Dimer models and homological mirror symmetry for triangles

TL;DR

This paper establishes a connection between dimer models, coamoebas, and vanishing cycles, leading to a torus-equivariant homological mirror symmetry for certain two-dimensional toric Fano stacks.

Contribution

It proves a conjecture linking dimer models and homological mirror symmetry for two-dimensional toric Fano stacks of Picard number one.

Findings

Proved a conjecture relating dimer models and mirror symmetry.

Derived a torus-equivariant version of homological mirror symmetry.

Connected coamoebas and vanishing cycles in the mirror symmetry context.

Abstract

We prove a conjecture on the relation between dimer models, coamoebas and vanishing cycles for the mirrors of two-dimensional toric Fano stacks of Picard number one. As a corollary, we obtain a torus-equivariant version of homological mirror symmetry for such stacks.