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Mirror maps, modular relations and hypergeometric series. II. (English) Zbl 0957.32501

Summary: As a continuation of [XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 163–184 (1995; Zbl 1052.14513), see also arXiv:hep-th/9507151], we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties.

In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a three-moduli family of Calabi-Yau toric varieties along a codimension one subfamily which can be described by the vanishing of certain Mori coordinate, corresponding to going to the “large volume limit” in a certain direction. Then we see that the deformation space of the subfamily is the same as a certain family of $K3$ toric surfaces. This family can in turn be studied by further degeneration along a subfamily which in the end is described by a family of elliptic curves. The periods of the $K3$ family (and hence the original Calabi-Yau family) can be described by the squares of the periods of the elliptic curves. The consequences include: (1) proofs of various conjectural formulas of physicists involving mirror maps and modular functions; (2) new identities involving multi-variable hypergeometric series and modular functions – generalizing [loc. cit.].

In Part B, we study for two-moduli families the perturbation series of the mirror map and the type A Yukawa couplings near certain large volume limits. Our main tool is a new class of polynomial PDEs associated with Fuchsian PDE systems. We derive the first few terms in the perturbation series. For the case of degree 12 hypersurfaces in ${P}^{4}\left[6,2,2,1,1\right]$, in one limit the series of the couplings are expressed in terms of the $j$ function. In another limit, they are expressed in terms of rational functions. The latter give explicit formulas for infinite sequences of “instanton numbers” ${n}_{d}$.

##### MSC:
 32G20 Period matrices, variation of Hodge structure; degenerations 14J32 Calabi-Yau manifolds 14N10 Enumerative problems (algebraic geometry) 32G81 Applications of deformations of analytic structures to physics 33C70 Other hypergeometric functions and integrals in several variables