category, and its relation to Chern-Simons theory was explored recently by Stirling [@St], inspired by earlier work by Belov-Moore [@BM]. In a different direction the Verlinde ring, which encodes the 2-dimensional reduction of Chern-Simons theory, was recognized by three of the authors as a twisted form of equivariant $K$-cohomology [@FHT]. That description, or rather its equivalent in $K$-homology, inspires the categories of skyscraper sheaves which we use to extend toral Chern-Simons to a 0-1-2-3 theory.

The gauge theories discussed in this paper have a classical description as pure gauge theories whose only field is a $G$-bundle with connection. On any fixed manifold $M$ the $G$-bundles with connection form a groupoid, and it is the underlying stack which should be regarded as the fields in the theory. When $M$ is a point, there is up to isomorphism a unique $G$-bundle with connection (the trivial one), but it has a group of automorphisms isomorphic to $G$. Thus the stack of $G$-bundles with connections is $*{/\!/}G=BG$. A cohomology class in $H^{n+1}(BG; {\mathbb Z})$, or rather the geometric manifestation which we describe (esp. \[subsec:1.4\] and \[subsec:3.1\]), defines the classical gauge theory on $M \simeq \operatorname{pt}$. We directly “quantize” this classical data over a point. 