arXiv subjects

B. H. Lian

Publications and source records attributed to B. H. Lian.

5 recordsLinked to original sources

arXiv1996

Maximal Degeneracy Points of GKZ Systems

Motivated by mirror symmetry, we study certain integral representations of solutions to the Gel'fand-Kapranov-Zelevinsky(GKZ) hypergeometric system. Some of these solutions arise as period integrals for Calabi-Yau manifolds in mirror symmetry. We prove that for a suitable compactification of the parameter space, there exists certain special boundary points, which we called maximal degeneracy points, at which all but one solutions become singular.

alg-geom↗

arXiv1996

Calabi-Yau Varieties and Pencils of K3 Surfaces

In this note, we give a list of Calabi-Yau hypersurfaces in weighted projective 4-spaces with the property that a hypersurface contains naturally a pencil of K3 variety. For completeness we also obtain a similar list in the case K3 hypersurfaces in weighted projective 3-spaces. The first list significantly enlarges the list of K3-fibrations of \KlemmLercheMayr~ which has been obtained on some assumptions on the weights. Our lists are expected to correspond to examples of the so-called heterotic-type II duality \KachruVafa\KachruSilverstein.

alg-geom↗

arXiv1996

GKZ Hypergeometric Systems and Applications to Mirror Symmetry

We analyze GKZ(Gel'fand, Kapranov and Zelevinski) hypergeometric systems and apply them to study the quantum cohomology rings of Calabi-Yau manifolds. We will relate properties of the local solutions near the large radius limit to the intersection rings of a toric variety and of a Calabi-Yau hypersurface. (Talk presented at "Frontiers in Quantum Field Theory", Osaka, Japan, Dec.1995)

hep-th↗

arXiv1995

GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces

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 to $h^{1,1}=3$. We also find and analyze several non Landau-Ginzburg models which are related to singular models.

alg-geom↗

arXiv1994

A Note on ODEs from Mirror Symmetry

We give close formulas for the counting functions of rational curves on complete intersection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the instanton corrected Yukawa coupling.

hep-th↗