Path integration, anticommuting variables, and supersymmetry

A geometric theory of path integration on supermanifolds is developed, and used to establish a path‐integral formula for the super heat kernel exp(−D/2t−D/τ) of the Dirac operator D/ on a Riemannian manifold. The earlier part of the paper describes an extension of Wiener measure, Brownian motion, and stochastic calculus to include paths in spaces parametrized by anticommuting variables; these constructions are then combined with the geometrical data of a Riemannian manifold to construct stochastic differential equations whose solutions provide appropriate superpaths for the study of the Dirac operator.