  Feynman's time-slicing construction approximates the path integral by a
product, determined by a partition of a finite time interval, of approximate
propagators. This paper formulates general conditions to impose on a short-time
approximation to the propagator in a general class of imaginary-time quantum
mechanics on a Riemannian manifold which ensure these products converge. The
limit defines a path integral which agrees pointwise with the heat kernel for a
generalized Laplacian. The result is a rigorous construction of the propagator
for supersymmetric quantum mechanics, with potential, as a path integral.
Further, the class of Laplacians includes the square of the twisted Dirac
operator, which corresponds to an extension of N=1/2 supersymmetric quantum
mechanics. General results on the rate of convergence of the approximate path
integrals suffice in this case to derive the local version of the Atiyah-Singer
index theorem.
