Classification of three-generation models by orbifolding magnetized T2 × T2

Abstract

We study orbifolding by the ℤ2(per) permutaion of T12 × T22 with magnetic fluxes and its twisted orbifolds. We classify the possible three generation models which lead to nonvanishing Yukawa couplings on the magnetized T12 × T22 and orbifolds including the ℤ2(per) permutation and ℤ2(t) twist. We also study the modular symmetry on such orbifold models. As an illustrating model, we examine the realization of quark masses and mixing angles.

Metaplectic flavor symmetries from magnetized tori

“…(4.37) and (4.38) are consistent with their discussion in [30, equation (126)]. However, as discussed in section 4.3, we disagree with the statement made in [26][27][28][29][30] that the modular transformed wave functions do not follow the appropriate boundary conditions. As we have shown, the T transformation can generally not be represented by a matrix multiplication of the set of wave functions, but necessarily goes beyond this.…”

“…(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.…”

“…(4.31). Nevertheless, the modular T transformation of the 2D compact wave functions for odd M was excluded in [26][27][28][29]. In [30] they were introduced through the so-called Scherk-Schwarz phases.…”

“…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].…”