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, ℂ) modular symmetries of Calabi-Yau threefolds with h being the number of moduli fields. Together with the $$ {\mathbb{Z}}_2^{\mathrm{CP}} $$ ℤ 2 CP CP symmetry, they are enhanced to GSp(2h + 2, ℂ) ≃ Sp(2h + 2, ℂ) ⋊ $$ {\mathbb{Z}}_2^{\mathrm{CP}} $$ ℤ 2 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.

On stringy origin of minimal flavor violation

“…In multi-moduli theories, the modular symmetry is enlarged to the symplectic modular symmetry Sp(2h, Z), which governs the flavor and CP structures. For instance, in the heterotic string theory with standard embedding on Calabi-Yau threefolds, Yukawa and higher-order couplings computed by taking an appropriate number of derivatives of the prepotential are tensor representations of a subgroup of Sp(2h, Z) in the Calabi-Yau moduli space (such as S 4 flavor symmetry [28]). These couplings depend on the complex structure and Kähler moduli for the fundamental and anti-fundamental representations of E 6 gauge group, respectively.…”

Time reversal and $\boldsymbol{CP}$ invariance in Calabi-Yau compactifications

“…For the supersymmetric vacua which preserve CP , we ask whether CP invariant fluxes can be chosen which stabilize the moduli at these points. Similar questions have also been pursued in [11][12][13], though we arrive at somewhat different conclusions: we argue that the fully corrected prepotential preserves CP invariance, and we derive a condition on the third cohomology of the Calabi-Yau manifold which determines whether a supersymmetric flux vacuum preserves CP symmetry. In the case of one-parameter models, we show that the condition is always satisfied.…”

Generalized Matter Parities from Finite Modular Symmetries

“…• We focus on the holomorphic modular transformation of the modulus τ , which parameterizes the modular group. When considering the anti-holomorphic modular transformation (τ → −τ ), that is the CP transformation Z CP 2 , the modular group is enlarged to the generalized modular group such as GL(2, Z) SL(2, Z) Z CP 2 [24,78] which is generalized to multimoduli cases [29]. Since the generalized matter parities are embedded in the subgroup of SL(2, Z), the generalized matter parities Z p are also enhanced to Z p Z CP 2 together with the CP transformation.…”

“…If the four-dimensional SUSY models are embedded in the string theory in a consistent way, the modular symmetry will be regarded as the stringy symmetry and protected by quantum gravity effects. Through the compactification, the modular symmetry will naturally appear in the four-dimensional low-energy effective action (see for heterotic string theory on toroidal orbifolds [21][22][23][24][25] and Calabi-Yau manifolds [26][27][28][29], and Type IIB superstring theory with magnetized D-branes [30][31][32][33][34][35][36]). It is known that the modular group includes non-Abelian discrete groups in the principal subgroups [37].…”