Mathematics – Algebraic Geometry
Scientific paper
1993-07-30
Commun. Math. Phys. 168 (1995) 493
Mathematics
Algebraic Geometry
45 pages, Latex 2.09
Scientific paper
10.1007/BF02101841
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 $V_1, \ldots, V_r$ in a toric variety ${\bf P}_{\Sigma}$ 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 ${\bf P}_{\Sigma}$ 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 $V'$ which is called Mirror of $V$. Using the generalized hypergeometric series, we propose a general construction for Mirrors $V'$ of $V$ and canonical $q$-coordinates on the moduli spaces of Calabi-Yau manifolds.
Batyrev Victor V.
Straten Duco van
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