White noise analysis and non-commutative differential geometry

Abstract:
We patch together the tools of white noise analysis and non-commutative differential geometry in order
- to define a Feynman path integral on a manifold;
- to rigorize the works of Bismut (after pionneering works of Atiyah)  relating the structure of the loop space and the Index theorem, for a single Dirac operator or a family of Dirac operators;
- to define the speed of the Brownian motion on a manifold;
- to define the J.L.O. cocycle for a family of Dirac operators as a white noise distribution.