Algebraic aspects of hypergeometric differential equations

Abstract

We review some classical and modern aspects of hypergeometric differential equations, including $A$-hypergeometric systems of Gel'fand, Graev, Kapranov and Zelevinsky. Some recent advances in this theory, such as Euler-Koszul homology, rank jump phenomena, irregularity questions and Hodge theoretic aspects are discussed with more details. We also give some applications of the theory of hypergeometric systems to toric mirror symmetry.

References in corpus (30)

• The intrinsic normal cone
• An integral structure in quantum cohomology and mirror symmetry for toric orbifolds
• Hypergeometric periods for a tame polynomial
• Irregularity of hypergeometric systems via slopes along coordinate subspaces
• Feynman integrals as A-hypergeometric functions
• Quantum D-modules and generalized mirror transformations
• Equivariant D-modules
• Hypergeometric D-modules and twisted Gauss-Manin systems
• Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems
• Resonance equals reducibility for A-hypergeometric systems
• Periods for flat algebraic connections
• GKZ-hypergeometric systems for Feynman integrals
• Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles
• The rank of a hypergeometric system
• Laurent Polynomials, GKZ-hypergeometric Systems and Mixed Hodge Modules
• Rigid irregular connections on P^1
• On the irregular Hodge filtration of exponentially twisted mixed Hodge modules
• Twistor property of GKZ-hypergeometric systems
• Binomial D-modules
• Mixed twistor D-modules
• Irreducible components of characteristic varieties
• Irregular Hodge theory
• A mirror construction for the big equivariant quantum cohomology of toric manifolds
• Irregular Hodge filtration of some confluent hypergeometric systems
• $A$-Hypergeometric Modules and Gauss--Manin Systems
• Examples of hypergeometric twistor $\mathcal{D}$-modules
• Systems of parameters and holonomicity of A-hypergeometric systems
• Periods of Feynman Diagrams and GKZ D-Modules
• Dualizing, projecting, and restricting GKZ systems
• On the local monodromy of A-hypergeometric functions and some monodromy invariant subspaces

Cited by in corpus (3)

• On Feynman graphs, matroids, and GKZ-systems
• Algebraic Analysis of the Hypergeometric Function 1F1 of a Matrix Argument
• Nonlinear Algebra and Applications