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

Abstract

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.

51 pages, 6 figures; corrected minor errors, updated references

References in corpus (5)

• Point-like bounding chains in open Gromov-Witten theory
• Relative quantum cohomology
• The moduli space of regular stable maps
• Moduli of Open Stable Maps to a Homogeneous Space
• Differential forms on orbifolds with corners

Cited by in corpus (7)

• Relative quantum cohomology
• Point-like bounding chains in open Gromov-Witten theory
• Family Floer program and non-archimedean SYZ mirror construction
• Involutions, obstructions and mirror symmetry
• Differential forms on orbifolds with corners
• Open Gromov-Witten invariants from the Fukaya category
• Relative quantum cohomology of the Chiang Lagrangian