Differential forms, Fukaya $A_\infty$ algebras, and Gromov-Witten axioms

TL;DR

This paper constructs a family of cyclic unital curved $A_infty$ structures on differential forms of a Lagrangian submanifold, satisfying Gromov-Witten axioms without virtual fundamental class techniques.

Contribution

It introduces a canonical family of $A_infty$ structures on $A^*(L)$ parameterized by relative cohomology, satisfying Gromov-Witten axioms, avoiding virtual fundamental class methods.

Findings

Constructs $A_infty$ structures on differential forms of Lagrangian submanifolds.

Satisfies properties analogous to Gromov-Witten axioms.

Operates without virtual fundamental class, assuming regular moduli spaces.

Abstract

Consider the differential forms $A^*(L)$ on a Lagrangian submanifold $L \subset X$. Following ideas of Fukaya-Oh-Ohta-Ono, we construct a family of cyclic unital curved $A_\infty$ structures on $A^*(L),$ parameterized by the cohomology of $X$ relative to $L.$ The family of $A_\infty$ structures satisfies properties analogous to the axioms of Gromov-Witten theory. Our construction is canonical up to $A_\infty$ pseudoisotopy. We work in the situation that moduli spaces are regular and boundary evaluation maps are submersions, and thus we do not use the theory of the virtual fundamental class.