Tropical counting from asymptotic analysis on Maurer-Cartan equations

Abstract

Let X = X Σ X = X_\Sigma be a toric surface and let ( X ˇ , W ) (\check {X}, W) be its Landau-Ginzburg (LG) mirror where W 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 ( X ˇ , W ) (\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 X with Maslov index 0 or 2, the latter of which produces a universal unfolding of W W . For X = P 2 X = \mathbb {P}^2 , our construction reproduces Gross’ perturbed potential W n W_n [Adv. Math. 224 (2010), pp. 169–245] which was proven to be the universal unfolding of W W written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of W n 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 = P 2 X = \mathbb {P}^2 ).