Dimer models and toric diagrams

We study some aspects of the recently discovered connection between dimer models and D-brane gauge theories. We argue that dimer models are also naturally related to closed string theories on non compact orbifolds of $\BC^2$ and $\BC^3$, via their twisted sector R charges, and show that perfect matchings in dimer models correspond to twisted sector states in the closed string theory. We also use this formalism to study the combinatorics of some unstable orbifolds of $\BC^2$.

A dimer model is a quiver with faces embedded in a surface. We define and investigate notions of consistency for dimer models on general surfaces with boundary which restrict to well-studied consistency conditions in the disk and torus case. We define weak consistency in terms of the associated dimer algebra and show that it is equivalent to the absence of bad configurations on the strand diagram. In the disk and torus case, weakly consistent models are nondegenerate, meaning...

We present a geometrical approach for studying dimers. We introduce a connection for dimer problems on bipartite and non-bipartite graphs. In the bipartite case the connection is flat but has non-trivial ${\bf Z}_2$ holonomy round certain curves. This holonomy has the universality property that it does not change as the number of vertices in the fundamental domain of the graph is increased. It is argued that the K-theory of the torus, with or without punctures, is the appropr...

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...

A dimer model is a bipartite graph described on the real two-torus, and it gives the quiver as the dual graph. It is known that for any three-dimensional Gorenstein toric singularity, there exists a dimer model such that a GIT quotient parametrizing stable representations of the associated quiver is a projective crepant resolution of this singularity for some stability parameter. It is also known that the space of stability parameters has the wall-and-chamber structure, and f...

This is an expanded version of a three-hour minicourse given at the winterschool Winterbraids IV held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometrically minded audience. We spoke neither of braids nor of knots, but tried to show how several geometrical tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistica...

A ghor algebra is the path algebra of a dimer quiver on a surface, modulo relations that come from the perfect matchings of its quiver. Such algebras arise from abelian quiver gauge theories in physics. We show that a ghor algebra $\Lambda$ on a torus is a dimer algebra (a quiver with potential) if and only if it is noetherian, and otherwise $\Lambda$ is the quotient of a dimer algebra by homotopy relations. Furthermore, we classify the simple $\Lambda$-modules of maximal dim...

We introduce new techniques based on brane tilings to investigate D3-branes probing orientifolds of toric Calabi-Yau singularities. With these new tools, one can write down many orientifold models and derive the resulting low-energy gauge theories living on the D-branes. Using the set of ideas in this paper one recovers essentially all orientifolded theories known so far. Furthermore, new orientifolds of non-orbifold toric singularities are obtained. The possible applications...

In this paper, we describe a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well studied case of dimer models on a disc. We prove that all Berenstein--Fomin--Zelevinsky quivers for Schubert cells in a symmetric Kac--Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten and Smith that the corresponding superpotentials ...

We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph. The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus. We show that the phase space of the latter...