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

Authors: Kwokwai Chan and Ziming Nikolas Ma
Journal: Trans. Amer. Math. Soc. 373 (2020), 6411-6450
MSC (2010): Primary 32G05, 14J33, 14T05; Secondary 14M25, 14N10, 53D37, 14N35
DOI: https://doi.org/10.1090/tran/8128
Published electronically: June 24, 2020
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Abstract: Let $X = X_\Sigma$ be a toric surface and let $(\check {X}, W)$ be its Landau-Ginzburg (LG) mirror where 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 with Maslov index 0 or 2, the latter of which produces a universal unfolding of . For $X = \mathbb{P}^2$ , our construction reproduces Gross' perturbed potential [Adv. Math. 224 (2010), pp. 169-245] which was proven to be the universal unfolding of written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of 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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