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A tropical view on Landau-Ginzburg models

Michael Carl, Max Pumperla, Bernd Siebert

Abstract

This paper, largely written in 2009/2010, fits Landau-Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001. This point of view transparently brings in tropical disks of Maslov index 2 via the notion of broken lines, previously introduced in two dimensions by Mark Gross in his study of mirror symmetry for $\mathbb{P}^2$. A major insight is the equivalence of properness of the Landau-Ginzburg potential with smoothness of the anticanonical divisor on the mirror side. We obtain proper superpotentials which agree on an open part with those classically known for toric varieties. Examples include mirror LG models for non-singular and singular del Pezzo surfaces, Hirzebruch surfaces and some Fano threefolds.

A tropical view on Landau-Ginzburg models

Abstract

This paper, largely written in 2009/2010, fits Landau-Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001. This point of view transparently brings in tropical disks of Maslov index 2 via the notion of broken lines, previously introduced in two dimensions by Mark Gross in his study of mirror symmetry for . A major insight is the equivalence of properness of the Landau-Ginzburg potential with smoothness of the anticanonical divisor on the mirror side. We obtain proper superpotentials which agree on an open part with those classically known for toric varieties. Examples include mirror LG models for non-singular and singular del Pezzo surfaces, Hirzebruch surfaces and some Fano threefolds.
Paper Structure (21 sections, 29 theorems, 87 equations, 17 figures)

This paper contains 21 sections, 29 theorems, 87 equations, 17 figures.

Table of Contents

  1. Landau-Ginzburg tropical data and scattering diagram
  2. The superpotential at $t$-order zero
  3. Scattering of monomials
  4. The superpotential via broken lines
  5. Fibers of the superpotential over $0,1,\infty$, and the LG mirror map
  6. Wall structures and broken lines via tropical disks
  7. Tropical disks
  8. Virtual tropical disks
  9. Deformations of $\Delta$
  10. Deformations of tropical disks
  11. Structures via virtual tropical disks
  12. Claim 1: Over $R^k$, the counting monomials arise via scattering.
  13. Claim 2: The counting functions \ref{['countfct']} can be lifted.
  14. Virtual counts of tropical Maslov index two disks
  15. Toric degenerations of del Pezzo surfaces and their mirrors
  16. ...and 6 more sections

Key Result

Lemma 1.3

Assume that $(B,\mathscr{P})$ is compactifiable (Definition Def: compactifiable (B,P)). Then there exists a sequence of compact subsets $\overline B_1\subseteq\overline B_2\subseteq \ldots\subseteq B$ with (1) $B=\bigcup_{\nu\ge 1} \overline B_\nu$, and (2) $(\overline B_\nu,\overline\mathscr{P}_\nu

Figures (17)

  • Figure 2.1: An intersection complex $(\check B,\check \mathscr{P})$ for $\mathbb{P}^2$ with straight boundary and its Legendre dual $(B,\varphi)$ for the minimal polarization, with a chart on the complement of the shaded region and a chart showing the three parallel unbounded edges.
  • Figure 3.1: Scattering diagram with perturbed trajectories (cuts and rays solid, perturbed trajectories dashed).
  • Figure 6.1: A tropical Maslov index zero disk bounding $x$ belonging to a moduli space of dimension $5$. The dashed lines indicate a part of the discriminant locus.
  • Figure 6.2: Disks near $\Delta$ (left) and their moduli cell complex (right).
  • Figure 7.1: Fans of the five toric del Pezzo surfaces
  • ...and 12 more figures

Theorems & Definitions (98)

  • Definition 1.2
  • Lemma 1.3
  • proof
  • Definition 1.4
  • Proposition 1.5
  • proof
  • Remark 1.6
  • Definition 2.1
  • Proposition 2.3
  • proof
  • ...and 88 more