# Topological Quantum Field Theory Seminar

### Differential geometry on the path space and applications

When trying to construct a Riemannian geometry on the path space of a Riemannian manifold several approaches could be thought about. The local chart approach, considering the path space as an infinite dimensional manifold and the basic tangent space the Cameron-Martin Hilbert space, leads to the study of the so-called Wiener-Riemann manifolds. Several difficulties appear in this study, namely the difficulty of finding an atlas such that the change of charts is compatible with the probabilistic structure (preserves the class of Wiener measures together with the Cameron-Martin type tangent spaces) and the non-availability of an effective computational procedure in the local coordinate system. Indeed, in infinite dimensions, the summation operators of differential geometry become very often divergent series. But the path space is more than a space endowed with a probability: time and the corresponding Itô filtration provide a much richer structure. In particular, the parallel transport over Brownian paths can be naturally defined by a limiting procedure from ODEs to SDEs. The stochastic parallel transport defines a canonical moving frame on the path space: the point of view we have adopted is the one of replacing systematically the machinery of local charts by the method of moving frames (as in Cartan theory). In this way it is possible to transfer geometrical quantities of the path space to the classical Wiener space and use Itô calculus to renormalize the apriori divergent expressions. An effective computational procedure is then achieved, where Stochastic Analysis and Geometry interact, not only on a technical level, but in a deeper way: Stochastic Analysis makes it possible to define geometrical quantities, Geometry implies new results in Stochastic Analysis. An application to assymptotics of the vertical derivatives of the heat kernel associated to the horizontal Laplacian on the frame bundle is discussed.
##### References
1. A. B. Cruzeiro and P. Malliavin -"Renormalized differential geometry on path space: structural equation, curvature", J.Funct. Anal. 139 (1996), p. 119 -181.
2. A. B. Cruzeiro, P. Malliavin and S. Taniguchi - "Ground state estimations in gauge theory", Bull Sci.Math. 125, 6-7 (2001), p. 623-640.