Gromov-Witten Invariants and Mirror Symmetry For Non-Fano Varieties Via Tropical Disks

Abstract

Under mirror symmetry a non-Fano variety $X$ corresponds to an instanton corrected Hori-Vafa potential $W$. The classical period of $W$ equals the regularized quantum period of $X$, which is a generating function for descendant Gromov-Witten invariants. These periods define closed mirror maps relating complex with symplectic parameters and open mirror maps relating coordinates on the mirror curves. We interpret the corrections to $W$ by broken lines in a scattering diagram, so that $W$ is the primitive theta function ${\vartheta }_1$. We show that, after wall crossing to infinity and application of the closed mirror map, $W={\vartheta }_1$ is equal to the open mirror map. By tropical correspondence, ${\vartheta }_1$ is a generating function for $2$-marked logarithmic Gromov-Witten invariants, which are algebraic analogues of counts of Maslov index $2$ disks. This generalizes the predictions of mirror symmetry to the non-Fano case.

38 pages, 18 figures