A tropical view on Landau-Ginzburg models

TL;DR

This paper integrates tropical geometry into Landau-Ginzburg models within mirror symmetry, revealing new insights into properness and smoothness, and providing explicit examples for various Fano varieties.

Contribution

It introduces a tropical perspective on Landau-Ginzburg models, connecting properness of potentials with smoothness of divisors, and constructs explicit mirror models for several Fano varieties.

Findings

Properness of Landau-Ginzburg potentials corresponds to smoothness of the anticanonical divisor.

Constructed explicit mirror LG models for del Pezzo surfaces, Hirzebruch surfaces, and Fano threefolds.

Established the role of tropical disks and broken lines in mirror symmetry context.

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.