Dimer models and toric diagrams

We initiate a systematic study of 2d (0,2) quiver gauge theories on the worldvolume of D1-branes probing singular toric Calabi-Yau 4-folds. We present an algorithm for efficiently calculating the classical mesonic moduli spaces of these theories, which correspond to the probed geometries. We also introduce a systematic procedure for constructing the gauge theories for arbitrary toric singularities by means of partial resolution, which translates to higgsing in the field theor...

It is shown that dimers is Yang-Baxter integrable as a six-vertex model at the free-fermion point with crossing parameter $\lambda=\tfrac{\pi}{2}$. A one-to-many mapping of vertex onto dimer configurations allows the free-fermion solutions to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by 45 degrees compared to their usual orientation. This dimer model is exactly solvable in geometries of arbitrary finite size. In this paper, we ...

We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi-Yau singularity. Our method combines information from the geometry and topology of Sasaki-Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3-branes probing a toric Calabi-Yau singularity can be deduced. The infinite family of toric singularities with ...

The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ${\mathbb Q}$-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, one can assign a lattice polygon called the perfect matching polygon. It is known th...

We study asymptotics of the dimer model on large toric graphs. Let $\mathbb L$ be a weighted $\mathbb{Z}^2$-periodic planar graph, and let $\mathbb{Z}^2 E$ be a large-index sublattice of $\mathbb{Z}^2$. For $\mathbb L$ bipartite we show that the dimer partition function on the quotient $\mathbb{L}/(\mathbb{Z}^2 E)$ has the asymptotic expansion $\exp[A f_0 + \text{fsc} + o(1)]$, where $A$ is the area of $\mathbb{L}/(\mathbb{Z}^2 E)$, $f_0$ is the free energy density in the bul...

We discuss some diverse open problems in the dimer model, motivated by a geometric viewpoint. This is part of a conference proceedings for the OPAC 2022 conference.

Toric Duality arises as an ambiguity in computing the quiver gauge theory living on a D3-brane which probes a toric singularity. It is reviewed how, in simple cases Toric Duality is Seiberg Duality. The set of all Seiberg Dualities on a single node in the quiver forms a group which is contained in a larger group given by a set of Picard-Lefschetz transformations. This leads to elements in the group (sometimes called fractional Seiberg Duals) which are not Seiberg Duality on a...

We study the behavior of a dimer model under the operation of removing a corner from the lattice polygon and taking the convex hull of the rest. This refines an operation of Gulotta, and the special McKay correspondence plays an essential role in this refinement. As a corollary, we show that for any lattice polygon, there is a dimer model such that the derived category of finitely-generated modules over the path algebra of the corresponding quiver with relations is equivalent...

In this note, we discuss some properties of the quiver BPS algebras. We consider how they would transform under different operations on the toric quivers, such as dualities and higgsing. We also give free field realizations of the algebras, in particular for the chiral quivers.

BPS sector in $\mathcal{N}=2$, four-dimensional toric quiver gauge theories has previously been studied using crystal melting model and dimer model. We introduce the Mahler measure associated to statistical dimer model to study large $N$ limit of these quiver gauge theories. In this limit, generating function of BPS states in a general toric quiver theory is studied and entropy, growth rate of BPS states and free energy of the quiver are obtained in terms of the Mahler measur...