Scattering diagrams from asymptotic analysis on Maurer-Cartan equations

TL;DR

This paper studies the asymptotic behavior of Maurer-Cartan solutions on semi-flat Calabi-Yau manifolds, showing that their Fourier modes produce scattering diagrams important in mirror symmetry reconstruction.

Contribution

It demonstrates that semi-classical limits of Maurer-Cartan solutions naturally generate scattering diagrams, linking asymptotic analysis to mirror symmetry tools.

Findings

Fourier modes of Maurer-Cartan solutions produce scattering diagrams.

Semi-classical limits give rise to tropical combinatorial objects.

Results connect asymptotic analysis with mirror symmetry reconstruction.

Abstract

Let $\check{X}_0$ be a semi-flat Calabi-Yau manifold equipped with a Lagrangian torus fibration $\check{p}:\check{X}_0 \rightarrow B_0$. We investigate the asymptotic behavior of Maurer-Cartan solutions of the Kodaira-Spencer deformation theory on $\check{X}_0$ by expanding them into Fourier series along fibres of $\check{p}$ over a contractible open subset $U\subset B_0$, following a program set forth by Fukaya in 2005. We prove that semi-classical limits (i.e. leading order terms in asymptotic expansions) of the Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to consistent scattering diagrams, which are tropical combinatorial objects that have played a crucial role in works of Kontsevich-Soibelman and Gross-Siebert on the reconstruction problem in mirror symmetry.