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

الأصل · EN

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 V₁,, Vᵣ in a toric variety PΣ and the system of differential operators annihilating the special hypergeometric function Φ₀ depending on the fan Σ. In this context, the Mirror Symmetry phenomenon can be interpreted as the following twofold characterization of the series Φ₀. First, Φ₀ is defined by intersection numbers of rational curves in PΣ with the hypersurfaces Vᵢ and their toric degenerations. Second, Φ₀ 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.