Scattering diagrams from asymptotic analysis on Maurer-Cartan equations

Abstract

Let ${\check{X}}_0$ be a semi-flat Calabi-Yau manifold equipped with a Lagrangian torus fibration $\check{p}:{\check{X}}_0\to B_0$. 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.

54 pages, 17 figures; v2: significantly shortened because the preliminary materials (Sections 2 and 3) have been made much more concise (see the survey article arXiv:1811.09042 for more background) and Section 6, the main result of which basically follows directly from previous sections, has been removed (and will be appear elsewhere); all the main results and their proofs remain unchanged

References in corpus (1)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor

Cited by in corpus (7)

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• Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program
• The extended tropical vertex group
• Smoothing, scattering, and a conjecture of Fukaya
• SYZ mirror symmetry from Witten-Morse theory