Fukaya’s conjecture on Witten’s twisted $A_\infty$ structure

Abstract

The wedge product on de Rham complex of a Riemannian manifold $M$ can be pulled back to $H^\ast (M)$ via explicit homotopy constructed by using Green’s operator which gives higher product structures. We prove Fukaya’s conjecture which suggests that Witten deformation of these higher product structures has semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of de Rham differential from a statement concerning homology to one concerning real homotopy type of $M$. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya in [6].