Tropical counting from asymptotic analysis on Maurer-Cartan equations

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 $X={\mathbb{P}}^2$).

arXiv comment34 pages, 3 figures; v3: final version to appear in Trans. AMS

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Cited by in corpus (1)

• Smoothing, scattering, and a conjecture of Fukaya