JournalsjemsVol. 24, No. 3pp. 773–849

Scattering diagrams from asymptotic analysis on Maurer–Cartan equations

  • Kwokwai Chan

    The Chinese University of Hong Kong, Hong Kong
  • Naichung Conan Leung

    The Chinese University of Hong Kong, Hong Kong
  • Ziming Nikolas Ma

    The Chinese University of Hong Kong, Hong Kong
Scattering diagrams from asymptotic analysis on Maurer–Cartan equations cover
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Abstract

Let Xˇ0{\check{X}_0} be a semi-flat Calabi–Yau manifold equipped with a Lagrangian torus fibration pˇ:Xˇ0B0{\check{p}:\check{X}_0 \rightarrow B_0}. We investigate the asymptotic behavior of Maurer–Cartan solutions of the Kodaira–Spencer deformation theory on Xˇ0{\check{X}_0} by expanding them into Fourier series along fibres of pˇ{\check{p}} over a contractible open subset UB0{U\subset 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.

Cite this article

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. 773–849

DOI 10.4171/JEMS/1100