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

TL;DR

This paper extends mirror symmetry to non-Fano varieties by relating Gromov-Witten invariants to tropical disks and theta functions, providing a new framework for understanding mirror maps and invariants.

Contribution

It introduces a tropical interpretation of the instanton corrected potential as a primitive theta function, linking it to Gromov-Witten invariants in non-Fano cases.

Findings

W=W=theta function equals open mirror map after wall crossing

Tropical correspondence relates theta function to logarithmic Gromov-Witten invariants

Generalizes mirror symmetry predictions to non-Fano varieties

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.