Differential forms, Fukaya A_∞ algebras, and Gromov-Witten axioms

Consider the differential forms A^∗(L) on a Lagrangian submanifold L ⊂ X. Following ideas of Fukaya-Oh-Ohta-Ono, we construct a family of cyclic unital curved A_∞ structures on A^∗(L), parameterized by the cohomology of X relative to L. The family of A_∞ structures satisfies properties analogous to the axioms of GromovWitten theory. Our construction is canonical up to A_∞ 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.