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

Consider the differential forms $A^{\ast }(L)$ on a Lagrangian submanifold $L \subset X$. Following ideas of Fukaya-Oh-Ohta-Ono, we construct a family of cyclic unital curved $A_{\mathrm{\infty }}$ structures on $A^{\ast }(L),$ parameterized by the cohomology of $X$ relative to $L\mathrm{.}$ The family of $A_{\mathrm{\infty }}$ structures satisfies properties analogous to the axioms of Gromov-Witten theory. Our construction is canonical up to $A_{\mathrm{\infty }}$ 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.