Mirror Maps, Modular Relations and Hypergeometric Series II

TL;DR

This paper explores the modular properties of periods, mirror maps, and Yukawa couplings in multi-moduli Calabi-Yau varieties, establishing new identities and proofs related to hypergeometric series and modular functions through degeneration techniques.

Contribution

It introduces a novel degeneration approach linking Calabi-Yau families to elliptic curves and K3 surfaces, leading to proofs of conjectural formulas and new identities involving hypergeometric series.

Findings

Proofs of physicists' conjectural formulas involving mirror maps and modular functions

New identities relating multi-variable hypergeometric series and modular functions

Explicit formulas for instanton numbers in specific Calabi-Yau hypersurfaces

Abstract

As a continuation of \lianyaufour, 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 \vk\lkm~ involving mirror maps and modular functions; (2) new identities involving multi-variable hypergeometric series and modular functions -- generalizing \lianyaufour. 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'' $n_d$.