Modular flavor symmetry on a magnetized torus

Abstract

We study the modular invariance in magnetized torus models. The modular invariant flavor model is a recently proposed hypothesis for solving the flavor puzzle, where the flavor symmetry originates from modular invariance. In this framework, coupling constants such as Yukawa couplings are also transformed under the flavor symmetry. We show that the low-energy effective theory of magnetized torus models is invariant under a specific subgroup of the modular group. Since Yukawa couplings as well as chiral zero modes transform under the modular group, the above modular subgroup (referred to as modular flavor symmetry) provides a new type of modular invariant flavor models with D 4 × Z 2 , ðZ 4 × Z 2 Þ ⋊ Z 2 , and ðZ 8 × Z 2 Þ ⋊ Z 2. We also find that conventional discrete flavor symmetries which arise in magnetized torus model are noncommutative with the modular flavor symmetry. Combining both symmetries, we obtain a larger flavor symmetry, which is the semidirect product of the conventional flavor symmetry and the modular flavor symmetry for the nonvanishing Wilson line. For the vanishing Wilson line, we have additional Z 2 symmetry, i.e., parity, which is the unique common element between the conventional flavor symmetry and the modular flavor symmetry.

Metaplectic flavor symmetries from magnetized tori

“…It has been stated in the literature [26,27] that the wave functions given by eq. (2.1) do not satisfy the boundary conditions given by the lattice periodicity when transformed under eq.…”

“…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].…”

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

“…In the former case, since the representations on the T 2 /Z 2 twist orbifolds satisfy the relation in eq. (3.34), the tensor product of these obeys 15) where m 1 , m 2 denote the Z 2 -twist eigenmodes 0, 1 on T 2 1 and T 2 2 , respectively. Therefore, the products of the same Z 2 -twist eigenmodes (m 1 = m 2 ) correspond to Γ 2lcm(|M 1 |,|M 2 |) , while the different Z 2 -twist eigenmodes (m 1 = m 2 ) correspond to Γ 2lcm(|M 1 |,|M 2 |) .…”

“…Zero-modes on such a geometry, corresponding to flavors of the SM quarks or leptons, transform under the modular transformation. It was investigated in magnetized D-brane models [12][13][14][15][16] and heterotic orbifold models [17][18][19][20][21][22][23][24][25]. (See also [26][27][28].)…”

Completing the eclectic flavor scheme of the ℤ2 orbifold

“…In more detail, the traditional flavor symmetry of ref. [1] is extended from [32,49] This picture reveals the fact that the top-down discussion of modular flavor symmetry constitutes an extremely restrictive scenario, which is confirmed in other top-down scenarios [38][39][40][41][42]. As in the case of the bottom-up discussion, firstly the role of (otherwise freely chosen) flavons is played by the moduli T and U , and secondly we arrive at a specific finite modular group, being…”