Brane dimers and quiver gauge theories

We describe a technique which enables one to quickly compute an infinitenumber of toric geometries and their dual quiver gauge theories. The centralobject in this construction is a ``brane tiling,'' which is a collection ofD5-branes ending on an NS5-brane wrapping a holomorphic curve that can berepresented as a periodic tiling of the plane. This construction solves thelongstanding problem of computing superpotentials for D-branes probing asingular non-compact toric Calabi-Yau manifold, and overcomes many difficultieswhich were encountered in previous work. The brane tilings give the largestclass of N=1 quiver gauge theories yet studied. A central feature of this workis the relation of these tilings to dimer constructions previously studied in avariety of contexts. We do many examples of computations with dimers, whichgive new results as well as confirm previous computations. Using our methods weexplicitly derive the moduli space of the entire Y^{p,q} family of quivertheories, verifying that they correspond to the appropriate geometries. Ourresults may be interpreted as a generalization of the McKay correspondence tonon-compact 3-dimensional toric Calabi-Yau manifolds.Comment: 56 pages, 36 figures, JHEP. v2: added references, corrected figures 9, 10, some minor change