Loop Space and Deformations of De Rham Cohomology

We show that the 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. A currently very active eld in Mathematics is the theory of quantum cohomol-ogy, which provides a special kind of deformations of the multiplication structure on the de Rham cohomology of symplectic manifolds. The mathematical formulation of quantum cohomology relies on the study of (pseudo-)holomorphic curves, but the origin of this theory lies in string theory. Classical physical theories treat a particle as a point moving in a spacetime which is a Lorentzian manifold, and trajectories of particles are given by one-dimensional objects such as geodesics. In string theory, a particle is described by a one-dimensional object: if it is an interval , then the theory is called open string theory; if it is a circle, then the theory is called closed string theory. The set of all closed strings consists of smooth maps from the circle to the background manifold, i.e., it is the free loop space of the manifold. The trajectory of a string is a smooth map from a Riemann surface to the background manifold which minimizes some action functional. When the background is a symplectic manifold, one is then led to the pseudo-holomorphic curves. From this rather crude description, it is clear that quantum cohomology is closely related to the geometry and topology of loop spaces. Indeed, from the loop spaces of symplectic manifolds, Floer cohomology, a version of Morse cohomology, can be deened, and it can be identiied with quantum cohomology. In this paper, we show that the deformations of the de Rham cohomology of a class of smooth manifolds (including closed simply connected KK ahler manifolds) are governed by the dual of real-valued cohomology of their free loop spaces. In general, the algebraic deformation theory of a real associative graded algebra A with unit is governed by its graded Hochschild cohomology HH (A). However, when A has a Frobenius algebra structure, i.e. a nondegenerate symmetric bilinear form, then we have an isomorphism A = A which induces an isomorphism HH (A) = HH (A) , where HH (A) is a version of graded Hochschild homology. If A is the cohomology of a diierential graded algebra (DGA) (B; d), then there is a version of DGA Hochschild homology HH (B; d) and there is a …