Smoothing, scattering, and a conjecture of Fukaya

TL;DR

This paper advances the understanding of mirror symmetry by reformulating Fukaya's conjecture using tropical geometry, connecting deformation theory, smoothing of Calabi-Yau varieties, and scattering diagrams.

Contribution

It provides a tropical geometric reformulation of Fukaya's second correspondence in mirror symmetry, utilizing asymptotic analysis and algebraic techniques.

Findings

Reformulation of Fukaya's conjecture using tropical geometry.

Construction of scattering diagrams from Maurer-Cartan solutions.

Application of asymptotic analysis to tropicalize algebraic structures.

Abstract

In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the multi-valued Morse theory on the base $\check{B}$ of an SYZ fibration $\check{p}: \check{X}\to \check{B}$, and the other between deformation theory of the mirror $X$ and the same multi-valued Morse theory on $\check{B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base $\check{B}$ is replaced by tropical geometry on the Legendre dual $B$. In the proof, we apply techniques of asymptotic analysis developed in our previous works to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in another of our recent work. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part $X_{\text{sf}} \subseteq X$ allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.