  Certain phase space path integrals can be evaluated exactly using equivariant
cohomology and localization in the canonical loop space. Here we extend this to
a general class of models. We consider hamiltonians which are {\it a priori}
arbitrary functions of the Cartan subalgebra generators of a Lie group which is
defined on the phase space. We evaluate the corresponding path integral and
find that it is closely related to the infinitesimal Lefschetz number of a
Dirac operator on the phase space. Our results indicate that equivariant
characteristic classes could provide a natural geometric framework for
understanding quantum integrability.
