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The proper Landau-Ginzburg potential is the open mirror map

Tim Gräfnitz, Helge Ruddat, Eric Zaslow

Abstract

The mirror dual of a smooth toric Fano surface $X$ equipped with an anticanonical divisor $E$ is a Landau-Ginzburg model with superpotential, W. Carl-Pumperla-Siebert give a definition of the the superpotential in terms of tropical disks using a toric degeneration of the pair $(X,E)$. When $E$ is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle $K_X$, in framing zero. As a consequence, the proper Landau-Ginzburg potential is a solution to the Lerche-Mayr Picard-Fuchs equation. Along the way, we prove a generalization of a result about relative Gromov-Witten invariants by Cadman-Chen to arbitrary genus using the multiplication rule of quantum theta functions. In addition, we generalize a theorem of Hu that relates Gromov-Witten invariants of a surface under a blow-up from the absolute to the relative case. One of the two proofs that we give introduces birational modifications of a scattering diagram. We also demonstrate how the Hori-Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram.

The proper Landau-Ginzburg potential is the open mirror map

Abstract

The mirror dual of a smooth toric Fano surface equipped with an anticanonical divisor is a Landau-Ginzburg model with superpotential, W. Carl-Pumperla-Siebert give a definition of the the superpotential in terms of tropical disks using a toric degeneration of the pair . When is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle , in framing zero. As a consequence, the proper Landau-Ginzburg potential is a solution to the Lerche-Mayr Picard-Fuchs equation. Along the way, we prove a generalization of a result about relative Gromov-Witten invariants by Cadman-Chen to arbitrary genus using the multiplication rule of quantum theta functions. In addition, we generalize a theorem of Hu that relates Gromov-Witten invariants of a surface under a blow-up from the absolute to the relative case. One of the two proofs that we give introduces birational modifications of a scattering diagram. We also demonstrate how the Hori-Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram.
Paper Structure (33 sections, 37 theorems, 154 equations, 16 figures)

This paper contains 33 sections, 37 theorems, 154 equations, 16 figures.

Table of Contents

  1. Introduction and Summary
  2. The open and closed mirror maps
  3. The case of $\mathbb{P}^2$: Picard--Fuchs equations and mirror maps
  4. Other smooth toric Fano surfaces
  5. The Landau-Ginzburg potential in the proper case
  6. Visualizing the polarized integral affine surface $(B,\mathscr{P},\varphi)$ that underlies the mirror dual of $\mathbb{P}^2$
  7. Review of the construction of $W\colon \widecheck{\frak{X}}\to\mathbb{A}^1$ from $(B,\mathscr{P},\varphi)$
  8. Properness of the Landau--Ginzburg potential
  9. Perfect pairing with curve classes
  10. A Landau-Ginzburg model as a mirror dual to a Fano variety
  11. $\boldsymbol{q}$-refined wall structures
  12. Definition of invariants and tropical correspondence
  13. Log Gromov-Witten invariants
  14. Tropical curves
  15. Tropical correspondence
  16. ...and 18 more sections

Key Result

Theorem 1.1

Assume $X=\mathbb{P}^2$. We take a torus fiber near $E$ and set $Q = -t^3y^{-3},$ the broken line expansion of the Landau--Ginzburg superpotential is

Figures (16)

  • Figure 3.1: $A,B,C$ and $T$ represent the images of regions of the underlying singular affine surface in various coordinate charts. The intersection of any two charts will be the disjoint union of the interiors of $T$ and one of $A,B,C.$ The transition function is the identity along $T$ and determined by the conditions it maps the singularity to itself, leaves the edge of the triangle invariant, and identifies the remaining regions labeled by the same letter. For example, if we take the upper right singularity to be the origin of the plane, the upper and right charts are related by the linear transformation $\binom{\;2\;\;1}{-1\;0}$.
  • Figure 3.2: In each local halfplane outside an edge, a shear identifies the two boundaries of the excised shaded region, while preserving the edge.
  • Figure 3.3: Another fundamental domain, with cut regions in the central triangle. The central singularity is at the origin, while the other two are at $(\pm1,- \frac{3}{2})$ on lines of slope $\pm 3$. The $\mathbb{Z}$ quotient identifying outer edges is effected by the affine transformation $(x,y)$$\mapsto$$(x+3,y-9x-\frac{27}{2})$.
  • Figure 6.1: Degeneration of $\widehat{X}$
  • Figure 6.2: Claim: a graph $\Gamma$ that contributes non-trivially takes this form.
  • ...and 11 more figures

Theorems & Definitions (91)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Conjecture 1.9: intrinsic mirror construction
  • Conjecture 1.10: symplectic geometry and FOOO theory
  • ...and 81 more