Characteristic Numbers and Generalized Path Integrals

This is an extended write-up of a talk given in April, 1993 in honor of Raoul Bott's 70th birthday. We first illustrate how some traditional topological and geometric invariants obey ``gluing laws'' inspired by those in classical and quantum field theory. Here we discuss characteristic numbers, particularly the Euler number of a complex line bundle over an oriented surface. In the second part of the paper we show how path integrals give rise to invariants which obey gluing laws. The argument is formal in general, but for a particular example of invariants related to finite groups it is rigorous. In that ``toy model'' we show how to go further and generalize the path integral to obtain more exotic gluing laws. This idea was used--at least in this toy model--to directly construct quantum groups from 3 dimensional Chern-Simons invariants.