Index theorem and equivariant cohomology on the loop space (Q1074932)

The paper deals with the description of the index of the Dirac operator on the spin complex twisted by a complex vector bundle as an integral of a form on the loop space of the manifold. A conjectural formula is deduced using methods of the author's paper on a probabilistic proof of the index theorem [J. Functional Anal. 57, 56-99 (1984; Zbl 0538.58033); ibid. 57, 329-348 (1984; Zbl 0556.58027)]. This extends ideas of \textit{M. Atiyah} [Astérisque 131, 43-59 (1985; Zbl 0578.58039)] and \textit{E. Witten} [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)]. In addition an extension of the Chern character to an equivariant cohomology class on the loop space of a manifold is constructed. It is shown that for a compact smooth manifold with finite fundamental group each non-zero homotopy class of free loops has measure 0 for the index measure associated to the index problem.