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There are two types of $J$-holomorphic spheres in a symplectic manifoldinvariant under an anti-symplectic involution: those that have a fixed pointlocus and those that do not. The former are described by moduli spaces of$J$-holomorphic disks, which are well studied in the literature. In this paper,we first study moduli spaces describing the latter and then combine the twotypes of moduli spaces to get a well-defined theory of counting real curves ofgenus 0. We use equivariant localization to show that these invariants (unlikethe disk invariants) are essentially the same for the two (standard)involutions on $\mathbb{P}^{4n-1}$.

Counting genus zero real curves in symplectic manifolds