Let be a semi-flat Calabi–Yau manifold equipped with a Lagrangian torus fibration . We investigate the asymptotic behavior of Maurer–Cartan solutions of the Kodaira–Spencer deformation theory on by expanding them into Fourier series along fibres of over a contractible open subset , 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.
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Kwokwai Chan, Naichung Conan Leung, Ziming Nikolas Ma, Scattering diagrams from asymptotic analysis on Maurer–Cartan equations. J. Eur. Math. Soc. 24 (2022), no. 3, pp. 773–849DOI 10.4171/JEMS/1100