Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties

Abstract

We formulate general conjectures about the relationship between the A-model connection on the cohomology of a $d$-dimensional Calabi-Yau complete intersection $V$ of $r$ hypersurfaces $V_1,\dots ,V_r$ in a toric variety ${\mathbf{P}}_{\Sigma }$ and the system of differential operators annihilating the special hypergeometric function ${\Phi }_0$ depending on the fan $\Sigma$. In this context, the Mirror Symmetry phenomenon can be interpreted as the following twofold characterization of the series ${\Phi }_0$. First, ${\Phi }_0$ is defined by intersection numbers of rational curves in ${\mathbf{P}}_{\Sigma }$ with the hypersurfaces $V_i$ and their toric degenerations. Second, ${\Phi }_0$ is the power expansion near a boundary point of the moduli space of the monodromy invariant period of the holomorphic differential $d$-form on an another Calabi-Yau $d$-fold $V^{\prime }$ which is called Mirror of $V$. Using the generalized hypergeometric series, we propose a general construction for Mirrors $V^{\prime }$ of $V$ and canonical $q$-coordinates on the moduli spaces of Calabi-Yau manifolds.

45 pages, Latex 2.09

References in corpus (1)

• Considerations of One-Modulus Calabi-Yau Compactifications: Picard-Fuchs Equations, Kähler Potentials and Mirror Maps

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