Mirror Symmetry for Two Parameter Models -- I

TL;DR

This paper explores the complex structure of the moduli space of Calabi-Yau manifolds with two parameters using mirror symmetry, computing instanton numbers and Yukawa couplings, revealing richer structures than one-parameter models.

Contribution

It provides a detailed analysis of two-parameter Calabi-Yau models, including instanton expansions, Yukawa couplings, and the structure of the moduli space, extending previous one-parameter studies.

Findings

Computed genus zero and one instanton numbers.

Described the moduli space structure and singular loci.

Identified models birational to one-parameter models at specific parameters.

Abstract

We study, by means of mirror symmetry, the quantum geometry of the K\"ahler-class parameters of a number of Calabi-Yau manifolds that have $b_{11}=2$. Our main interest lies in the structure of the moduli space and in the loci corresponding to singular models. This structure is considerably richer when there are two parameters than in the various one-parameter models that have been studied hitherto. We describe the intrinsic structure of the point in the (compactification of the) moduli space that corresponds to the large complex structure or classical limit. The instanton expansions are of interest owing to the fact that some of the instantons belong to families with continuous parameters. We compute the Yukawa couplings and their expansions in terms of instantons of genus zero. By making use of recent results of Bershadsky et al. we compute also the instanton numbers for instantons of genus one. For particular values of the parameters the models become birational to certain models with one parameter. The compactification divisor of the moduli space thus contains copies of the moduli spaces of one parameter models. Our discussion proceeds via the particular models $\P_4^{(1,1,2,2,2)}[8]$ and $\P_4^{(1,1,2,2,6)}[12]$. Another example, $\P_4^{(1,1,1,6,9)}[18]$, that is somewhat different is the subject of a companion paper.