Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On SYZ mirrors of Hirzebruch surfaces

Honghao Jing

Abstract

The Strominger-Yau-Zaslow (SYZ) approach to mirror symmetry constructs a mirror space and a superpotential from the data of a Lagrangian torus fibration on a Kähler manifold with effective first Chern class. For Kähler manifolds whose first Chern class is not nef, the SYZ construction is further complicated by the presence of additional holomorphic discs with non-positive Maslov index. In this paper, we study SYZ mirror symmetry for two of the simplest non-Fano toric examples: the Hirzebruch surfaces F_3 and F_4. Our approach is to regularize moduli spaces of stable holomorphic discs using obstruction sections arising from infinitesimal deformations of the complex structure. For F_3, we determine the SYZ mirror associated to generic regularizing perturbations of the complex structure, and demonstrate that the mirror depends on the choice of perturbation. For F_4, we determine the SYZ mirror for a specific regularizing perturbation, where the mirror superpotential is an explicit infinite Laurent series. Finally, we relate this superpotential to those arising from other perturbations of F_4, as determined in the literature \cite{CPS24, BGL25}, via a scattering diagram.

On SYZ mirrors of Hirzebruch surfaces

Abstract

The Strominger-Yau-Zaslow (SYZ) approach to mirror symmetry constructs a mirror space and a superpotential from the data of a Lagrangian torus fibration on a Kähler manifold with effective first Chern class. For Kähler manifolds whose first Chern class is not nef, the SYZ construction is further complicated by the presence of additional holomorphic discs with non-positive Maslov index. In this paper, we study SYZ mirror symmetry for two of the simplest non-Fano toric examples: the Hirzebruch surfaces F_3 and F_4. Our approach is to regularize moduli spaces of stable holomorphic discs using obstruction sections arising from infinitesimal deformations of the complex structure. For F_3, we determine the SYZ mirror associated to generic regularizing perturbations of the complex structure, and demonstrate that the mirror depends on the choice of perturbation. For F_4, we determine the SYZ mirror for a specific regularizing perturbation, where the mirror superpotential is an explicit infinite Laurent series. Finally, we relate this superpotential to those arising from other perturbations of F_4, as determined in the literature \cite{CPS24, BGL25}, via a scattering diagram.

Paper Structure

This paper contains 14 sections, 28 theorems, 60 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background and Main Results
  3. Related Work
  4. SYZ Mirrors of the Third Hirzebruch Surface $\mathbb{F}_3$
  5. Obstruction Bundle
  6. Maslov Index $0$ Holomorphic Discs and Walls
  7. Maslov Index $2$ Holomorphic Discs and Superpotential
  8. An SYZ Mirror of the Fourth Hirzebruch Surface $\mathbb{F}_4$
  9. Obstruction Bundle
  10. Maslov Index $0$ Holomorphic Discs and Walls
  11. Maslov Index $2$ Holomorphic Discs and Superpotential
  12. Comparison with Other Constructions of Mirrors for $\mathbb{F}_4$
  13. Another Mirror Superpotential for $\mathbb{F}_4$
  14. A Scattering Diagram

Key Result

Theorem 1.1

With a generic perturbation of the complex structure on $\mathbb{F}_3$, the SYZ mirror of $\mathbb{F}_3$ is the Landau-Ginzburg model $(\mathbb{F}_3^\vee,W)$, where the mirror space The superpotential $W:\mathbb{F}_3^\vee\to\Lambda$ restricts to on the $(x,y)$-coordinate chart, where $(x,y)=(v^{-1},w^{-1})\in(\Lambda^*)^2$. Analogously, $W$ restricts to on the $(x',y')$-coordinate chart, where

Figures (9)

  • Figure 1: The $k^{\text{th}}$ Hirzebruch surface $\mathbb{F}_k$
  • Figure 2: The $3^{\text{rd}}$ Hirzebruch surface $\mathbb{F}_3$
  • Figure 3: Disc classes in $\pi_2(\mathbb{F}_3,L)$
  • Figure 4: Disc classes in $\pi_2(\mathbb{F}_4,L)$
  • Figure 5: The map $f:\mathbb{C}\times\mathbb{C}\to\mathbb{C}$ as a fibration
  • ...and 4 more figures

Theorems & Definitions (42)

  • Remark
  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • ...and 32 more