Open WDVV equations for non-commutative families of bounding cochains

Abstract

We derive the noncommutative open WDVV equations for the open Gromov-Witten superpotential introduced in joint work of the author and Solomon, extending the Solomon-Tukachinsky equations to noncommutative differential graded algebras. The resulting recursive formulas facilitate the computation of open Gromov-Witten invariants with multiple boundary point constraints for even-dimensional Lagrangians. We apply the open WDVV equations to compute all open Gromov-Witten invariants for the real loci of even-dimensional complex quadrics of dimension greater than two. These computations show that open Gromov-Witten invariants with multiple boundary point constraints can be nontrivial in arbitrary dimensions. We also construct a character for a natural subalgebra of big quantum cohomology associated to a Lagrangian endowed with a point-like bounding cochain. The character is in the spirit of the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono. Applying the character construction to the even-dimensional quadric illustrates how point-like bounding cochains refine the information about closed quantum cohomology obtained from weak bounding cochains of Floer theory. Open Gromov-Witten invariants with three boundary point constraints determine the two extensions of the character from the subalgebra to big quantum cohomology.