Open Gromov-Witten invariants in dimension six

TL;DR

This paper establishes the invariance of certain counts of holomorphic discs with boundary on a Lagrangian in a six-dimensional symplectic manifold, defining open Gromov-Witten invariants under specific conditions.

Contribution

It proves the existence of well-defined open Gromov-Witten invariants in dimension six for Lagrangian submanifolds under particular topological and geometric assumptions.

Findings

Invariance of disc counts under generic almost-complex structures

Definition of open Gromov-Witten invariants in six dimensions

Conditions ensuring independence from auxiliary choices

Abstract

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X, \omega)$. We assume that the first homology group $H_1 (L ; A)$ with coefficients in a commutative ring $A$ injects into the group $H_1 (X ; A)$ and that $X$ contains no Maslov zero pseudo-holomorphic disc with boundary on $L$. Then, we prove that for every generic choice of a tame almost-complex structure $J$ on $X$, every relative homology class $d \in H_2 (X, L ; \Z)$ and adequate number of incidence conditions in $L$ or $X$, the weighted number of $J$-holomorphic discs with boundary on $L$, homologous to $d$, and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of $J$, provided that at least one incidence condition lies in $L$. These numbers thus define open Gromov-Witten invariants in dimension six, taking values in the ring $A$.