by the field theory $F$ and the self-duality of $A$ provided by the bilinear form $b$, we can also identify $F(S^1)$ with the orthogonal $[A,A]^{\perp}$ of $[A,A]$ with respect to $b$: that is, with the center of the algebra $A$.

In particular, if $G$ is a finite group and $\eta \in H^3(G; {{\mathbb Z}})$ is a cohomology class, then the twisted group algebra ${{{\mathbb C}}^\tau [G]}$ admits a canonical trace (obtained by taking the coefficient of the unit in $G$ and dividing by the order of $G$). Applying Theorem \[bdo\] in this case, we obtain another construction of the topological field theory described in \[subsec:1.3\].

Finite path integrals {#sec:5}
=====================

We sketch an idea to construct extended field theories via finite sums.

\[thm:20\] Integration over a finite field, such as a gauge field with finite gauge group, sometimes occurs in a quantum field theory with other fields. These cases also fit into this framework. Examples include orbifolds and orientifolds in string theory.

As in [@L §3.2] let ${\operatorname{Fam}_n}$ denote the $n$-category whose objects are finite groupoids $X$. (A groupoid $X$ is finite if there is a finite number of inequivalent objects and each object has a finite automorphism group.) A 