Open Gromov-Witten invariants in dimension six

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X,\omega )$. We assume that the first homology group $H_1 (L ; A)$withcoefficientsinacommutativering$A$injectsintothegroup$H_1 (X ; A)$andthat$X$ 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 $d \in H_2 (X, L ; \Z)$andadequatenumberofincidenceconditionsin$L$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 $A$.