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.
Abstract. This text is based on lectures by the author in the Summer School Algebraic Geometry and Hypergeometric Functions in Istanbul in June 2005. It gives a review of some of the basic aspects of the theory of hypergeometric structures of Gelfand, Kapranov and Zelevinsky, including Differential Equations, Integrals and Series, with emphasis on the latter. The Secondary Fan is constructed and subsequently used to describe the 'geography' of the domains of convergence of the Γ-series. A solution to certain Resonance Problems is presented and applied in the context of Mirror Symmetry. Many examples and some exercises are given throughout the paper.
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