Building SO${}_{10}$- models with ${\mathbb{D}}_4$ symmetry

Abstract

Using characters of finite group representations and monodromy of matter curves in F-GUT, we complete partial results in literature by building SO 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 $S{O}_{10}\times {\mathbb{D}}_4$ models depending on the ways the ${\mathbb{S}}_4$-triplets break down in terms of irreducible ${\mathbb{D}}_4$- representations: $\left(\alpha \right)$ as $1_{{}_{+,-}}\oplus 1_{{}_{+,-}}\oplus 1_{{}_{-,+}};$ or $\left(\beta \right)1_{{}_{+,+}}\oplus 1_{{}_{+,-}}\oplus 1_{{}_{-,-}};$ or also $\left(\gamma \right)$ $1_{{}_{+,-}}\oplus 2_{{}_{0,0}}$. Superpotentials and other features are also given.

Comments: 20 pages, Nuclear Physics B (2015)