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

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 $Sp(2h+2,\mathbb{C})$ 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 $GSp(2h+2,\mathbb{C})\simeq Sp(2h+2,\mathbb{C})⋊{\mathbb{Z}}_2^{\mathrm{CP}}$ generalized symplectic modular symmetry. We exemplify the $S_3,S_4,T^{\prime },S_9$ non-Abelian flavor symmetries on explicit toroidal orbifolds with and without resolutions and ${\mathbb{Z}}_2,S_4$ 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 threefolds. These flavor symmetries are further enlarged to non-Abelian discrete groups by the CP symmetry.

Comments: 36 pages, v2: published version