path integral reduces to an integral over $G$. The 2-dimensional case (§\[sec:2\]) exhibits more features. In §\[sec:3\] we discuss the 3-dimensional gauge theory with finite gauge group. On the one hand this is an application of the finite path integral and on the other a warm-up for the treatment in later sections of torus groups. Section \[sec:c1\] is a set of variations on ‘algebra’ of increasing (categorical) complexity. It provides some underpinnings for our discussion of the anomaly theory ${\mathscr{A}}$ and Chern-Simons theory.

As befits a conference proceedings, our presentation here emphasizes the big picture and our speculation is uninhibited; many details are not worked out.

The recent paper [@BN] generalizes the 3-dimensional untwisted Chern-Simons theory for finite groups in a different direction and gives applications to representation theory. Also, Bartels, Douglas, and Henriques have recently announced a construction of Chern-Simons theory for 1-connected compact Lie groups [@D] which uses conformal nets and the cobordism hypothesis.

Raoul Bott was an inspiration to each of us, both personally and through his mathematics. We offer this paper as a tribute.

The cohomology group $H^2(BG; {\mathbb Z})$ and one-dimensional theories {#sec:1}
========================================================================

To begin, we recall the definition of a topological quantum field 