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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 \url{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” $n_d$.

32G20Period matrices, variation of Hodge structure; degenerations
14J32Calabi-Yau manifolds
14N10Enumerative problems (algebraic geometry)
32G81Applications of deformations of analytic structures to physics
33C70Other hypergeometric functions and integrals in several variables
Full Text: DOI
[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
[3] Klemm, A.; Lerche, W.; Mayr, P.: K3 fibrations and heterotic type II string duality
[4] Hosono, S.; Klemm, A.; Theisen, S.; Yau, S. -T.: Comm. math. Phys.. 167, 301 (1995)
[5] Vafa, C.; Witten, E.: Dual string pairs with N = 1 and N = 2 supersymmetry in four dimensions · Zbl 0957.81590
[6] Lian, B.; Yau, S. -T.: Arithmetic properties of mirror maps and quantum couplings
[7] Yonemura, T.: Tohoku math. J.. 42, 351-380 (1990)
[8] Candelas, P.; De La Ossa, X.; Green, P.; Parkes, L.: Nucl. phys.. 359, 21 (1991)
[9] Candelas, P.; De La Ossa, X.; Font, A.; Katz, S.; Morrison, D.: Nucl. phys.. 416, 481 (1994)
[10] Kaplunovsky, V.; Louis, J.; Theisen, S.: Aspects of duality in N=2 string vacua