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Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Sam Bardwell-Evans, Man-Wai Mandy Cheung, Hansol Hong, Yu-Shen Lin

Abstract

We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-Siebert and the canonical scattering diagrams of Gross-Hacking-Keel. With an additional assumption on the non-negativity of boundary divisors, we compute the disc potentials of the Lagrangian torus fibres via a holomorphic/tropical correspondence. As an application, we provide a version of mirror symmetry for rank two cluster varieties.

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Abstract

We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-Siebert and the canonical scattering diagrams of Gross-Hacking-Keel. With an additional assumption on the non-negativity of boundary divisors, we compute the disc potentials of the Lagrangian torus fibres via a holomorphic/tropical correspondence. As an application, we provide a version of mirror symmetry for rank two cluster varieties.

Paper Structure

This paper contains 36 sections, 37 theorems, 129 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tropical Geometry
  4. Scattering diagrams
  5. The scattering diagrams of Gross-Pandharipande-Siebert
  6. Scattering Diagrams from Simple Blowups
  7. Scattering Diagrams from Orbifold Blowups
  8. Canonical Scattering Diagram of Gross-Hacking-Keel
  9. Lagrangian Floer theory
  10. Bulk-Deformed $A_{\infty}$ Structures and Bulk-Deformed Superpotentials
  11. Pseudo-isotopies and wall-crossing
  12. Floer theory on toric surfaces
  13. Construction of tropical discs from holomorphic discs in toric surfaces
  14. Tropical interpretation of the potentials for toric semi-Fano surfaces
  15. SYZ fibrations on Log Calabi-Yau Surfaces and wall-crossing
  16. ...and 21 more sections

Key Result

Theorem 1.1

One can recover the Gross-Hacking-Keel canonical scattering diagram from the Lagrangian Floer theory of the admissible SYZ fibres in $Y\setminus D$.

Figures (10)

  • Figure 1: $A_2$ scattering diagram
  • Figure 2: $\mathbb{Z}_2$-symmetry of a tropical curve
  • Figure 3: The areas of affine line segments between $K_i$ and $K_j$ are bounded below.
  • Figure 4: sub-tropical discs of $\mathcal{T}$
  • Figure 5: Toric model for (a toric blowup of) the pair $(\mathbb{P}^2, D)$
  • ...and 5 more figures

Theorems & Definitions (77)

  • Theorem 1.1: see Theorem \ref{['correspondence: scattering diagram']} and discussion in Section \ref{['sec: can']}
  • Corollary 1.2: Corollary \ref{['open-closed']}
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • ...and 67 more