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

Abstract

A bstract 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(2 h + 2 , ℂ) modular symmetries of Calabi-Yau threefolds with h being the number of moduli fields. Together with the $$ {\mathbb{Z}}_2^{\mathrm{CP}} $$ <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> CP symmetry, they are enhanced to G Sp(2 h + 2 , ℂ) ≃ Sp(2 h + 2 , ℂ) ⋊ $$ {\mathbb{Z}}_2^{\mathrm{CP}} $$ <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> generalized symplectic modular symmetry. We exemplify the S 3 , S 4 , T ′ , S 9 non-Abelian flavor symmetries on explicit toroidal orbifolds with and without resolutions and ℤ 2 , S 4 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.