Orbifold family unification using vectorlike representation on six dimensions

In orbifold family unification on the basis of $SU(N)$ gauge theory on the six-dimensional space-time $M^4\times T^2/Z_m$ ($m=2, 3, 4, 6$), enormous numbers of models with three families of the standard model matter multiplets are derived from a massless Dirac fermion with a vectorlike representation $[N, 3] + [N, N-3]$ of $SU(N)$ ($N = 8, 9$). They contain models with three or more than three neutrino singlets and without any non-Abelian continuous flavor gauge symmetries. The relationship between flavor numbers from a fermion with $[N, N-k]$ and those from a fermion with $[N, k]$ are studied from the viewpoint of charge conjugation.


Introduction
One of the most intriguing riddles in particle physics is the origin of the family replication in the standard model (SM) matter multiplets. Various investigations have been performed, using models on the four-dimensional Minkowski space-time M 4 [1,2,3,4,5,6,7], but, in most cases, we encounter difficulties relating to the chiralness of fermions. Concretely, chiral fermions do not, in general, come from a fermion in an anomaly free representation of a large gauge group, e.g., 2 n−1 for SO(2n) (n ≥ 6), or a vectorlike (non-chiral) set of representations, e.g., N + N for SU (N ), as an extension of grand unified theories (GUTs). In most cases, particles with opposite quantum numbers under the SM gauge group SU (3) C × SU (2) L × U (1) Y , called mirror particles, appear and the survival hypothesis is adopted to get rid of them from the low-energy spectrum. Then, the SM family members can also disappear. Here, the survival hypothesis is stated such that if a symmetry is broken down into a smaller one at a scale M S , then any fermion mass terms invariant under the smaller group induce fermion masses of O(M S ) and such heavy fermions disappear from the low-energy spectrum [3,8]. 3 The above difficulty can be overcome by extending the structure of space-time. That is, extra particles including mirror ones can be eliminated using orbifold breaking mechanism, as originally proposed in superstring theory [9,10,11]. Hence, a candidate realizing the family unification is an extension of GUTs defined on a higher-dimensional space-time including an orbifold. 4 These studies have been carried out intensively [14,15,16,17,18,19,20,21,22,23,24,25], and three replicas of matter multiplets are derived from characteristics of extra dimensions. For instance, three replication SU (5) multiplets have been derived from a single bulk fermion in the rank k totally antisymmetric tensor representation [N , k] (N ≥ 9) of SU (N ) on M 4 ×S 1 /Z 2 [20]. Enormous numbers of models with three families of the SM matter multiplets have been obtained from a single massless Dirac fermion in [N , k] (N ≥ 9) of SU (N ) on M 4 × T 2 /Z m (m = 2, 3, 4) [23]. The relationship between the flavor numbers of chiral fermions and the Wilson line phases has been studied in these models [26]. Using models originated from SU (9) gauge theory on M 4 × T 2 /Z 2 , their reality has been examined from the structure of the Yukawa interactions [27].
In Ref. [23], we find that the number of neutrino singlets is less than three, the smallest gauge group is SU (9), and most models contain extra non-Abelian continuous gauge group relating to a flavor symmetry, under the precondition that three SM families are derived from a massless Dirac fermion in a chiral representation [N , k] of SU (N ). Then, we need extra neutrino singlets to produce massive neutrinos and extra scalar fields to break extra gauge symmetries. By changing the precondition into that three SM families are derived from a massless Dirac fermion in a vectorlike representation [N , k] + [N , N − k] of SU (N ), there is a possibility that some models possess features such that the number of neutrino singlets is three or more than three, the smallest gauge group is less than SU (9), and all extra gauge symmetries are Abelian. Furthermore, extra gauge symmetries could be broken down by the vacuum expectation values of superpartners of neutrino singlets.
In this paper, we study the possibility of family unification on the basis of SU (8) and SU (9) gauge theory on M 4 ×T 2 /Z m , using the method in Ref. [20,23]. We investigate whether or not three families of the SM matter multiplets are derived from a single massless Dirac fermion in a vectorlike representation [8, k] + [8, 8 − k] or [9, k] + [9, 9 − k], through the orbifold breaking mechanism. We clarify the relationship between flavor numbers from a fermion in [N , N − k] and those from a fermion in [N , k] from the viewpoint of charge conjugation.
The contents of this paper are as follows. In Sec. 2, we provide general arguments on the orbifold breaking based on two-dimensional orbifold T 2 /Z m . In Sec. 3, we give formulae for numbers of the SM matter multiplets. In Sec. 4, we study a possibility of the family unification in six-dimensional SU (8) and SU (9) gauge theories containing a massless Dirac fermion in a vectorlike representation. Section 5 is devoted to conclusions and discussions.

Z m orbifold breaking, fermions and decomposition of field
We explain the orbifold T 2 /Z m (m = 2, 3, 4, 6), a six-dimensional fermion and a decomposition of field in [N , k].

Z m orbifold breaking
On a two-dimensional lattice T 2 , the points z + e 1 and z + e 2 are identified with the point z, where e 1 and e 2 are basis vectors and z takes a complex value. The orbifold T 2 /Z m is obtained by dividing T 2 by the Z m transformation z → ρz, where ρ is the m-th root of unity (ρ m = 1). Then, z is identified with ρz, or z is identified with ρ k z + ae 1 + be 2 , where k, a and b are integers. For more details, see Appendix A.
We explain the Z m transformation properties of a six-dimensional scalar field Φ(x, z, z), using T 2 /Z 3 whose basis vectors are given by e 1 = 1 and e 2 = i . The extension of other fields (fermions and gauge bosons) and other orbifolds is straightforward. From the requirement that the Lagrangian density L should be invariant under the Z 3 transformations s 0 : z → ωz and s 1 : z → ωz + 1 (ω = e 2πi /3 ) or it should be a single-valued function, the boundary conditions of fields on T 2 /Z 3 are determined up to some overall Z 3 factors, which we refer to as intrinsic Z 3 elements of fields and denote as η aΦ corresponding to the Z 3 transformations s a (a = 0, 1). When Φ is a multiplet of some transformation group G concerning some internal symmetries (including gauge symmetries), L should be invariant under the transformation where T Φ is a representation matrix of G on Φ. For instance, if a theory has SU (N ) gauge symmetry, L is, in general, invariant under a (global) U (N ) transformation, i.,e., G = U (N ). From (2.1) and (2.2), the following boundary conditions on Φ are allowed, where T Φ [U 0 , η 0Φ ] and T Φ [U 1 , η 1Φ ] represent appropriate representation matrices, which are ele- ] (a = 0, 1), using representation matrices U a for the fundamental representations of G and the intrinsic Z 3 elements η aΦ (see (2.13)), and some relations can appear among the intrinsic Z 3 elements (see (2.6) or (B.20)). Arbitrary U 0 and U 1 can be diagonalized simultaneously by a global unitary transformation and a local gauge transformation or each equivalence class of boundary conditions contains diagonal representatives [28]. Hence we use diagonal ones later.
We list basis vectors and the transformations relating to identifications of points on T 2 /Z m , and denote its representation matrices for the fundamental representation as U a (a = 0, 1, 2 for T 2 /Z 2 and a = 0, 1 for T 2 /Z 3 and T 2 /Z 4 and a = 0 for T 2 /Z 6 ), in Table 1 [29,30]. Note that there is a choice in transformations independently of each other.
Components of Φ possess discrete charges associated with eigenvalues of T Φ [U a , η aΦ ]. When the eigenvalues are given as e 2πi l /m (l = 0, 1, · · · , m − 1), the discrete charges are assigned as numbers l /m. We refer to e 2πi l /m as Z m elements. In the absence of contributions from the Wilson line phases, the massless six-dimensional fields whose Z m elements for all a are equal to 1 contain zero modes, but those including a Z m element different from 1 do not contain zero modes. 5 Here, zero modes mean four-dimensional massless fields. If the size of extra dimensions is small enough, massive modes called Kaluza-Klein modes do not appear in low-energy theories. Unless all components of non-singlet field have a common Z m charge, a symmetry reduction occurs upon compactification. This type of symmetry breaking mechanism is called "orbifold breaking mechanism". 6 5 In the presence of non-vanishing Wilson line phases, gauge symmetries and particle spectrum are rearranged via the Hosotani mechanism [31,32,33,34]. 6 The Z 2 orbifolding was used in superstring theory [35] and heterotic M-theory [36,37]. In field theoretical models, it was applied to the reduction of global supersymmetry (SUSY) [38,39], which is an orbifold version of Scherk-Schwarz mechanism [40,41], and then to the reduction of gauge symmetry [42].

Fermions
We explain fermions in six dimensions. For more details, see the Appendix B. A massless Weyl fermion on six dimensions is regarded as a Dirac fermion or a pair of Weyl fermions with opposite chiralities on four dimensions. The six-dimensional Dirac fermion consists of two six-dimensional Weyl fermions such that where Ψ + and Ψ − are fermions with positive and negative chirality, respectively, and Γ 7 is the chirality operator on six dimensions. Here and hereafter, the subscript ± and L(R) stand for the chiralities on six and four dimensions, respectively. The charge conjugation of a six-dimensional Dirac fermion Ψ is defined as where Γ M (M = 0, 1, 2, 3, 5, 6) are six-dimensional gamma matrices, B = −i Γ 7 Γ 2 Γ 5 up to a phase factor, and the asterisk * means the complex conjugation. 7 Note that the chirality in six dimensions does not flip under the charge conjugation, as shown in (B.12) and (B.13).
From the Z m invariance of kinetic term and the transformation property of the covariant derivatives D z → ρD z and D z → ρD z with ρ(= ρ * ) = e −2πi /m and ρ = e 2πi /m , we have the relations: where z ≡ x 5 + i x 6 and z ≡ x 5 − i x 6 , and η a±L(R) are the intrinsic Z m elements of ψ ±L(R) . For the derivation of (2.6), see from (B.14) to (B.20).
Chiral gauge theories including Weyl fermions on even dimensional space-time become, in general, anomalous in the presence of gauge anomalies, gravitational anomalies, mixed anomalies and/or global anomaly [44,45]. Here we consider a non-supersymmetric model for simplicity. In SU (N ) gauge theories on six dimensions, the global anomaly is absent because of π 6 (SU (N )) = 0 for N ≥ 4. Here, π 6 (SU (N )) is the six-th homotopy group of SU (N ). Other anomalies must be canceled out by the contributions from several fermions. For instance, they are canceled out by the contributions from fermions with different chiralities such as (Ψ r + , Ψ r − ), where r stands for r -dimensional representation of SU (N ). Each pair in (Ψ r + , Ψ r − ), (Ψ r + , Ψ r − ) and (Ψ r + , Ψ r − ) does not contribute to the anomalies, where r stands for the complex conjugate representation of r. The cancellation on six dimensions is understood that the gauge anomaly is proportional to a group-theoretical factor such as where Str stands for the trace over the symmetrized product of the gauge group generators T a i , and this trace is invariant under the exchange between T a i and −(T a i ) * , corresponding to the exchange 7 In this paper, the complex conjugation is also represented by the overlined one. between a fermion in r and one in r. The gravitational anomaly is canceled out, if the following condition is fulfilled, (2.8) where N ± is the numbers (including degrees of freedom) of Ψ ± .

Decomposition of representation
With suitable diagonal representation matrices U a , the SU (N ) gauge group is broken down into its subgroup such that 9) where N = p 1 + p 2 + · · · + p n . Here and hereafter, SU (1) unconventionally stands for U (1), SU (0) means nothing and n ′ is a sum of the number of SU (0). A concrete form of U a will be given in the next section.
After the breakdown of SU (N ), the rank k totally antisymmetric tensor representation [N , k], whose dimension is N C k , is decomposed into a sum of multiplets of the subgroup SU (p 1 )×SU (p 2 )× · · · × SU (p n ) as where l n = k −l 1 −· · ·−l n−1 and our notation is that n C l = 0 for l > n and l < 0. Here and hereafter, we use n C l instead of [n, l ] in many cases. We sometimes use the ordinary notation for representations too, e.g., N and N in place of N C 1 and N C N−1 .
The [N , k] is constructed by the antisymmetrization of k-ple product of the fundamental repre- where a tiny subscript A means the antisymmetrization. For Weyl fermions Ψ ± in [N , k], the boundary conditions are given by a± ] stand for appropriate representation matrices, which are elements of U (N ) on Ψ ± , U a are the representation matrices for the fundamental representation and η (k) a± are the intrinsic Z m elements of Ψ ± in [N , k]. We omit the subscripts L and R on η (k) a± , for simplicity. Note that there are relations such as (2.6) between η (k) a±L and η (k) a±R . Using (2.11) and (2.12), the Z m transformation property of [N , k] can be expressed by (2.13) By definition, η (k) a± take values of Z m elements, i.e., e 2πi l /m (l = 0, 1, · · · , m − 1). Note that η (k) a+ are not necessarily same as η (k) a− , and the chiral symmetry is still respected.
In the same way, the [N , N −k] is constructed by the antisymmetrization of (N −k)-ple product of N : (2.14) or it is also constructed by the antisymmetrization of k-ple product of the complex conjugate representation N : Using (2.15), the Z m transformation property is given by a± . Strictly speaking, in this case, the relations are written asη (k)

Formulae for numbers of SM species
Let us investigate the family unification with the breaking pattern: where SU (3) and SU (2) are identified with SU (3) C and SU (2) L in the SM gauge group. After the breakdown of SU (N ), [N , k] is decomposed into a sum of multiplets as The flavor numbers of down-type anti-quark singlets (d R ) c , lepton doublets l L , up-type anti-quark singlets (u R ) c , positron-type lepton singlets (e R ) c , and quark doublets q L are denoted as nd , n l , nū, nē and n q . Using the survival hypothesis and the equivalence on charge conjugation in four dimensions, we define the flavor number of each SM chiral fermion as where ♯ represents the number of zero modes for each multiplet. The SM singlets are regarded as the right-handed neutrinos, which can obtain heavy Majorana masses among themselves as well as the Dirac masses with left-handed neutrinos. Some of them can be involved in see-saw mechanism [46,47,2]. The total number of (heavy) neutrino singlets (ν R ) c and/or ν R is denoted by nν and defined as , the number of zero modes for each multiplet is given by the formulae: where the P mk±L(R) (m = 2, 3, 4, 6) are projection operators to pick out zero modes of ψ ±L(R) in [N , k], and they are listed in Table 2. In Table 2, ϕ = e i π/3 and ϕ = e −i π/3 , and each operator is defined by .
where n 0 , n 1 and n 2 are integers, P (k) a± are the Z m elements determined by U a and η (k) a±L(R) , as will be given below. For instance, P (ω n 0 ,ω n 1 ) 3k± is an projection operator to pick out modes with P (k) 0± = ω n 0 and P (k) (3.9) and (3.10), we obtain following formulae for the SM species and neutrino singlets derived from a pair of six-dimensional Weyl fermions ( where P mk± and P (ν) mk± are defined by respectively. By the insertion of (−1) l 1 +l 2 , we obtain ♯( 3 C l 1 , 2 C l 2 ) L(R) for l 1 + l 2 = even integer and −♯( 3 C l 1 , 2 C l 2 ) L(R) for l 1 + l 2 = odd integer. Although the above formulae (3.15) - (3.19) are derived with no consideration for the Wilson line phases, they still hold for the case with non-vanishing Wilson line phases relating to extra gauge symmetries, thanks to a hidden quantum-mechanical supersymmetry [26].
We explain how the Z m elements P (k) a± of multiplets in 3 C l 1 , 2 C l 2 , · · · , p n C l n decomposed from Ψ ± in [N , k](= N C k ) are determined by the intrinsic Z m elements η (k) a± and the representation matrices a± ] act multiplets in 3 C l 1 , 2 C l 2 , · · · , p n C l n . The components of Ψ ± are writ- where a tiny subscript A means the antisymmetrization, and the operation of and k = l 1 +l 2 . From the observation that p 1 C l 1 , p 2 C l 2 is multiplied by +1 l 1 times and multiplied by −1 l 2 times through the operation of T Ψ± [U 0 , η (k) 0± ] on [N , k], we see the Z 2 element of p 1 C l 1 , p 2 C l 2 as P (k) 0± = η (k) 0± (+1) l 1 (−1) l 2 = (−1) l 1 −k η (k) 0± where we use k = l 1 + l 2 and (−1) n = (−1) −n (n is an integer).

SU (N ) Representations
for the ψ ±L are written, and those for the ψ ±R can be seen from (3.30).
In the third column, l i not on the list are zero. In the fourth column, the subscripts L and R are omitted on the intrinsic Z 3 elements.
In the same way, we can obtain particle contents with just three SM families and three neutrino singlets as zero modes from ψ [8,3] ±L + ψ [8,3] ±R + ψ [8,5] ±L + ψ [8,5] ±R with intrinsic Z 3 elements assigned in Table  8, after the survival hypothesis works. Table 8: Another assignment of intrinsic Z 3 elements and the particle contents as zero modes obtained from 56 and 56.
Finally, we point out that the classification of our models has not yet been completed in our setup.
Concretely, we consider the breaking pattern (2.9) with the identification of SU (p 1 ) = SU (3) C and SU (p 2 ) = SU (2) L , and take the diagonal representation matrices (3.22), (3.24), (3.26) and (3.28). Based on the representation matrices given above, there is a choice to take p i = 3 and p j = 2 with (i , j ) = (1, 2) as SU (3) C ×SU (2) L . Or provided that p 1 = 3 and p 2 = 2, we can choose different diagonal representation matrices, that are obtained by the exchange of components in the above ones. Same results are obtained from most of them, but there are independent choices to generate models different from those mentioned in this section. Complete analysis and classification will be reported, including results from a fermion in [N , k] ≥ 4), in a forthcoming paper [48].

Conclusions
We It is meaningful to study phenomenological implications relating to the breakdown of extra U (1) gauge symmetries, D-term contributions to scalar (squark, slepton and Higgs) masses and the generation of realistic fermion masses and family mixing, based on SU (8) models illustrated in Sect. 4. The SU (8) models are attractive, because there is no non-Abelian continuous gauge group, and extra U (1) gauge bosons can be massive by the vacuum expectation values of the SM singlets scalar fields. Moreover, superpartners of neutrino singlets can be candidates of such scalar fields. In SUSY models, there appear D-term contributions to scalar masses after the breakdown of extra gauge symmetries, if soft SUSY breaking terms have a non-universal structure, and its contributions lift the mass degeneracy [49,50,51,52,53]. Under assumptions that SUSY is broken down by the dynamics on a brane and non-universal soft SUSY breaking terms are induced, the D-term contributions have been studied in the framework of SU (N ) orbifold GUTs [54,55,56], and they can become useful probes to specify a realistic model in GUTs. Then we need to reconsider the anomaly cancellations on a construction of SUSY models, because various fermions exist there. Fermion mass hierarchy and family mixing can occur through the Froggatt-Nielsen mechanism [57] on the breakdown of extra U (1) gauge symmetries and/or the suppression of brane-localized Yukawa coupling constants among brane weak Higgs doublets and bulk fermions with the volume suppression factor [58].
It would be interesting to reconstruct our models in the framework of E 8 gauge theory or superstring theory. Various 4-dimensional string models including three families have been constructed from several methods, see e.g. [59] and references therein for useful articles. 8 It has been pointed out that SO(1, D − 1) space-time symmetry can lead to family structure [62,63], and hence it would offer a hint to explore the family structure in our models.
Furthermore, it would be intriguing to study cosmological implications of the class of models presented in this paper, see e.g. [64] and references therein for useful articles toward this direction.
The orbifold T 2 /Z 2 is obtained by identifying z +e 1 , z +e 2 and −z with z. Here e 1 = 1 and e 2 = i . The resultant space is depicted in Figure 1. Fix points z fp satisfy z fp = −z fp + ae 1 + be 2 where a and b are integers. There are four kinds of fixed points 0, e 1 /2, e 2 /2, (e 1 +e 2 )/2. Around these points, we define six kinds of transformations: s 0 : z → −z, s 1 : z → −z + e 1 , s 2 : z → −z + e 2 , s 3 : z → −z + e 1 + e 2 , t 1 : z → z + e 1 , t 2 : z → z + e 2 (A.1) and they satisfy the relations: where I is the identity operation.
The boundary conditions of six-dimensional bulk fields are specified by representation matrices (U 0 ,U 1 ,U 2 ,U 3 ,V 1 ,V 2 ) and intrinsic Z 2 elements (η 0 , η 1 , η 2 , η 3 , ξ 1 , ξ 2 ) corresponding to the above transformations. These matrices and Z 2 elements satisfy the relations: as the consistency conditions. Here, we omit the subscripts specifying fields and/or chiralities such as Φ, ±, L and/or R. Note that η 1 η 0 η 2 = η 2 η 0 η 1 and ξ 1 ξ 2 = ξ 2 ξ 1 hold automatically because intrinsic Z m elements are numbers. From (A.2) and (A.3), we find that any three transformations are independent and others are constructed as combinations of them. We choose the transformations s 0 : z → −z, s 1 : z → 1 − z and s 2 : z → i − z and the corresponding matrices U 0 , U 1 and U 2 .

A.2 T 2 /Z 3
The orbifold T 2 /Z 3 is obtained by identifying z + e 1 , z + e 2 and ωz with z. Here e 1 = 1 and e 2 = ω = e 2πi /3 . The resultant space is depicted in Figure 2. Fixed points satisfying z fp = ωz fp + ae 1 + be 2 The boundary conditions of bulk fields are specified by matrices (U 0 ,U 1 ,U 2 ,V 1 ,V 2 ) and intrinsic Z 3 elements (η 0 , η 1 , η 2 , ξ 1 , ξ 2 ) satisfying the relations: where we omit the subscripts specifying fields and/or chiralities such as Φ, ±, L and/or R. Because two of these matrices are independent, we choose representation matrices U 0 and U 1 corresponding to the transformations s 0 : z → e 2πi /3 z and s 1 : z → e 2πi /3 z + 1. The Z 4 transformations s 0 and s 1 are independent of each other and those representation matrices are denoted as U 0 and U 1 , respectively. Other representation matrices are determined uniquely, if U 0 and U 1 are given.

B Fermions on six dimensions
We explain gamma matrices, charge conjugation of fermions and Z m transformation properties on six dimensions [43]. We use the metric η M N = diag(1, −1, −1, −1, −1, −1) (M, N = 0, 1, 2, 3, 5, 6), and the following representation for six-dimensional gamma matrices: where µ = 0, 1, 2, 3, σ i (i = 1, 2, 3) are Pauli matrices, and I 4×4 is the 4 × 4 unit matrix. We take the chiral representation on four-dimensional space-time for γ µ such that where I 2×2 is the 2 × 2 unit matrix. The Γ M satisfy the anti-commutation relations of the Clifford algebra such that {Γ M , Γ N } = 2η M N where η M N is the inverse of η M N . The chirality operator Γ 7 for six-dimensional fermion Ψ is defined by where γ 5 is the chirality operator on four dimensions defined by Six-dimensional fermions with a definite chirality is called Weyl fermions on six dimensions. The Weyl fermion (Ψ + ) with positive chirality and that (Ψ − ) with negative chirality are given by respectively. Here, the subscript ± and L(R) stand for the chiralities on six and four dimensions, respectively. Using Weyl fermions ξ ± and η * ± on four dimensions, Ψ and ψ ±L(R) are expressed as The charge conjugation of Ψ is defined as where B is a 8 × 8 matrix which satisfies the relation The B is given by up to a phase factor and, using it, we derive the charge conjugation of ξ ± and η * ± , From (B.12) and (B.13), we find that the chirality in six dimensions does not flip under the charge conjugation.
In terms of ψ ±L(R) , the kinetic terms for Ψ + and Ψ − are rewritten as where Ψ + , Ψ − , Γ z and Γ z are defined by Here, z ≡ x 5 +i x 6 and z ≡ x 5 −i x 6 . The Kaluza-Klein masses are generated from the terms including D z and D z upon compactification.
The Z m elements are the eigenvalues of the representation matrices T Ψ ± [U a , η a± ] for the Z m trans- where U a represent the representation matrices for the fundamental representation, η a± are the intrinsic Z m elements and the subscript L and R are omitted on η a± . Let the intrinsic Z m elements of ψ ±L(R) be η a±L(R) . Then, the intrinsic Z m elements of ψ † ±L(R) are η a±L(R) (complex conjugations of η a±L(R) ). From the Z m invariance of the kinetic term (B.14) and (B.15) and the Z m transformation property of the covariant derivative D z → ρD z and D z → ρD z under z → ρz and z → ρ z (ρ = e 2πi /m , ρ = e −2πi /m ), the following relations are derived: (−1) l 1 +l 2P mk± p 3 C l 3 · · · p n C l n , (C.13) n q [N,k] = ± (l 1 ,l 2 )=(1,1),(2,1) (−1) l 1 +l 2P mk± p 3 C l 3 · · · p n C l n , (C.14) mk p 3 C l 3 · · · p n C l n , (C. 15) whereP mk± andP (ν) mk± are defined bỹ P mk± ≡P mk±R −P mk±L ,P (ν) mk± ≡P mk±R +P mk±L , (C. 16) respectively. TheP mk±R(L) are projection operators to pick out zero modes of ψ ±R(L) in [N , k], and they are listed in Table 9. In Table 9, each operator is defined by  is an projection operator to pick out modes withP (k) 0± = ω n 0 andP (k) 1± = ω n 1 in Ψ ± . By the insertion of (−1) l 1 +l 2 , we obtain ♯( 3 C l 1 , 2 C l 2 ) R(L) for l 1 + l 2 = even integer and −♯( 3 C l 1 , 2 C l 2 ) R(L) for l 1 + l 2 = odd integer.
TheP (k) a± of 3 C l 1 , 2 C l 2 , · · · , p n C l n are given bỹ P (k) 0± = (−1) l 1 +l 2 +l 3 +l 4 −kη(k) 0± ,P (k) 1± = (−1) l 1 +l 2 +l 5 +l 6 −kη(k) 1± ,P (k) 2± = (−1) l 1 +l 3 +l 5 +l 7 −kη(k) In the last equality in the above second relation, we use the fact that the projection operators take a real number 1 or 0. From (C.27), we find that the flavor numbers derived from the projection by (−1) l 1 +l 2P mk± are equal to those from that by (−1) l 1 +l 2 P mk± = (−1) l 1 +l 2 P mk± . In this way, we have a feature that each flavor number from a fermion in [N , k] with intrinsic Z m elements η (k) a± is equal to that from a fermion in [ The equivalence based on the relations (C.31) and (C.32) is illustrated with the particle contents listed in Table 7 and 8.