Brane Dimers and Quiver Gauge Theories

TL;DR

This paper introduces a brane tiling technique to efficiently compute an extensive class of toric geometries and their dual quiver gauge theories, solving longstanding problems in superpotential calculations for D-branes.

Contribution

The work presents a novel brane tiling method that enables rapid computation of toric geometries and quiver gauge theories, expanding the class of theories studied and linking to dimer models.

Findings

Derived the moduli space for Y^{p,q} quiver theories.

Confirmed the correspondence between brane tilings and toric geometries.

Provided new computational results for dimer-based gauge theories.

Abstract

We describe a technique which enables one to quickly compute an infinite number of toric geometries and their dual quiver gauge theories. The central object in this construction is a ``brane tiling,'' which is a collection of D5-branes ending on an NS5-brane wrapping a holomorphic curve that can be represented as a periodic tiling of the plane. This construction solves the longstanding problem of computing superpotentials for D-branes probing a singular non-compact toric Calabi-Yau manifold, and overcomes many difficulties which were encountered in previous work. The brane tilings give the largest class of N=1 quiver gauge theories yet studied. A central feature of this work is the relation of these tilings to dimer constructions previously studied in a variety of contexts. We do many examples of computations with dimers, which give new results as well as confirm previous computations. Using our methods we explicitly derive the moduli space of the entire Y^{p,q} family of quiver theories, verifying that they correspond to the appropriate geometries. Our results may be interpreted as a generalization of the McKay correspondence to non-compact 3-dimensional toric Calabi-Yau manifolds.