Brane Tilings

TL;DR

This paper reviews and extends the understanding of toric quiver gauge theories associated with D3-branes on toric Calabi-Yau cones, emphasizing the role of brane tilings and dimer models in their structure and analysis.

Contribution

It introduces a comprehensive framework using brane tilings and dimer models to determine gauge theories without requiring the Sasaki-Einstein metric.

Findings

Brane tilings encode gauge theories from toric Calabi-Yau cones.

Dimer models facilitate the construction of the moduli space.

Mirror symmetry relates brane tilings to fractional D6-branes.

Abstract

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 world-volume of a stack of D3-branes placed at the tip of a toric Calabi-Yau cone, at an ``orbifold point'' in Kaehler 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 world-volumes of the fractional D6-branes that are mirror to the stack of D3-branes at the tip of the cone.