Mirror Symmetry of Minimal Calabi-Yau Manifolds

TL;DR

This paper explores mirror symmetry transformations of minimal Calabi-Yau manifolds with small Hodge numbers, deriving Picard-Fuchs equations, analyzing monodromy, and computing instanton numbers to understand their geometric and physical properties.

Contribution

It provides explicit derivations of Picard-Fuchs equations and monodromy behaviors for minimal Calabi-Yau models, including (1,4), (1,3), and (1,1), using computational tools.

Findings

Derived Picard-Fuchs equations for the models.

Determined monodromy matrices consistent with mirror symmetry.

Computed genus 0 and 1 instanton numbers, proposing a weighted discriminant.

Abstract

We perform the mirror transformations of Calabi-Yau manifolds with one moduli whose Hodge numbers $(h^{11}, h^{21})$ are minimally small. Since the difference of Hodge numbers is the generation of matter fields in superstring theories made of compactifications, minimal Hodge numbers of the model of phenomenological interest are (1,4). Genuine minimal Calabi-Yau manifold which has least degrees of freedom for K\"ahler and complex deformation is (1,1) model. With help of {\it Mathematica} and {\it Maple}, we derive Picard-Fuchs equations for periods, and determine their monodromy behaviors completely such that all monodromy matrices are consistent in the mirror prescription of the model (1,4), (1,3) and (1,1). We also discuss to find the description for each mirror of (1,3) and (1,1) by combining invariant polynomials of variety on which (1,5) model is defined. The genus 0 instanton numbers coming from mirror transformations in above models look reasonable. We propose the weighted discriminant for genus 1 instanton calculus which makes all instanton numbers integral, except (1,1) case.