is the subalgebra of central functions, a description which persists for all $G$.

Associated to a central extension ${\mathbb T}\to {G^\tau }\to G$ is a twisted complex group algebra ${{{\mathbb C}}^\tau [G]}$, the algebra of sections of ${K^\tau }\to G$ under convolution. Its center is again the commutative algebra of central functions.

Gauge theory with finite gauge group {#subsec:1.3}
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In an $n$-dimensional gauge theory with finite gauge group $G$ the path integral over a closed $n$-manifold $X$ reduces to a finite sum, and this sum defines the quantum invariant $F(X)$. The starting point of [@F2] is an expansion of this standard picture: the classical action can be extended to manifolds of dimension $<n$ and a higher categorical version of the path integral determines the quantum invariants. For any manifold $M$ of dimension $\le n$, the groupoid of $G$-bundles may be identified with the fundamental groupoid of the mapping space $\operatorname{Map}(M,BG)$. One can obtain a larger class of theories by replacing $BG$ with a topological space which has a finite number of nonzero homotopy groups, each of which is a finite group [@Q]. At least heuristically, the path integrals on manifolds of dimension $\le n$ define an extended field theory 