Completing the eclectic flavor scheme of the $\boldsymbol{\mathbb Z_2}$ orbifold

TL;DR

This paper thoroughly analyzes the flavor symmetry structure of the two-dimensional $oldsymbol{ ext{Z}_2}$ orbifold, revealing how modular and mirror symmetries constrain particle physics models with predictive flavor patterns.

Contribution

It provides a comprehensive understanding of the eclectic flavor structure in the $ ext{Z}_2$ orbifold, including mirror symmetry and $R$-symmetries, and explores implications for flavor model building.

Findings

Complete picture of local flavor unification in $(T,U)$ moduli space.

Identification of restrictions from modular, mirror, CP, and $R$-symmetries.

Insights into limited flavor representations and their impact on model predictivity.

Abstract

We present a detailed analysis of the eclectic flavor structure of the two-dimensional $\mathbb Z_2$ orbifold with its two unconstrained moduli $T$ and $U$ as well as $\mathrm{SL}(2,\mathbb Z)_T\times \mathrm{SL}(2,\mathbb Z)_U$ modular symmetry. This provides a thorough understanding of mirror symmetry as well as the $R$-symmetries that appear as a consequence of the automorphy factors of modular transformations. It leads to a complete picture of local flavor unification in the $(T,U)$ modulus landscape. In view of applications towards the flavor structure of particle physics models, we are led to top-down constructions with high predictive power. The first reason is the very limited availability of flavor representations of twisted matter fields as well as their (fixed) modular weights. This is followed by severe restrictions from traditional and (finite) modular flavor symmetries, mirror symmetry, CP and $R$-symmetries on the superpotential and Kaehler potential of the theory.