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

We study the modular symmetry of zero-modes on T12×T22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {T}_1^2\times {T}_2^2 $$\end{document} and orbifold compactifications with magnetic fluxes, M1, M2, where modulus parameters are identified. This identification breaks the modular symmetry of T12×T22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {T}_1^2\times {T}_2^2 $$\end{document}, SL(2, ℤ)1× SL(2, ℤ)2 to SL(2, ℤ) ≡ Γ. Each of the wavefunctions on T12×T22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {T}_1^2\times {T}_2^2 $$\end{document} 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.


Introduction
The origin of the flavor structure such as the masses and the mixing angles of quarks and leptons is one of the significant mysteries of the standard model (SM). Many ideas have been proposed to understand the flavor structure. Among them, non-Abelian discrete flavor models [1][2][3][4][5][6][7][8][9] are attractive. In such flavor models, various non-Abelian discrete symmetries such as S N , A N , ∆(3N 2 ), ∆(6N 2 ) are assumed as symmetries of quark and lepton flavors. In those models, the realistic masses and mixing angles of quarks or leptons are obtained through breaking the flavor symmetries by vacuum expectation values (VEVs) of gauge singlet scalars, so-called flavons. However, a complicate vacuum alignment is required.
It is also interesting that the finite modular groups Γ N ≡ Γ/Γ(N ) for N = 2, 3, 4, 5 are isomorphic to S 3 , A 4 , S 4 , A 5 [29], respectively. Note that Γ(N ) (Γ(N )) is a normal subgroup of Γ (Γ), so-called the principal congruence subgroup of level N . Recently, a lot of bottom-up approaches of flavor models inspired by these aspects have been studied . In those models, the Yukawa couplings as well as higher order couplings are treated as modular forms of even-number weights. Furthermore, ref. [72] shows modular forms of odd-number weights are representations of Γ N ≡ Γ/Γ(N ), which is the double covering group of Γ N . In the latest studies [73,74], flavor models of Γ 4 S 4 with modular forms of weight integer were studied. Thus, it is important to study the modular flavor symmetry, Γ N and its covering groups Γ N from both top-down and bottom-up approaches.
The moduli stabilization is a key issue. Three-form fluxes can stabilize complex structure modulus [75] as well as the dilaton. In ref. [28], for example, T 2 1 × T 2 2 × T 2 3 with three-form fluxes have been considered and the complex modulus of T 2 1 , τ 1 and that of T 2 2 , τ 2 have to be related each other, such as τ 1 = τ 2 ≡ τ due to the three-form fluxes. In other words, the modular symmetry on T 2 1 × T 2 2 , Γ × Γ is broken to Γ due to the three-form fluxes below the heavy mass scale of the stabilized moduli. A similar breaking Γ × Γ → Γ can be realized by imposing a permutation symmetry between T 2 1 and T 2 2 . Such a setup is quite interesting as follows. The zero-mode wavefunctions on T 2 with magnetic flux M behaves as the modular forms of the weight 1/2 representing Γ 2|M | . Thus we expect that the above setup would lead to zero-mode wavefunctions behaving the modular forms of the weight 1 and representing double covering groups of Γ N , i.e., Γ N .
Our purpose in this paper is to study the modular symmetry of zero-modes on T 2 1 × T 2 2 with magnetic fluxes, where the complex structure moduli are identified as τ 1 = τ 2 = τ . Furthermore, we also study its orbifolding by the Z 2 twist, the Z N shift, and also the Z 2 permutation, which interchanges T 2 1 and T 2 2 . These orbifoldings decompose a representation to smaller representations such as irreducible representations. This paper is organized as follows. In section 2 we briefly review the zero-mode wavefunctions on T 2 with magnetic flux. In section 3, we give a review on the modular symmetry of zero-modes on T 2 and its orbifolding. In section 4, we apply them on a magnetized T 2 1 × T 2 2 and its orbifolding by the Z 2 twist, the Z N shift, and the Z 2 permutation. Here we identify τ 1 = τ 2 ≡ τ . We find that the wavefunctions on the T 2 1 × T 2 2 behave as modular forms of weight 1 for Γ(N ). The zore-modes are multiplets of Γ N and Γ N . In section 5, we conclude this study. In appendix A, we explain a problem in models with odd magnetic fluxes.

JHEP11(2020)101
2 Zero-mode wavefunctions on magnetized T 2 First, we review ten-dimensional (10D) N = 1 super Yang-Mills theory on M 4 ×T 2 1 ×T 2 2 ×T 2 3 with magnetic fluxes, which is the low energy effective field theory of superstring theory. Also, our setup in section 4 is applicable to D7-brane models on M 4 × T 2 1 × T 2 2 . The 10D Lagrangian is given by with M, N = 0, 1, . . . , 9. By Kaluza-Klein decomposition, ten-dimensional vector potential A M and Majorana-Weyl spinors λ can be written as Here ψ n i (z i ) is the n i -th excited mode of 2D Weyl spinors on the i-th torus, T 2 i , and satisfies the following Dirac equation, Since m n i gives the compact scale mass of the four-dimensional Weyl spinor, we consider massless mode ψ 0 (z i ). In this section, we focus on one torus T 2 , hence, we consider zeromode wavefunctions on the magnetized T 2 . For simplicity, we study the wavefunctions on the torus T 2 with U(1) magnetic flux [76]. The torus can be regarded as the complex plane C divided by a two-dimensional lattice Λ, that is T 2 C/Λ. Then, the lattice Λ is characterized by the complex modulus parameter τ ≡ e 2 /e 1 (Imτ > 0), where e 1 , e 2 are the basis that spans the lattice Λ. This torus has the metric such as (2.5) and the U(1) magnetic flux is given by This flux leads to the vector potential one-form,

JHEP11(2020)101
where ζ is a Wilson line phase. Since the complex coordinate z on T 2 is identified with z + 1 and z + τ , the vector potential A(z,z) obeys the following boundary conditions, which correspond to U(1) gauge transformation. Here, χ 1 (z,z) and χ 2 (z,z) are given by (2.10) Here and in what follows, we consider the 2D spinor with the U(1) charge q = 1. To preserve 2D Dirac equation, eq. (2.4), under U(1) gauge transformation, the boundary conditions for 2D Weyl spinors, (2.11) are obtained by Then, considering contractible loops on T 2 , we obtain the Dirac quantization condition, (2.14) To determine the zero-mode of ψ(z), we need to compute the 2D Dirac operator, where γ z , γz are given by so-called the Jacobi theta function. We can show that eq. (2.18) satisfies [77] ψ j,|M | 0 as well as and, later we will use these relations. We can extend the above U(1) theory to U(N) super Yang-Mills theory. We introduce magnetic fluxes along the diagonal direction of U(N), diag(M, M , · · · ). Then, our theory has several zero-modes, whose Dirac equations include various magnetic fluxes, although their zero-mode wavefunctions are written by the above wavefunctions with corresponding magnetic fluxes. The product of such wavefuncitons, ψ j,|M | ±,0 (z, τ )ψ k,|M | ±,0 (z, τ ) can be expanded by the wavefunctions ψ Similarly, n-point couplings, are also written by products of C jk T 2 (τ ) [79].

Modular symmetry in magnetized T 2
In this section, we review the modular symmetry of the zero-mode wavefunctions on the magnetized T 2 and its orbifolding by Z N twist and shift [16]. To simplify our analysis, we consider the torus with the magnetic U(1) flux and no Wilson lines. However, we can extend this analysis to the models with any flux and non-vanishing Wilson lines without any difficulty.

T 2 models
First, we briefly review modular transformation of zero-mode wavefunctions on T 2 . (See for the modular symmetry, e.g., [80][81][82][83].) The torus T 2 is constructed by C/Λ, where Λ is spanned by the basis e 1 , e 2 and characterized by the modulus parameter τ = e 2 /e 1 (Imτ > 0). Then, the same lattice with different modulus parameter is given by the following basis, This SL(2, Z) transformation is generated by two generators, which satisfy the algebra: This gives following transformations: where u is the complex coordinate of C and z is that of T 2 . Note that the Wilson line ζ is transformed as in z. These are also generated by two generators S and T , Since Z = −I leaves τ invariant, Z(z, τ ) = (−z, τ ), the transformation group for τ is isomorphic toΓ ≡ Γ/{±I}. Here, we introduce the principal congruence subgroup of level N defined by This is the normal subgroup of Γ, e.g., Γ(1) Γ. Its quotient group is given by Similarly, we can also introduceΓ(N ) ≡ Γ(N )/{±I} and obtain its quotient group as follows: The quotient Γ N is isomorphic to Γ 2 S 3 , Γ 3 A 4 , Γ 4 S 4 , and Γ 5 A 5 . In addition, Γ N is the double covering group of Γ N . (See e.g., [72][73][74].)

JHEP11(2020)101
We are now ready to construct the holomorphic functions of τ , the modular forms f (τ ) of integer weight k for Γ(N ). First, we define the automorphy factor J k (γ, τ ) as It is straightforward to show it satisfies Then, the modular forms f (τ ) are defined as the functions satisfying the following relation: and therefore ρ is a unitary representation of the quotient group Γ Here, we can extend the modular forms to the half integer weight k/2. (See e.g., [82,84,85].) We define the double covering group of Γ = SL(2, Z), This SL(2, Z) group is generated by two generators, which satisfy the algebra: The normal subgroup of Γ, Γ(N ) corresponding to Γ(N ) of Γ is defined by Then, the new automorphy factor J k/2 ( γ, τ ) is given by where we take (−1) k/2 = e −iπk/2 . In this extension, the modular forms f (τ ) of half integer weight k/2 are defined as follows,

JHEP11(2020)101
where ρ(h) = I, h ∈ Γ(N ), that is, ρ is a unitary representation of the quotient group Γ N ≡ Γ/ Γ(N ). The algebra of Γ N is given by eq. (3.17) added the further relation T N = I. Next, we consider the modular transformation of zero-modes. Under S and T transformations in eq. (3.6), the equation of motion for 2D Weyl spinor ψ(z), eq. (2.4), is preserved. The boundary conditions for ψ(z), eqs. (2.12) and (2.13), however, are not preserved under T transformation unless M = even. In appendix A, we show the inconsistency between the boundary conditions and the T transformation for M = odd. Here and hereafter, we treat only M = even case. Under S and T , the zero-modes in eq. (2.18) are transformed as By using the modular forms of half integer weight in eq. (3.20), we can rewrite them as Thus, the zero-mode wavefunctions on T 2 behave as the modular forms of weight 1/2 for Γ(2|M |).

T 2 /Z N twist orbifold models
Here, we review the modular symmetry for the wavefunctions on the magnetized T 2 /Z N twist orbifolds. The T 2 /Z N twist orbifolds are obtained by further identifying the complex coordinate of T 2 , z with the Z N discrete rotated points α k N z, where In this identification, the wavefunctions on the magnetized , are required to satisfy the following boundary condition, and, therefore, can be expressed by liner combinations of the wavefunctions on T 2 as [12,77,86,87] ψ j,|M | where N t N is the normalization factor. There exist only four consistent orbifolds such as N = 2, 3, 4, 6. However, except for N = 2, the modulus τ must be fixed to be a certain value for N = 3, 4, 6. Thus, any value of τ is allowed for N = 2, that is the full modular symmetry remains for only N = 2. Now, we focus on the T 2 /Z 2 twist orbifold although we can also consider others with the broken modular symmetry.
The zero-mode wavefunctions on , satisfy the following relation, Thus, using eq. (2.21), we can write ψ The unitary representation ρ T 2 /Z m 2 is given by where ρ T 2 /Z 0 2 ( S) is multiplied by a further factor 1/ √ 2 for j or k = 0, |M |/2. We can directly check that they satisfy the algebra of Γ 2|M | , and the further algebraic relation, Thus, the representations on the T 2 /Z 2 twist orbifold satisfy the same algebra with T 2 . Note that we have not necessarily obtained the irreducible representation of Γ 2|M | . Actually, we will see the further decomposition in the end of this section.

T 2 /Z N shift orbifold models
As another example, we now review the magnetized T 2 /Z N shift orbifolds. The T 2 /Z N shift orbifolds are obtained by further identifying the complex coordinate of T 2 , z with the

JHEP11(2020)101
Since the full modular symmetry remains only on the T 2 /Z N shift orbifolds obtained by further identifying z with z + ke (m,n) N for all m, n ∈ Z N , we consider such full shift orbifolds. Any Z N shift is generated by two shifts, e (1,0) N = 1/N and e (0,1) N = τ /N . We should consider the identifications with these shifts. In these identifications, the wavefunctions on the magnetized T 2 /Z N shift orbifolds, ψ j,|M | , are required to satisfy the following further boundary conditions, The exponential factor e iχ (m,n) N (z) is required to be consistent with the torus boundary conditions in eqs. (2.12) and (2.13). Moreover, to generate any shift from these two shifts, it should be satisfied that This shows the conditions for both the magnetic flux and the Z N phase as follows: .

(3.40)
Remembering the assumption M ∈ 2Z, the case of s ∈ 2Z + 1, N ∈ 2Z + 1 is rejected from the above. Taking these into account , the boundary condition for any Z N shift is induced as According to eq. (3.40), for s ∈ 2Z + 1, N ∈ 2Z, the extra factor mnN/2 is added to . Then, the consistency of the contractible loops on T 2 gives the further magnetic flux condition M/N ≡ t ∈ Z, but it has been already satisfied. Thus, the eigenfunctions for ∃ e (m,n) N -shift can be expressed by liner combinations of the wavefunctions on T 2 as where N s N is the normalization factor. Since e −ikχ (m,n) where M and s can only take the values allowed in eq.
for γ ∈ Γ, and unitary matrices are represented by for s ∈ 2Z, we can directly show that the Z N shifts invariant modes behave as modular forms for Γ(2|M |/N 2 ) and variant modes are that for Γ(2|M |). Although it may seem these give the same result for s ∈ 2Z + 1, the modular transformation does not close in the Z N shift invariant modes, but they transform to Z N shift variant modes. Note that the invariant modes correspond to modes on T 2 C/Λ , Λ ≡ Λ/N with the magnetic flux M/N 2 = s. Thus, we can understand this from the fact that the wavefunctions on torus with the magnetic flux M = 2Z + 1 are not consistent with T transformation τ → τ + 1.
Instead, there are the |s| number of the Z N shift ( 1 , 2 ) = (N/2, N/2) modes and the other (|M | − |s|) modes transformed independently under the modular transformation. Also we can check that the former modes behave as the modular forms for Γ(8|M |/N 2 ) and they satisfy the further algebraic relation, (3.49) The latter modes just behave as that for Γ(2|M |).

T 2 /Z N twist and shift orbifold models
As the end of the review of the T 2 /Z N orbifolds, we study the T 2 /Z N twist and shift orbifold models. The full modular symmetry remains only on the combination of the T 2 /Z 2 twist orbifold and the full T 2 /Z 2 shift orbifold, that is the full T 2 /Z 2 twist and shift orbifold, since the full T 2 /Z 2 shift orbifold only satisfies the consistency condition with the T 2 /Z 2 twist orbifold, i.e., N − 1,2 ≡ 1,2 (mod N ). For M/4 = s ∈ 2Z, the wavefunctions on the above orbifold are given by Ψ r,|s| Then, we can directly show that the Z 2 shifts invariant modes (m; 1 , 2 ) = (m; 0, 0) behave as the modular forms for Γ(|M |/2) and variant modes are that for Γ(2|M |). Moreover, they satisfy the further algebraic relation such as eq. (3.34).

JHEP11(2020)101
4 Modular symmetry in magnetized T 2 1 × T 2 2 In the previous section, we have seen the modular symmetry on the magnetized T 2 and its orbifolds by the Z N twist and shift. In this section, let us consider the modular symmetry of the zero-mode wavefunctions on the magnetized T 2 1 ×T 2 2 and orbifolds where the complex modulus parameters are identified as τ 1 = τ 2 ≡ τ . As in the previous analyses on T 2 , we assume the even magnetic fluxes and focus on the zero-mode wavefunctions on the orbifolds. 4.1 T 2 1 × T 2 2 models Since the wavefunctions on T 2 behave like the modular forms of weight 1/2 for Γ(2|M |), we can treat the wavefunctions on T 2 1 × T 2 2 as the modular forms of weight 1 as follows: where the lower indices 1 and 2 of the coordinates z and the magnetic fluxes M denote the tori T 2 1 and T 2 2 , respectively. Note that the modular symmetry on T 2 1 × T 2 2 , Γ × Γ, is broken to Γ by the identification τ 1 = τ 2 ≡ τ . Similarly, the unitary representation ρ(γ) (jk)(mn) is broken from Γ 2|M 1 | × Γ 2|M 2 | to its subgroup. Since ρ(γ) (jk)(mn) is given by the tensor products of the representations on T 2 1 and T 2 2 such as above, by multiplying both algebraic relations in eq. (3.25) for T 2 1 and T 2 2 , we can obtain the following relations for T 2 1 × T 2 2 :

(4.4)
This is just the algebra of Γ 2lcm(|M 1 |,|M 2 |) . 2 Thus, the zero-mode wavefunctions on T 2 1 × T 2 2 behave as the modular forms of weight 1 for Γ(2lcm(|M 1 |, |M 2 |)). This argument on the algebraic relations is valid for other orbifolds unless we study orbifolding across the tori T 2 1 and T 2 2 . Such orbifolding by permutation may affect the algebraic relations since their representations cannot be written as the tensor products. The remaining of this section, we consider orbifolding T 2 1 × T 2 2 by the Z 2 twist, the full Z N shifts and the Z 2 permutation that interchanges the two tori coordinates, z 1 ↔ z 2 .
The n-point couplings are also obtained by products of C (j,j ,j ),(k,k ,k ) (τ ), and they are modular forms of weight (n − 2).

(T 2 1 × T 2 2 )/Z N twist and shift orbifold models
First of all, we consider orbifolding by the Z 2 twist and the full Z N shift, where the algebraic relations in the previous section is valid. On the (T 2 1 × T 2 2 )/Z N twist and shift orbifolds, in general, the tensor product of the representations of Γ 2|M 1 | and Γ 2|M 2 | gives the representation of Γ 2lcm(|M 1 |,|M 2 |) .
Since the wavefunctions on the above orbifolds are obtained by the tensor products of each orbifold, for example, the wavefunctions on the pair of T 2 1 and T 2 2 with M 1 = M 2 = 2 are obtained as while the Z N -shift even modes on the pair of T 2 1 and the T 2 (4.9) -14 -

JHEP11(2020)101
Then, the representations of the S, T transformations are same on both wavefunctions, (4.10) They generate the group Γ 4 S 4 which has the order 48. From the correspondence of the Z N -shift even modes to the 1/N torus with the magnetic flux M 2 /N 2 = 2, we can understand these equalities. Now, it is straightforward to confirm that they satisfy the above general rule for the algebraic relations on the (T 2 1 × T 2 2 )/Z N twist and shift orbifolds, hence, the product representation of Γ 4 and Γ 4 gives the representation of Γ 4 . The matrices ρ(S) and ρ(T ) in eq. (4.10) correspond to a reducible representation Γ 4 S 4 . They can be decomposed into a triplet and a singlet. The triplet corresponds to where S and T are expressed as follows: In addition, the singlet corresponds to where S and T are expressed as 14) The wavefunction of the singlet vanishes at z 1 = z 2 = 0, while the other do not vanish. Thus, the singlet is trivial as the conventional modular form f (τ ). Similarly, we can study other types of orbifolding. However, there are exceptions on the pair of the Z 2 twist orbifolds and that of the Z N shift orbifolds. 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 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 |) .

JHEP11(2020)101
In the latter case, only on the pair of the Z N shift orbifolds with the magnetic fluxes M = N 2 s, s ∈ 2Z + 1, N ∈ 2Z, there are the further relation in eq. (3.49) for the Z N -shift ( 1 , 2 ) = (N/2, N/2) modes. Then, the tensor products of the algebraic relations obey (4. 16) This means that the products of the Z N -shift ( 1 , 2 ) = (N/2, N/2) modes correspond to Γ 4lcm(|s 1 |,|s 2 |) , where 4lcm(a, b) denotes 4 times the least common multiple of a and b.
With these in mind, we show the algebraic relations for the unitary representation on the (T 2 1 × T 2 2 )/Z N twist and shift orbifolds in table 1. The dimension of each normal subspace (eigenmode) is given by the products of the number of modes on each T 2 /Z N orbifold discussed in section 3. It is shown in table 2. For simplicity, we omit the results of the Z N shift orbifolds with the magnetic fluxes M = N 2 s, s ∈ 2Z + 1, N ∈ 2Z, but we can obtain them from the arguments up to now.

Magnetic flux
Algebra for each mode Table 1. The algebraic relations for the unitary representation on the (T 2 1 × T 2 2 )/Z N twist and shift orbifolds. The first (second) column shows the types of orbifolds from T 2 1 (T 2 2 ), which also include T 2 1 (T 2 2 ) itself. The third column shows the flux condition on each orbifold. The last column shows the algebraic relations that the unitary representation on the orbifolds at least satisfies. The sign (+/− t/s 1/2 ) means the invariant (+)/variant (−) modes under the Z 2 twist ( t )/Z N shift ( s ) on

JHEP11(2020)101
where it is multiplied by further factor 1/2 for m = . It satisfies the same algebraic relations with Γ 2|M | . Thus, the representations on the (T 2 1 × T 2 2 )/Z 2 permutation orbifold obey the same algebra with that on T 2 1 × T 2 2 , but their dimensions are smaller. For example, the Z 2 -permutation even modes on the (T 2 1 × T 2 2 )/Z 2 permutation orbifold with the magnetic flux M = 2 is given by (4.21) The unitary representations of the S and T are expressed as follows: These matrices are the same as those in eq. (4.12). They generate the group Γ 4 S 4 which has the order 48. The three zero-modes correspond to a triplet of Γ 4 S 4 . Thus, orbifolding by twist, shift, and permutation can decompose reducible representations into smaller one such as a reducible representation by their eigenvalues.

(T 2
1 × T 2 2 )/Z N twist, shift and permutation orbifold models Now, we are ready to write down the algebraic relations for the unitary representations of the zero-mode wavefunctions on the (T 2 1 × T 2 2 )/Z N twist, shift, and permutation orbifolds. As we saw in the previous subsection, the Z 2 permutation does not affect the algebraic relations for the representations, although it is the permutation across the two tori T 2 1 and T 2 2 . Thus, the algebraic relations in section 4.1 is valid for orbifolds including the Z 2 permutation. Note that to identify two tori T 2 1 and T 2 2 , the only pairs of the same Z 2twist (Z N -shift) eigenmodes are allowed on the (T 2 1 × T 2 2 )/Z N twist, shift and permutation orbifolds.
As shown in the previous subsection, we can construct the wavefunctions on the (T 2 1 × T 2 2 )/Z N twist, shift and permutation orbifolds as where ψ j,|M |

JHEP11(2020)101
Orbifolds Magnetic flux Algebra for each mode Table 3. The algebraic relations for the unitary representation on the (T 2 1 × T 2 2 )/Z N twist, shift and permutation orbifolds. The first column shows the types of orbifolds. The second column shows the magnetic fluxes. The last column shows the algebraic relations for the unitary representation. The notation is same with table 1.

Conclusion
In this paper, we have discussed the modular symmetry on (T 2 1 × T 2 2 )/Z N assuming the modulus parameters are identified. This identification allows us to regard the zero-mode wavefunctions on (T 2 1 × T 2 2 )/Z N as the modular forms of weight 1. Moreover, the modular symmetry on T 2 1 ×T 2 2 , Γ×Γ, is broken to Γ. Zero-modes are multiplets of the favor symmetry Γ 2lcm(|M 1 |,|M 2 |) and Γ 2lcm(|M 1 |,|M 2 |) depending on orbifolding. Only if we consider the pair of the T 2 /Z 2 twist orbifolds, the flavor symmetry of zero-modes are Γ 2lcm(|M 1 |,|M 2 |) . In the case of other orbifolds obtained by the Z 2 twist, the full Z N shift and the Z 2 permutation, the flavor symmetries are given by Γ 2lcm(|M 1 |,|M 2 |) . Especially, we have shown the realization of the double cover of S 4 , i.e., S 4 . That would be interesting from the recent bottom-up approach of model building [73,74]. Also the flavor symmetry Γ 2lcm(|M 1 |,|M 2 |) would be interesting.
Orbifolding decomposes zero-modes by eigenvalues of the Z 2 twist, the Z N shift and the Z 2 permutation, and reduce the number of zero-modes, namely the generation number of quarks and leptons. Three-generation models on twist orbifolds T 2 /Z 2 with magnetic fluxes have been classified in refs. [89,90]. Combinations of orbifolding by the Z 2 twist, the Z N shift and the Z 2 permutation provide us with the further possibility to construct three-generation models. We would study such model building and its phenomenological aspects elsewhere.

JHEP11(2020)101
These lead to the following conditions for ψ(z, τ + 1): ψ(z + 1, τ + 1) = e iπM Im(z+ζ) Imτ ψ(z, τ + 1), (A.4) ψ(z + τ, τ + 1) = e iπM · e iπM Imτ (z+ζ) Imτ ψ(z, τ + 1). (A.5) Here, the only second condition is different with the boundary conditions for ψ(z, τ ) by the exponential factor e iπM . If we focus on M = even, e iπM drops and the boundary conditions for ψ(z, τ + 1) are the same with ψ(z, τ ). While if we focus on M = odd, e iπM equals to -1. Then, ψ(z, τ + 1) cannot be expanded by ψ(z, τ ) since it obeys the same equation of motion and the different boundary conditions with ψ(z, τ ). This inconsistency is caused from the gaps of the phase factor between the wavefunctions at z and z + nτ + m. Thus, for M = odd, we cannot define the T transformation as the map among the wavefunctions at the point z, ψ(z, τ ). (See for details [16].) Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.