Dimer Models from Mirror Symmetry and Quivering Amoebae

TL;DR

This paper uses mirror symmetry to explain how dimer models encode gauge theories from D-branes on toric Calabi-Yau singularities, revealing their geometric origin via D6-branes on a T^2 subspace.

Contribution

It provides a string-theoretic geometric interpretation of dimer models through mirror symmetry and D6-branes, clarifying their role in encoding quiver gauge theories.

Findings

Dimer models are realized on a T^2 subspace in mirror symmetry.

D6-branes wrapped on this T^2 encode the same gauge theories as D3-branes.

Mirror symmetry offers a geometric explanation for dimer models in string theory.

Abstract

Dimer models are 2-dimensional combinatorial systems that have been shown to encode the gauge groups, matter content and tree-level superpotential of the world-volume quiver gauge theories obtained by placing D3-branes at the tip of a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the quiver graph. However, the string theoretic explanation of this was unclear. In this paper we use mirror symmetry to shed light on this: the dimer models live on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the singular point, and geometrically encode the same quiver theory on their world-volume.