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Zbl 0957.32501
Lian, Bong H.; Yau, Shing-Tung
Mirror maps, modular relations and hypergeometric series. II. (English)
[J] Nucl. Phys., B, Proc. Suppl. 46, 248-262 (1996). ISSN 0920-5632

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 2000:: *32G20 Period matrices
14J32 Calabi-Yau manifolds
14N10 Enumerative problems (classical algebraic geometry)
32G81 Appl. of deformations of analytic structures to physics
33C70 Other hypergeometric functions and integrals in several variables

Citations: Zbl 1052.14513

Cited in: Zbl 1211.11053

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