Modular symmetry by orbifolding magnetized T2 × T2: realization of double cover of ΓN

Abstract

We study the modular symmetry of zero-modes on $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 and orbifold compactifications with magnetic fluxes, M1, M2, where modulus parameters are identified. This identification breaks the modular symmetry of $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 , SL(2, ℤ)1× SL(2, ℤ)2 to SL(2, ℤ) ≡ Γ. Each of the wavefunctions on $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 and orbifolds behaves as the modular forms of weight 1 for the principal congruence subgroup Γ(N), N being 2 times the least common multiple of M1 and M2. Then, zero-modes transform each other under the modular symmetry as multiplets of double covering groups of ΓN such as the double cover of S4.

Metaplectic flavor symmetries from magnetized tori

“…(E.3)). Specifically, we need ∆z = 1 2 for the T transformation also for even M , in which case our results differ from [26][27][28][29] by phase factors which are absent in eq. (4.31).…”

“…(4.38a), representation matrices of the generators of finite metaplectic modular groups. They are compatible with [27][28][29][30], but (4.38a) differs from [26] by the e iπ/4 phase. For even M , the T transformation can rather be represented as in eqs.…”

“…symmetry based (SB), i.e. impose the modular flavor symmetry to construct the Lagrange density [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], and 2. torus based (TB), in which one derives the symmetries from an underlying torus or related setup [18][19][20][21][22][23][24][25][26][27][28][29][30].…”

Automorphic forms and fermion masses

“…An important part of the theoretical activity has indeed been devoted to the study of Yukawa couplings in realistic string theory compactifications [4][5][6][7][8][9][10][11][12][13][14] and their modular properties [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. Nevertheless, the above considerations suggest a complementary analysis where the flexibility of the bottom-up approach can be combined with the clues coming from string theory.…”

Spontaneous CP violation by modulus τ in A4 model of lepton flavors

“…The level 7 finite modular group Γ 7 PSL(2, Z 7 ) was also presented for the lepton mixing [40]. Based on those works, phenomenological studies have been developed in many works while theoretical investigations have been also proceeded [81][82][83][84][85][86].…”