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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 \(P^4[6, 2, 2, 1, 1]\), 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 (algebro-geometric aspects)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
32G81 Applications of deformations of analytic structures to the sciences
33C70 Other hypergeometric functions and integrals in several variables
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References:
[1] Lian, B.H.; Yau, S.-T., Mirror maps, modular relations and hypergeometric series I
[2] Kachru, S.; Vafa, C., Exact results for N=2 compactifications of heterotic strings · Zbl 0957.14509
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