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

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>We examine a common origin of four-dimensional flavor, CP, and U(1)<jats:sub><jats:italic>R</jats:italic></jats:sub> symmetries in the context of heterotic string theory with standard embedding. We find that flavor and U(1)<jats:sub><jats:italic>R</jats:italic></jats:sub> symmetries are unified into the Sp(2<jats:italic>h</jats:italic> + 2<jats:italic>,</jats:italic> ℂ) modular symmetries of Calabi-Yau threefolds with <jats:italic>h</jats:italic> being the number of moduli fields. Together with the <jats:inline-formula><jats:alternatives><jats:tex-math>$$ {\mathbb{Z}}_2^{\mathrm{CP}} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>CP</mml:mi> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula> CP symmetry, they are enhanced to <jats:italic>G</jats:italic>Sp(2<jats:italic>h</jats:italic> + 2<jats:italic>,</jats:italic> ℂ) ≃ Sp(2<jats:italic>h</jats:italic> + 2<jats:italic>,</jats:italic> ℂ) ⋊ <jats:inline-formula><jats:alternatives><jats:tex-math>$$ {\mathbb{Z}}_2^{\mathrm{CP}} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>CP</mml:mi> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula> generalized symplectic modular symmetry. We exemplify the <jats:italic>S</jats:italic><jats:sub>3</jats:sub><jats:italic>, S</jats:italic><jats:sub>4</jats:sub><jats:italic>, T</jats:italic>′<jats:italic>, S</jats:italic><jats:sub>9</jats:sub> non-Abelian flavor symmetries on explicit toroidal orbifolds with and without resolutions and ℤ<jats:sub>2</jats:sub><jats:italic>, S</jats:italic><jats:sub>4</jats:sub> 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.</jats:p>