Gromov-Witten Invariants and Mirror Symmetry for Non-Fano Varieties Using Scattering Diagrams

Abstract

Gromov-Witten invariants arise in the topological A-model as counts of worldsheet instantons. On the A-side, these invariants can be computed for a Fano or semi-Fano toric variety using generating functions associated to the toric divisors. On the B-side, the same invariants can be computed from the periods of the mirror. We utilize scattering diagrams (aka wall structures) in the Gross-Siebert mirror symmetry program to extend the calculation of Gromov-Witten invariants to non-Fano toric varieties. Following the work of Carl-Pumperla-Siebert, we compute corrected mirror superpotentials \(\vartheta_1(\mathbb{F}_m)\) and their periods for the Hirzebruch surfaces \(\mathbb{F}_m\) with \(m \ge 2\).