Brane tilings

We review and extend the progress made over the past few years in understanding the structure of toric quiver gauge theories; those which are induced on the worldvolume of a stack of D3-branes placed at the tip of a toric Calabi-Yau cone, at an orbifold point in Kahler moduli space. These provide an infinite class of four-dimensional N = 1 superconformal field theories which may be studied in the context of the AdS/CFT correspondence. It is now understood that these gauge theories are completely specified by certain two-dimensional torus graphs, called brane tilings, and the combinatorics of the dimer models on these graphs. In particular, knowledge of the dual Sasaki-Einstein metric is not required to determine the gauge theory, only topological and symplectic properties of the toric Calabi-Yau cone. By analyzing the symmetries of the toric quiver theories we derive the dimer models and use them to construct the moduli space of the theory both classically and semiclassically. Using mirror symmetry the brane tilings are shown to arise in string theory on the worldvolumes of the fractional D6-branes that are mirror to the stack of D3-branes at the tip of the cone.