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

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