Hypergeometric functions and toral manifolds

Gel'fand, I. M.; Zelevinskii, A. V.; Kapranov, Mikhail

doi:10.1007/bf01078777

Cited by 224 publications

(287 citation statements)

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“…In particular, there exist relations of order n based on the use of the ideal ∂ y ip , 7 , i . This brings the system of differential equations in the form, first discussed by Gelfand-Kapranov and Zelevinskii [44]. 7 There are in general many ways of obtaining relations such as the one below.…”

Section: General Construction Of Picard-fuchs Equations For Calabi-yamentioning

confidence: 99%

Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties

1995

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The moduli dependence of (2, 2) superstring compactifications based on CalabiYau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with c = 9 whose potential is a sum of A-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at c = 9. We use mirror symmetry to derive the dependence of the models on the complexified Kähler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic ("twisted") deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent work of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactified on Calabi-Yau manifolds. CERN-TH-7528/956/95

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Section: General Construction Of Picard-fuchs Equations For Calabi-yamentioning

confidence: 99%

Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties

1995

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“…The systems of differential equations (5)- (7), (8)- (10) and (14)- (16) found in the preceding examples are special cases of systems of differential equations discovered by Gelfand, Kapranov and Zelevinsky [10,12,14]. The general GKZ system for functions Φ of N variables u 1 , .…”

Section: General Gkz Systems Of Differential Equationsmentioning

confidence: 99%

“…times the usual Euclidean volume. In [11] it is pointed out that the proof in [10] requires an additional condition on A. …”

Section: Dimension Of the Solution Space Of A Gkz Systemmentioning

confidence: 99%

“…Theorems 2 and 5 in [10] state that the dimension of the space of (local) solutions of (19)- (20) near a generic point is equal to the normalized volume of the k-dimensional polytope ∆ A := convex hull(A) ; here 'normalized volume' means k! times the usual Euclidean volume.…”

Section: Dimension Of the Solution Space Of A Gkz Systemmentioning

confidence: 99%

“…GKZ stands for Gelfand, Kapranov and Zelevinsky, who discovered fascinating generalizations of the classical hypergeometric structures of Euler, Gauss, Appell, Lauricella, Horn [10,12,14]. The main ingredient for these new hypergeometric structures is a finite subset A ⊂ Z k+1 which generates Z k+1 as an abelian group and for which there exists a group homomorphism h : Z k+1 → Z such that h(A) = {1}.…”

Section: Introductionmentioning

confidence: 99%

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GKZ Hypergeometric Structures

Stienstra

Arithmetic and Geometry Around Hypergeometric Functions

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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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Small Flux Superpotentials for Type IIB Flux Vacua Close to a Conifold

Álvarez-García

Blumenhagen

Brinkmann

et al. 2020

Fortschritte der Physik

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We generalize the recently proposed mechanism by Demirtas, Kim, McAllister and Moritz [1] for the explicit construction of type IIB flux vacua with |W 0 | ≪ 1 to the region close to the conifold locus in the complex structure moduli space. For that purpose tools are developed to determine the periods and the resulting prepotential close to such a codimension one locus with all the remaining moduli still in the large complex structure regime. As a proof of principle we present a working example for the Calabi-Yau manifold ℙ 1,1,2,8,12. [24]

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Hypergeometric functions and toral manifolds

Cited by 224 publications

References 4 publications

Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties

Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties

GKZ Hypergeometric Structures

Small Flux Superpotentials for Type IIB Flux Vacua Close to a Conifold

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