Tropicalization of the mirror curve near the large radius limit

TL;DR

This paper explores the tropical geometry of mirror curves associated with toric Calabi-Yau 3-folds, demonstrating their structure as cyclic-M curves near the large radius limit and describing their decomposition into basic geometric pieces.

Contribution

It defines the tropical spine of mirror curves for toric Calabi-Yau 3-folds and shows these curves are cyclic-M curves near the large radius limit, revealing their geometric decomposition.

Findings

Tropical spine of mirror curves calculated for smooth toric Calabi-Yau 3-folds.

Mirror curves are cyclic-M curves near the large radius limit.

Mirror curves are constructed from tubes and pairs of pants.

Abstract

The mirror of a toric orbifold is an affine curve called the mirror curve. In this paper, firstly, we recall the basic tools in tropical geometry and give a definition of the mirror curve. Then we calculate the tropical spine of the mirror curve for a smooth toric Calabi-Yau 3-fold near the large radius limit. Finally, we recall a special kind of real algebraic curves called the cyclic-M curve and show that under some special choices of parameters near the large radius limit, the mirror curve is a cyclic-M curve. Applying Mikhalkin's result of cyclic-M curves, we show that the mirror curve is glued from tubes and pairs of pants.