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

Abstract

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