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Smoothing, scattering, and a conjecture of Fukaya

Kwokwai Chan, Naichung Conan Leung, Ziming Nikolas Ma

TL;DR

This work globalizes Fukaya’s mirror-symmetry program by replacing the base Morse-theoretic data with tropical geometry on the Legendre dual base $B$. It tropicalizes the pre-dgBV algebra governing smoothing of the maximally degenerate Calabi–Yau log variety $igl<X^{0}igr>$ and relates it to the semi-flat dgBV algebra that governs deformations of the semi-flat part $X_{ ext{sf}}$. From Maurer–Cartan solutions, the authors extract consistent scattering diagrams that encode wall-crossing factors in a tropical vertex group, tying together the Gross–Siebert cone construction, tropical differential forms, and L-infinity/ dgBV formalisms. The results provide a global, rigorous link between smoothing of maximally degenerate CY spaces and tropical geometry, offering a pathway toward a local-to-global understanding of genus-$0$ mirror symmetry within the SYZ framework and Gross–Siebert program.

Abstract

In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the multi-valued Morse theory on the base $\check{B}$ of an SYZ fibration $\check{p}: \check{X}\to \check{B}$, and the other between deformation theory of the mirror $X$ and the same multi-valued Morse theory on $\check{B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base $\check{B}$ is replaced by tropical geometry on the Legendre dual $B$. In the proof, we apply techniques of asymptotic analysis developed in our previous works to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in another of our recent work. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part $X_{\text{sf}} \subseteq X$ allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.

Smoothing, scattering, and a conjecture of Fukaya

TL;DR

This work globalizes Fukaya’s mirror-symmetry program by replacing the base Morse-theoretic data with tropical geometry on the Legendre dual base . It tropicalizes the pre-dgBV algebra governing smoothing of the maximally degenerate Calabi–Yau log variety and relates it to the semi-flat dgBV algebra that governs deformations of the semi-flat part . From Maurer–Cartan solutions, the authors extract consistent scattering diagrams that encode wall-crossing factors in a tropical vertex group, tying together the Gross–Siebert cone construction, tropical differential forms, and L-infinity/ dgBV formalisms. The results provide a global, rigorous link between smoothing of maximally degenerate CY spaces and tropical geometry, offering a pathway toward a local-to-global understanding of genus- mirror symmetry within the SYZ framework and Gross–Siebert program.

Abstract

In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold and the multi-valued Morse theory on the base of an SYZ fibration , and the other between deformation theory of the mirror and the same multi-valued Morse theory on . In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base is replaced by tropical geometry on the Legendre dual . In the proof, we apply techniques of asymptotic analysis developed in our previous works to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in another of our recent work. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.
Paper Structure (29 sections, 24 theorems, 238 equations, 10 figures)

This paper contains 29 sections, 24 theorems, 238 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Gross--Siebert's cone construction of maximally degenerate Calabi--Yau varieties
  3. Integral tropical manifolds
  4. Monodromy, positivity and simplicity
  5. Cone construction by gluing open affine charts
  6. Log structures
  7. A generalized moment map and the tropical singular locus on $B$
  8. A generalized moment map
  9. Construction of charts on $B$
  10. The tropical singular locus $\mathscr{S}$ of $B$
  11. Contraction of $\mathcal{A}$ to $\mathscr{S}$
  12. Monodromy invariant differential forms on $B$
  13. Smoothing of maximally degenerate Calabi--Yau varieties via dgBV algebras
  14. Good covers and local smoothing data
  15. An explicit description of the sheaf of log de Rham forms
  16. ...and 14 more sections

Key Result

Theorem 1.1

There exists a solution $\phi$ to the classical Maurer--Cartan equation eqn:classical_maurer_cartan_equation giving rise to a smoothing of the maximally degenerate Calabi--Yau log variety $\prescript{0}{}{X}^{\dagger}$ over $\mathbb{C}[[q]]$, from which a consistent scattering diagram $\mathscr{D}(\

Figures (10)

  • Figure 1: The polyhedral decomposition
  • Figure 2: Affine coordinate charts
  • Figure 3: The polyhedral decomposition on a facet
  • Figure 4: Contraction map $\mathscr{C}$ when $\dim_{\mathbb{R}}(B) = 3$
  • Figure 5: Contraction at $\rho$
  • ...and 5 more figures

Theorems & Definitions (90)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Definition 2.2: gross2011real, Def. 1.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5: Gross-Siebert-logI, Def. 1.43
  • Definition 2.6: Gross-Siebert-logI, Def. 1.45 and 1.47
  • Definition 2.7: Gross-Siebert-logI, Def. 1.48
  • ...and 80 more