Families of Quintic Calabi-Yau 3-Folds with Discrete Symmetries

TL;DR

This paper explores special loci in Calabi-Yau 3-fold families with large discrete symmetries, deriving Picard-Fuchs equations for their periods and examining the relation to Yukawa couplings, enhancing understanding of their mathematical and physical properties.

Contribution

It introduces a generalized technique for computing Picard-Fuchs equations for Calabi-Yau hypersurfaces with large symmetry groups at special loci.

Findings

Derived Picard-Fuchs equations for new families of Calabi-Yau manifolds.

Found a correlation between symmetry group size and Yukawa couplings.

Provided accessible mathematical exposition for physicists.

Abstract

At special loci in their moduli spaces, Calabi-Yau manifolds are endowed with discrete symmetries. Over the years, such spaces have been intensely studied and have found a variety of important applications. As string compactifications they are phenomenologically favored, and considerably simplify many important calculations. Mathematically, they provided the framework for the first construction of mirror manifolds, and the resulting rational curve counts. Thus, it is of significant interest to investigate such manifolds further. In this paper, we consider several unexplored loci within familiar families of Calabi-Yau hypersurfaces that have large but unexpected discrete symmetry groups. By deriving, correcting, and generalizing a technique similar to that of Candelas, de la Ossa and Rodriguez-Villegas, we find a calculationally tractable means of finding the Picard-Fuchs equations satisfied by the periods of all 3-forms in these families. To provide a modest point of comparison, we then briefly investigate the relation between the size of the symmetry group along these loci and the number of nonzero Yukawa couplings. We include an introductory exposition of the mathematics involved, intended to be accessible to physicists, in order to make the discussion self-contained.