Brane Dimers and Quiver Gauge Theories

TL;DR

The paper develops brane tilings and dimer models as a unified, computational framework to derive both quiver gauge theories and the toric geometries of non-compact Calabi–Yau 3-folds. Central to the approach is the Kasteleyn-matrix formalism, which links perfect matchings to GLSM fields and encodes toric data via the characteristic polynomial P(z,w); this yields a fast, scalable alternative to the Forward Algorithm. The authors demonstrate this method across infinite families (notably Y^{p,q} and X^{p,q}) and numerous explicit examples, including non-unique GLSM multiplicities and Higgsing via partial resolution, uncovering both powerful insights and subtle caveats. The work significantly broadens the toolbox for AdS/CFT duals, enabling systematic enumeration of toric phases and providing new perspectives on Seiberg duality, Higgsing, and the geometry–gauge theory correspondence. Overall, brane tilings offer a concrete physical realization of toric McKay-type correspondences and a practical path to mapping large classes of toric quiver theories to their geometric moduli spaces.

Abstract

We describe a technique which enables one to quickly compute an infinite number of toric geometries and their dual quiver gauge theories. The central object in this construction is a ``brane tiling,'' which is a collection of D5-branes ending on an NS5-brane wrapping a holomorphic curve that can be represented as a periodic tiling of the plane. This construction solves the longstanding problem of computing superpotentials for D-branes probing a singular non-compact toric Calabi-Yau manifold, and overcomes many difficulties which were encountered in previous work. The brane tilings give the largest class of N=1 quiver gauge theories yet studied. A central feature of this work is the relation of these tilings to dimer constructions previously studied in a variety of contexts. We do many examples of computations with dimers, which give new results as well as confirm previous computations. Using our methods we explicitly derive the moduli space of the entire Y^{p,q} family of quiver theories, verifying that they correspond to the appropriate geometries. Our results may be interpreted as a generalization of the McKay correspondence to non-compact 3-dimensional toric Calabi-Yau manifolds.

Figures (36)

• Figure 1: A finite region in the infinite brane tiling and quiver diagram for Model I of ${\bf dP}_3$. We indicate the correspondence between: gauge groups $\leftrightarrow$ faces, bifundamental fields $\leftrightarrow$ edges and superpotential terms $\leftrightarrow$ nodes.
• Figure 2: The logical flowchart.
• Figure 3: The quiver gauge theory associated to one of the toric phases of the cone over ${\bf F}_0$. In the upper right the quiver and superpotential (\ref{['superpot.f0']}) are combined into the periodic quiver defined on $T^2$. The terms in the superpotential bound the faces of the periodic quiver, and the signs are indicated and have the dual-bipartite property that all adjacent faces have opposite sign. To get the bottom picture, we take the planar dual graph and indicate the bipartite property of this graph by coloring the vertices alternately. The dashed lines indicate edges of the graph that are duplicated by the periodicity of the torus. This defines the brane tiling associated to this $\mathcal{N}=1$ gauge theory.
• Figure 4: a) Brane tiling for Model I of ${\bf dP}_3$ with flux lines indicated in red. b) Unit cell for Model I of ${\bf dP}_3$. We show the edges connecting to images of the fundamental nodes in green. We also indicate the signs associated to each edge as well as the powers of $w$ and $z$ corresponding to crossing flux lines.
• Figure 5: Toric diagram for Model I of ${\bf dP}_3$ derived from the characteristic polynomial in (\ref{['pol_dP3_1']}).