Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces

Hosono, Shinobu; Klemm, Albrecht; Theisen, Stefan; Yau, Shing–Tung

doi:10.1007/bf02100589

Cited by 388 publications

(900 citation statements)

References 47 publications

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“…The −-signs in (107) and the factor 3 in (108) are needed to match the calculations below with the results in [16] Appendix B2.…”

Section: The Hypersurface Of Degreementioning

confidence: 98%

“…Note the appearance of the set A = 1 a ∈ Z k+1 | a ∈ A in (13) and (16). Notice also the torus action (1) on the left hand side of (15).…”

Section: Integral With K-variable Laurent Polynomial Integrandmentioning

confidence: 99%

“…We leave it as an exercise in Mathematica, Maple or PARI programming. A table of the numbers N j1,j2 for this example appears in [16] Appendix B2 under the name X (3|3) (1, 1, 1|1, 1, 1). In [16] one finds many more 2-parameter models.…”

Section: The Hypersurface Of Degreementioning

confidence: 99%

“…First note that, since dim R A,C ⊗ C < dim R A,C ⊗ C = volume ∆ A , the components of Φ L,γ,ε (u) with respect to some basis of R A,C ⊗ C can not suffice as a basis for the solution space of the GKZ system. They do however constitute a basis for the solution space of some extension of the GKZ system (see [16]). So, strictly speaking we are not talking about the Schwarz map for the GKZ system, but for an extension thereof.…”

Section: The Schwarz Map For Some Extended Gkz Systemsmentioning

confidence: 99%

“…The converse is, however, not true: the GKZ system can have solutions which are not solutions to the system of differential equations in Mirror Symmetry. This means that the latter system contains extra differential equations in addition to those of the underlying GKZ system (see [16] §3.3). On the other hand, the solutions to the differential equations which one encounters in Mirror Symmetry, can all be obtained by a few differentiations from solutions to extremely resonant GKZ hypergeometric systems with c = 0.…”

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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“…The −-signs in (107) and the factor 3 in (108) are needed to match the calculations below with the results in [16] Appendix B2.…”

Section: The Hypersurface Of Degreementioning

confidence: 98%

“…Note the appearance of the set A = 1 a ∈ Z k+1 | a ∈ A in (13) and (16). Notice also the torus action (1) on the left hand side of (15).…”

Section: Integral With K-variable Laurent Polynomial Integrandmentioning

confidence: 99%

Section: The Hypersurface Of Degreementioning

confidence: 99%

Section: The Schwarz Map For Some Extended Gkz Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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GKZ Hypergeometric Structures

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N = 2 String-String Duality and Holomorphic Couplings

Curio

1998

Fortschr. Phys.

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We review aspects of N = 2 duality between the heterotic and the type IIA string. After a description of string duality intended for the non‐specialist the computation of the heterotic prepotential and the F1 function for the ST, STU and STUV model (V a Wilson line) and the matching with the Calabi‐Yau instanton expansions are given in detail. Relations with BPS spectral sums in various connections are pointed out.

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Black Hole Entropy, Special Geometry and Strings

Mohaupt

2001

Fortschritte der Physik

183

251

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We review work done over the last years on the macroscopic and microscopic entropy of supersymmetric black holes in fourdimensional N = 2 supergravity and in N = 2 compactifications of string theory and M-theory. Particular emphasis is put on the crucial role of higher curvature terms and of modifications of the area law in obtaining agreement between the macroscopic entropy, which is a geometric property of black hole solutions and the microscopic entropy, which is computed by state counting in Calabi-Yau compactifications of string or M-theory. We also discuss invariance properties of the entropy under stringy T-duality and S-duality transformations in N = 2, 4 compactifications in presence of higher curvature terms.In order to make the paper self-contained we review the laws of black hole mechanics in higher derivative gravity, the definition of entropy as a surface charge, the superconformal offshell description of N = 2 supergravity, special geometry, and N = 2 compactifications of heterotic and type II string theory and of M-theory.

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Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces

Cited by 388 publications

References 47 publications

GKZ Hypergeometric Structures

GKZ Hypergeometric Structures

N = 2 String-String Duality and Holomorphic Couplings

Black Hole Entropy, Special Geometry and Strings

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