Modular symmetry at level 6 and a new route towards finite modular groups

Abstract

We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ(N′)/Γ(N″), and the modular group SL(2, ℤ) is ex- tended to a principal congruence subgroup Γ(N′). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry are discussed and some example models are studied numerically. The case of N′ = 2 and N″ = 6 is considered, the finite modular group is Γ(2)/Γ(6) ≅ T′, and a benchmark model is constructed.

Modular flavor symmetry and vector-valued modular forms

“…Although we have focused on the modular group SL(2, Z), the modular invariance framework can be extended to more general Fuchsian group of genus zero, such as Γ(2) [41] and Γ 0 (2) [102], even the symplectic modular group Sp(2g, Z) [42] and so on. The well-defined vector-valued modular forms exist for these group.…”

“…If the integer weight modular forms of level N are used as the building block, the finite modular group is Γ N ≡ Γ/ Γ(N ) or its double covering group Γ N ≡ Γ/Γ(N ) [4]. Modular flavor symmetry has been exploited to explain the hierarchical masses and mixing patterns in the lepton and quark sectors, and many models have been built by using the groups Γ 2 ∼ = S 3 [5,6], Γ 3 ∼ = A 4 [3,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], Γ 4 ∼ = S 4 [14,[23][24][25][26][27][28][29][30][31], Γ 5 ∼ = A 5 [28,32,33], Γ 7 ∼ = PSL(2, Z 7 ) [34], Γ 3 ∼ = T [4,35], Γ 4 ∼ = S 4 [36,37], Γ4 ∼ = S4 [38], Γ 5 ∼ = A 5 [39,40], Γ 6 ∼ = S 3 × T [41] and Γ5 ∼ = A 5 × Z 5 [40]. The possible role of fractional weight modular forms has been explored, and the finite modular group should be extend to the metaplectic cover Γ ...…”

Subcritical regime of hybrid inflation with modular A4 symmetry

“…[2] pointed out that it gives a consistent pattern of the neutrino mixing data [3][4][5][6][7][8]. Variety types of symmetries have been studied so far; for instance, the modular S 3 [9], A 4 [2,, A 5 [53][54][55], and other modular groups [56][57][58][59]. Quark masses and mixings are investigated in refs.…”

A minimal modular invariant neutrino model

“…The model construction is based the inhomogeneous finite modular groups Γ N ≡ Γ/ Γ(N ) [5] or homogeneous finite modular groups Γ N ≡ Γ/Γ(N ) [6]. For finite modular groups of small order, many lepton and quark mass models have been constructed and discussed, for example Γ 2 ∼ = S 3 [7,8], Γ 3 ∼ = A 4 [5,, Γ 4 ∼ = S 4 [16,[28][29][30][31][32][33][34][35][36], Γ 5 ∼ = A 5 [33,37,38], Γ 7 ∼ = PSL(2, Z 7 ) [39], Γ 3 ∼ = T [6,40,41], Γ 4 ∼ = S 4 [42,43], Γ 5 ∼ = A 5 [44][45][46] and Γ 6 ∼ = S 3 × T [47]. In the modular invariant models, the Yukawa couplings are integer weight modular forms of the principal congruence subgroup Γ(N ).…”