The eclectic flavor symmetry of the $\boldsymbol{\mathbb{Z}_2}$ orbifold

Abstract: Modular symmetries naturally combine with traditional flavor symmetries and $\mathcal{CP}$, giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the two-dimensional $\mathbb{Z}_2$ orbifold, which is equipped with two modular symmetries $\mathrm{SL}(2,\mathbb{Z})_T$ and $\mathrm{SL}(2,\mathbb{Z})_U$ associated with two moduli: the K\"ahler modulus $T$ and the complex structure modulus $U$. The resulting finite modular group is $((S_3\times S_3)\rtimes \mathbb{Z}_4)\times\mathbb{Z}_2$ including mirror symmetry (that exchanges $T$ and $U$) and a generalized $\mathcal{CP}$-transformation. Together with the traditional flavor symmetry $(D_8\times D_8)/\mathbb{Z}_2$, this leads to a huge eclectic flavor group with 4608 elements. At specific regions in moduli space we observe enhanced unified flavor symmetries with as many as 1152 elements for the tetrahedral shaped orbifold and $\langle T \rangle = \langle U \rangle = \exp(\pi\,\mathrm{i}\,/\,3)$. This rich eclectic structure implies interesting (modular) flavor groups for particle physics models derived form string theory.