Spinor Fields over Stochastic Loop Spaces

We give the construction of a line bundle over the based Brownian bridge, as well as the construction of spinor fields over the based and the free Brownian bridge. Introduction The Dirac operator over the free loop space is a very important object for the algebraic topology [24]; its index gives the Witten genus and it can predict the rigidity theorem of Witten: the index of some classical operator is rigid under a geometrical action of the circle over the manifold. Unfortunately, the Dirac operator over the free loop space is an hypothetical object. In [13], we have constructed an approximation of it by considering the Brownian measure over the loop space: why is a measure important? It is to compute the adjoint of the Dirac operator, the associated Laplacian, and the Hilbert space of spinors where it acts; the choice of physicists gives a hypothetical measure over the loop space. The purpose of [13] is to replace the formal measure of physicists by a well-defined measure, that is the Brownian bridge measure. The fiber of the Dirac operator is related to the Fourier expansion. After [25, 13] the Fourier expansion has extended in an invariant by rotation way for the natural circle action over the free loop space. Unfortunately, this works only for small loops. The problem to construct the spin bundle over the free loop space is now a wellunderstood problem in mathematics (see [14, 24, 26, 7, 6, 20]). In order to construct a suitable stochastic Dirac operator over the free loop space, it should be reasonable to define a Hilbert space of spinor fields over the free loop space where the operator acts. It should be nice to extend the previous work mentioned in the references above in the stochastic context. It is the subject of [17, 18] and [19]. The goal of this paper is to do a review of the results of [17, 18, 19]. In the first part, we study the problem to construct stochastic line bundles over the stochastic loop space: their transition functions are only almost surely defined. Therefore, we define the line bundle by its sections. If we consider the path space as a family of Brownian bridges, we meet the problem to glue together all the line bundles over the Brownian bridge into a line bundle over the Brownian motion. There is an obstruction which is measured in [5] for smooth loops. When the criterium of this obstruction is satisfied, the tools of the quasi-sure analysis allow one to restrict a smooth section of the line bundle over the Brownian motion into a section of the line bundle over the Brownian bridge. Moreover, if we consider the bundle associated to a given curvature (we neglect