Mirror Symmetry of Minimal Calabi-Yau Manifolds

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.