GKZ Hypergeometric Structures

Abstract

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.

Hypergeometric series representations of Feynman integrals by GKZ hypergeometric systems

“…The following recapitulation is adapted for the application to Feynman integrals and will miss some generality in order to keep the discussion short. Since the theory of general hypergeometric functions involves many mathematical aspects inter alia algebraic geometry, combinatorics, number theory and Hodge theory, we refer for more detailed studies to [2,19,26,31,62,71,75].…”

Towards open string mirror symmetry for one-parameter Calabi-Yau hypersurfaces

“…We will argue that the solutions to the differential operator L B are so-called semi-periods. These are solutions to the GKZ hypergeometric system of differential equations (see [55] for a nice review). This system arises naturally in the extension of the Greene-Plesser mirror construction to arbitrary Calabi-Yau hypersurfaces in toric varieties found by Batyrev [56,57].…”

“…This means that σ is necessarily an integral of Ω over a three-chain with nontrivial boundary. Let us therefore briefly recall the relation between the GKZ hypergeometric system and the PF system [55][56][57]. A weighted projective space is a toric variety, and toric varieties can be encoded in terms of fans of cones in a lattice polytopes (for a concise review in the context of Calabi-Yau threefolds see [74]).…”

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

“…In the late 1980's Gelfand, Kapranov and Zelevinsky discovered fascinating generalizations of the classical hypergeometric structures of Euler, Gauss, Appell, Lauricella, Horn [8,9,10,11,25]. The main ingredient for these new hypergeometric structures is a finite sequence A = (a 1 , .…”

“…The beautiful insight of Gelfand, Kapranov and Zelevinsky was that hypergeometric structures greatly simplify if one introduces extra variables and balances this with an appropriate torus action [8,9,10,11,25]. In order to profit from the simplication they developed tools like the secondary fan, secondary polytope and principal A-determinant.…”