Modular flavour symmetry and orbifolds

Abstract

We develop a bottom-up approach to flavour models which combine modular symmetry with orbifold constructions. We first consider a 6d orbifold ${\mathbb{T}}^2/{\mathbb{Z}}_N$, with a single torus defined by one complex coordinate $z$ and a single modulus field $\tau$, playing the role of a flavon transforming under a finite modular symmetry. We then consider 10d orbifolds with three factorizable tori, each defined by one complex coordinate $z_i$ and involving the three moduli fields ${\tau }_1,{\tau }_2,{\tau }_3$ transforming under three finite modular groups. Assuming supersymmetry, consistent with the holomorphicity requirement, we consider all 10d orbifolds of the form $({\mathbb{T}}^2{)}^3/({\mathbb{Z}}_N\times {\mathbb{Z}}_M)$, and list those which have fixed values of the moduli fields (up to an integer). The key advantage of such 10d orbifold models over 4d models is that the values of the moduli are not completely free but are constrained by geometry and symmetry. To illustrate the approach we discuss a 10d modular seesaw model with $S_4^3$ modular symmetry based on $({\mathbb{T}}^2{)}^3/({\mathbb{Z}}_4\times {\mathbb{Z}}_2)$ where ${\tau }_1=i,{\tau }_2=i+2$ are constrained by the orbifold, while ${\tau }_3=\omega$ is determined by imposing a further remnant $S_4$ flavour symmetry, leading to a highly predictive example in the class CSD$(n)$ with $n=1-\sqrt{6}$.

arXiv comment22 pages, 3 figures. v3: Matches published version

References in corpus (31)

• The fate of hints: updated global analysis of three-flavor neutrino oscillations
• Neutrino Mass and Mixing with Discrete Symmetry
• Computing Yukawa Couplings from Magnetized Extra Dimensions
• Lepton Masses and Mixing from Modular $S_4$ Symmetry
• Unified Models of Neutrinos, Flavour and CP Violation
• Tri-bimaximal Neutrino Mixing from Orbifolding
• Unification of Flavor, CP, and Modular Symmetries
• Modular $S_4$ Models of Lepton Masses and Mixing
• Modular $S_4$ and $A_4$ Symmetries and Their Fixed Points: New Predictive Examples of Lepton Mixing
• Spontaneous CP violation from vacuum alignment in $S_4$ models of leptons
• Double Cover of Modular $S_4$ for Flavour Model Building
• A note on the predictions of models with modular flavor symmetries
• Trimaximal TM${}_1$ mixing with two modular $S_4$ groups
• Multiple modular symmetries as the origin of flavour
• A SUSY GUT of Flavour with S4 x SU(5) to NLO
• Eclectic Flavor Groups
• Lessons from eclectic flavor symmetries
• Modular Invariant Quark and Lepton Models in Double Covering of $S_4$ Modular Group
• Landscape of Modular Symmetric Flavor Models
• Eclectic flavor scheme from ten-dimensional string theory -- I. Basic results
• Classification of symmetric toroidal orbifolds
• The eclectic flavor symmetry of the ${\mathbb{Z}}_2$ orbifold
• Symplectic modular symmetry in heterotic string vacua: flavor, CP, and $R$-symmetries
• Non-Abelian Discrete Flavor Symmetries from T^2/Z_N Orbifolds
• Flavored Orbifold GUT - an SO(10) x S4 model
• An $S_4\times SU(5)$ SUSY GUT of flavour in 6d
• Neutrino Mass and Mixing Models with Eclectic Flavor Symmetry $\Delta (27)⋊T^{\prime }$
• Two A4 modular symmetries for Tri-Maximal 2 mixing
• Littlest Modular Seesaw
• Two A5 modular symmetries for Golden Ratio 2 mixing
• Classification of 1D and 2D Orbifolds

Cited by in corpus (2)

• Neutrino mixing sum rules and the Littlest Seesaw
• Orbifold Modular GUT of Flavour