Institut de Mathematiques,

Faculte des Sciences,

Universite de Bourgogne,

21000 Dijon,

FRANCE

E-mail: Remi.leandre@u-bourgogne.fr

**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.