Scattering diagrams from asymptotic analysis on Maurer–Cartan equations

Abstract

Let X0 be a semi-flat Calabi-Yau manifold equipped with a Lagrangian torus fibration p : X0 → B0. We investigate the asymptotic behavior of Maurer-Cartan solutions of the Kodaira-Spencer deformation theory on X0 by expanding them into Fourier series along fibres of p over a contractible open subset U ⊂ B0, following a program set forth by Fukaya [21] 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 [37] and Gross-Siebert [28] on the reconstruction problem in mirror symmetry.1 Except in the semi-flat case when B = B0 where there are no singular fibres; see [40].5 Indeed Assumptions I and II (or more precisely Assumptions 5.48 and 5.49) are extracted from properties of the MC solutions we constructed.

Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties

“…Scattering diagrams are combinatorial structures used by and as quantum-corrected gluing data to solve the important reconstruction problem in mirror symmetry. What we showed in [5,6] is that scattering diagrams are indeed hidden in and encoded by Maurer-Cartan solutions.…”

“…On the other hand, our earlier work [5] and [6], where we attempted to realize Fukaya's asymptotic approach [18], showed how asymptotic expansions of Maurer-Cartan solutions for the Kodaira-Spencer dgLa Ω 0, * (X, * T 1,0 X ), where X is a torus bundle over a base B, give rise to scattering diagrams and tropical disk counts as the torus fibers shrink. Scattering diagrams are combinatorial structures used by and as quantum-corrected gluing data to solve the important reconstruction problem in mirror symmetry.…”

Deformations of holomorphic pairs and 2d-4d wall-crossing

“…Our construction is a variant of the remarkable recent results of Chan, Conan Leung and Ma [2]. That work shows how consistent scattering diagrams, in the sense of Kontsevich-Soibelman and Gross-Siebert (see e.g.…”

Log Calabi-Yau surfaces and Jeffrey-Kirwan residues

“…Note that the generalised monodromy approach is already contained implicitly in the construction of the GHK family, because of the fundamental identities satisfied by the canonical regular functions of the Gross-Siebert model X o → S (see Remarks 5.1 and 5.7, and the remarks in [14], p. 27). We expect that a similar result, relating GHK mirrors to JK residues, could be established by using instead the approach based on scattering for the Maurer-Cartan equation introduced in [9].…”