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

TL;DR

The paper develops a non-Fano extension of mirror symmetry by introducing the primitive theta function ϑ_1 as a corrected mirror potential for log Calabi–Yau pairs (X,D). It proves that the classical period π_{ϑ_1} matches the regularized quantum period G_X and that, after wall-crossing to infinity, ϑ_1 encodes the open mirror map and counts of 2-marked logarithmic Gromov–Witten invariants via tropical disks, extending tropical correspondence beyond nef cases. The framework relies on toric degenerate models, scattering diagrams, and broken lines to relate GW data (relative, descendant, and open invariants) to tropical geometry; mutations connect different chambers and ensure invariants’ consistency across birational models. Concrete examples include Hirzebruch surfaces and blowups of the plane, illustrating mutation relations among theta functions and their potentials, and highlighting the breakdown of PF equations in non-semi-Fano settings. Overall, the work broadens mirror symmetry to non-Fano geometries and provides a robust tropical toolkit for computing open/relative invariants and their generating functions.

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.

Figures (18)

• Figure 1: The scattering diagram for the Hirzebruch surface $\mathbb{F}_3$ and broken lines (red) defining the corrected potential $W=\vartheta_1$.
• Figure 2: The fan of a $\mathbb{Q}$-Gorenstein degeneration of a smooth cubic.
• Figure 3: The fan $\Sigma$ with piecewise linear function $\varphi$ (left) and the dual quasi-polytope $\Delta$ (right) for $\mathbb{F}_3$ with anticanonical divisor.
• Figure 4: The dual intersection complex $(B,\mathcal{P})$ for $(\mathbb{P}^2,3L)$.
• Figure 5: The intersection complex of $\mathbb{F}_1$ (left) and the dual intersection complex of $\mathbb{F}_1$ in two different charts (middle and right).

Theorems & Definitions (103)

• Example 1.1
• Definition
• Theorem 1: Theorem \ref{['thm:theta']}
• Theorem 2: Corollary \ref{['cor:theta']}, Theorem \ref{['thm:mirmap']}
• Remark 1.2
• Definition 2.1
• Remark 2.2
• Example 2.3
• Definition 2.4
• Example 2.5