Tropical counting from asymptotic analysis on Maurer-Cartan equations

TL;DR

This paper connects asymptotic analysis of Maurer-Cartan equations in Landau-Ginzburg models of toric surfaces to tropical geometry, revealing how tropical disks encode deformations and wall-crossing phenomena of the mirror potential.

Contribution

It introduces a novel approach linking asymptotic solutions of Maurer-Cartan equations to tropical disks, providing new insights into mirror symmetry and potential deformations.

Findings

Semi-classical limits produce tropical disks with Maslov index 0 or 2.

Reproduces Gross' universal unfolding of the mirror potential for .

Explains wall-crossing phenomena via tropical disk scattering diagrams.

Abstract

Let $X = X_\Sigma$ be a toric surface and $(\check{X}, W)$ be its Landau-Ginzburg (LG) mirror where $W$ is the Hori-Vafa potential. We apply asymptotic analysis to study the extended deformation theory of the LG model $(\check{X}, W)$, and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in $X$ with Maslov index 0 or 2, the latter of which produces a universal unfolding of $W$. For $X = \mathbb{P}^2$, our construction reproduces Gross' perturbed potential $W_n$ which was proven to be the universal unfolding of $W$ written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of $W_n$ across walls of the scattering diagram formed by the Maslov index 0 tropical disks originally observed by Gross (in the case of $X = \mathbb{P}^2$).