Description
We explore the interplay between non-Fano toric varieties and their mirror Landau-Ginzburg (LG) superpotentials. The mirror LG model is described by the Hori-Vafa potential, modified by corrections arising from tropical disks. These corrections are such that the classical period of the LG superpotential is equal to the quantum period of the non-Fano toric variety, extending mirror symmetry to this broader setting (arXiv:2404.16782). The corrected LG superpotential satisfies a new Picard-Fuchs system, offering new insights into the arithmetic properties of non-Fano toric varieties. We introduce ToricZeta, a software package in development for computing zeta functions of Calabi-Yau hypersurfaces in (possibly non-Fano) toric varieties. Finally, we outline new research directions in arithmetic and modularity made possible by these advancements.
