CHERN-SIMONS THEORY WITH FINITE GAUGE GROUP

We construct in detail a 2 + 1 dimensional gauge field theory with fin ite gauge group. In this case the path integral reduces to a finite su m, so there are no analytic problems with the quantization. The theory was originally introduced by Dijkgraaf and Witten without details. Th e point of working it out carefully is to focus on the algebraic struc ture, and particularly the construction of quantum Hilbert spaces on c losed surfaces by cutting and pasting. This includes the ''Verlinde fo rmula.'' The careful development may serve as a model for dealing with similar issues in more complicated cases.