Symplectic modular symmetry in heterotic string vacua: flavor, CP, and R-symmetries

We examine a common origin of four-dimensional flavor, CP, and U(1)_R symmetries in the context of heterotic string theory with standard embedding. We find that flavor and U(1)_R symmetries are unified into the Sp(2h + 2, ℂ) modular symmetries of Calabi-Yau threefolds with h being the number of moduli fields. Together with the ℤ2CP CP symmetry, they are enhanced to GSp(2h + 2, ℂ) ≃ Sp(2h + 2, ℂ) ⋊ ℤ2CP generalized symplectic modular symmetry. We exemplify the S₃, S₄, T′, S₉ non-Abelian flavor symmetries on explicit toroidal orbifolds with and without resolutions and ℤ₂, S₄ flavor symmetries on three-parameter examples of Calabi-Yau threefolds. Thus, non-trivial flavor symmetries appear in not only the exact orbifold limit but also a certain class of Calabi-Yau three-folds. These flavor symmetries are further enlarged to non-Abelian discrete groups by the CP symmetry.