$Z_N$ twisted orbifold models with magnetic flux

We propose new backgrounds of extra dimensions to lead to four-dimensional chiral models with three generations of matter fermions, that is $T^2/Z_N$ twisted orbifolds with magnetic fluxes. We consider gauge theory on six-dimensional space-time, which contains the $T^2/Z_N$ orbifold with magnetic flux, Scherk-Schwarz phases and Wilson line phases. We classify all the possible Scherk-Schwarz and Wilson line phases on $T^2/Z_N$ orbifolds with magnetic fluxes. The behavior of zero modes is studied. We derive the number of zero modes for each eigenvalue of the $Z_N$ twist, showing explicitly examples of wave functions. We also investigate Kaluza-Klein mode functions and mass spectra.


Introduction
Extra-dimensional field theories play an important role in particle physics. Phenomenological aspects of four-dimensional low energy effective field theory strongly depend on geometrical aspects of compactification of extra dimensions and some gauge backgrounds as well as other backgrounds. For example, one of the simplest compactifications is a torus compactification. However, the toroidal compactification without any non-trivial gauge backgrounds leads to a four-dimensional non-chiral theory, and that is not realistic. In general a more complicated geometrical background can lead to a four-dimensional chiral theory, but it is difficult to solve zero-mode equations in generic background and derive the four-dimensional low energy effective field theory.
Torus compactifications with some magnetic fluxes are quite interesting extra-dimensional backgrounds [1,2]. 1 That can realize chiral spectra in four-dimensional low energy effective field theory. One can solve zero-mode equations analytically and their zero-mode profiles are nontrivially quasi-localized.
The number of zero modes depends on the magnitude of magnetic flux, and threegeneration chiral models can be obtained by choosing properly magnetic fluxes. 2 In addition, since zero modes are quasi-localized, their couplings in the four-dimensional low energy effective field theory are non-trivial. That is, when they are quasi-localized far away from each other, their couplings can be suppressed. Thus, magnetic flux backgrounds are quite interesting. Indeed, several studies have been carried out to derive four-dimensional realistic models and study their phenomenological aspects, e.g., Yukawa couplings [2], 3 realization of quark/lepton masses and their mixing angles [6], higher order couplings [7], flavor symmetries [8][9][10][11][12], massive modes [13], and so on [14][15][16][17][18][19][20].
The T 2 /Z 2 twisted orbifold compactification with magnetic flux is also interesting [21,22]. 4 The zero modes and their wave functions on T 2 are classified by Z 2 charges, that is, Z 2 even and odd. Then, either Z 2 even or odd eigenstates are projected out exclusively by the orbifold boundary conditions. Thus, the number of zero modes on T 2 /Z 2 with magnetic flux is different from one on T 2 with the same magnetic flux. Also each of zeromode wave functions on T 2 /Z 2 can be derived analytically and it has a non-trivial profile. One can construct three-generation orbifold models, which are different from models on T 2 with magnetic flux. Such analysis can be extended into higher dimensional models such as T 6 /Z 2 and T 6 /(Z 2 × Z ′ 2 ) orbifolds with magnetic fluxes [21,22]. In addition to T 2 /Z 2 , there are other two-dimensional orbifolds, T 2 /Z N for N = 3, 4, 6 [25]. (See for their geometrical aspects [26][27][28]. 5 ) Moreover, there are various orbifolds in six dimensions like T 6 /Z 7 , T 6 /Z 8 , T 6 /Z 12 , etc. Obviously, geometrical aspects of T 2 /Z N with N = 3, 4, 6 are different from those of T 2 /Z 2 . Thus, one can derive interesting models on T 2 /Z N for N = 3, 4, 6 with magnetic fluxes, which are different from those on T 2 /Z 2 . Hence, it is our purpose to study these orbifold models with magnetic fluxes. In addition, non-trivial (discrete) Scherk-Schwarz phases [34] and Wilson line phases are possible on orbifolds [27,35,36]. 6 Such backgrounds have not been taken into account in [21,22] for the study on T 2 /Z 2 with magnetic flux. Here, we also consider these phases.
In this paper, we study T 2 /Z N orbifold models for N = 2, 3, 4, 6 with magnetic flux, Scherk-Schwarz phases and Wilson lines. We clarify possible Scherk-Schwarz phases as well as Wilson lines on T 2 /Z N orbifolds for N = 2, 3, 4, 6 with magnetic fluxes. Then, we study the behavior of zero modes on T 2 /Z N for each eigenvalue under the Z N twist. We will show that one can obtain three-generation models in various cases and model building becomes rich. Furthermore, we show Kaluza-Klein mode functions and its interesting mass spectrum.
This paper is organized as follows. In section 2, we review the U (1) gauge theory on a two-dimensional torus with magnetic fluxes. In section 3, we study the general formalism of Z N twisted orbifolds with magnetic flux. Especially, we investigate the form of the eigenfunctions for each Z N eigenvalue and the allowed values of important parameters such as Scherk-Schwarz phases and Wilson lines on each orbifold with magnetic flux. In section 4, we analyze the number of zero-mode eigenfunctions, that is, the number of generations for matter fermions on each orbifold. In section 5, we also show Kaluza-Klein mode functions and their mass spectrum. Section 6 is devoted to the conclusions and discussions. In appendix A our notation is summarized. In appendix B we show the relations between Wilson lines and Scherk-Schwarz phases. In appendix C we show some examples of calculations on zero-mode wave functions on T 2 /Z N with magnetic fluxes.
2 Gauge field theory on M 4 × T 2 with magnetic flux Let us study the behavior of gauge and matter fields on six-dimensional space-time, which contains four-dimensional Minkowski space-time M 4 and an extra two-dimensional torus T 2 . We denote coordinates on M 4 by x µ (µ = 0, 1, 2, 3) and we use the complex coordinate z on T 2 . We consider a theory containing the torus with magnetic flux. Then, one can obtain an attractive feature that chiral zero-mode fermions appear and their number is determined by the magnitude of the magnetic flux. We will see it below.
First of all, we consider the Lagrangian density based on a U (1) gauge theory on M 4 × T 2 such as where M, N = µ(= 0, 1, 2, 3), z,z and D M = ∂ M −iqA M (x, z) 7 with a U (1) charge q. Here, Ψ ± are six-dimensional Weyl fermions, and are obtained by projection operators 1±Γ 7 2 such as In this paper, we use a notation as in appendix A. Note that fields such as AM (x, z), A z (z), Ψ±(x, z), ψ±,n(z) and so on are written by functions depending on not only z but alsoz.
where n means the label of mass eigenstates. Ψ is a six-dimensional Dirac fermion, ψ 4R/L,n and ψ ′ 4R/L,n are four-dimensional right/left-handed fermions, and ψ 2±,n are twodimensional Weyl fermions. For convenience, we also use the following notation: .
Then, the action for the one Weyl fermion Ψ + and its gauge interaction can be written as M (x) are higher modes. Here, we used ψ 4,n ≡ ψ 4R,n + ψ 4L,n and the mass equations with background gauge fields A We have also used the orthonormality condition Our first interest is the feature of zero modes for a fermion with m n = 0, and its interaction with magnetic flux. We will investigate the zero-mode parts of the two-dimensional Weyl fermion ψ 2±,n (z) and the background gauge fields A

Magnetic flux quantization on T 2
We review the U (1) gauge theory on a two-dimensional torus with magnetic flux following ref. [2,40]. The complex coordinate z on one-dimensional complex plane satisfies the iden- , the vector potential A (b) can be written as where a w is a complex Wilson line phase. From eq.(2.8), we obtain where χ 1 (z + a w ) and χ τ (z + a w ) are given by 9 Im[τ (z + a w )]. (2.10) Moreover, we require the Lagrangian density L 6D (2.1) to be single-valued, i.e., Then, this field Ψ + (x, z) should satisfy the pseudo periodic boundary conditions i.e., ψ ±,n (z + 1) = U 1 (z)ψ ±,n (z), ψ ±,n (z + τ ) = U τ (z)ψ ±,n (z), (2.13) with where α 1 and α τ are allowed to be any real number, and are called Scherk-Schwarz phases. The consistency of the contractible loops, e.g., z → z + 1 → z + 1 + τ → z + τ → z, requires the magnetic flux quantization condition, Then, U 1 (z) and U τ (z) satisfy It should be emphasized that all of the Wilson line phase and the Scherk-Schwarz phases can be arbitrary, but are not physically independent because the Wilson line phase can be absorbed into the Scherk-Schwarz phases by a redefinition of fields and vice versa (see appendix B). This fact implies that we can take, for instance, the basis of vanishing Wilson line phases, without any loss of generality. It is then interesting to point out that allowed Scherk-Schwarz phases are severely restricted for T 2 /Z N orbifold models, as we will see in the next section, while there is no restriction on the Scherk-Schwarz phases for T 2 models.

Twisted orbifolds with magnetic flux
In the previous section, we reviewed the U (1) gauge theory on a two-dimensional torus T 2 with magnetic flux. Then, we found that the number of zero-mode fermions is given by the magnitude of magnetic flux |M |. In this section, we study the U (1) gauge theory on twisted orbifolds T 2 /Z N with magnetic flux, and investigate the degeneracy of zero-mode solutions and the allowed values of the Wilson line phase a w and the Scherk-Schwarz phases α 1 and α τ .

T 2 /Z N twisted orbifold
A two-dimensional twisted orbifold T 2 /Z N is defined by dividing a one-dimensional complex plane by lattice shifts t 1 , t τ and a Z N discrete rotation (twist) s such as Thus, the orbifold obeys the identification It has already been known that there exist only four kinds of the orbifolds such as T 2 /Z N (N = 2, 3, 4, 6). We would like to note the relation between the moduli τ and the rotation ω for each orbifold. For N = 2, there is no limitation on τ except for Imτ > 0, but for N = 3, 4, 6, τ should be equivalent to ω because of the analysis by crystallography [28]. For convenience, we still use both τ and ω as a base vector on the lattice and the Z N twist, respectively below, though τ = ω for N = 3, 4, 6. Moreover, an important feature is the existence of fixed points z fp defined by Since each fixed point is specified by the Z N twist ω and the shift m + nτ , we define z fp as (ω, m + nτ ) with the language of space group. On the complex plane, there exist an infinite number of fixed points because the possible combinations of (m, n) exist countlessly. On the torus, however, z fp and z fp + m ′ + n ′ τ ( ∀ m ′ , n ′ ∈ Z) should be identified with the torus identification z ∼ z + m + nτ . Then, it follows from eq.(3.4) that since z fp + m ′ + n ′ τ satisfies the relation i.e., For example, let us consider the case of . From these identifications, one can find three fixed points, i.e., Actually, the three fixed points on the fundamental region are given by In the same way, 10 on T 2 /Z 2 , four fixed points exist, which are z fp = 0, 1 2 , τ 2 , 1+τ 2 for (m, n) = (0, 0), (1, 0), (0, 1), (1, 1), respectively. On T 2 /Z 4 , two fixed points exist, which are z fp = 0, 1+τ 2 for (m, n) = (0, 0), (1, 0), respectively. On T 2 /Z 6 , only one fixed point exists, which is z fp = 0 for (m, n) = (0, 0). The fundamental region and the fixed points for each orbifold are depicted in figure 1. As we will see below, the number of fixed points correspond to the variety of allowed Scherk-Schwarz phases on each orbifold.

Field theory on orbifold
Next, we study a field theory on the orbifold. Let us consider the following Lagrangian density on six-dimensional space-time with the orbifold T 2 /Z N , Here and hereafter, we take the gauge with a w = 0 because the Wilson line phase can be absorbed into the 10 See in detail ref. [27]. Scherk-Schwarz phases (see appendix B). In order to define a field theory on the orbifold, we need to specify what are boundary conditions under the lattice shifts t 1 , t τ and the Z N twist s for the fermion. Then, we define the boundary conditions for Ψ T 2 /Z N + (x, z) as , and V (z) is a transformation function for the Z N twist. The transformation functions U 1 (z) and U τ (z) for the lattice shifts t 1 and t τ are given in eq.(2.14) with a w = 0. However, the Scherk-Schwarz phases α 1 and α τ cannot be freely chosen, and are allowed to be certain discrete values on the orbifold, as we will see later.
In a way similar to eq.(2.2), we can expand the Weyl fermion Ψ .
Then, the boundary conditions (3.10) for Ψ T 2 /Z N + (x, z) are replaced by those for ψ T 2 /Z N ±,n (z), i.e., Here, it is worthwhile to note that the wave functions ψ T 2 /Z N ±,n (z) on the orbifold T 2 /Z N can be constructed from certain linear combinations of ψ ±,n (z) on the torus T 2 . This is because the orbifold T 2 /Z N is obtained by dividing the torus T 2 by the Z N discrete rotation. 14) The transformation function V (z) is given by where β is a real number. From the requirement that N times the twisted transformation in eq.(3.1) should be identical to the identity operation, i.e., s N = 1, β has to satisfy When we require that L Weyl 6D is single-valued under the lattice shifts and the Z N twist, the boundary conditions for the gauge fields A M (x, z) can be obtained as Moreover, let us investigate the boundary conditions for general lattice shifts m + nτ (m, n ∈ Z) and Z N twists ω k (k ∈ Z). To this end, we define the transformation function U m+nτ (z) through the relation Then, we obtain because ω k (m+nτ ) for ∀ k, m, n ∈ Z can be equivalently expressed as a lattice shift m ′ +n ′ τ for ∃ m ′ , n ′ ∈ Z.
From the above definition (3.18), U m+nτ (z) turns out to satisfy where we have used the relations U 1 (z + m) = U 1 (z) and U τ (z + nτ ) = U τ (z), which will be derived from eqs.(2.10) and (2.14). We can further show that from eq.(3.10), U m+nτ (z) should obey the relation It follows that we find

Scherk-Schwarz phases with magnetic flux
Next, let us investigate the Scherk-Schwarz phases with magnetic flux. Then, U 1 (z) and U τ (z) depend on z, The allowed Scherk-Schwarz phases are shown in table 1. It is found that the variety of the Scherk-Schwarz phases still corresponds to the number of fixed points even with non-zero magnetic flux. 11 However, it is remarkable that the non-zero magnetic flux with M = odd affects the values of the Scherk-Schwarz phases for N = 3, 6, and especially does not permit them to vanish.
It should be emphasized that ψ (j+α 1 ,ατ ) T 2 /Z N ±,n (z) ω ℓ are mutually independent for different values of ℓ, i.e., eigenvalues ω ℓ under the Z N twist, but are not always linearly independent for different values of j. We will see this feature in the next section explicitly.

Wilson line phase
In section 3.2, we have investigated the variety of Scherk-Schwarz phases in the gauge with a w = 0. Here, we would like to consider the case of non-zero Wilson line phases. From the results given in appendix B, let us transform the a w = 0 gauge into the gauge, where they satisfy Mã w = α 1 τ − α τ ,α 1 = 0,α τ = 0, (3.42) whereã w and (α 1 ,α τ ) are the redefined Wilson line phase and the redefined Scherk-Schwarz phases, respectively. Substituting the value of (α 1 , α τ ) (mod 1) of  T 2 /Z N ±,n (z) ω ℓ on T 2 /Z N and found the allowed values for the Wilson line phase a w and the Scherk-Schwarz phases α 1 and α τ on each orbifold. Here, we focus on the zero-mode eigenstates for each Z N eigenvalue with a w = 0, and study their number for each M . In particular, we will pay attention to the cases that the number of zero-mode eigenstates is given by around three, because we would like to construct a three generation model.

Kaluza-Klein mode functions and mass spectra
Thus, Kaluza-Klein mode functions are given by (z)) on the zero-mode functions are well-defined for M > 0 (M < 0). The masses squared of ψ (j+α 1 ,ατ ) T 2 ±,n (z) are found to be of the form Note that these are masses squared for spinor fields, while eigenvalues of ∆ correspond to masses squared for scalar fields as m 2 n = 4π|M | A (n + 1 2 ) for n ∈ {0, N}. As an illustrative example, the mass spectra of ψ (j,0) T 2 ±,n (z) (j = 0, 1) for M = 2 and (α 1 , α τ ) = (0, 0) are depicted in figure 3. The red crosses mean the absence of zero-mode solutions, and the blue (green) filled circles correspond to a zero mode and its Kaluza-Klein modes of ψ (0,0) T 2 ±,n (z) (ψ (1,0) T 2 ±,n (z)). The blue (green) arrows mean thatâ † operates on the Then, the Kaluza-Klein modes ψ (j+α 1 ,ατ ) T 2 /Z N ±,n (z) η for ∀ η possess the masses squared Here, let us show an illustrative example. Figure 4 shows the zero-mode eigenstates ψ (j,0) T 2 /Z 3 +,0 (z) η (j = 0, 1) for M = 2 in Table 4 and its Kaluza-Klein modes. The meaning of symbols in figure 4 is the same as in figure 3. The important difference between figures 3 and 4 is how Kaluza-Klein modes grow up. In the orbifolds, they grow up as changing the Z N eigenstates. T 2 /Z3±,n (z) η (j = 0, 1) for M = 2 in table 4. The red crosses mean the absence of zero-mode solutions and Kaluza-Klein modes, and the blue (green) filled circles correspond to a zero mode and its Kaluza-Klein modes. The blue (green) arrows mean that a † operates on nth modes ψ (j,0) T 2 /Z3+,n (z) and the next modes ψ (j,0) T 2 /Z3+,n+1 (z) are made by it. Two modes in each black oval make a pair to have a mass term.
We have studied the U (1) gauge theory on the T 2 /Z N orbifolds with magnetic fluxes, Scherk-Schwarz phases and Wilson line phases. We have shown all of the possible Scherk-Schwarz and Wilson line phases. It is remarkable that the allowed Scherk-Schwarz phases as well as Wilson line phases depend on the magnitude of magnetic flux for the T 2 /Z 3 and T 2 /Z 6 orbifolds, in particular, whether M is even or odd. At any rate, the variety of possible Scherk-Schwarz and Wilson line phases corresponds to the number of fixed points on each orbifold with any value of magnetic flux. Under these backgrounds, we have studied the behavior of zero modes. We have derived the number of zero modes for each eigenvalue of the Z N twist. This result was obtained by showing explicitly and analytically wave functions for some examples and also by studying numerically Z N -eigenfunctions for many models. The exactly same results will be derived by another approach for the generic case [41]. The Kaluza-Klein modes were also investigated.
Our results show that one can derive models with three generations of matter fermions in various backgrounds, i.e., the T 2 /Z N orbifolds for N = 2, 3, 4, 6 with various magnetic fluxes and Scherk-Schwarz phases. Using these results, one could construct realistic threegeneration models. The toroidal compactification can lead to three zero modes only when M = 3, and such a model leads to ∆(27) flavor symmetry [8][9][10][11]. 16 On the other hand, three generations can be realized in various orbifold models and that would lead to a rich flavor structure. Couplings among zero modes in the four-dimensional low energy effective field theory are obtained by overlap integrals of their wave functions. Our analysis shows that zero-mode wave functions on the orbifold with magnetic flux can be obtained as linear combinations of zero-mode wave functions on the torus with the same magnetic flux. Since overlap integrals and couplings of zero-mode wave functions on T 2 were calculated in [2,7], such couplings can be similarly computed for generic orbifold models. These analyses on realistic model building for three generations and their low energy effective field theory will be studied elsewhere. We have focused on the bulk modes originated from higher dimensions. However, the orbifolds have fixed points. Then, we can put any localized modes with δ-function like profile on such fixed points, if that is consistent from the viewpoint of four-dimensional field theory.
At any rate, our results can become a starting point for these studies. Also, our study is applicable to more general twisted orbifold models in higher-dimensional theory more than six-dimensional one, e.g., T 6 /Z N , T 6 /(Z N × Z ′ N ) and so on.
Government of India for the Regional Centre for Accelerator-based Particle Physics (RE-CAPP), Harish-Chandra Research Institute.

B Redefinition of fields
We consider the relation between the redefinition of a field Φ(z) interacting with a U (1) gauge field A(z) and Scherk-Schwarz phases α 1 and α τ which are real numbers. Here, we denote Φ(z) and A(z) as Φ(z; a w ) and A(z; a w ) to emphasize the Wilson line phase a w .