Modular flavor symmetries of three-generation modes on magnetized toroidal orbifolds

We study the modular symmetry on magnetized toroidal orbifolds with Scherk-Schwarz phases. In particular, we investigate finite modular flavor groups for three-generation modes on magnetized orbifolds. The three-generation modes can be the three-dimensional irreducible representations of covering groups and central extended groups of $\Gamma_N$ for $N=3,4,5,7,8,16$, that is, covering groups of $\Delta(6(N/2)^2)$ for $N=$ even and central extensions of $PSL(2,\mathbb{Z}_{N})$ for $N=$odd with Scherk-Schwarz phases. We also study anomaly behaviors.


Introduction
The origin of the flavor structure such as quark and lepton masses and their mixing angles is one of the most significant mysteries in particle physics. Non-Abelian discrete flavor symmetries [1][2][3][4][5][6] such as S N , A N , ∆(3N 2 ), and ∆(6M 2 ) for the three generations of quarks and leptons are attractive candidates to realize the flavor structure. However, in order to obtain the realistic masses and mixing angles of the quarks and leptons, the complicated vacuum alignment of gauge singlet scalars, the so-called flavons, is required.
The geometries of compact spaces predicted in higher dimensional theories such as superstring theory can be candidates of the origin of the flavor structure. (See Refs. [7,8].) For example, a torus and its orbifold have the complex structure modulus τ , which decides the shape of the torus and the orbifold. There is the modular symmetry Γ ≡ SL(2, Z) as well asΓ ≡ SL(2, Z)/Z 2 as the geometrical symmetry on a torus and some of orbifolds. Under the modular transformation, chiral zero-modes on the torus and the orbifolds, corresponding to the flavors of quarks and leptons, are transformed. That is, the modular symmetry can be regarded as the flavor symmetry. In addition, Yukawa coupligns as well as higher order couplings can be functions of the modulus τ and then they also transform under the modular transformation since they can be obtained by overlap integrals of the zero-mode profiles on the torus and the orbifolds. Instead of flavons, a vacuum expectation value of the modulus τ breaks the flavor symmetry, and characterizes the flavor structure. These features are different from ones in the conventional flavor models. The modular transformation behavior of zero-modes was investigated in magnetized D-brane models [9][10][11][12][13][14][15] and heterotic orbifold models [16][17][18][19][20].
(See also Refs. [21][22][23].) In particlular, on magnetized T 2 with the magnetic flux M, there are M-number of chiral zero-modes [24] and in recent work [13], it was shown that the zero-modes with M = even and vanishing Scherk-Schwarz (SS) phases behave as modular forms of weight 1/2 and then they transform as M-dimensional representations of the finite modular subgroup Γ 2M , which is the quadruple covering group of Γ 2M . There also exists the modular symmetry on the magnetized T 2 /Z (t) 2 twisted orbifold. The number of zero-modes on the magnetized T 2 /Z (t) 2 twisted orbifold was investigated in Refs. [25][26][27][28]. Similarly, in Ref. [14], it was shown that zero-modes on the magnetized T 2 1 × T 2 2 with the magnetic fluxes M (i) (i = 1, 2) on T 2 i and its orbifolds 1 behave as modular forms of weight 1 and they transform under the finite modular subgroup Γ ′ 2lcm(M (1) ,M (2) ) , which is the double covering group of Γ 2lcm(M (1) ,M (2) ) . The number of zero-modes was investigated in Ref. [15]. The modular transformation for Yukawa couplings has also studied in Ref. [15]. Thus, it is important to study the modular flavor symmetries, particularly in magnetized orbifold models.
In this paper, we study modular flavor groups of the three-generation modes on magnetized orbifolds. We study non-vanishing SS phases, although previous studies on the modular symmetry did not include SS phase. We find that the three-generation modes are the threedimensional representations of corresponding covering groups and central extended groups of the above finite modular subgroups provided in Ref. [29].
After this paper, relevant papers appeared [33,34]. In Ref. [33], it was claimed that violation of the modular symmetry in models with odd magnetic fluxes is strange and it is inconsistent. To preserve the modular symmetry, a certain shift of the coordinate was introduced in the models with odd magnetic fluxes in Ref. [33]. That is one class of compacfitication. However, the modular symmetry can break when we impose further boundary conditions on wavefunctions by geometry and/or gauge background, that is, a generic compactification. For example, T 2 /Z N orbifolds with N = 3, 4, 6 break the modular symmetry, while some residual symmetries remain. The full modular symmetry remains in wavefunctions on T 2 and T 2 /Z 2 with even magnetic fluxes and vanishing Wilson lines (WLs), which are equivalent to SS phases. However, nonvanishing SS phases can break the modular symmetry for even magnetic fluxes. Indeed, the number of zero-modes depends on SS phases [26,28]. On the other hand, the modular symmetry is broken in wavefunctions for odd magnetic fluxes and vanishing Wilson lines and SS phases, but the modular symmetry remains for odd magnetic fluxes and non-vanishing WLs, which is a discrete shift of the coordinate. This result is consistent with Ref. [33]. At any rate, a general class of compactifications can be decomposed into two classes. One class of compactifications preserves the modular symmetry, while the other class breaks the modular symmetry. Both are consistent compactifications. Thus, one can concentrate on the compactification preserving the modular symmetry, or one can discuss generic compactification including breaking of the modular symmetry. In Ref. [34], SS phases were also studied. This paper is organized as follows. In section 2, we review the modular symmetry on magnetized T 2 and T 2 /Z (t) 2 twisted orbifold without the SS phases. In section 3, we study the modular symmetry on magnetized T 2 and T 2 /Z (t) 2 twisted orbifold with the SS phases. We can consider the modular symmetry of not only wavefunctions with the magnetic flux M =even and the vanishing SS phases but also ones with the magnetic flux M =odd and the certain SS phases. In section 4, we show the specific modular flavor groups for three-generation modes on magnetized T 2 /Z (t) 2 twisted orbifold with the SS phases. We can find the threegeneration modes are the three-dimensional representations of the quadruple covering groups and Z 8 central extended groups of the corresponding modular flavor groups provided in Ref. [29].
We also extend the analyses to the modular symmetry on magnetized T 2 1 /Z 2 ) orbifold in sections 5 and 6. We can obtain three-dimensional representations of all the double covering groups of Γ N for N = 4, 8, 16, i.e. covering groups of ∆(6N ′2 ) with N ′ = N/2, and Z 4 central extended groups of Γ N for N = 3, 5, 7, i.e. Z 4 extensions of P SL(2, Z N ). In section 7, we conclude this study. In Appendix A, we review that the SS phases can be replaced by the WLs through gauge transformation and we show that the modular transformation for them are consistent. In Appendix B, we also show that the Z N SS phases are related to the Z N shift modes. In Appendix C, we prove that ∆(6M 2 ), which is the quadruple covering group of ∆(6M 2 ), can be obtained. In Appendix D, we express three-dimensional modular forms obtained from the wavefunctions on magnetized orbifolds.
2 Modular symmetry on magnetized T 2 and T 2 /Z (t) 2 twisted orbifold without the Scherk-Schwarz phases In this section, we review the modular symmetry on magnetized T 2 and T 2 /Z (t) 2 twisted orbifold without the SS phases.
First, we review the moular symmetry of T 2 [35][36][37][38]. A two-dimensional torus T 2 can be constructed as T 2 ≃ C/Λ, where Λ is a two-dimensional lattice spanned by lattice vectors e k (k = 1, 2). The torus is characterized by the complex structure modulus τ ≡ e 2 /e 1 (Imτ > 0). We also define the complex coordinate of C as u and one of T 2 as z ≡ u/e 1 , so that z + 1 and z + τ are identified with z. The metric on T 2 is given by and then the area of T 2 is A = |e 1 | 2 Imτ . Here, we can consider the same lattice spanned by the following lattice vectors transformed by SL(2, Z) ≡ Γ, The SL(2, Z) is generated by and they satisfy the following algebraic relations, Under the SL(2, Z) transformation, the complex coordinate of the torus z and the complex structure modulus τ are transformed as The above transformation for the modulus τ is called the (imhomogeneous) modular transformation and alsoΓ ≡ Γ/{±I} is called the (imhomogeneous) modular group since τ is invariant under Z = −I. We define the principal congruence subgroup, Γ(N), of level N by Then, the modular forms, f (τ ), of the (integral) weight k for Γ(N) is the holomorphic functions of τ which transform under the modular transformation in Eq. (5) as Here, ρ(γ) denotes the unitary representation of the quotient group Γ ′ N ≡ Γ/Γ(N) satisfying the following algebraic relations, For even weight k, in particular, ρ(γ) becomes the unitary representation of the quotient group Γ N ≡Γ/Γ(N), whereΓ(N) ≡ Γ(N)/{±I} for N = 1, 2 2 andΓ(N) ≡ Γ(N) for N > 2. Note that Γ N for N = 2, 3, 4, and 5 are isomorphic to S 3 , A 4 , S 4 , and A 5 , respectively [29], and also Γ ′ N for N = 3, 4, and 5 are isomorphic to the corresponding double covering groups: T ′ , S ′ 4 , and A ′ 5 , respectively [30]. In what follows, we review the wavefunctions of (z, τ ) on a magnetized torus and then review their behavior as modular forms under the modular transformation in Eq. (5).
First, let us review the wavefunctions, particularly the zero-mode wavefunctions of the twodimensional spinor, on the torus with U(1) magnetic flux [24]. Here, we do not consider the WLs or the SS phases. In the next section, we will study the case with the non-vanishing SS phases 3 . The U(1) magnetic flux is given by which satisfies the quantization condition, (2π) −1 This flux is induced by the vector potential, This vector potential transforms under lattice translations as which correspond to U(1) gauge transformation. Thereby, the two-dimensional spinor with U(1) unite charge q = 1, should satisfy the following boundary conditions, ψ(z + τ, τ ) = e iχ 2 (z) ψ(z, τ ) = e πiM Imτ z Imτ ψ(z, τ ).
Under these boundary conditions, we can solve the zero-mode Dirac equation, and then only ψ + (z, τ ) (ψ − (z, τ )) has |M|-number of degenerate zero-modes when M is positive (negative). In what follows, we consider the positive flux M. The j-th zero-mode wavefunction on the torus with the flux M is expressed as where ϑ denotes the Jacobi theta function defined as We take the following normalization condition, Now, we can see the wavefunctions for ∀j in Eq. (18) behave as modular forms of weight 1/2 4 under the modular transformation in Eq. (5) [13] as follows. In order to see that, we first introduce the double covering group of Γ, where ρ T 2 ( γ) satisfies Eqs. (27)-(31) with k/2 = 1/2 and N = 2M although I jk = δ j,k in Eq. (27) is modified into δ M −j,k , derived from Note that the above modular transformation for the wavefunctions without the SS phases can be valid only if the magnetic flux M is even because of the consistency of the boundary conditions in Eqs. (15) and (16) under the T transformation. That is, the wavefunctions after the T transformation satisfy while the wavefunctions before the T transformation satisfy In the next section, however, we will show that when we take the SS phases into account, we can also consider the modular transformation for wavefunctions with the flux M =odd. Thus, the wavefunctions on T 2 with the magnetic flux M ∈ 2Z and vanishing the SS phases behave as the modular forms of weight 1/2 for Γ(2M). They seem to be a M-dimensional representation. However, they can be reducible representation. Their concrete flavor symmetry depends on irreducible representations. For example, they can not be faithful. Thus, we will study concrete flavor symmetries of zero-modes in the following sections. Finally, we also review the zero-mode wavefunctions on the magnetized T 2 /Z (t) 2 twisted orbifold without the SS phases [25] and the modular transformation for them [13]. (See also Ref. [9,10].) T 2 /Z (t) 2 twisted orbifold can be obtained by further identifying Z (t) 2 twisted point −z with z. Note that the modulus τ is not restricted by Z (t) 2 twist orbifolding, which means we can also consider the modular transformation on the T 2 /Z (t) 2 twisted orbifold. Then, the wavefunctions on the magnetized T 2 /Z (t) 2 twisted orbifold should also satisfy the following boundary condition, in addition to the boundary conditions on the magnetized T 2 in Eqs. (15) and (16). Actually, their boundary conditions are satisfied by the following linear combination of the wavefunctions on the magnetized T 2 as where N j (t) denotes the normalization factor determined by the normalization condition in Eq. (20). Since the wavefunctions on the T 2 without the SS phases satisfy Eq. (34), ones on the T 2 /Z (t) 2 twisted orbifold without the SS phases can be expanded by In this case without the SS phases, there are M/2 + 1-number of Z where 2 -even and odd modes, respectively. That is, the representations on the magnetized T 2 can be decomposed into smaller representations on the magnetized T 2 /Z (t) 2 twisted orbifold. We will study their concrete flavor symmetries in the following sections.
3 Modular symmetry on magnetized T 2 and T 2 /Z (t) 2 twisted orbifold with the Scherk-Schwarz phases In this section, we review the wavefunctions on magnetized T 2 and T 2 /Z (t) 2 twisted orbifold with the SS phases [26] and then we study the modular symmetry for them.
The wavefunctions on T 2 with the flux M and the SS phases (α 1 , α 2 ) (0 ≤ α 1 , α 2 < 1) 5 satisfy the following boundary conditions, instead of Eqs. (15) and (16). Then, the j-th zero-mode wavefunction is expressed as Note that Eq. (18) corresponds to Eq. (44) with (α 1 , α 2 ) = (0, 0). 5 The wavefunction on the magnetized T 2 ≃ C/Λ with the Z N SS phases is related to the Z N -eigenmode wavefunction on the magnetized Z N full shifted orbifold of T 2 ≃ C/ Λ ( Λ = N Λ) withouth the SS phases [13,41], as shown in Appendix B. The analyses for the wavefunctions on the magnetized T 2 with the (Z N ) SS phases are consistent with ones for the wavefunctions on the magnetized T 2 /Z N full shifted orbifold without the SS phases in Ref. [13].
Let us study the modular transformation for the wavefunction in Eq. (44). First, we check the consistency of the boundary conditions under the modular transformation. For example, the wavefunctions after the T transformation satisfy while the wavefunctions before the T transformation satisfy Thus, in order to see the modular symmetry, particularly the T symmetry, of the wavefunctions, Then, the modular transformation in Eqs. (32) and (33) are deformed as However, Eq. (31) is not obtained in the general SS phases. Note that under the modular transformation, in general, the wavefunctions with the SS phases (α 1 , α 2 ) transform into ones with the different SS phases (α ′ 1 , α ′ 2 ). Conversely, when M is even, only the wavefunctions with (α 1 , α 2 ) = (0, 0) are closed under the modular transformation. This case is reviewed in previous section. Similarly, when M is odd, only the wavefunctions with (α 1 , α 2 ) = (1/2, 1/2) are closed under the modular transformation. In this case, ρ T 2 ( T ) satisfies Thus, the wavefunctions on T 2 with the magnetic flux M ∈ 2Z+ 1 and the SS phases (α 1 , α 2 ) = (1/2, 1/2) behave as the modular forms of weight 1/2. They transform as M-dimensional representations, but they can be reducible. Furthermore, we consider the magnetized T 2 /Z (t) 2 twisted orbifold with the SS phases 6 . In this case, we can only consider the Z 2 SS phases, (α 1 , α 2 ) = (ℓ 1 /2, ℓ 2 /2), (ℓ 1 , ℓ 2 ∈ Z 2 ), which are derived from (52) 6 Similarly, the wavefunctions on the magnetized T 2 /Z (t) 2 twisted orbifold with the SS phases are related to ones on the magnetized T 2 /Z 2 twisted and full shifted orbifold without the SS phases in Ref. [13].
The wavefunctions on the magnetized T 2 /Z (t) 2 twisted orbifold with the Z 2 SS phases can be expanded by ones on the magnetized T 2 in Eq. (44) as where we use Eq. (50) instead of Eq. (34). Then, the modular transformation for the wavefunctions in Eq. (53) is similarly obtained by replacing Eqs. (48) and (49) with In particular, when M =even and (α 1 , α 2 ) = (0, 0), they correspond to Eqs. (40) and (41). When M =odd and (α 1 , α 2 ) = (1/2, 1/2), they become where Eqs.       Table 1, the threegeneration modes are obtained from the Z First, Γ N satisfy On the other hand, ∆(96) ≃ (Z 4 ×Z ′ 4 )⋊Z 3 ⋊Z 2 ≃ ∆(48)⋊Z 2 and ∆(384 [2,3,42]. In order to obtain ∆(96) and ∆(384) from the above algebra (62) for N = 8 and 16, respectively, the following relation, should be also satisfied. Actually, we can show that if S and T satisfy Eq. (64) in addition to Eq. (62) for N = 2M, M ∈ 4Z, the following generators 8 , which means the generators in Eq. (66) are ones of ∆( Note that here and hereafter (as well as in section 6), we omit ρ. Both of the above S and T matrices are the same forms as and we can check that Eq. (70) satisfies Eq. (64) in general. Thus, the three-generation Z We also comment on the modular flavor anomaly. As discussed in Ref. [22,44], the transformation g can be anomalous if det(g) = 1. Then, let us see the anomaly of the modular flavor group ∆(6M 2 ). From Eqs.
which satisfy Eqs. (27)- (30) and (51) with k/2 = 1/2 and replacing I in Eq. (27) with (−1) m=1 I = −I. When we define the following generators, from the above S and T in Eq. (72), they satisfy which mean they are the generators of A 5 × Z 8 . Thus, the three-generational Z

Modular symmetry on magnetized orbifolds of T 2 × T 2
In this section, we extend the analyses to the modular symmetry on magnetized orbifolds of T 2 1 ×T 2 2 , where both of the modulus on T 2 i (i = 1, 2), τ i , are identified each other, i.e. τ 1 = τ 2 ≡ τ . (See Ref. [14].) First, let us consider the modular transformation for the wavefunctions on the with the magnetic flux M (i) =even and the SS phases (α where ρ T corresponding to Eq. (9), where s (1) , s (2) ∈ Z and we omit the Z symmetries in the next section. Also note that when m 1 + m 2 = 1, S 2 = I is satisfied even though the modular weight k = 1. 12 We can further consider the Z permutation orbifold can be considered by identifying z 1 and z 2 . Hence, the wavefunctions on the Z 2 ) orbifold, are expressed as and they satisfy the following boundary condition, where it satisfies Eq. (8) with k = 1 and also satisfies corresponding to Eq. (9). Thus, the wavefunctions on the (T 2 2 ) orbifold with the magnetic flux M ∈ 2Z and the SS phases (α 1 , α 2 ) = (0, 0) behave as the modular forms of weight 1 and then they transform as N (m,n) (M)-dimensional representations, as shown in Table 3. Similarly, the wavefunctions with the magnetic flux M ∈ 2Z + 1 and the SS phases (α 1 , α 2 ) = (1/2, 1/2) also behave as the modular forms of weight 1 and then they transform as N (m,n) (M)-dimensional representations, as shown in Table 4.
In the next section, we show the specific modular flavor groups of the three-generation modes on the magnetized orbifolds of T 2 × T 2 .
6 Modular flavor groups of three-generation modes on magnetized orbifolds of T 2 × T 2 Firstly, we consider the three-generation modes on the magnetized (T 2 2 ) orbifold in Tables 3 and 4.

Other
Finally, we consider the three-generation modes on the magnetized T 2 1 /Z are not identified. In order to obtain N m 1 (M (1) )N m 2 (M (2) ) = 3 on the magnetized T 2 1 /Z orbifold, we can only consider N m 1 (M (1) ) = 3 and N m 2 (M (2) ) = 1. Then, from Tables 1 and 2, we can consider twelve patterns, listed in Table 5. The corresponding finite modular subgroups which can be found by considering Z = −(−1) m 1 +m 2 1 and Eqs. (84)-(87) 14 are also listed in Table 5. The S and T transformation matrices for the Z while ones for N m 1 (M (1) ) = 3 modes are expressed in section 4. Then, we can find the specific modular flavor groups as shown in Table 5. We also show the anomaly-free groups of them in Table 5.
orbifold. The anomaly-free subgroups are also shown.

Conclusion
We have studied the modular symmetry of wavefunctions on magnetized orbifolds: 2 ) orbifold, with the Scherk-Schwarz phases. It has been found that we can consider the modular symmetry of not only wavefunctions with the magnetic flux M =even and the vanishing SS phases (α 1 , α 2 ) = (0, 0) but also ones with the magnetic flux M =odd and the SS phases (α 1 , α 2 ) = (1/2, 1/2).
Moreover, we have investigated the specific modular flavor groups for three-generation modes on the magnetized orbifolds. The three-generation modes on the magnetized T 2 /Z Note that since the anomalous Z 8 symmetry is discrete subgroup of U(1), it can be canceled by the Green-Schwarz mechanism. The three-generation modes on the magnetized T 2 /Z (t) 2 twisted orbifold with the magnetic flux M = 5, 7 are three-dimensional representations of A 5 × Z 8 , P SL(2, Z 7 ) × Z 8 , respectively. Among them, only Z 8 symmetries can be anomalous and then A 5 and P SL(2, Z 7 ) are anomaly free, respectively. Similarly, the three-generation modes on the magnetized (T 2 2 ) orbifold are the corresponding three-dimensional representations of the double covering groups of Γ N for N = 4, 8, 16 and Z 4 central extended groups of Γ N for N = 3, 5, 7, provided in Ref. [29]. Among them, only Z 8 symmetries can be anomalous and then ∆(3M 2 ) for N = 2M = 4, 8, 16, A 4 for N = 3, A 5 for N = 5, P SL(2, Z 7 ) for N = 7 are anomaly free. We have also showed the specific modular flavor groups of the three-generation modes on the other distinguishable magnetized T 2 1 /Z orbifolds in Table 5. Our results on flavor symmetries of three generations are useful to understand quarks and lepton masses and their mixing angles. Also, anomaly behaviors are useful. (See e.g. [45].) We would investigate the realistic model building considering the obtained modular flavor groups in magnetized orbifold models elsewhere.

B Z N Scherk-Schwarz phases and Z N shift modes
Here, we also show that the wavefunctions on magnetized T 2 ≃ C/Λ with the Z N SS phases are related to the Z N -eigenmode wavefunctions on magnetized full Z N shifted orbifold of T 2 ≃ C/ Λ ( Λ = NΛ) without the SS phases as follows.
The T 2 /Z N full shifted orbifold [13], on which the full modular symmetry remains, can be obtained by furthre identifying any Z N shifted points z + (r + s τ )/N (∀r, s ∈ Z N ) with z.
(See also Ref. [41].) Then, the boundary conditions of the wavefunction on the T 2 /Z N full shifted orbifold with the magnetic flux M and the vanishing SS phases are just the following two conditions, where ℓ 1 , ℓ 2 ∈ Z N are the Z N -eigenvalues. From the above boundary conditions, M /N 2 ≡ M ∈ Z should be satisfied. The above wavefunction on the magnetized T 2 /Z N full shifted orbifold without the SS phases, ψ j,M , can be expanded by the wavefunction on the magnetized T 2 without the SS phases as Furthermore, by considering the relation, ( z, τ ) = (z/N, τ ), the boundary conditions in Eqs. (128) and (129) correspond to ones in Eqs. (42) and (43) with the Z N SS phases, (α 1 , α 2 ) = (ℓ 1 /N, ℓ 2 /N) (ℓ 1 , ℓ 2 ∈ Z N ). Actually, the above wavefunction with the Z N -eigenvalue, (ℓ 1 , ℓ 2 ), on the T 2 /Z N full shifted orbifold with the magnetic flux M and the vanishing SS phases is related to the wavefunction on T 2 with the magnetic flux M and the Z N SS phases, (α 1 , α 2 ) = (ℓ 1 /N, ℓ 2 /N), as The analyses of the modular transformation are also consistent. Similarly, the wavefunction on the magnetized T 2 /Z 2 twisted and full shifted orbifold without the SS phases is related to one on the magnetized T 2 with the Z 2 SS phases. Their behavior of the modular transformation are consistent each other.
Then, we can prove

D Three-dimensional modular forms
Here, we express three-dimensional modular forms obtained from the wavefunctions on magnetized orbifolds at z = 0, which means the modular forms can be obtained from Z 2 -even (m = n = 0) modes.