Equivariant Cohomology of Loop Space and Frobenius Deformations of De Rham Cohomology

We show that the graded Frobenius formal deformations of the de Rham cohomology of a closed simply connected KK ahler manifold are governed by the dual of real-valued cohomology of its free loop space. This is a sequel to 9] in which the author showed that the deformations of the de Rham cohomology of a closed simply connected KK ahler manifold are governed by the dual of the real-valued cohomology of its free loop space. We refer to that paper for motivations and background. In this paper, we are going to examine a restricted class of deformations. Recall that the de Rham cohomology of a closed oriented smooth manifold has a Frobenius algebra structure. Deformations which preserve this structure are called Frobenius deformations. As pointed out by Connes-Flato-Sternheimer 3] in the context of deformations of the function algebra of a symplectic manifold, Frobenius deformations are governed by the cyclic cohomology. See also Penkava-Schwarz 8]. As in 9], we will relate the cyclic cohomology of the de Rham cohomology to the DGA cyclic homology of the de Rham algebra. Then by a result due to Getzler-Jones-Petrack 5], the latter is related to the equivariant cohomology of the free loop space of the manifold. For this paper to be more self-contained, we review a lot of preliminary deeni-tions and results. The rest of the paper is arranged as follows. We review in x1 the deenitions of cyclic cohomology of an algebra. The notion of cyclic modules due to Connes 2] is reviewed in x2. The deenition of graded Frobenius formal deformations and their relationship with graded cyclic cohomology and cyclic homology are reviewed in x3. A key result in x3 is the identiication of the graded cyclic cohomology used in Penkava-Schwarz 8], with the cyclic cohomology of Getzler-Jones-Petrack 5]. In x4, DGA cyclic homology is studied. In x5, we prove our main results (Theorem 5.2 and Corollary 5.1).