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Mathematics > Algebraic Geometry

arXiv:1807.08145 (math)
[Submitted on 21 Jul 2018 (v1), last revised 7 Jan 2019 (this version, v2)]

Title:Scattering diagrams from asymptotic analysis on Maurer-Cartan equations

Authors:Kwokwai Chan, Naichung Conan Leung, Ziming Nikolas Ma
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Abstract:Let $\check{X}_0$ be a semi-flat Calabi-Yau manifold equipped with a Lagrangian torus fibration $\check{p}:\check{X}_0 \rightarrow B_0$. 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.
Comments: 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
Subjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph); Symplectic Geometry (math.SG)
Cite as: arXiv:1807.08145 [math.AG]
  (or arXiv:1807.08145v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1807.08145
Journal reference: J. Eur. Math. Soc. (JEMS) 24 (2022), no. 3, 773-849
Related DOI: https://doi.org/10.4171/JEMS/1100

Submission history

From: Kwokwai Chan [view email]
[v1] Sat, 21 Jul 2018 13:00:23 UTC (661 KB)
[v2] Mon, 7 Jan 2019 07:48:38 UTC (567 KB)
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