GKZ systems, Gröbner fans, and moduli spaces of Calabi-Yau hypersurfaces. (English) Zbl 0955.14030

Kashiwara, Masaki (ed.) et al., Topological field theory, primitive forms and related topics. Proceedings of the 38th Taniguchi symposium, Kyoto, Japan, December 9-13, 1996 and the RIMS symposium with the same title, Kyoto, Japan, December 16-19, 1996. Boston, MA: Birkhäuser. Prog. Math. 160, 239-265 (1998).
From the paper: Mirror symmetry of Calabi-Yau manifolds has been playing a central role in revealing non-perturbative aspects of the type II string vacua, i.e., the moduli spaces for a family of Calabi-Yau manifolds. Since the success in determining the quantum geometry on the IIA moduli space made by P. Candelas, X. C. de la Ossa, P. S. Green, L. Parkes [in: Essays on mirror manifolds, 31-95 (1992; Zbl 0826.32016)], there has been much progress and a lot of communications between physics and mathematics on these topics.
In this article we present a detailed analysis of the GKZ hypergeometric systems [see I. M. Gel’fand, A. V. Zelevinskij and M. M. Kapranov, Funct. Anal. Appl. 23, No. 2, 94-106 (1989); translation from Funkts. Anal. Prilozh. 23, No. 2, 12-26 (1989; Zbl 0721.33006)] in the context of mirror symmetry of Calabi-Yau hypersurfaces in toric varieties. As an application, we will derive a concise formula for the prepotential about large complex structure limits.
For the entire collection see [Zbl 0905.00081].


14J32 Calabi-Yau manifolds (algebro-geometric aspects)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
83E30 String and superstring theories in gravitational theory
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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