Dimer Models from Mirror Symmetry and Quivering Amoebae

TL;DR

The paper addresses how dimer models encode the data of toric quiver gauge theories realized by D3-branes at toric Calabi–Yau singularities. It develops a mirror-symmetry framework in which D3-branes map to intersecting D6-branes wrapped on a T^3 in the mirror, with the quiver data arising from their intersections; a T^2 projection of this setup yields the dimer model, while an untwisting map relates the dimer on Σ to the D6-brane configuration. By employing amoeba and alga projections, the authors connect geometric features of the mirror curve P(z,w)=W to the dimer's faces, edges, and vertices, and demonstrate Seiberg duality as PL-transformations within this geometric picture. They illustrate the construction through examples (C^3, conifold, C^3/Z_3, Y^{3,1}) and discuss how zig-zag paths and (p,q) webs encode gauge data and superpotentials. This approach provides a direct string-theoretic origin for dimer-quiver correspondences and a robust toolkit for analyzing toric gauge theories and their moduli.

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.