SUPERSYMMETRY AND BROWNIAN MOTION ON SUPERMANIFOLDS

Abstract

An anticommuting analogue of Brownian motion, corresponding to fermionic quantum mechanics, is developed, and combined with classical Brownian motion to give a generalised Feynman-Kac-Itô formula for paths in geometric supermanifolds. This formula is applied to give a rigorous version of the proofs of the Atiyah-Singer index theorem based on supersymmetric quantum mechanics. After a discussion of the BFV approach to the quantization of theories with symmetry, it is shown how the quantization of the topological particle leads to the supersymmetric model introduced by Witten in his study of Morse theory.