A simple proof of a theorem of Fukaya and Oh

TL;DR

This paper provides a straightforward proof connecting the $A_ ablafty$ structure of the zero section in a cotangent bundle to the Morse $A_ ablafty$ structure of the base manifold, using moduli space analysis.

Contribution

It offers a simple proof of a theorem by Fukaya and Oh, explicitly relating holomorphic disk moduli spaces to Morse graphs, clarifying the $A_ ablafty$ structures involved.

Findings

The $A_ ablafty$ structure of the zero section matches the Morse $A_ ablafty$ structure of the base.

Explicit description of solutions via Morse graphs.

Perturbation techniques enable explicit solution characterization.

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.