Stringy origin of non-Abelian discrete flavor symmetries

TL;DR

This paper addresses the origin and classification of non-Abelian discrete flavor symmetries in heterotic orbifold string compactifications. By analyzing the interplay between orbifold geometry and space-group selection rules, it identifies how couplings acquire flavor symmetries such as $D_4$ and $\Delta(54)$, and how these arise from the building blocks of the internal space. It provides a systematic enumeration of symmetry realizations for key orbifold blocks (e.g., $S^1/Z_2$, $T^2/Z_3$, $T^2/Z_4$, $T^6/Z_7$, and their factorizable combinations), and discusses symmetry enhancements at special moduli values as well as breaking patterns via Wilson lines and singlet VEVs. The results offer a geometric framework linking orbifold structure to flavor, with potential implications for Yukawa textures, FCNC suppression, and SUSY-breaking terms in string-inspired models, and motivate extensions to non-factorizable geometries and Gepner/Calabi–Yau regimes.

Abstract

We study the origin of non-Abelian discrete flavor symmetries in superstring theory. We classify all possible non-Abelian discrete flavor symmetries which can appear in heterotic orbifold models. These symmetries include D_4 and Delta(54). We find that the symmetries of the couplings are always larger than the symmetries of the compact space. This is because they are a consequence of the geometry of the orbifold combined with the space group selection rules of the string. We also study possible breaking patterns. Our analysis yields a simple geometric understanding of the realization of non-Abelian flavor symmetries.

Figures (4)

• Figure 1: $\mathbbm{S}^1/\mathbbm{Z}_2$ orbifold. Points which are related by a reflection on the dashed line are identified. The fundamental region of the orbifold is an interval with the fixed points sitting at the boundaries.
• Figure 2: (a) $\mathbbm{T}^2_{\mathrm{SO(5)}}$ is defined by two orthonormal vectors $e_1$ and $e_2$. There are two $\theta$ fixed points, which are indicated by (blue) squares. In addition one has two $\theta^2$ quasi-fixed points (red bullets). The fundamental region of the torus consists of the shaded region, the fundamental region of the orbifold is one quarter thereof. One can fold the fundamental region along the dashed line and identify adjacent edges to obtain a triangle with a fore- and a backside (b).
• Figure 3: $\mathbbm{T}^2/\mathbbm{Z}_2$. Points which are related by a reflection at the origin are identified. The fundamental region of the orbifold (dark gray region) is half of the fundamental region of the torus (gray region). By folding it along the dashed line and identifying the edges one obtains a 'ravioli' (or 'pillow') with the fixed points being the corners.
• Figure 4: If the $\mathbbm{T}^2$ lattice vectors have equal length and enclose $120^\circ$, one can also fold the fundamental region of $\mathbbm{T}^2/\mathbbm{Z}_2$ to a tetrahedron.