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.

arXiv comment70 pages, 1 figure, 1 table; some updates and modifications in exposition. arXiv admin note: text overlap with arXiv:1809.08919

References in corpus (6)

• Gromov-Witten classes, quantum cohomology, and enumerative geometry
• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Counting genus zero real curves in symplectic manifolds
• Point-like bounding chains in open Gromov-Witten theory
• Relative quantum cohomology
• Enumeration of real curves in CP^{2n-1} and a WDVV relation for real Gromov-Witten invariants

Cited by in corpus (9)

• Intersection theory on moduli of disks, open KdV and Virasoro
• Relative quantum cohomology
• WDVV-Type Relations for Welschinger's Invariants: Applications
• Steenrod Pseudocycles, Lifted Cobordisms, and Solomon's Relations for Welschinger's Invariants
• Solomon-Tukachinsky's vs. Welschinger's Open Gromov-Witten Invariants of Symplectic Sixfolds
• A construction of open descendant potentials in all genera
• Relative quantum cohomology of the Chiang Lagrangian
• A Geometric Depiction of Solomon-Tukachinsky's Construction of Open GW-Invariants
• Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds