of $G$-bundles over ${\ensuremath{S^1}}$ is equivalent to $G{/\!/}G$, where $G$ acts on itself by conjugation. The value of the classical theory $I$ is given by the central extension ${G^\tau }\to G$, viewed as an equivariant principal $G$-bundle. In other words, the value $I({\ensuremath{S^1}},P)$ of this theory at a bundle with holonomy $x\in G$ is the circle torsor ${G^\tau }_x$ (together with the action of the centralizer of $x$, the automorphisms of $P$, by conjugation in the central extension.) In the quantum theory we make a sum of the corresponding hermitian lines $K_P$ analogous to  and so compute that $F({\ensuremath{S^1}})$ is the vector space of central sections of ${K^\tau }\to G$: $$\label{eq:32}
     F({\ensuremath{S^1}}) = \bigoplus\limits_{P}\,\frac{1}{\#\operatorname{Aut}P}\,(K_P)^{\operatorname{Aut}P}$$ In this expression the metric on the hermitian line $K_x$ is scaled by the prefactor.

The quantum “path integrals”  and  may be expressed in categorical language. Let ${\textnormal{Bord}^{SO}_2(G)}$ denote the bordism category whose objects are closed oriented 1-manifolds equipped with a principal $G$-bundle and whose morphisms are compact oriented 2-manifolds equipped with a principal $G$-bundle. Then the classical topological theory $I$ is a 2-dimensional topological quantum field theory in the sense of Definition \[thm:2\] with a special feature: its values are *invertible*. Thus the numbers attached 