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

TL;DR

This paper constructs a model for real sphere maps in symplectic manifolds with anti-symplectic involutions, establishing orientability conditions and defining open Gromov-Witten invariants without dimension restrictions.

Contribution

It introduces a new model for moduli spaces of real sphere maps and provides explicit conditions for orientability, enabling the definition of open Gromov-Witten invariants in broader settings.

Findings

Many real moduli spaces are orientable, including all odd-dimensional projective spaces.

Explicit expression for the first Stiefel-Whitney class of the moduli space.

Open Gromov-Witten invariants are defined without restrictions on dimension or constraints, under certain conditions.

Abstract

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. 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.