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

A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold