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.
Highlights
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
Our results indicate that a certain class of CY threefolds has the S4 flavor symmetry on matter fields associated with the moduli fields
We find that non-trivial flavor symmetries on three-parameter examples of CY threefolds are given by S4 and Z2 symmetries, enlarging to S4 ZC2 P and Z2 ZC2 P taking into account the ZC2 P symmetry
Summary
The CP transformations exchanging the fundamental and anti-fundamental representations of E6 are regarded as the anti-holomorphic transformations of moduli fields. The symplectic modular, flavor, CP, and U(1)R symmetries are treated in a unified manner
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