Rogers Alice 
Stochastic Calculus in Superspace II: Differential forms, supermanifolds and the Atiyah-Singer
index theorem
(58K, TeX)

ABSTRACT.  Starting with vector bundles over manifolds, supermanifolds are
constructed whose function algebras correspond to twisted differential
forms. Stochastic calculus for bosonic and fermionic Brownian
paths is used to provide a geometric construction of Brownian paths on
these supermanifolds. A Feynman-Kac formula for the heat kernel of the
Laplace-Beltrami operator is then derived. This is used to provide a
simple, rigorous version of the supersymmetric proofs of the
Atiyah-Singer index theorem.