Open Gromov–Witten disk invariants in the presence of an anti-symplectic involution

For a symplectic manifold with an anti-symplectic involution having non-empty fixed locus, we construct a model of the moduli space of real sphere maps out of moduli spaces of decorated disk maps and give an explicit expression for its first Stiefel–Whitney class. As a corollary, we obtain a large number of examples, which include all odd-dimensional projective spaces and many complete intersections, for which many types of real moduli spaces are orientable. For these manifolds, we define open Gromov–Witten invariants with no restriction on the dimension of the manifolds or the type of the constraints if there are no boundary marked points; a WDVV-type recursion obtained in a sequel computes these invariants for many real symplectic manifolds. If there are boundary marked points, we define the invariants under some restrictions on the allowed boundary constraints, even though the moduli spaces are not orientable in these cases.