Brane dimers and quiver gauge theories

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.

Dimers and orientifolds

“…They are related to each other by removing/adding an edge on top of an orientifold plane, see section 3.3. This operation is well known and corresponds to Higgsing/unHiggsing in the parent theory [18].…”

“…or dimer diagrams [17,18]. This is a tiling of T 2 defined by a bipartite graph, namely one whose nodes can be colored black and white, with no edges connecting nodes of the same color.…”

“…Another simpler derivation for the prescription is described in section 3.4. Higgsing in the parent theory corresponds to removal of edges associated with the fields acquiring vevs in the dimer diagram [18]. In the presence of orientifold quotients, removal of an edge should be accompanied by the removal of its orientifold image, since both correspond to a single field in the orientifold theory.…”

Properly ordered dimers,R-charges, and an efficient inverse algorithm

“…A dimer model [4][5][6][7][8][9][10] consists of a graph whose vertices are colored black or white, and every edge connects a white vertex to a black vertex, i. e. the graph is bipartite. We will use dimer models embedded on the torus T 2 to describe toric quiver gauge theories.…”

“…With this definition, each node of the dimer becomes a face of the zigzag path diagram, with all edges oriented counterclockwise for a white node, or clockwise for a black node. The other faces of the zigzag path diagram correspond to faces of the dimer, and 1 The original definition of the Kasteleyn matrix imposes constraints on c for the purpose of counting perfect matchings [5][6][7][8]. However, these constraints are not necessary for determining the Newton polygon.…”