Tropical mirror for toric surfaces

TL;DR

This paper constructs the tropical mirror for complex toric surfaces, providing explicit formulas for mirror states, their enumerative forms, and applications to tropical Gromov-Witten invariants and Landau-Ginzburg-Saito theory.

Contribution

It offers an explicit tropical mirror construction for toric surfaces, linking holomorphic germs to enumerative geometry and deformation interpretations.

Findings

Explicit formulas for mirror states in tropical geometry

Derivation of divisor relations for tropical Gromov-Witten invariants

Interpretation of deformations as blow-ups on toric surfaces

Abstract

We describe the tropical mirror for complex toric surfaces. In particular we provide an explicit expression for the mirror states and show that they can be written in enumerative form. Their holomorphic germs give an explicit form of good section for Landau-Ginzburg-Saito theory. We use an explicit form of holomorphic germs to derive the divisor relation for tropical Gromov-Witten invariants. We interpret the deformation of the theory by a point observable as a blow up of a point on the toric surface. We describe the implication of such interpretation for the tropical Gromov-Witten invariants.