Differential forms on loop spaces and the cyclic bar complex

Abstract

In this article, we present a model for the differential graded algebra (dga) of differential forms on the free loop space LX of a smooth manifold X and show how to construct certain important differential forms in terms of this model. Our motivation is an observation of Witten (described by Atiyah in [2]) that the index theorem for the Dirac operator can be thought of as an application of the localization (or fixed point) theorem in T-equivariant homology, suitably generalised to the infinite dimensional case of the free loop space; here, T is the circle group. We can summarise our main results as follows. We are really concerned with equivariant differential forms and equivariant currents and we show how to reformulate these geometric objects as cyclic chains and cochains over the differential graded algebra Ω(X) of differential forms on X. In fact, the cyclic chain complex of Ω(X), if it is normalized correctly, is a sub-complex of the complex of equivariant differential forms on LX, and is a good enough approximation that it allows us to compute the ordinary and equivariant cohomology of LX, and to write down explicitly certain important differential forms and currents. We begin by explaining the motivation more carefully. Let S be a Clifford module on X with Dirac operator D. Witten observed that it should be possible to associate to D an equivariantly closed (inhomogeneous) current μD on the free loop space of X. The basic property of this current is that the index of D is given by pairing μD with the differential form 1 ∈ Ω(LX):