We establish a homology relation for the Deligne–Mumford moduli spaces of real curves which lifts to a Witten–Dijkgraaf–Verlinde–Verlinde (WDVV)-type relation for a class of real Gromov–Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus real invariants with conjugate pairs of constraints to genus invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all cases when they are nonzero and thus provide nontrivial lower bounds in high-dimensional real algebraic geometry. We also show that the real invariants of the -dimensional projective space with conjugate point constraints are congruent to their complex analogues modulo .
Penka Georgieva. Aleksey Zinger. "Enumeration of real curves in and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants." Duke Math. J. 166 (17) 3291 - 3347, 15 November 2017. https://doi.org/10.1215/00127094-2017-0023