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Earlier results show that the N = 1/2 supersymmetric path integral on aclosed even dimensional Riemannian spin manifold (X,g) can be constructed in amathematically rigorous way via Chen differential forms and techniques fromnon-commutative geometry, if one considers it as a current on the smooth loopspace of X. This construction admits a Duistermaat-Heckman localizationformula. In this note, fixing a topological spin structure on X, we prove thatany smooth family of Riemannian metrics on X canonically induces a Chern-Simonscurrent which fits into a transgression formula for the supersymmetric pathintegral. In particular, this result entails that the supersymmetric pathintegral induces a differential topological invariant on X, which essentiallystems from the A-hat-genus of X.

A Chern-Simons transgression formula for supersymmetric path integrals on spin manifolds