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 , . . . , V r in a toric variety P Σ and the system of differential operators annihilating the special generalized hypergeometric function Φ 0 depending on the fan Σ. In this context, the mirror symmetry phenomenon can be interpreted as the twofold characterization of the series Φ 0 . First, Φ 0 is defined by intersection numbers of rational curves in P Σ with the hypersurfaces V i and their toric degenerations. Second, Φ 0 is the power expansion near a boundary point of the moduli space of the monodromy invariant period of the homolomorphic differential d-form on an another Calabi-Yau d-fold V ′ called the mirror of V . Using this 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.
Quintics in P 4In this section we give a review of the (conjectural) computation of the Gromov-Witten invariants Γ d and predictions n d for numbers of rational curves of degree d on quintics V in P 5 due to P. Candelas, X. de la Ossa, P.S. Green, and L. Parkes [8]. The main ingedients of this computations were considered in papers of D. Morrison [28,29]. The purpose of this review is to stress that this computation needs knowing only properties of the special generalized hypergeometric function Φ 0 (z). We begin with the computational algorithm for computing the coefficients in the q-expansion of the Yukawa coupling and the predictions for number of rational curves.
We provide certain unusual generalizations of Clausen's and Orr's theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present several examples of algebraic transformations of Calabi-Yau differential equations.
In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds F (n 1 , . . . , n l , n). This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of F (n 1 , . . . , n l , n) to a certain Gorenstein toric Fano variety P (n 1 , . . . , n l , n) which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of P (n 1 , . . . , n l , n) and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of P (n 1 , . . . , n l , n).
In the process of studying the ζ-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill, for which the quartic numerator of the ζ-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in ℚ, so independent of any particular prime. Such factorisations are expected to be modular with each quadratic factor associated to a modular form. If the parameter is defined over ℚ this modularity is assured by the proof of the Serre Conjecture. We identify three values of the parameter that give rise to persistent factorisations, one of which is defined over ℚ, and identify, for all three cases, the associated modular groups. We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. The values of the periods and their covariant derivatives, at the attractor points, are identified in terms of critical values of the L-functions of the modular groups. Thus the critical L-values enter into the calculation of physical quantities such as the area of the black hole in the 4D spacetime. In our search for additional rank two attractor points, we perform a statistical analysis of the numerator of the ζ-function and are led to conjecture that the coefficients in this polynomial are distributed according to the statistics of random USp(4) matrices.
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