Tropical mirror for toric surfaces
Andrey Losev, Vyacheslav Lysov

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.
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Taxonomy
TopicsNonlinear Waves and Solitons · Algebraic Geometry and Number Theory · Polynomial and algebraic computation
