Scattering diagrams from asymptotic analysis on Maurer-Cartan equations

Let X-0 be a semi-flat Calabi-Yau manifold equipped with a Lagrangian torus fibration p : X-0 -> B-0. We investigate the asymptotic behavior of Maurer-Cartan solutions of the KodairaSpencer deformation theory on X-0 by expanding them into Fourier series along fibres of pL over a contractible open subset U subset of B-0, following a program set forth by Fukaya [Graphs and Patterns in Mathematics and Theoretical Physics (2005)] 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 and Soibelman [The Unity of Mathematics (2006)] and Gross and Siebert [Ann. of Math. (2) 174 (2011)] on the reconstruction problem in mirror symmetry.