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GKZ-Generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces

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Abstract

We present a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of Calabi-Yau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Gröbner basis for the so-called toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the special point called the large radius limit, we find a close relationship between the principal parts of the operators in the GKZ system and the intersection ring of a toric variety. As applications, we analyze general three dimensional hypersurfaces of Fermat and non-Fermat types with Hodge numbers up toh 1,1=3. We also find and analyze several non-Landau-Ginzburg models which are related to singular models.

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Authors and Affiliations

  1. Department of Mathematics, Toyama University, 930, Toyama, Japan

    S. Hosono

  2. Department of Mathematics, Brandeis University, 02154, Waltham, MA, USA

    B. H. Lian

  3. Department of Mathematics, Harvard University, 02138, Cambridge, MA, USA

    S. -T. Yau

Authors
  1. S. Hosono
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  2. B. H. Lian
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  3. S. -T. Yau
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Communicated by R.H. Dijkgraaf

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Hosono, S., Lian, B.H. & Yau, S.T. GKZ-Generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces. Commun.Math. Phys. 182, 535–577 (1996). https://doi.org/10.1007/BF02506417

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  • DOI: https://doi.org/10.1007/BF02506417

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