Tropical counting from asymptotic analysis on Maurer-Cartan equations
Publication in refereed journal


摘要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 Maslov index 0 tropical disks originally observed by Gross (in the case of $X = \mathbb{P}^2$).
著者Kwokwai Chan, Ziming Nikolas Ma
期刊名稱Transactions of the American Mathematical Society
出版社American Mathematical Society
關鍵詞tropical geometry, deformation theory, mirror symmetry, toric variety, Landau-Ginzburg model

上次更新時間 2020-12-09 於 23:50