Hypergeometric systems in two variables, quivers, dimers and dessins d’enfants

Abstract

This paper presents some parallel developments in Quiver/Dimer Models, Hypergeometric Systems and Dessins d'Enfants. It demonstrates that the setting in which Gelfand, Kapranov and Zelevinsky have formulated the theory of hypergeometric systems, provides also a natural setting for dimer models. The Fast Inverse Algorithm of [14] and the untwisting procedure of [4] are recasted in this more natural setting and then immediately produce from the quiver data the Kasteleyn matrix for dimer models, which is best viewed as the biadjacency matrix for the untwisted model. Some perfect matchings in the dimer models are direct reformulations of the triangulations in GKZ theory and the rule which maps triangulations to the vertices of the secondary polygon extends to a rule for mapping perfect matchings to lattice points in the secondary polygon. Finally it is observed in many examples and then conjectured to hold in general, that the determinant of the Kasteleyn matrix with suitable weights becomes after a simple transformation equal to the principal A-determinant in GKZ theory. Illustrative examples are distributed throughout the text.

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

“…Our construction of the perfect matchings is similar to Theorem 7.2 of [10]. Let R be any ray whose direction does not coincide with that of the winding number of any zigzag path.…”

“…As we mentioned in section 3.3, Hanany and Vegh [8] and Stienstra [10] have previously made proposals for drawing a dimer with given zigzag winding numbers, but their procedures are impractical for large dimers because of the large amount of trial and error required.…”

“…There are algorithms for solving the former problem by taking the determinant of the Kasteleyn matrix [4][5][6][7][8][9][10] and by counting the windings of zigzag paths [8][9][10]. The latter problem can be solved by the "Fast Inverse Algorithm" [8][9][10], although the algorithm is computationally infeasible for all but very simple toric varieties due to the large amount of trial and error required. We resolve this problem by eliminating the need for trial and error.…”

Remarks on quiver gauge theories from open topological string theory

“…Let Z be a weak toric Fano surface, and denote by X = tot(ω Z ) the total space of its canonical bundle. As discussed in [44], there is a consistent brane tiling corresponding to Z (see also [36,71]). Furthermore, the arguments of [45] imply that M θ coincides with X for a particular choice of the GIT parameter θ.…”

Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories

“…For example, straightforward application of our techniques should yield infinitely many, rich set of examples of dual pairs of 3d Chern-Simons-matter theories and Calabi-Yau 4-folds. Using these results, it would be possible to study the issue of Seiberg(-like) duality [32] for Chern-Simons matter theories (see [33] for recent discussions), or the inverse problem of obtaining bipartite graphs from the toric polytope of C(Y 7 ) (see [21,34] for AdS 5 /CF T 4 case).…”