Open Gromov-Witten invariants in dimension six

Let L be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold (X,ω). We assume that the first homology group H1(L;A) with coefficients in a commutative ring A injects into the group H1(X;A) and that X contains no Maslov zero pseudoholomorphic 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 ∈ H2(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 GromovWitten invariants in dimension six, taking values in the ring A.