Point-like non-commutative families of bounding cochains
Elad Kosloff, Jake P. Solomon
TL;DR
This work develops a framework for genus-zero open Gromov-Witten invariants with boundary and interior constraints for even-dimensional Lagrangians by introducing point-like, non-commutative families of bounding cochains. It builds a robust algebraic backbone via A$_\infty$ algebras over associative (possibly noncommutative) bases, extending scalars with a non-commutative parameter while preserving cyclicity and unitality. An obstruction theory in twisted cohomology is developed, with a novel pseudo-completeness property ensuring convergence of the Maurer–Cartan equation in these extended settings; a spectral sequence then computes the twisted cohomology that controls obstructions. The paper also analyzes real settings with anti-symplectic involutions, showing how the resulting invariants relate to Welschinger counts in dimension two and providing gauge-invariance and pseudoisotopy tools to define well-behaved open GW invariants in general. Together, these advances enable nonvanishing open invariants with arbitrary numbers of boundary constraints and pave the way for computational approaches (including a real-setting theory) in higher dimensions.
Abstract
We define genus zero open Gromov-Witten invariants with boundary and interior constraints for a Lagrangian submanifold of arbitrary even dimension. The definition relies on constructing a canonical family of bounding cochains that satisfy the point-like condition of the second author and Tukachinsky. Since the Lagrangian is even dimensional, the parameter of the family is odd. Thus, to avoid the vanishing of invariants with more than one boundary constraint, the parameter must be non-commutative. The invariants are defined either when the Lagrangian is a rational cohomology sphere or when the Lagrangian is fixed by an anti-symplectic involution, has dimension $2$ modulo $4$, and its cohomology is that of a sphere aside from degree $1$ modulo $4$. In dimension $2$, these invariants recover Welschinger's invariants. We develop an obstruction theory for the existence and uniqueness of bounding cochains in a Fukaya $A_\infty$ algebra with non-commutative coefficients. The obstruction classes belong to twisted cohomology groups of the Lagrangian instead of the de Rham cohomology of the commutative setting. A spectral sequence is constructed to compute the twisted cohomology groups. The extension of scalars of an $A_\infty$ algebra by a non-commutative ring is treated in detail. A theory of pseudo-completeness is introduced to guarantee the convergence of the Maurer-Cartan equation, which defines bounding cochains, even though the non-commutative parameter is given zero filtration.