Brane Tilings and Quiver Gauge Theories

Abstract

This work presents recent developments on brane tilings and their vacuum moduli spaces. Brane tilings are bipartite periodic graphs on the torus and represent 4d N = 1 supersymmetric worldvolume theories living on D3-branes probing Calabi-Yau 3-fold singularities. The graph and combinatorial properties of brane tilings make the set of supersymmetric quiver theories represented by them one of the largest and richest known so far. The aim of this work is to give a concise pedagogical introduction to brane tilings and a summary on recent exciting advancement on their classification, dualities and construction. At first, particular focus is given on counting distinct Abelian orbifolds of the form C3/[gamma]. The presented counting of Abelian orbifolds of C3 and in more general of CD gives a first insight on the rich combinatorial nature of brane tilings. Following the classification theme, the work proceeds with the identification of all brane tilings whose mesonic moduli spaces as toric Calabi-Yau 3-folds are represented by reflexive polygons. There are 16 of these special convex lattice polygons. It is shown that 30 brane tilings are associated with them. Some of these brane tilings are related by a correspondence known as toric duality. The classification of brane tilings with reflexive toric diagrams led to the discovery of a new correspondence between brane tilings which we call specular duality. The new correspondence identifies brane tilings with the same master space - the combined mesonic and baryonic moduli space. As a by-product, the new correspondence paves the way for constructing brane tilings which are not confined to the torus but are on Riemann surfaces with arbitrary genus. We give the first classification of genus 2 brane tilings, illustrate the corresponding supersymmetric quiver theories and analyse their vacuum moduli spaces.