Open Gromov-Witten invariants in dimension four

Abstract

Given a closed orientable Lagrangian surface L in a closed symplectic four-manifold X together with a relative homology class d in H_2 (X, L; Z) with vanishing boundary in H_1 (L; Z), we prove that the algebraic number of J-holomorphic discs with boundary on L, homologous to d and passing through the adequate number of points neither depends on the choice of the points nor on the generic choice of the almost-complex structure J. We furthermore get analogous open Gromov-Witten invariants by counting, for every non-negative integer k, unions of k discs instead of single discs.