In the first part of the talk, I will explain the connections between the heat equation proof of the index theorem for Dirac operators, and the localization formulas of Duistermaat-Heckman. In particular, I will show how to pass from integration with respect to the Brownian measure on the loop space of a Riemannian manifold to integration of differential forms on this loop space.

In the above situation, the geometry of the loop space is associated with its L2-Riemannian metric. In a second part of the talk, I will show how replacing the L2-metric by a H1-metric determines a new measure on the loop space, which corresponds to a geometric Langevin process, whose generator is a hypoelliptic operator on the total space of the tangent bundle.

I will naturnally explain how the above suggests the possibility of deforming the elliptic Dirac operator to a family of hypoelliptic Dirac operators.