Tropical counting from asymptotic analysis on Maurer-Cartan equations

Ziming Nikolas Ma The Chinese University of Hong Kong Kwokwai Chan The Chinese University of Hong Kong

Algebraic Geometry mathscidoc:1808.01001

Transactions of the American Mathematical Society, 373, 6411-6450, 2020.6
Let 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).
Tropical counting, topic surface, scattering diagram, Maurer-Cartan equation
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  • https://doi.org/10.1090/tran/8128
@inproceedings{ziming2020tropical,
  title={Tropical counting from asymptotic analysis on Maurer-Cartan equations},
  author={Ziming Nikolas Ma, and Kwokwai Chan},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180801185333865554121},
  booktitle={Transactions of the American Mathematical Society},
  volume={373},
  pages={6411-6450},
  year={2020},
}
Ziming Nikolas Ma, and Kwokwai Chan. Tropical counting from asymptotic analysis on Maurer-Cartan equations. 2020. Vol. 373. In Transactions of the American Mathematical Society. pp.6411-6450. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180801185333865554121.
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