Anticommuting Variables, Fermionic Path Integrals and Supersymmetry

TL;DR

This paper develops a rigorous mathematical framework for fermionic paths using anticommuting variables, applying stochastic calculus on supermanifolds to prove the Atiyah-Singer index theorem via supersymmetric methods.

Contribution

It introduces fermionic Brownian paths and stochastic calculus on supermanifolds, providing a rigorous proof of the Atiyah-Singer index theorem inspired by physics approaches.

Findings

Defined fermionic Brownian paths with anticommuting variables

Developed stochastic calculus on supermanifolds

Provided a rigorous proof of the Atiyah-Singer index theorem

Abstract

(Replacement because mailer changed `hat' for supercript into something weird. The macro `\sp' has been used in place of the `hat' character in this revised version.) Fermionic Brownian paths are defined as paths in a space para\-metr\-ised by anticommuting variables. Stochastic calculus for these paths, in conjunction with classical Brownian paths, is described; Brownian paths on supermanifolds are developed and applied to establish a Feynman-Kac formula for the twisted Laplace-Beltrami operator on differential forms taking values in a vector bundle. This formula is used to give a proof of the Atiyah-Singer index theorem which is rigorous while being closely modelled on the supersymmetric proofs in the physics literature.