A simple proof of a theorem of Fukaya and Oh

Abstract

We study the moduli space of pseudo pointed holomorphic disks with boundaries mapped in the zero section of the cotangent bundle of a manifold. We define perturbations of the equation for which it is possible to describe explicitly all the solutions of the problem in terms of Morse graphs on the manifold. In particular, this proves that the $A_\infty$ structure of the zero section of the cotangent bundle is equivalent to the Morse $A_\infty$ structure of the base manifold.