WDVV-type relations for disk Gromov–Witten invariants in dimension 6

Abstract

The first author’s previous work established Solomon’s WDVV-type relations for Welschinger’s invariant curve counts in real symplectic fourfolds by lifting geometric relations over possibly unorientable morphisms. We apply her framework to obtain WDVV-style relations for the disk invariants of real symplectic sixfolds with some symmetry, in particular confirming Alcolado’s prediction for \({\mathbb {P}}^3\) and extending it to other spaces. These relations reduce the computation of Welschinger’s invariants of many real symplectic sixfolds to invariants in small degrees and provide lower bounds for counts of real rational curves with positive-dimensional insertions in some cases. In the case of \({\mathbb {P}}^3\), our lower bounds fit perfectly with Kollár’s vanishing results.