The eclectic flavor symmetries of \( {\mathbbm{T}}^2/{\mathbb{Z}}_K \) orbifolds

Only four ${\mathbb{T}}^2/{\mathbb{Z}}_K$ \( {\mathbbm{T}}^2/{\mathbb{Z}}_K \) orbifold building blocks are admissible in heterotic string compactifications. We investigate the flavor properties of all of these building blocks. In each case, we identify the traditional and modular flavor symmetries, and determine the corresponding representations and (fractional) modular weights of the available massless matter states. The resulting finite flavor symmetries include Abelian and non-Abelian traditional symmetries, discrete R symmetries, as well as the double-covered finite modular groups (S₃ × S₃) ⋊ ℤ₄, T ^′, 2D₃ and S₃ × T ^′. Our findings provide restrictions for bottom-up model building with consistent ultraviolet embeddings.