Revisiting modular symmetry in magnetized torus and orbifold compactifications

We study the modular symmetry in $T^2$ and orbifold comfactifications with magnetic fluxes. There are $|M|$ zero-modes on $T^2$ with the magnetic flux $M$. Their wavefunctions as well as massive modes behave as modular forms of weight $1/2$ and represent the double covering group of $\Gamma \equiv SL(2,\mathbb{Z})$, $\widetilde{\Gamma} \equiv \widetilde{SL}(2,\mathbb{Z})$. Each wavefunction on $T^2$ with the magnetic flux $M$ transforms under $\widetilde{\Gamma}(2|M|)$, which is the normal subgroup of $\widetilde{SL}(2,\mathbb{Z})$. Then, $|M|$ zero-modes are representations of the quotient group $\widetilde{\Gamma}'_{2|M|} \equiv \widetilde{\Gamma}/\widetilde{\Gamma}(2|M|)$. We also study the modular symmetry on twisted and shifted orbifolds $T^2/\mathbb{Z}_N$. Wavefunctions are decomposed into smaller representations by eigenvalues of twist and shift. They provide us with reduction of reducible representations on $T^2$.


Introduction
The standard model (SM) is now well established. However, the origin of the flavor structure of quarks and leptons is still one of mysteries of the SM. Various studies have been carried out to understand the flavor structure. One of the interesting approaches is to impose some non-Abelian discrete flavor symmetries [1][2][3][4][5][6][7][8][9] on the flavors of quarks and leptons. Various discrete symmetries such as S N , A N , ∆(6N 2 ) are used. Then, these flavor symmetries are broken by vacuum expectation values (VEVs) of gauge singlet scalars, the so-called flavons in order to realize the masses and the mixing angles of quarks and leptons. However, a complicated vacuum alignment is required.
Superstring theory predicts six-dimensional (6D) compact space in addition to our fourdimensional (4D) space-time. Such a compact space may provide us with origins of non-Abelian discrete flavor symmetryies. (See, e.g. [10,11].) In particular, the torus as well as orbifolds has the modular symmetry as geometrical symmetry. Zero-modes transform under the modular symmetry. That is, the modular symmetry is a flavor symmetry in a sense. These transformation behavior has been studied in magnetized D-brane models [12][13][14][15] and heterotic orbifold modelds [16][17][18][19][20]. (See also [21][22][23].) However, the modular flavor symmetry is different from the conventional flavor symmetries. Yukawa couplings as well as higher order couplings are not singlets, but transform under the modular symmetry.
As mentioned above, the modular symmetry is quite important from both top-down and bottom-up approaches. That could become a bridge to connect high and low energy scales. Our purpose of this paper is to study the modular symmetry in more detail. We study how wavefunctions on T 2 with magnetic flux transform under the modular symmetry. Furthermore, we also study twisted and shifted orbifolds. Orbifold twist and shift decompose wavefunctions by their eigenvalues. That provides us with reduction of reducible representations. Also, it provides us with a new approach to construct three-generation models from the phenomenological viewpoint.
This paper is organized as follows. In section 2, we briefly review the modular symmetry on T 2 and modular forms. After reviewing T 2 with magnetic flux in section 3.1, we study the modular symmetry on the magnetized T 2 in section 3.2. We find that the wavefunctions on the magnetized T 2 are transformed under the modular transformations like modular forms of weight 1/2 for Γ(2|M|). We also study the modular symmetry on various magnetized T 2 /Z N orbifolds in sections 4-6. In section 4, we study that on the T 2 /Z N twisted orbifolds. In section 5, we study that on the T 2 /Z N shifted orbifolds. Furthermore, in section 6, we study that on the T 2 /Z N twisted and shifted orbifolds. In those sections, we find that the modular symmetry remains on the T 2 /Z 2 twisted orbifold and the "full" shifted T 2 /Z N orbifolds. In particular, the "full" T 2 /Z 2 shifted orbifold is consistent with the T 2 /Z 2 twisted orbifold. Section 7 is conclusion. Appendix A shows the extension for the generalized CP symmetry with the modular symmetry on the magnetized T 2 . Appendix B shows the detail calculation of discussion in section 3.2. Appendix C shows examples of the magnetized T 2 /Z 2 twisted and shifted orbifold models.

Modular symmetry and modular forms
In this section, we briefly review the modular symmetry on T 2 and the modular forms. (See e.g [42][43][44][45]. See also [34].) First, we review the modular symmetry on T 2 . The torus T 2 can be constructed as division of the complex plane C by a two-dimensional (2D) lattice Λ, i.e. T 2 ≃ C/Λ. The lattice Λ is spanned by two lattice vectors e i (i = 1, 2). We denote the complex coordinate of C as u and that of the T 2 as z ≡ u/e 1 . We also introduce the complex modulus parameter τ ≡ e 2 /e 1 (Imτ > 0). However, there is some ambiguity in choice of the lattice vectors. The lattice spanned by the following lattice vectors e ′ is the same lattice spanned by the lattice vectors e i (i = 1, 2). Under the above SL(2, Z) transformation, the coordinate of T 2 and the modulus are transformed as This is the modular transformation. The group Γ ≡ SL(2, Z) is generated by two generators, They satisfy the following algebraic relations 1 , Under S and T , the coordinate of T 2 and the modulus, (z, τ ), are transformed as Note that −I : (z, τ ) → (−z, τ ). Next, we review the modular forms. The principal congruence subgroup of level N, Γ(N) is the normal subgroup of Γ defined by where we have Γ(1) ≃ Γ. The modular forms f (τ ) of weight k for Γ(N) are holomorphic functions of τ which transform as Here, k is integer while k is even for N = 1, 2 because of −I ∈ Γ(N). The above modular forms of weight k for Γ(N) transform as under Γ transformation, where ρ is a unitary representation of the quotient group Γ ′ N ≡ Γ/Γ(N). Thus, the representation of Γ ′ N , ρ, satisfies the following relations, Note that since the relation (−1) k ρ(−I) = I should be satisfied, it is required that ρ(−I) = I (ρ(−I) = −I) when k is even (odd). Consequently, when k is even, ρ becomes a representation of Γ N ≡ Γ/Γ(N), where we define Γ ≡ Γ/{±I} and Γ(N) ≡ Γ(N)/{±I} (N = 1, 2) while Γ ≡ Γ(N) (N > 2). As mentioned in section 1, Γ N are isomorphic to Γ 2 ≃ S 3 , Γ 4 ≃ A 4 , Γ 4 ≃ S 4 , and Γ 5 ≃ A 5 . Furthermore, we define the automorphy factor as which satisfies In the next section, we study the modular symmetry for wavefunctions on the magnetized T 2 . In the following analysis, we extend the above discussion on modular forms.

Modular symmetry in magnetized T 2 model
We consider ten-dimensional N = 1 supersymmetric Yang-Mills theory, as an effective field theory of magnetized D-brane models of superstring theory, compactified on T 2 × T 2 × T 2 with non-vanishing magnetic fluxes. Magnetic fluxes induce degenerate zero-modes corresponding to flavors of quarks and leptons as well as massive modes. In particular, we focus on one T 2 with a magnetic flux, and start with 6D theory. In this case, the wavefunction of the fermion on 6D space-time, λ(X), is decomposed into the wavefunction on 4D space-time, ψ j n (x), and the wavefunction on T 2 , ψ j n (z) as follows, where we chose ψ j n (z) as the eigenstate of 2D Dirac operator i D 2 as i D 2 ψ j n (y) = m n ψ j n (z).
Here, we denote the n-th excited and j-th degenerate wavefunction as ψ j n , and also we denote the coordinates of 6D space-time, 4D space-time, and T 2 as X, x, and z, respectively. Then, the 6D action for massless fermion λ(X) is reduced to 4D action as In this section, we study the modular symmetry for the wavefunctions on the magnetized T 2 .

Wavefunctions on magnetized T 2
First, we briefly review the wavefunctions on T 2 ≃ C/Λ with U(1) magnetic flux 2 [46]. The metric on T 2 is given by The U(1) magnetic flux on T 2 , should satisfy the quantization condition, 2 The following analysis in this paper can be applied for U (N ) magnetic fluxes.
It is induced from the following vector potential, where ζ is a Wilson line. The above vector potential satisfies the following boundary conditions, which correspond to U(1) gauge transformation. Then, the wavefunctions on the T 2 with the above gauge background satisfy the following boundary conditions, where we consider the unit U(1) charge, q = 1. Note that the quantization condtion M ∈ Z originates from the above boundary conditions. The zero-mode wavefunction of the twodimensional spinor with charge q = 1, is obtained by solving the zero-mode Dirac equation, where γ z , γz are written by satisfying {γ z , γz} = 2h zz , and we denote the covariant derivative as D z = ∂ z − iA z (Dz = ∂z − iAz). From Eq. (26), each componet of Eq. (25) is described by When the magnetic flux M is positive, ψ M + (z) and ψ −M − (z) = ψ M + (z) have |M| number of degenerate zero-modes described by with ∀j ∈ Z |M | = {0, 1, 2, ..., |M| − 1}, where A = |e 1 | 2 Imτ is the area of T 2 and ϑ denotes the Jacobi theta function defined as Similarly, when M is negative, ψ M − (z) has |M| degenerate zero-modes, whose wavefuctions are the same as the above. Thus, we can realize a chiral theory.
Furthermore, the wavefunctions of the n-th excited-modes [47], whose squared masses are m 2 n = 4πM A n, can be described by where we use the creation and annihilation operators, which satisfy [a, a † ] = 1, and H n (x) is the Hermite function. We note that the wavefunctions in Eqs. (29) and (31) are normalized by From Eq. (33) and Eq. (15), we can obtain the following 4D kinetic terms, which means that the wavefunctions on the 4D space-time, ψ j,|M | n (x), have modular weight −k = −1/2. Thus, the modular symmetry in the 4D low-energy effective field thoery is determined by behaviors of wavefunctions on the magnetized T 2 . In the next section, we study the modular symmetry on the magnetized T 2 . Before ending of this section, we also note that the wavefunctions in Eqs. (29) and (31) satisfy the following relation,

Modular symmetry in magnetized T 2 model
Here, we study how the fields on the magnetized T 2 are transformed under the modular transformation, Eq. (6). The transformation of the Wilson line ζ is the same as the coordinate z, i.e. Γ ∋ γ : ζ → ζ/(cτ + d). The fields F in Eq. (17) and A in Eq. (19) are modular invariant. The equations of motions for ψ M (z) with any excited-modes, including zero-modes, are also modular invariant. On the other hand, while the boundary conditions for ψ M (z) in Eqs. (22) and (23) are consistent with the S transformation, they are consistent with the T transformation only if M is even. In general, the boundary conditions are consistent with the modular transformation in Eqs. (2) and (3) only if M is even or both ab and cd are even, where a, b, c, d are elements of γ ∈ Γ 1,2 ⊂ Γ. (See Ref. [15].) Here, we focus on the models with M=even.
The wavefunctions of the n-th excited-modes in Eq. (31), including the zero-modes in Eq. (29), are transformed as under the modular transformation, Eq. (6). Note that the creation operator is modular invariant and commutative with the above coefficients. Thus, the wavefunctions transform like modular forms of weight 1/2. It is consistent with Eq. (34). Modular forms of weight 1/2 is relevant to the double covering group of Γ = SL(2, Z), Γ ≡ SL(2, Z). (See e.g. [44,48,49].) The double The multiplication of arbitrary two elements, where A(γ 1 , γ 2 ) is called as Kubota's twisted 2-cocycle for Γ, defined as follows. We first introduce Kubota's function χ : Γ → Z, defined by We also introduce the Hirbert symbol, defined by Then, A(γ 1 , γ 2 ) is defined by In particular, we set They satisfy the following algebraic relations, Note that inverses of S, T , Z are written by Hereafter, we often denote an element of Γ, [γ, ǫ], as γ, where γ is, in general, independent of γ.
Due to the above extension by ǫ ∈ {±1}, the definition of the automorphy factor in Eq.(11) is also extended by ǫ ∈ {±1} as follows, where we take (−1) k/2 = e −iπk/2 . From Eqs. (12) and (39), Eq. (46) satisfies the following relation, where the extension by ǫ ∈ {±1} dose not affect the modular transformation, i.e. γ(z, τ ) = γ(z, τ ). It allows us to study modular forms of weight k/2, where k is integer. Considering the above extension, the wavefunctions on the magnetized T 2 transform as under the modular transformation. Note that Z(z, τ ) = (−z, τ ) and Z 2 (z, τ ) = (z, τ ) require J 1/2 ( Z, τ )ρ( Z) = δ |M |−j,k and J 1/2 ( Z 2 , τ )ρ( Z) 2 = δ j,k , respectively. Actually, we can check that the following relations, are satisfied 3 . Therefore, the wavefunctions on the magnetized T 2 transform under the modular transformation like modular forms of weight 1/2 for Γ(2|M|) 4 , which is the normal subgroup of Γ, defined as Then ρ is a unitary representation of the quotient group Γ ′ 2|M | ≡ Γ/ Γ(2|M|). That is, the group generated by ρ is homomorphic with Γ ′ 2|M | . For examples, when M = 2, the S and T transformations are represented as They generate the group G 2 whose order is 384, and it is isomorphic to When M = 4, the S and T transformations are represented as They generate the group G 4 , whose order is 96, and it is isomorphic to 3 The following relations are also satisfied In Appendix A, we study the extension for the generalized CP symmetry with the modular symmetry on the magnetized T 2 . So far, we have considered the Wilson line ζ, which transforms as ζ → ζ/(cτ + d). In that case, the modular transformation is restrictive due to the consistency with the boundary conditions for ψ M (z). Before ending this section, we comment about another possibility. In particular, if a Wilson line ζ is also changed to ζ + 1, which is the gauge transformation, simultaneously with the T transformation, the boundary conditions for ψ M (z) are consistent with the modular transformation even if M is odd, although the equations of motions for ψ M (z) are modified. In this case, the zero-mode wavefunction for j after the T transformation can be expanded by the all excited-mode wavefunctions for j before the T transformation as follows, where we use the following the generating function of the Hermite function, The detail calculation is shown in Appendix B. Similarly, the n-th excited-mode wavefunction for j after the T transformation can be also expanded by the all excited-mode wavefunctions for j before the T transformation.
In this section, we have discussed the modular symmetry on magnetized T 2 . In the following sections, we study the modular symmetry on various magnetized T 2 /Z N orbidolds.
4 Modular symmetry in magnetized T 2 /Z N twisted orbifold models In this section, we study the modular symmetry of the wavefunctions on the magnetized T 2 /Z N twisted orbifolds [12,[50][51][52]. Here and hereafter, we often omit the KK index n, because each KK level satisfies the same relations in what follows. For simplicity, we do not introduce nonvanishing discrete Wilson lines, although we can discuss models with non-vanishing Wilson lines similarly. The The wavefunction on the magnetized T 2 /Z N twisted orbifold, ψ j,|M | T 2 /Z m N (z) must satisfy the following boundary condition, Hence, such wavefunctions can be written by linear combinations of wavefunctions on the magnetized T 2 as where N t N is the normalization factor determined by remaining Eq. (33). Furthermore, ψ j,|M | where the normalization factor N t 2 is determined by N t 2 = 1, 1/2, and 1/ √ 2 for j = 0, |M|/2, and the others, respectively. Note that there are no Z 2 -odd modes, ψ j,|M | T 2 /Z 1 2 (z, τ ), for j = 0, |M|/2. When M is even, the numbers of Z 2 -even (m = 0) and -odd (m = 1) modes are (|M|/2 + 1) and (|M|/2 − 1), respectively 5 . On the T 2 /Z 2 twisted orbifold basis, Eq. (49) is deformed by where we need to multiply ρ T 2 /Z 0 2 ( S) further by 1/ √ 2 when j or k is 0 or |M|/2. The above deformations induce deformation of the relation in Eq. (50) as while the other relations are same with the T 2 basis. Thus, the representations on the T 2 /Z 2 twisted orbifold basis satisfy the same algebraic relations with that on the T 2 basis, although the dimensions of the representations are different. For example, when M = 4, the wavefunctions on the T 2 /Z 2 twisted orbifold basis are expressed as The representations of the S and T transformations for Z 2 -even modes are expressed as which are the generators of the group G 4 0 . The group G 4 0 has the order 96 and is isomorphic to which is the same as the group on T 2 in Eq.(60). The above wavefunctions in Eq. (69) correspond to a triplet under G 4 0 ≃ ∆(48) ⋊ Z 8 . The representations of the S and T transformations for Z 2 -odd mode, on the other hand, are expressed as which are the generators of the group G 4 1 . The group G 4 1 is nothing but which is a subgroup of G 4 0 ≃ ∆(48)⋊Z 8 . The above representation in Eq. (70) is a representation of this Z 8 symmetry and it also corresponds to a singlet under G 4 0 ≃ ∆(48) ⋊ Z 8 . Thus, the T 2 /Z 2 twisted orbifold is consistent with the modular symmetry. Furthermore, the wavefuncitons on T 2 are decomposed into smaller representations by Z 2 eigenvalues, even and odd, that is, the T 2 /Z 2 twisted orbifold basis. For smaller |M|, this basis of wavefunctions provide us with irreducible representations of Γ ′ 2|M | ≡ Γ/ Γ(2|M|). For larger |M|, wavefunctions on the T 2 /Z 2 twisted orbifold basis could be decomposed further. We will study it in section 6.
Next, we comment about the other magnetized T 2 /Z N twisted orbifolds. In the case of T 2 /Z 4 , since the wavefunction ψ On the T 2 /Z 4 twisted orbifold basis, the representation of S transformation is diagonalized as That is the Z 8 symmetry.
In the case of T 2 /Z N for N = 3, 6, however, the wavefunction ψ On the T 2 /Z 3 twisted orbifold base, the representation of ST transformation is diagonalized as That is the Z 6 symmetry. On the T 2 /Z 6 twisted orbifold base, the representation of ST −1 transformation is diagonalized as That is the Z 12 symmetry. Thus, there remain Z 2N symmetries in ρ(γ) on the magnetized Z N twisted orbifolds for N = 3, 4, 6. Remaining ρ(γ) represent a spinor representation under Z N twist. Obviously, ρ(γ) on the T 2 and Z 2 bases also correspond to spinor representations under the 2D (discrete) rotation.

Modular symmetry in magnetized T 2 /Z N shifted orbifold models
In this section, we study the modular symmetry for the wavefunctions on the magnetized T 2 /Z N shifted orbifolds [53]. The T 2 /Z N shifted orbifold can be obtained by further identifying the points on T 2 ≃ C/Λ which are shifted by ke {0, 1, 2, ..., N − 1}. Then, the wavefunctions on the T 2 /Z N shifted orbifold have to also satisfy the following boundary condition, (81) with ℓ ∈ Z N , which is consistent with the boundary condition for z → z + m + nτ , in addition to Eq. (22) and Eq. (23). Furthermore, these boundary conditions constrain the magnetic flux M to be M = Nt, t ∈ Z. The above wavefunction can be written by linear combinations of wavefunctions on the magnetized T 2 as where N s N is the normalization factor determined by remaining Eq. (33). Furthermore, since which can be obtained from Eqs. (29) and (31) directly. The normalization factor N s N is determined as N s N = 1/ √ N for n = 0 or N s N = 1/N for n = 0. We discuss the modular symmetry on the magnetized T 2 /Z N shifted orbifolds. There is the modular symmetry on the T 2 /Z N shifted orbifold only if the points on T 2 ≃ C/Λ which are shifted by e The above boundary conditions are consistent with Eq. (22), Eq. (23), and Eq. (81) for ∀ m, ∀ n ∈ Z N , where we denote ℓ in Eq. (81) as ℓ (m,n) , determined by ℓ (m,n) ≡ mℓ 1 + nℓ 2 (mod N). From the above boundary conditions, we obtain s ≡ M/N 2 ∈ Z. Furthermore, when s is even and ℓ 1 ≡ j (mod N), the following wavefunctions, Ψ r,|s| for γ ∈ Γ, where The above deformations induce the deformation of the relation in Eq. (50) as We should modify several terms in the following particular case. Since N − ℓ 1 ≡ ℓ 1 (mod N) is satisfied when ℓ 1 = 0 or N = 2, δ |s|−r−1,r ′ should be modified into δ |s|−r,r ′ when ℓ 1 = 0 or δ |s|−r−ℓ 1 ,r ′ when N = 2. Furthermore, when r = 0 in addition to ℓ 1 = 0 or N = 2, e 2πiℓ 2 /N dose not appear even if ℓ 2 = 0. Note that Eq. (90) leads to the following relation 6 , Ψ r,|s| 6 The following calculation is useful to confirm Eq. (91), The other relations except for Eq. (90) are same with the T 2 basis, where we note that the representation of T N transformation is diagonalized. However, the Z N -shift invariant modes on the full T 2 /Z N shifted orbifold, i.e. (ℓ 1 , ℓ 2 ) = (0, 0), in particular, correspond to the modes on the T 2 /N ≃ T 2 ′ with magnetic flux s = M/N 2 ∈ 2Z. In other words, the Z N -shift invariant modes behave like modular forms for Γ(2|M|/N 2 ), while the other modes corrspond to modular forms for Γ(2|M|).
6 Modular symmetry in magnetized T 2 /Z N twisted and shifted orbifold models In this section, we study the modular symmetry for the wavefunctions on the magnetized T 2 /Z N twisted and shifted orbifolds. The modular symmetry remains on the T 2 /Z 2 twisted orbifold.

Conclusion
We have studied the modular symmetry of wavefunctions on the magnetized T 2 ≃ C/Λ. When the magnetic flux M is even, the wavefunctions behave as modular forms of weight 1/2 and represent the double covering group of Γ ≡ SL(2, Z), Γ ≡ SL(2, Z). Each wavefunction on T 2 with the magnetic flux M transforms under Γ(2|M|). Then, |M| zero-modes as well as massive modes are representations of the quotient group Γ ′ 2|M | ≡ Γ/ Γ(2|M|). If we change the Wilson line ζ → ζ + 1 simultaneously with the T transformation of the modular transformations, T 2 with any magnetic flux M is consistent with the modular transformations. However, the zero-mode wavefunctions after the T transformation are expanded by the all excited-mode wavefunctions before the T transformation.
We have also studied the modular symmetry for the wavefunctions on various magnetized T 2 /Z N orbidolds. The T 2 /Z N twisted orbifold can be constructed for N = 2, 3, 4, 6. However, the modulus τ ≡ e 2 /e 1 is fixed as τ = e 2πi/N for N = 3, 4, 6 while any τ is allowed for N = 2. It means that the only ST, S, ST −1 transformations of the modular transformations remain for N = 3, 4, 6, respectively. They correspond to Z 2N symmetries. On the other hand, there remains the full modular symmetry for N = 2. The representations of the modular transformations on the T 2 /Z 2 twisted orbifold basis satisfy the same algebraic relations with the representations on the T 2 basis. However, the representations on the T 2 basis are decomposed into smaller representations on the T 2 /Z 2 twisted orbifold basis.
In order for the T 2 /Z N shifted orbifold to be consistent with the modular transformations, all Z N -shifted points should be identified, where we call it as the full T 2 /Z N shifted orbifold. The full T 2 /Z N shifted orbifold with the magnetic flux M corresponds to T 2 ′ ≃ C/Λ ′ , Λ ′ ≡ Λ/N with the magnetic flux s ≡ M/N 2 ∈ 2Z. In particular, the Z N -shift invariant modes corresponds to the modes on T 2 /N ≃ T 2 ′ . Therefore, the Z N -shift invariant modes behave like modular forms for Γ(2|M|/N 2 ), while the other modes behave as modular forms for Γ(2|M|).
Furthermore, the only full T 2 /Z 2 shifted orbifold is consistent with the T 2 /Z 2 twisted orbidfold. On that T 2 /Z 2 twisted and shifted orbifold, the Z 2 -shift invariant modes correspond to the modes on (T 2 /N)/Z 2 twisted orbifold with the magnetic flux s ≡ M/4 ∈ 2Z.
The wavefunctions on T 2 are decomposed into smaller representations by the Z 2 twist and shift. They provides us with a reduction of representations towards irreducible representations. Also, the combination of the Z 2 twist and shift provides us with a new approach to realize three generations from the phenomenological viewpoints 7 . It is interesting to study three-generation models by a combination of Z 2 twist and shfit. We would study elsewhere.

Appendix
A Extension for generalized CP symmetry with the modular symmetry on the magnetized T 2 Here, we study the extension for generalized CP symmetry with the modular transformations on the magnetized T 2 8 . The CP transfomration for the mudulus τ is defined as CP : τ → −τ , where it remains Im(−τ ) > 0. It is derived from The CP matrix in Eq. (101) satisfies the following relations, When we also consider the avobe CP transformation in addition to the modular transformations, the modular group Γ = SL(2, Z) is extended to Γ * ≡ SL(2, Z) ⋊ Z CP , where the above in Eq. (105) is just modular transformation and the below in Eq. (105) contains odd numbers of CP transformation. It leads to redefine the automorphy factor as where it satisfies Eq. (12). Note that γ * dose not mean the complex conjugate of γ but an element of Γ * . In order to see how the wavefunctions on the magnetized T 2 transform under the extended modular transformation by γ * ∈ Γ * , furthermore, we consider the double covering group of Γ * ≃ GL(2, Z), Γ * ≃ GL(2, Z), similar to Eq. (38). Note that only Eq. (42) is redefined as , (See Ref. [49].) In particular, we set Then, Eqs. (108), (43), and (45) leads to the following relations, in addition to Eq. (44). The automorphy factor is the same with Eq. (46) and satisfies Eq. (47), where we should apply Eq. (106) and Eq. (107).
Here, we study the CP transformation of the fields on the magnetized T 2 . In addition to Eq. (102) and Eq. (103), it is also needed that the magnetic flux M is flipped as In this case, any field in section 3.1 after the CP transformation corresponds to the complex conjugate of the field. In particular, the wavefunctions of the n-th excited-modes in Eq. respectively.