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Equivariant de Rham cohomology is extended to the infinite-dimensional setting of a loop subgroup acting on a loop group, using Hida supersymmetric Fock space for the Weil algebra and Malliavin test forms on the loop group. The Mathai–Quillen isomorphism (in the BRST formalism of Kalkman) is defined so that the equivalence of various models of the equivariant de Rham cohomology can be established.

The standard BRST procedure for canonical quantization of a system with symmetry generated by a closed constraint algebra is extended to the case where the algebra is partly open. The method is illustrated by a specific topological model.

Various analytic results which combine fermionic Brownian motion with stochastic integration are described, and it is shown that a wide class of stochastic differential equations in superspace have solutions. Such solutions are then used to derive a Feynman-Kac formula for a supersymmetric system in terms of the supercharge whose square is the Hamiltonian of the system. This is achieved by introducing superpaths parametrized by a commuting and an anticommuting time variable.

A new $(1,1)$-dimensional super vector bundle which exists on any super Riemann surface is described. Cross-sections of this bundle provide a new class of fields on a super Riemann surface which closely resemble holomorphic functions on a super Riemann surface, but which (in contrast to the case with holomorphic functions) form spaces which have a well defined dimension which does not change as odd moduli become non-zero., 12pp, kcl-th-94-7