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

TL;DR

This paper explores how flavor, CP, and R-symmetries in heterotic string theory originate from symplectic modular symmetries of Calabi-Yau threefolds, revealing unified structures and explicit examples of non-Abelian flavor groups.

Contribution

It demonstrates the unification of flavor and R-symmetries into symplectic modular groups and provides explicit examples of non-Abelian flavor symmetries in string compactifications.

Findings

Flavor and R-symmetries unify into symplectic modular groups.

Explicit realization of non-Abelian flavor symmetries on orbifolds and Calabi-Yau threefolds.

CP symmetry enlarges flavor groups to non-Abelian discrete groups.

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 $\mathbb{Z}_2^{\rm CP}$ CP symmetry, they are enhanced to $GSp(2h+2, \mathbb{C})\simeq Sp(2h+2, \mathbb{C})\rtimes \mathbb{Z}_2^{\rm CP}$ 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.