  We give a rigorous construction of the path integral in N=1/2 supersymmetry
as an integral map for differential forms on the loop space of a compact spin
manifold. It is defined on the space of differential forms which can be
represented by extended iterated integrals in the sense of Chen and
Getzler-Jones-Petrack. Via the iterated integral map, we compare our path
integral to the non-commutative loop space Chern character of G\"uneysu and the
second author. Our theory provides a rigorous background to various formal
proofs of the Atiyah-Singer index theorem using supersymmetric path integrals,
as investigated by Alvarez-Gaum\'e, Atiyah, Bismut and Witten.
