(G, h) with May structures, establishing a framework for quantizing Verlinde modules via geometric K-homology.

Findings

Defined the monoidal categories C(G, h) and D(G, h) with May structures.

Established a functorial relationship linking geometric K-cycles to Verlinde modules.

Proved an algebra isomorphism between geometric equivariant twisted K-homology and the Verlinde algebra.

Abstract

Given a compact simple Lie group G and a primitive degree 3 twist h, we define a monoidal category C(G, h) with a May structure. An object in the category C(G, h) is a pair (X, f), where X is a compact G-manifold and f a smooth G-map from X to G with respect to the conjugation action of G on itself. Such an object determines a module, the equivariant twisted K-homology K^G(X, f^*(h)), for the Verlinde algebra, termed a Verlinde module, where the module action is induced by the G-action on X. In order to understand which objects in C(G, h) can be quantized, we define the closely related monoidal category D(G, h) consisting of equivariant twisted geometric K-cycles, which also has a May structure. There is a forgetful functor from D(G, h) to C(G, h), showing that an object in D(G, h) determines a Verlinde module. Every object in the category D(G, h) also has a quantization, valued in the Verlinde algebra. Finally, an equivalence relation ~ is defined on objects in D(G, h) such that the quantization functor determines an algebra isomorphism between the geometric equivariant twisted K-homology groups K^G_{geo}(G, h) = D(G, h))/~, and the Verlinde algebra.