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Transactions of the American Mathematical Society

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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 $ 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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Additional Information

Kwokwai Chan
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
MR Author ID: 821162
Email: kwchan@math.cuhk.edu.hk

Ziming Nikolas Ma
Affiliation: The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
MR Author ID: 1001651
Email: zmma@ims.cuhk.edu.hk

DOI: https://doi.org/10.1090/tran/8128
Received by editor(s): January 31, 2019
Received by editor(s) in revised form: January 2, 2020
Published electronically: June 24, 2020
Additional Notes: The work of the first author was supported by grants of the Hong Kong Research Grants Council (Project No. CUHK14302015 $&$ CUHK14314516).
The work of the second author was partially supported by the Institute of Mathematical Sciences (IMS) and Department of Mathematics at The Chinese University of Hong Kong.
Article copyright: © Copyright 2020 American Mathematical Society