Abstract
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 \( {\mathbb{Z}}_2^{\mathrm{CP}} \) CP symmetry, they are enhanced to GSp(2h + 2, ℂ) ≃ Sp(2h + 2, ℂ) ⋊ \( {\mathbb{Z}}_2^{\mathrm{CP}} \) 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.
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Ishiguro, K., Kobayashi, T. & Otsuka, H. Symplectic modular symmetry in heterotic string vacua: flavor, CP, and R-symmetries. J. High Energ. Phys. 2022, 20 (2022). https://doi.org/10.1007/JHEP01(2022)020
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DOI: https://doi.org/10.1007/JHEP01(2022)020