Tropical counting from asymptotic analysis on Maurer-Cartan equations

Chan, Kwokwai; Ma, Ziming Nikolas

doi:10.1090/tran/8128

Tropical counting from asymptotic analysis on Maurer-Cartan equations
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by Kwokwai Chan and Ziming Nikolas Ma PDF

Trans. Amer. Math. Soc. 373 (2020), 6411-6450 Request permission

Abstract:

Let $X = X_\Sigma$ be a toric surface and let $(\check {X}, W)$ be its Landau-Ginzburg (LG) mirror where $W$ is the Hori-Vafa potential as shown in their preprint. 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 $W_n$ [Adv. Math. 224 (2010), pp. 169–245] 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 Maslov index 0 tropical disks originally observed by Gross in the same work (in the case of $X = \mathbb {P}^2$).

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