Classifications of magnetized T4 and T4/Z2 orbifold models

We study constructions and classifications of three-generation models based on magnetized T4 and T4/Z2 orbifold as candidates of the compact space. We focus on chiral fermion zero-mode wave functions in the extra dimensions. Freedoms of constant gauge fields, called Scherk-Schwarz phases are taken into account. We classify three-generation models in a systematic manner, clarifying the relationship between different models. The Higgs sector is also studied by analyzing possible assignments of the magnetic flux and Scherk-Schwarz phases, etc. to left- and right-handed fermions.


Introduction
The Standard Model (SM) of particle physics leave some fundamental questions of the universe unanswered. We wonder why three generations of quarks and leptons appear. The observed parameters such as masses and mixing angles among the generations of particles show some remarkable features without clear reasons. For instance, we can see large mass hierarchies in both lepton and quark sectors. It is also mysterious why the degree of mixings is so different between the two sectors. The SM cannot explain these behaviors. Furthermore, the origin of the gauge group structure SU(3) × SU(2) × U(1) is unclear. The SM is a chiral theory that left-and right-handed fermions have different quantum numbers and representations.
In order to answer these questions, one should look for a high-energy underlying theory. Superstring theory is a promising candidate. This theory predicts 10 dimensional spacetime, so the extra 6 dimensional space needs to be compactified. The gauge group structure originates from D-brane configurations in the compact space. (See for reviews refs. [1,2].) In addition, the geometry of the compact space determines the flavor structures and the patterns of the above-mentioned parameters in the four dimensional (4D) theory.
For quantitative evaluations of the relationship between the SM and superstring theory, one should first consider a relatively simple setup allowing analytical computations. An explicit way to compute Yukawa couplings was proposed based on the supersymmetric Yang-Mills theory (SYM) in higher dimensions known as a low-energy effective field theory of superstring theory. Yukawa couplings can be obtained by overlap integrals of zero-mode wave functions corresponding to the matter modes and the Higgs boson over the compact space. The torus compactifiction with magnetic fluxes is rather simple, but quite interesting [3][4][5][6][7]. That can lead to a 4D chiral theory, and one can realize realistic gauge symmetry breaking depending on configurations of the background magnetic fluxes. Moreover, the strength of the magnetic flux determines the generation number. For the factorizable torus T 2 × T 2 × T 2 with magnetic fluxes, one can obtain explicit forms of the wave functions in terms of the Jacobi theta-function, and then one can compute Yukawa couplings and higher order couplings analytically [7,8]. The flavor structure such as mass hierarchies and mixing anlges is originated from wave functions on one of T 2 's, because the generation number, 3, is a prime number, although wave functions on other T 2 × T 2 contribute to only the overall factor of Yukawa matrices.
These studies were extended to the T 2 /Z N orbifolds with magnetic fluxes [9][10][11][12]. We have various possibilities to construct three-generation models on the T 2 /Z N orbifolds,

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we explain how our way of classification is reflected in the flavor structure of the Yukawa coupling constants in magnetized T 4 models. Section 7 is our conclusion. In appendix A, we give a derivation of the theta identity, and show Yukawa couplings in magnetized T 4 models including Scherk-Schwarz phases. In appendix B, we give proofs of some facts we used in section 6.

Compactification with magnetic fluxes
Here we review zero-mode wave functions on magnetized T 4 and T 4 /Z 2 [7,30]. In order to simplify the discussion we focus on the case when the gauge symmetry is just U(1).

Magnetic flux on T 4
We begin with writing down the magnetic field and the corresponding gauge potential on T 4 . T 4 can be defined as follows. We denote the four-dimensional coordinates by z = (z 1 , z 2 ) ∈ C 2 . Let Λ be a lattice spanned by 4 vectors where Ω is a 2 × 2 complex matrix and satisfies Im Ω > 0, namely its eigenvalues are greater than zero. Then we define T 4 C 2 /Λ. This means we have the following identifications on C 2 , z ∼ z + e i ∼ z + Ω e i . (2.2) The background magnetic flux on T 4 is written by In terms of the complex coordinates, (2.4) the flux is given by where From the requirement that the supersymmetry survives, the flux needs to be a (1, 1)-form. This can be satisfied by demanding the following condition Ω T p xx Ω + p yy + p T xy Ω − Ω T p xy = 0. (2.7)

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As a result, F can be written as . (2.8) From this, we see the hermiticity of the flux: F z izj = F † z izj . The expression is simplified further by assuming p xx = p yy = 0, (2.9) The SUSY condition is also simplified (p T xy Ω) T = p T xy Ω. (2.10) The flux must be quantized due to the consistency with the boundary conditions we take. Thus, we can write p xy in terms of a 2×2 real integer matrix N, referred as the intersection matrix, p xy = 2πN T . (2.11) This is referred to as Dirac's quantization condition. Then we obtain with the SUSY condition rewritten as (NΩ) T = NΩ. (2.13) The corresponding gauge potential is given by where ζ is a complex constant 2-component vector known as the Wilson lines. The boundary conditions are where We first consider the case under the positive definite condition N · Im Ω > 0. (2.17) This means we have det N > 0 and Tr(N · Im Ω) > 0. Then the following wave functions on T 4 : (2.19) is known as the Riemann theta-function. Note that j is defined up to integer because ψ j = ψ j+ e k . N is just a normalization constant. ψ j are the zero-mode wave functions of the Dirac equation, The wave functions ψ j satisfy the following boundary conditions: (2.21) We can clearly see consistencies with the boundary conditions of the gauge potential shown by eq. (2.15). Finally, it is important to note that the degeneracy of the zero-modes is given by the determinant of N. When N · Im Ω < 0, we have det N > 0 and Tr(N · Im Ω) < 0. Then, wave functions in eq. (2.18) are not defined. In this case, we need to take When det N < 0, we need to replace Ω by the effective complex structure Ω eff. =ΩΩ as stated in ref. [30],Ω where q is given by the following relation N = n 11 n 12

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In order to make the following discussions simple, let us concentrate on the case when wave functions are given by eq. (2.18), namely positive chirality modes with the positive definite condition N · Im Ω > 0.

T 4
Here we write down zero-mode wave functions including Scherk-Schwarz phases on T 4 . Boundary conditions eq. (2.21) are generalized, and we demand (2.29) where α and β are two dimensional real vectors and can take arbitrary values. We call them Scherk-Schwarz phases. The zero-mode wave functions satisfying the above boundary conditions are given by (2.30) where j · N ∈ Z 2 . JHEP08(2022)256

Three-generation models on T 4
In order to obtain three-generation models, we need 3 degenerated zero-modes. Thus, we demand det N = 3. (3.1) Obviously we have infinite number of N satisfying this condition. Stringy tadpole cancellation conditions may constrain the size of N, although it would also depend on other sectors. At any rate, we study the possibility of N from the viewpoint of effective field theory. In order to get better understanding we define Types in the intersection matrix N. Two matrices N 1 and N 2 are in the same Type if and only if they are related by Here, Γ = SL(2, Z) is defined as

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This group is generated by We get 4 different Types whose representative matrices are given by Type I : 3 0 0 1 , Type II : 3 1 0 1 , Type III :

Types
Here we give a proof of eq. (3.5), the classification of N into Types. For later convenience, let us consider the general case that D := det N is not only 3, but can also take an arbitrary integer. We explicitly show the representative matrices for each Type and the number of Types is given by the sum of divisors of D.
We write the matrix elements explicitly as Let us denote the greatest common divisor of n 3 and n 4 by x(≥ 1), Note that x must be a divisor of D. We can relate n 3 and n 4 by n 4 = qn 3 + r, (3.8) where q and r are both integers. We can in general make |r| ≤ |n 3 | 2 . We can verify x = gcd(n 3 , r). Then we multiply S ∈ Γ to the right From eq. (3.9), we can write − n 3 = q r + r , (3.12) where gcd(r, r ) = x and |r | ≤ |r| 2 . Thus, we can repeat the procedure shown by eqs. (3.10) and (3.11). After some iterations and the multiplication of −I ∈ Γ if necessary, we reach

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where * denotes an unspecified integer. The above Γ = SL(2, Z) multiplications do not change the determinant. Thus, we can write (3.14) It can be easily seen that if and only if k ≡ l, (mod D/x), the following two matrices (3.15) are in the same Type. Consequently, we have verified that intersection matrices N are classified into Types. The number of Types is given by which is nothing but the sum of all positive divisors of D = det N.
This way of classification into Types is helpful. This can be seen when we plot j of zero-modes on (j 1 , j 2 ) plane. Each Type corresponds to a distinct alignment as j 2 = kj 1 for all of the three vectors j, where k is constant in a model. We refer to the k as the gradient. This shows that Types we defined in eq. (3.2) is consistent with ref. [31]. There, types were defined in terms of the mod 3 structure of N without reference to the group theoretical approach we took. Types are shown in table 1. We define M (3) by The alignment of j is shown below.
Type I:

Type and alignment of zero-modes
We saw that zero-modes are labelled by two dimensional real vectors j. Under the intersection matrix N 1 , j must satisfy Suppose that N 1 and N 2 are in the same Type, namely N 2 = N 1 γ, γ ∈ Γ. We multiply γ from the right to both sides, This shows that all j yielded in the N 1 case are also present in the N 2 case and vice versa. Consequently, intersection matrices in the same Type produce the same j.

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Type N gradient First, let us consider the action of T . We can rewrite the condition eq. (3.22) as Thus, zero-modes labelled by j · T −1 are yielded under the intersection matrix T N, This shows that zero-modes under T N are aligned with gradient k−1. Next, let us consider the action of S in the same way. Zero-modes under SN are written as showing that their alignment is perpendicular to that of j. One more comment should be made. There is an arbitrariness in the gradient value by mod D. For example, we could say that Type II in table 1 corresponds to gradient 2 instead of −1. To understand this, let us consider (3.28) It is easily verified that T D N and N are in the same Type, therefore the gradient is fixed up to D. This also explains why we do not get {T 3 M (3) Γ} as an independent Type.

Consistency with SUSY
We have classified all N whose determinant is 3 without careful considerations of the SUSY condition. We would like to clarify whether it is possible to find Ω satisfying (NΩ) T = NΩ for any given N. In fact, we can always find such Ω as it will be shown in this section.

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However, if we further demand Ω T = Ω, we have severe restrictions. Set of such symmetric Ω are referred to as Siegel upper half space H g=2 . When Ω ∈ H 2 , T 4 is known as Jacobian variety [33]. In this case, the SUSY condition is not always satisfied and we study this point as well in detail. This analysis may be useful when considering modular transformations described by Sp(2g, Z).

Demanding Ω T = Ω
Let us first analyze the SUSY condition under the requirement Ω ∈ H 2 . Let us parametrize N and Ω by N = n 11 n 12 n 21 n 22 , Then the SUSY condition is equivalent to Im Ω must be positive definite. Thus, the following two conditions need to be satisfied simultaneously, Im We notice that Im ω 1 > 0, Im ω 4 > 0 must hold. Then, eq. (3.34) is automatically satisfied, and we focus on eq. (3.35). Dividing both sides by (Im ω 1 ) 2 gives us where x = Im ω 4 /Im ω 1 (> 0). If n 12 = 0, we get x > ∆ 2 n 2 21 showing the region of allowed Ω ∈ H 2 . If n 12 = 0, we can further divide both sides by −∆ 2 n 2 12 to get

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Thus, the allowed region of Ω is given by the intersection of x bounded by eq. (3.38) and x > 0. In order to get the non-vanishing intersection, we need Unless this inequality is met, there is no Ω ∈ H 2 satisfying the SUSY condition.
(II) n 11 = n 22 : the SUSY condition is simplified to Then, it is easy to show the following results. If N is of the form n 12 n 21 < 0, or n 12 = 0, n 21 = 0, or n 12 = 0, n 21 = 0, there is no Ω ∈ H 2 satisfying the SUSY condition. If n 12 n 21 > 0, Ω is bounded by

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First, let us consider the case when Im ω 2 > 0. Then we get where X = Im ω 1 / Im ω 2 and Y = Im ω 4 / Im ω 2 . We can see that Ω such that X + Y 0 and XY 0 can satisfy the SUSY condition regardless of N. To be more preceise, the curve given by and the line Y = −X become the boundaries of Ω satisfying the SUSY condition. The case with Im ω 2 < 0 can be analyzed in the same way, and again we always get an allowed region of Ω. The case with Im ω 2 = 0 is simple, and the region corresponds to Im ω 1 > 0, Im ω 4 > 0.

T /Z 2
We constructed Z 2 even wave functions on T 4 /Z 2 by the sum of two zero-modes on T 4 as shown in eq. (2.31). For some particular j, the two modes are identical,

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Thus, the corresponding Z 2 even mode ψ T 4 /Z 2 ,+ is written by a single wave function ψ on T 4 . We denote the number of such invariant modes by C + . Similarly, for some particular j we have Then from eq. (2.32), we notice that the corresponding Z 2 odd mode ψ T 4 /Z 2 ,− is written by a single wave function ψ on T 4 . We denote the number of such modes by C − . The degeneracies of the zero-modes for Z 2 even and odd parities are given by the following formulae respectively This motivates us to study how C + and C − are determined. As it will be shown later, they depend on Scherk-Schwarz phases and the mod 2 structure of N. The mod 2 structure is simply given by taking mod 2 of each matrix element of N. For example, if we have the corresponding mod 2 structure is and we often denote this by writing (n 11 , n 12 , n 21 , n 22 ) ≡ (1, 1, 0, 1), (mod 2). Table 2 summarizes the result when α = (0, 0).

15)
C + = 1 : (4.16) The remaining 7 patterns of the mod 2 structure correspond to the C + = 0 case. Next, we consider non-vanishing β. Let us look at the case when Scherk-Schwarz This shows that zero-modes (J 1 , J 2 ) = 0, 1 2 , 1 2 , 1 2 are no longer Z 2 invariant modes, but are Z 2 odd modes by themselves. The remaining zero-mode (J 1 , J 2 ) = 1 2 , 0 is still Z 2 invariant. This suffices to complete the table 3 because other setups with different Scherk-Schwarz phases can be analyzed in the same way.

Three-generation models of T 4 /Z 2
In order to get 3 degenerated zero-modes in the Z 2 even sector, D(= det N) must satisfy In the Z 2 odd sector, D must satisfy We summarize the conditions to get 3 degeneracy by presenting tables 4 and 5. Given α = (0, 0), table 4 shows the required assignments of mod 2 structure, determinant, Z 2 parity, and β. For non-vanishing α, table 5 summarizes the result. Note that when the mod 2 structure is of the form (0, 0, 0, 0), the determinant of N is a multiple of 4. This fact explains why we cannot produce three-generation models under the mod 2 structure (0, 0, 0, 0). From these tables, we find that we can realize three-generation models when D = 4, 5, 6, 7, 8. We will study these cases in what follows. Note that we still get a number of N even after both D and mod 2 structure are specified. Therefore, let us classify them. For that purpose, it is helpful to study the relationship between mod 2 structure and group actions on N.

Mod 2 structure of N
Here we consider actions of γ ∈ Γ on an intersection matrix N such that Nγ ≡ N, (mod 2). (4.20) In other words, we would like to identify the subgroups of Γ whose actions conserve the mod 2 structure of N. We summarize the results in what follows. Following 6 patterns of mod 2 structure are invariant only under the actions of Γ(2) ⊂ Γ, where where is invariant under Γ = SL(2, Z).

D = 5
Here we show the classifications of N when D = 5. This can be done in two steps. First, we classify them into Types as we did before. Then we further classify each Type in terms of the mod 2 structure. When D = 5, we get 6(= 1 + 5) Types as shown in table 6. We define M (5) by Next, let us consider classifications in terms of the mod 2 structure. First, we concentrate on the set of N which belongs to Type I and has mod 2 structure (1, 0, 0, 1). In general, we can write such N as follows, N = 5m 10n * * , gcd(m, 2n) = 1, m = 0.

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Type N gradient We notice that m is an odd number. We act Γ(2) to the right of N as where k ∈ Z. By the appropriate choice of k, we can write We consecutively act another element of Γ(2) to the right as We can repeat the process in eq. (4.32), and obtain 5m 10n * * , |n | < |n |. After some iterations, we reach Nγ = 5m 0 * * ,m ∈ Z, γ ∈ Γ(2).  Type I  Type II  Type III  Type IV  Type V  Type VI (1, 0, 0, 1) Table 7. Classification by the mod 2 structure for D = 5.

(4.41)
It is now easy to write similar expressions for other patterns of the mod 2 structure. For example, set of N ∈ (Type I) with mod 2 structure (1, 1, 0, 1) can be written as {M (5) T Γ(2)}. In order to move to different Types, we just need to act Γ to the left of N. We summarize the results in table 7. Only the representative matrix is shown for each type and mod 2 structure. For example, by writing M (5) T , we mean the infinite set given by {M (5) T Γ(2)}. One can understand the reason why we obtain 6 different patterns of mod 2 structure for each Type in the following way. Note that mod 2 structure of N with D = (odd) is invariant only under Γ(2) when we consider group actions from the right. Thus, S 3 Γ/Γ(2) corresponds to the change of mod 2 structure. Since the order of S 3 is 6, there should exist 6 different patterns in each Type.

D = 7
Here we show the classifications of N when D = 7. We get 8(= 1 + 7) Types as shown in table 8. We define M (7) by Classifications in terms of the mod 2 structure are straightforward. Results are shown in table 9. Type IV-VI are omitted, but it is easy to recover the full results.

D = 4
Here we show the classifications of N when D = 4. We get 7(= 1 + 2 + 4) Types as shown in table 10. We define M (4) and M (4) by Type VII is peculiar because one cannot align j in a straight line, instead they form a square given by j = (0, 0), (0, 1/2), (1/2, 0), (1/2, 1/2). (4.44) Next, let us consider classifications in terms of the mod 2 structure. The most simple one is Type VII because its mod 2 structure is always (0, 0, 0, 0). For other Types, we need JHEP08(2022)256 a careful treatment. First, we concentrate on the set of N which belongs to Type I and have mod 2 structure (0, 0, 0, 1). In general, we can write such N as follows, We act an element of Γ 1 (2) to the right of N as We know m = 0 because of the conservation of mod 2 structure. We consecutively act another element of Γ 1 (2) to the right as 4m 4n * * (0, 0, 0, 1)  where m = 0 is the new (1, 1) element of NΓ 1 (2). After some iterations and the multiplication of −I ∈ Γ 1 (2) if necessary, we reach Here we used the fact that actions of Γ 1 (2) do not change the mod 2 structure. Finally, we act 1 0 −2s 1 ∈ Γ 1 (2), (4.52) from the right, and we get N = M (4) γ, γ ∈ Γ 1 (2). (4.53) Consequently, set of all N ∈ (Type I) with mod 2 structure (0, 0, 0, 1) is given by {M (4) Γ 1 (2)}. It is straightforward to write similar expressions for other patterns of mod 2 structure. For example, set of N ∈ (Type I) with mod 2 structure (0, 0, 1, 0) is given by Similarly, set of N ∈ Type I with mod 2 structure (0, 0, 1, 1) is given by In order to move to different Types, we just need to act Γ to the left of N. Results are shown in tables 11, 12, and 13. One can understand the reason why we obtain 3 different patterns of mod 2 structure for each Type (except Type VII) in the following way. Let us focus on Type I as an JHEP08(2022)256 mod 2 structure (n 11 , n 12 , n 21 , n 22 ) Type V Type VI example. The mod 2 structure of the form (0, 0, 0, 1) is invariant only under Γ 1 (2) ∈ Γ. We can classify Γ into cosets. The index is given by [Γ : Γ 1 (2)] = 3 which is the reason of 3 patterns of mod 2 structure [34].

D = 6
Here we show the classifications of N when D = 6. We get 12(= 1 + 2 + 3 + 6) Types as shown in table 14. We define M (6) by We further classify in terms of mod 2 structure. Results are shown in tables 15, 16, 17.

D = 8
Here we show the classifications of N when D = 8. We get 15(= 1 + 2 + 4 + 8) Types as shown in table 18. We define M (8) and M (8) by We further classify in terms of mod 2 structure. Results are shown in tables 19, 20, 21, 22. One can understand the reason why we obtain 3 different Types corresponding to the mod 2 structure (0, 0, 0, 0) as shown in table 22. This is because the situation is just equivalent to the classifications of N with D = 2.

Higgs sector
Here we study the Higgs bosons which can couple to fermions under the T 4 /Z 2 compactification models. We require that the generation numbers for both left-and right-handed fermions are three as in the SM. For this purpose, let us consider Yukawa couplings among left-and right-handed fermions with Higgs boson,

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Type N gradient    Type N gradient      Here we concentrate on the case when all wave functions have positive chirality as in eq. (2.18). From the gauge invariance we need Moreover, note that the following conditions need to be satisfied to obtain non-vanishing Yukawa couplings,

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where p L , p R , and p H denote the Z 2 parity (±1) of left, right, and Higgs sector wave functions. For example, these conditions can be understood in the following way. Consider z + e n , then we find from which we obtain eq. (5.3). Similarly, taking z to z +Ω e n or − z provides the conditions eqs. (5.4) and (5.5).
It is a good starting point to suppose that the zero-modes of the left-handed fermions in T 4 /Z 2 are under the magnetic flux given by where k = {0, 1, . . . ., D − 1}. Then we find In order to satisfy the SUSY condition, we need On the other hand, we denote the magnetic flux in Dirac equation of the right-handed fermions as Then we get In order to satisfy the SUSY condition, we need From eqs. (5.9) and (5.12), we obtain Note that m i , (i =, 1, 2, 3, 4) are integers whereas ω i are complex numbers. It is then natural to demand m 1 = Dm 4 , m 2 = km 4 . (5.14) Otherwise the complex structure matrix Ω must satisfy some very special relations such as ω i /ω 1 ∈ Z, (i = 2, 3, 4) which strongly constrain the continuous degrees of freedom in Ω.
Here we analyze the Higgs sector under the conditions eq. (5.14). Then we get

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showing that N R is simply given by m 4 N L . It is obvious from table 4 and 5 that m 4 needs to be 1, i.e., It seems clear that we get the same conclusion even if N L is not given by eq. (5.7). As a result, N H is obtained as Now, we have all the information necessary to determine the Higgs sector.
Let us first look at the case when D = det N L = 4(= det N R ). Assignments of quantum numbers yielding 3 degenerated zero-modes of fermions are summarized in tables 23 and 24. Only in this section, let us count the number of models in the following way. When following four conditions for each three sector (L, R, H) are specified, we count it as one model although there exist infinite number of different N in it. The four conditions are: Z 2 parity p, Scherk-Schwarz phases α, β and the mod 2 structure. From table 23, we notice that there are 36 ways of assigning the quantum numbers in the left-handed sector. From eq. (5.16), we notice that N R must have the same mod 2 structure as N L . Therefore, the number of models is much less than 36 × 36 = 1296, and we only have 144 models. Once the four conditions for left-and right-handed fermion are specified, we automatically obtain the conditions for the Higgs sector. It is important to note that the mod 2 structure of N H is always (0, 0, 0, 0). This makes it easy to read the number of Higgs modes from tables 2, 3, and eq. (4.3). We notice that C + − C − is nonzero only when its Scherk-Schwarz phases are vanishing. Having noted this point, table 23 suggests that both left-and righthanded fermions must have the identical quantum number assignments in order to produce non-vanishing C + − C − in the Higgs sector. Therefore, we have only 36 models yielding non-vanishing C + − C − (= 4). This explains why we get 36 models yielding 10 Higgs modes as shown in table 25. All other models correspond to the cases C + − C − = 0, yielding 8 Higgs modes. It is not difficult to extend this kind of analysis for other values of D. Results are summarized in tables 26, 27, 28, 29. Based on this simple analysis, we found that the number of Higgs modes can be 8, 10, 12, 14, 16, or 18. We can observe similarities between D L = 4 and D L = 8 cases, as well as between D L = 5 and D L = 7 cases. This is traced back to tables 4 and 5 showing both D + and D − values realizing 3 degeneracy in the fermion sector. We can clearly observe the correspondence.
At last, we give the explicit form of Yukawa coupling constant in eq. (5.1) in terms of Riemann theta-functions. Yukawa coupling constants in T 4 /Z 2 models are given by the linear combinations of those computed in T 4 models. Yukawa coupling constants in T 4 models are written as

16
(1, 1, 1, 0)  combination number of ψ L and ψ R 108 0 36  Theta identities proven in ref. [30] allow us to evaluate explicitely. 3 In general, we can write 19) where l in the Kronecker's delta denotes possible two dimensional integer vectors. This means we take mod ( e n N H ). This kind of selection rule is also present in T 2 cases [7,14,17]. We can simplify the expression if N L = N R as where m is summed over integer points in the cell spanned by e n = e n (2D L ) · P N −1 L , (n = 1, 2). (5.22) JHEP08(2022)256

Yukawa couplings and types
Here, we discuss how the classification we carried out is reflected in physical quantities obtained through the magnetized T 4 models. When D L = D R and Scherk-Schwarz phases are vanishing, we can write the Yukawa coupling constant as Firstly, let us look at the characteristic written as This shows the significance of Types of N L and N R since i and j are present. Next, notice that by a suitable choice of P ∈ GL(2, Z), we can always bring N H P −1 into the representative form of our Type. This allows us to understand the flavor structure in a simple way. One may question m will be affected by the choice of P and our discussion is ruined. However, m can be chosen independent of P . For its proof, readers should refer to appendix B.1. Based on this fact, we can further simplify the alignment of m by knowing the Type of N H . We discussed about this in appendix B.2. Secondly, let us look at the selection rule, In addition to our Type defined in eq. (3.2), it is worth defining L-Type concerning the term lN H . If N 1 and N 2 are related by we call N 1 and N 2 are in the same L-Type. Intersection matrices in the same L-Type produce the same mod ( e n N) structure. This is because γ in eq. (6.6) can be interpreted as the change of lattice vectors which does not alter the lattice points. . Above analysis implies that our Types and L-Types will be important to understand the pattern of the Yukawa couplings.

Conclusion
We have studied T 4 and T 4 /Z 2 compactifications with magnetic fluxes. We found that Scherk-Schwarz phases can take 16 different values on T 4 /Z 2 . We wrote down zero-mode wave functions including these phases. In T 4 models, the degeneracy of zero-modes is equal to the determinant of the intersection matrix N. N is a 2 × 2 integer matrix which determines the orientations and strength of the background magnetic field. In T 4 /Z 2 models, the degeneracy is determined by the following four information: |det N|, Scherk-Schwarz phases, Z 2 parity, and the mod 2 structure of N. The mod 2 structure refers to the mod 2 value of each matrix element. There are a number of three-generation models in both T 4 and T 4 /Z 2 cases. Thus, we proposed a method to classify them. We used some group theoretical languages to do it. This method offers a unified understanding over a number of models including physical observables such as the Yukawa coupling constants. Moreover, we studied the number of Higgs modes under the requirement of the SUSY condition. Models in each Type have the same aliment of the vectors j on T 4 . That leads to the same flavor symmetry, which was studied in ref. [31]. Similarly, models in each Type have the same number of zero-modes on T 4 /Z 2 when their mod 2 structure are the same. In this sense, models in one Type are equivalent to each other when we see only the left-handed fermion sector or the right-handed fermion sector. However, phenomenological aspects, e.g. their Yukawa couplings, depend on models even if each of the left-handed sector and the right handed sector belongs to the same Type. We need information such as the flux of the Higgs sector in order to discuss the phenomenological aspects as we saw in section 6.
The obtained results may be a starting point to extend our analysis to more phenomenological studies in the future. One of interesting applications is to study the flavor structure in T 4 /Z 2 orbifold models as well as T 4 models. It is intriguing to study the realization of quark and lepton masses and their mixing angles similar to analyses on T 2 /Z 2 orbifold models [15][16][17][18][19]. Also, it is important to study the flavor symmetry of T 4 /Z 2 orbifold models. The modular symmetry of T 4 and T 4 /Z 2 is the symplectic modular symmetry Sp(4, Z), which is larger than the modular symmetry of T 2 and T 2 /Z 2 . 4 We would study behaviors of zero-mode wave functions under Sp(4, Z) elsewhere.

A Computing Yukawa couplings
Here we give a proof of eqs. (5.19) and (5.20) based on ref. [30].

A.1 Theta identity
When computing Yukawa couplings in magnetized T 4 models, it is convenient to use the following theta identity given by where P ∈ GL(2, Z). The sum of m is taken over integer points in the cell spanned by e n = e n lcm(D 1 , D 2 ) · P N −1 1 (N 1 + N 2 )N −1 2 , (n = 1, 2).
Proof. The left-hand side of eq. (A.1) is where we defined We introduce a transformation matrix:
(III) l 1 = l 2 + lαN −1 2 (N 1 + N 2 )N −1 1 + m , ( m = 0): lastly, we consider the case when the difference of l 1 and l 2 cannot be given by eq. (A.10). We write the difference as where l ∈ Z 2 and m ∈ Z 2 /{0}. Then the correspondence with eq. (A.8) is The sum over m is finite. To understand it, suppose m is large enough such that we have Then we notice that this is what we have already considered in (II). Thus, we can understand eq. (A.2). Let us summarize what we have looked at. (I) and (III) show that we have corresponding l 3 , l 4 ∈ Z 2 , and m for ∀ l 1 , l 2 ∈ Z 2 . Conversely, (II) and (III) show that we have corresponding l 1 , l 2 ∈ Z 2 for ∀ l 3 , l 4 ∈ Z 2 , and ∀ m. Consequently, we obtain eq. (A.1).

A.2 Computation of Yukawa couplings
We evaluate the overlap integral of the wave functions including Scherk-Schwarz phases as shown in eq. (5.19). First, we note the following property of the Riemann theta-function:

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By use of eq. (A.1), we get Then we can express the product of two zero-mode wave functions of left-and right-handed fermions as where l ∈ Z 2 [7]. By applying eq. (A.22), we have (A.23) where β H = β L + β R and l ∈ Z 2 . From the above results, we immediately obtain eqs. (5.19) and (5.20). which is trivial for the Riemann theta-function. The selection rule is also unaffected. This is because the replacement only induces a change which can be absorbed by the term lN H . Eq. (6.3) may be useful to show this. Thus, we can choose m independent of P because Yukawa coupling constants are invariant.

B.2 Alignment of m
Here, we explain why knowing the Type of N H can determine the alignment of m in eq. (6.5). We consider N H parameterized as In the above argument, we used the condition gcd(n 21 , n 22 ) = 1. As we showed in appendix B.3, intersection matrices in the same Type have the same gcd(n 11 , n 12 ) and gcd(n 21 , n 22 ). Thus, once we identified the Type of N H , we have information to determine possible alignments of m modulo e n (det N H )N −1 H . Although we have only considered the case when gcd(n 21 , n 22 ) = 1, we can extend the similar analysis for general cases.

B.3 Greatest common divisor and type
Here, we prove intersection matrices N in the same Type have the same gcd(n 11 , n 12 ) and gcd(n 21 , n 22  It is obvious that g := gcd(n 11 , n 11 + n 12 ) ≥ gcd(n 11 , n 12 ) holds. To show that it must be an equality, suppose g := gcd(n 11 , n 11 + n 12 ) > gcd(n 11 , n 12 ). Then we have n 11 + n 12 g = n 11 g + n 12 g , (B.12)

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where n 11 +n 12 g ∈ Z and n 11 g ∈ Z. Thus, n 12 g ∈ Z must be true and n 11 and n 12 have a common factor g . However, this is a contradiction. Therefore, we proved g = gcd(n 11 , n 11 + n 12 ) = gcd(n 11 , n 12 ). (B.13) Similarly, we can verify gcd(n 21 , n 21 + n 22 ) = gcd(n 21 , n 22 ). It is also possible to show that intersection matrices N in the same L-Type have the same gcd(n 11 , n 21 ) and gcd(n 12 , n 22 ).
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