Notes on genus one real Gromov-Witten invariants

In this paper, we propose a definition of genus one real Gromov-Witten invariants for certain symplectic manifolds with real a structure, including Calabi-Yau threefolds, and use equivariant localization to calculate certain genus one real invariants of the projective space. For this definition, we combine three moduli spaces corresponding to three possible types of involutions on a symplectic torus, by gluing them along common boundaries, to get a moduli space without codimension-one boundary and then study orientation of the total space. Modulo a technical conjectural lemma, we can prove that the result is an invariant of the corresponding real symplectic manifold. In the aforementioned example, our main motivation is to show that the physicists expectation for the existence of separate Annulus, Mobius, and Klein bottle invariants may not always be true.