Building SO$_{10}$- models with $\mathbb{D}_{4}$ symmetry

TL;DR

This paper constructs SO(10) models with dihedral D4 symmetry using finite group characters and monodromy, extending previous work on symmetric groups, and classifies three types of models based on irreducible representations.

Contribution

It extends the construction of SO(10) models with discrete symmetries from S4 and S3 to D4, providing a detailed classification and superpotential features.

Findings

Identified three types of SO(10)×D4 models based on irreducible representations.

Extended the group-theoretic construction from S4 and S3 to D4.

Provided superpotentials and features for each model type.

Abstract

Using characters of finite group representations and monodromy of matter curves in F-GUT, we complete partial results in literature by building SO$% _{10}$ models with dihedral $\mathbb{D}_{4}$ discrete symmetry. We first revisit the $\mathbb{S}_{4}$-and $\mathbb{S}_{3}$-models from the discrete group character view, then we extend the construction to $\mathbb{D}_{4}$.\ We find that there are three types of $SO_{10}\times \mathbb{D}_{4}$ models depending on the ways the $\mathbb{S}_{4}$-triplets break down in terms of irreducible $\mathbb{D}_{4}$- representations: $\left({\alpha} \right) $ as $\boldsymbol{1}_{_{+,-}}\oplus \boldsymbol{1}_{_{+,-}}\oplus \boldsymbol{1}_{_{-,+}};$ or $\left({\beta}\right) \boldsymbol{\ 1}_{_{+,+}}\oplus \boldsymbol{1}_{_{+,-}}\oplus \boldsymbol{1}_{_{-,-}};$ or also $\left({\gamma}\right) $ $\mathbf{1}_{_{+,-}}\oplus \mathbf{2}_{_{0,0}}$. Superpotentials and other features are also given.