Differential forms, Fukaya $A_\infty$ 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 $A_\infty$ structures on $A^\ast (L)$, 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.

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Received 9 December 2020

Accepted 26 October 2021

Published 16 March 2023