Towards unity of families: anti-SU(7) from Z(12-I) orbifold compactification

The problem of families,"Why are there three families of fermions?", is a long awaited question to be answered within a reasonable framework. We propose anti-SU($N$) groups for the unification of families in grand unification (GUT) groups, where the separation of color and weak gauge groups in the GUT is achieved by antisymmetric tensor Brout-Englert-Higgs boson instead of an adjoint representation. Theories of anti-SU($N$)'s are proposed for the unification of families. The minimal model is found as \antiSD \, GUT with the fermion representation [3]+2[2]+8[\bar-1]. We present an example in a Z(12-I) orbifold compactification, where the missing partner mechanism is also realized.


I. INTRODUCTION
The family problem, repeating fifteen chiral fields three times in the standard model (SM), has been known for almost four decades without an accepted theory so far. One family consists of fifteen chiral fields. In the Georgi-Glashow(GG) grand unification(GUT) model, the fifteen chiral fields in the first family are grouped into 10 (u c , u, d, e + ) and 5 (d c , ν e , e) of SU(5) GG [1]. The family problem posited in GUTs is how 10 + 5 repeats exactly three times in an extended GUT. The gauge coupling unification [2] is the underlying principle for a GUT. A gauge model based on a simple group is a GUT. A gauge model based on a semi-simple group rendering one coupling constant (by some discrete symmetry) is a GUT also [3].
A chiral model with fifteen fields in a GUT is achieved by assigning fifteen in some complex representation of a GUT gauge group. Toward a solution of the family problem, Georgi formulated a principle on the unification of GUTfamilies with gauge groups containing SU(5) as a subgroup [4]. Absence of gauge anomaly is required. Then, any gauge group except SU(N ) can be used for a GUT if they allow complex representations. In this try of anomaly-free groups, SO(2N ) with N = 2n + 1 allows complex representaions which are spinors of SO(2N ). The simplest case is SO (10) where the fifteen chiral fields plus a singlet neutrino are assigned in the spinor 16 [5]. It is one family model. One may try SO(12) which however is not considered to be complex because the spinor 32 branches to 16 and 16 of SO (10). So, the next step is a two family model where the spinor 64 of SO (14) is used [6,7] where however non-standard charges must appear. 1 If we exclude the possibility beyond the SM charges, 64 of SO (14) is vectorlike and no chiral family is obtained. A scheme toward unification of GUT-families (UGUTF) came to a dead end within SO(2N ) groups.
To open a gate from the dead end alley toward the meadow, Georgi proposed UGUTFs in SU(N ) groups [4], where the condition on the anomaly-freedom plays a central role. There have been some attempts along this line [10,11]. After the string revolution with heterotic strings [12], the low energy gauge groups and spectra are computed through the compactification process [13][14][15], and UGUTFs did not attain much interest because three SM families could have been obtained through the compactification process [16]. In this string scenario, the compactification schemes (such as orbifolds) basically choose what is the number of families. The drawback is that there are too many parameters to predict the fermion mass spectra.
In this paper, we attempt to realize Georgi's UGUTFs in string compactification. If possible, we will try to introduce some structure among three families so that the difference of the third family from the the first two is understood. Among heterotic strings, usually the E 8 × E ′ 8 hetrotic string has been used because it contains spinors in the adjoint representation 248 of E 8 . Then, the E 8 × E ′ 8 heterotic string has been favored subconciously because of the embedding chain of the SO(10) spinor 16 in SO(10)→E 6 →E 7 →E 8 . But, the SO(32) hetrotic string is also useful for phenomenology as we will briefly argue in this paper. The most severe obstacle in obtaining a realistic SU(5) GUT from heterotic string has been the difficulty, at the level 1 construction, of obtaining the adjoint representation 24 of SU (5) which is needed to break SU (5) down to the SM gauge group.
The first example without an adjoint representation for the Brout-Englert-Higgs(BEH) boson can be traced back to anti-SU (7) where antisymmetric tensor fields for BEH bosons, denoted as two and three index anti-symmetric tensor fields, Φ [αβ] , Φ [αβ] , Φ [αβγ] , and Φ [αβγ] , are used to reduce the rank of the SU(7)×U(1) GUT gauge group and separate the color and the weak parts [6,7]. 2 The sixteen chiral fields of the first family are grouped into 10 1 (d c , u, d, N 0 1 ) and 5 −3 (d c , ν e , e) and e + 5 in SU (5) flip . The structure is included in SU (7) anti2 . If it is applied to SO(2N ) gauge groups, we can call it anti-SU(N ) where anti means that separation of color SU(3) c from weak SU(2) W is by the anti-symmetric tensor fields instead of the adjoint representation. For N = 5, it is now known as the flipped SU(5) [18]. It is obvious that the essential feature is included in the word 'anti-SU(N )'. Anti-SU(5) GUTs were obtained in string compactification [19,20]. But, SU(5) GUTs are not UGUTFs.
In addition to the restriction on the number of families for n f = 3, the R-parity is used for proton longevity and weakly interacting massive particle possibility for cold dark matter candidate. The R-parity can be a discrete subgroup of a U(1) gauge group which has been discussed in Ref. [21,22]. Another issue in supersymmetric GUTs is the problem of doublet-triplet splitting in the BEH multiplets containing H u and H d . We will realize the doublet-triplet splitting mechanism in SU(7) anti2 anticipated in Ref. [8].
In Sec. II, we recapitulate the old UGUTF scenario, and continue in Sec. III to present a rationale that anti-SU(N )s are theories for UGUTFs. In Sec. IV, we summarize an SU(7) realization of UGUTF in the Z 12−I orbifold compactification. Here, we present some details on how the massless spectra are obtained in Z 12−I so that line by line can be followed up in other orbifold constructions. We also point out how the missing-partner mechanism is realized in SU(7) anti2 . Sec. V is a conclusion. In Appendix, we present the spectra not included in Sec. IV.

II. FAMILIES UNIFIED IN GRAND UNIFICATION
Let the fundamental representations (or anti-symmetric representations) of SU(N ) are bounded by square brackets. Representations [1] and [2] have the following matrix forms, where [1] means the dimension N, [2] means the dimension N(N−1)

2!
with two antisymmetric indices, [3] means the with three antisymmetric indices, etc. We do not consider the symmetric indices such as because they will contain color sextets. [1] contains one 5, and [2] contains one 10 of SU (5). The number of the SU(5) GG families, i.e. that of 10 plus 5, is counted by the number of 10 minus the number of 10.
The anomaly-freedom condition chooses the matching number of 5's. The numbers n 1 and n 2 for the vectorlike pairs n 1 (5 ⊕ 5) + n 2 (10 ⊕ 10) are not constrained by the anomaly freedom. Thus, we count the number of families just by the net number of two index fermion representations in the SU(5) GG subgroup. For several SU(N )'s, we have the following family number n f by counting the number of 10's, . Negative n f gives the chiral family number |n f |, which is also allowed. The anomaly units in SU(N ) are where α 1 , α 2 , · · · , and α m are completely antisymmetrized. Theory of families in GUTs does not allow repetition of the representation and gauge anomalies [4]. I. I. Rabi's terse question, "Who ordered that?" quipped after the 1936 discovery of muon, eloquently states the essence of the family problem. Let the matter representation be where the elements of the integer set {c i ,c j } do not have a common divisor. For three standard families of Georgi-Glashow GUT SU(5) GG , Georgi required that all integers are 1 and found an SU(11) unification of three SU(5) GG families [4], which was the first exampe of UGUTF. Allowing possiblities of c i > 1 andc j > 1 but requiring no common divisor for all of them is a non-repetition of a set, which is a reasonable requirement for a solution of the family problem. Then, there are many possibilities [23]. If one requires renormalizable Yukawa couplings, in addition, with one irreducible type of representation containing the BEH doublet, an SU(9) model seems minimal [11]. However, at the GUT scale it is possible to have some Planck mass suppressed nonrenormalizable Yukawa couplings. Thus, constraining models by renormalizable couplings is not warranted at this stage. The minimal chiral choice plus vectorlike representations for UGUTF by the rules of (2) is SU (7) : where n i = 0 for i = 1, 2 are needed for the BEH mechanism. For Eq. (5) [6]. It was the first example of anti-SU(N ) where antisymmetric tensor BEH bosons, Φ [αβ] , Φ [αβ] , Φ [αβγ] , and Φ [αβγ] are used to reduce the rank of the GUT gauge group. It allows two quark families and three lepton families, but it had a drawback due to the appearance of non-standard quarks and a doubly charged lepton. Note here that the rank 2n + 1 of SO(4n + 2) is reduced down to the rank 2n via the vacuum expectation values (VEVs) of Φ [αβ] and Φ [αβ] representations of the SU(2n + 1)×U(1) subgroup of SO(4n + 2).
Following the philosophy of Ref. [6], we define SU(N )×U(1) subgroups of SO(2N ) by SU(N ) anti2 , SU(N ) anti3 , SU(N ) anti4 , etc., which means that the spontaneous symmetry breaking of SO(2N ) takes the SU(N )×U(1) 2,3,4 subgroup route if Φ [AB] , Φ [ABC] , Φ [ABCD] are used for the BEH bosons, respectively. The symmetry breaking of Ref. [6] was the intersection of SU(7) anti2 and SU(7) anti3 . In this language, SU(5) flip is identical to SU(5) anti2 , since there is no higher order anti-symmetric tensor field beyond Φ [AB] for N = 5. 3 In this vain, we can define SU(N ) GG as the symmetry breaking route of the BEH field Φ A B of SU(N )×U(1) subgroup of SO(2N ). It looks like that the SM families are only possible from the spinor of SO (10), which has been a reason for a GUT extension chain, containing the spinor 16 of SO(10), or Of course, SO(2N ) contains spinors, and spinors for N > 5 can produce 16's of SO (10). In string theory, gauge groups E 8 × E ′ 8 and SO (32) are allowed only with their adjoint representations. This observation did not favor the SO(32) hetrotic string theory for the realistic purpose of obtaining 16 of SO(10), because the adjoint representation of SO(32) cannot produce any spinor representation in its subgroups.
Because no non-SM charges have been found up to the TeV scale, the SU(5) anti2 rather than [3] ⊕ [2] ⊕ [1] in SU(7) seems to contain some truth in it. But, for a UGUTF one should consider groups larger than SO (10). Here, we go beyond the SU(5) anti2 . Then, we can use anti-SU(N )'s for a theory of UGUTF. In particular, GUT symmetry breaking is easier in supersymmetric GUTs if we assign VEVs to Φ [45] (= α 45 of Eq. (1)) and Φ [45] , i.e. in SU(N ) anti2 . In addition, the flipped SU(5) GUTs have been obtained from string compactification [19,20]. These string-derived antior flipped-SU(5)'s are assuming the symmetry breaking chain through the SO(10) route assuming the appearance of the spinor 16, but the SO(10)-spinor chain is not possible in Eq. (2). Therefore, for UGUTFs we follow the extension chain In these chains, we do not follow the groups allowing only spinor representations. So, anti-SU(N ) by anti-symmetric tensor fields [6] is the key in string compactification. The UGUTF of Eq. (6) may be obtained in this way. The SO(32) heterotic string is useful for phenomenology and UGUTF. Symmetry breaking of SO(32) through SU(N ) anti2 is possible because 496 of SO(32) contains the following SU (16) representations , respectively, of its SU (16) subgroup. In the orbifold compactification of SO(32), it will be easy to realize the representation Φ [AB] and Φ [AB] even at level 1, and the key UGUTF breaking, i.e. the separation of color SU(3) c and weak SU(2) W , to the SM is possible by Φ [45] and Φ [45] of Eq.
(1). Because we allow only the SM fields, the fundamental representations Φ [A] and Φ [A] are also used to reduce the rank further by the VEVs at the locations f 6 , · · · , f N of Eq. (1).

III. TOWARD A THEORY OF FAMILY UNIFLICATION IN ORBIFOLD COMPACTIFICATION
As noted in the previous section, a minimal UGUTF needs a GUT allowing SU(7) as a subgroup. Compactification of heterotic string frequently needs an anti-SU(N ) or SU(N ) anti2 . The E 8 × E ′ 8 heterotic string allows the rank 8 SU(9) which is considered to be a subgroup of E 8 . We will try to realize Eq. (6) from the string compactification. In addition, no degeneracy between families is left below the compactification scale. This means that we will introduce all the needed Wilson lines. As we will see, it is more difficult to obtain multi-index tensor fields in the twisted sectors. In particular, the three-index tensor field Ψ [ABC] cannot be obtained in the twisted sectors. So, Ψ ABC] , if they appear, is required to come from U . In Table I, we list the number of fixed points for the non-prime orbifolds. For example, the cental number in Z 6−II and Z 12−I are 2 and 4, respectively, which mean that they have Z 6/2 and Z 12/4 symmetries, i.e. both have the Z 3 symmetry in the second torus. They correspond to the gauge group phase in the untwisted sector matter P · V = 2 6 and 4 12 , respectively, and the untwisted sector multiplicity is 3. This is the easiest way to obtain three families from the U sector, i.e. 3 (10 ⊕ 5). In this case, the three families are not distinguished, and there must be an S 3 discrete symmetry. It may be broken by Higgsing at a GUT scale. Here, we do not follow this line of argument because there are too many possibilities.
As an example, we show the U sector multiplicity of Ψ [ABC] in Z 12−I in Table II. If it is the representation of an SU(8) GUT, then there are two 10's (due to Eq. (2)) from U . If it is the representation of an SU(7) GUT, then there is a possibility of one 10 from U . Thus, we choose SU(7) anti2 . The remaining 10's of SU(5), i.e. 21's of SU(7) anti2 , come from the twised sectors. However, it is not so easy to obtain two Ψ [AB] 's of SU(7) anti2 from T .

Matter Ψ [AB]
must not appear in the untwisted sector.
3. Matter Ψ [AB] must be present in a twisted sector with the chirality that of Ψ [ABC] .
Matter in the untwisted sector U i occurs with P · V = Ni N . For example, N i = 5, 4, 1 for Z 12−I is shown in Table I. In addition, one matter Ψ [ABC] is allowed only in U 1 and U 3 in Z 12−I . We require that it has the spinor form, by choosing an appropriate V , where the underline means permutations. Here, the torus or untwisted sector are called U 3 and U 1 , respectively. Their CTP conjugates also appear in U as (− − − + + + +; −)(0 8 ) ′ . Representations (12) satisfies Condition 1. To satisfy Condition 2, Wilson lines can be used if needed. Condition 3 requires a full construction method. Ψ [AB] located in a twisted sector is the key part in this paper. In the Z N orbifold, multiplicities in the k-th twisted sector P k need to be calculated, which is given by 4 where Θ 0 will be defined later. The chirality is given by the first entry of s, denoted as L-or R-movers, with the even number of total '−'s, Mult. In Table III, multiplicities are presented for Z 6 , Z 8 and Z 12 orbifolds. The multiplicities in T N/4 are colored red. Matter Ψ [AB] can appear, if they do, only at T N/4 . From the red numbers in Table III, we note that Z 8−II and Z 12−I are the only allowed possibilities of obtaining multiplicity 2 at T N/4 , due to the large vacuum energy 2c N/4 = 13 8 , as presented below in Eqs. (24) and (25). In other models, it is impossible to house Ψ [AB] . Out of Z 8−II and Z 12−I , we choose the simpler case Z 12−I because we need to specify only one Wilson line. Note that in Z 8−II one has to specify two Wilson lines to specify the model completely without degeneracy. Then, at T N/4 , we must obtain two Ψ [AB] 's. This condition is very restrictive and may rule out the Z 12−I possibility. Luckily, we find a model, satisfying this condition.
In the twisted sector, the masslessness conditions are satisfied for the phases contributed by the left-and rightmovers [20], 4χ (θ k , θ l ) are presented in Ref. [24].
where j denotes the coordinate of the 6-dimensional compactified space running over {1,1}, {2,2}, {3,3}, andφ j = φ j s · sign(φ j ) with sign(φj) = −sign(φ j ). In calculating the multiplicities in Eq, (13), we use the phase Θ 0 with ∆ k , where V a is V distinguished by Wilson lines, and We choose 0 <φ j ≤ 1 mod integer and oscillator contributions due to (N L − N R ) to the phase can be positive or negative, with N L,R ≥ 0. But each contribution to the vacuum energy N j L,Rφ j is nonnegative. One oscillation contributes one number in φ s . With the oscillator, the vacuum energy is shifted to In Eq. (20), instead of p vec a four entry quantity p spin of ± 1 2 's with even number of −'s is possible, but we do not find any example with p spin . The vacuum energy contributions in the twisted sectors in Z 6,8,12 are given by 5 5 Typos of Appendix D of Ref. [24] are corrected here.
Note that 2c k − 2c k = 1 which is the required condition for N = 1 supersymmetry in the 4 dimensional spectra. After Ψ [ABC] and Ψ [AB] are obtained, the number of families is fixed. Namely, the number of Ψ [A] of SU (7) (7) anti2 can be achieved in many ways.
Not to allow any left-over degeneracy, one must assign all possible Wilson lines. For Z 3 and Z 4 , one must specify three Wilson lines, a 1 = a 2 , a 3 = a 4 , a 5 = a 6 . So, they have the most complicated Wilson line structures. For Z 6−II , Z 8−I and Z 8−II , one must specify two Wilson lines. Specifying one Wilson line is enough in Z 12−I , i.e. only a 3 = a 4 in the 2nd torus.
The Wilson loop integral is basically the Bohm-Aharanov effect in the internal space of two-torus, If the B-field (i.e. ∇ × V ) at the orbifold singularity is present, the phase through ∆ 0 contributes in the multiplicity. For Z 12−I , this is the case in T 1,2,4,5 . The complication arises at the points with 3a 3 = 0 mod. integer, i.e. at T 3,6 [20], 6 where the Bohm-Aharanov phase has to be taken into account explicitly. If there is no B-field at the orbifold singularity, there is no Bohm-Aharanov phase, but then there for the (internal space) gauge symmetry we must require explicitly This case applies to T 0,+,− 3 , T 6 , and of course at U also. We distinguish T 3 by 0, + and − because the phase ∆ 0 k of Eq. (19) contains an extra k 2 factor. Namely, Eq. (28) is applied only at U, T 3 and T 6 . We will comment more on this in Subsecs. T 3 and T 6 .

IV. SU(7)anti2 SPECTRA
We calculate the SU (7) non-singlet spectra of SU(7) anti2 in the Z 12−I orbifold. We choose the following model, Here, a 3 (= a 4 ) is chosen to allow and/or forbid some spectra, and is composed of fractional numbers with the integer multiples of We anticipated to achieve the key spectra needed for SU (7) anti2 where the sectors they appear are marked as subscripts. At this point, we do not fix how many vectorlike pairs appear in T 3 . The chiral representations, the candidates of fermion families, are represented by Ψ, and vectorlike reresentations, candidates for the BEH bosons, are represented by Φ. The orbifold conditions, toward a low energy 4 dimensional (4D) effective theory, remove some weights of the original ten dimensional E 8 × E ′ 8 weights. The remaining ones constitute the gauge multiplets and matter fields in the untwisted sector in the low energy 4D theory. Therefore, the weights in the U sector must satisfy P 2 = 2 as in the original E 8 × E ′ 8 weights. Orbifold conditions produce singularities. They are typically represented in three two-dimensional tori. A loop of string can be twisted around these singularities and define twisted sectors T k (k = 1, 2, · · · , 12). Twisting can introduce additional phases. Since T 12−k provides the anti-particles of T k , we consider only T k for k = 1, 2, · · · , 6. T 6 contains both particles and anti-particles. T 6 , not affected by Wilson lines, is like an untwisted sector. It contains the antiparticles also as in U . Since the Wilson lines can affect only in noncontractible loops as the hidden sector Aharanov-Bohm effect, the Wilson lines can affect only around the singularities in the twisted sectors, but have no effects in the untwisted sector.
In this section we consider two twisted sectors, T 3 and T 6 , explicitly. The other twisted sectors, T 1 , T 2 , T 4 , and T 5 will be listed in Appendix.

A. Untwisted sector U
In U , we find the following nonvanishing roots of SU (7) In addition, there exists U(1) 2 ×U(1) ′5 symmetry. The non-singlet SU(7) matter fields are E 8 matter multiplet : P · V = 5 12 , mod. integer SU (7) : Ψ [ABC] contains one 10 of SU(5) GG family, which belongs to the first family, and is R-handed as shown in Table II.
It is simple to find the E ′ 8 hidden sector matter in U ,

B. Twisted sectors T
In the twisted sectors, we list only SU (7) or SU(4) ′ non-singlets. Wilson lines distinguish three fixed points in the second torus, and the shift vectors we consider at T k are split into three cases where V + and V − are given in Eq. (30). Because 3a 3 = 0 mod. integer and due to Eq. (18), T 6 sector is not distingushed by the Wilson lines but T 3,9 are distinguished.
We select only the even lattices shifted from the untwisted lattices. They form even numbers for the sum of entries of each elements of P .
To obtain non-trivial number of families, we need two index tensor fields, Ψ [AB] and/or Ψ [AB] . This possibility arises only in T 3 because at T 3 there appear fractional number with integer times 1 4 . In other twisted sectors, the entries are not multiples of 1 4 in which case we cannot fulfil the masslessness condition with 2c k given below. In the k-th twisted sector, the masslessness condition to raise the tachyonic vacuum energy to zero is where 2c k and 2c k are given in Eq. (25), and the brackets must be taken into account when oscillators contribute. When the conditions (36,37) are satisfied, we obtain the SUSY spectra for which the chirality and multiplicity are calculated from Θ 0 in the k-th twisted sector, where We choose 0 <φ j ≤ 1 mod integer and oscillator contributions due to (N L − N R ) can be in principle positive or negative. As an example, consider the T 3 sector for the N j R contribution. Here, 3φ s = ( 5 4 , 4 4 , 1 4 ), needing p = (−1, −1, 0). So, (p + 3φ s ) 2 = 1 8 , needing the N R contribution 4 8 = 2 · 3 12 to make up 2c 3 = 5 8 [20]. Thus, the R-handed oscillator contribution is 6 12 = 2 · 3 12 . Namely, δ 3 is +3 12 which is included in the tables. We will select only the even lattices shifted from the untwisted lattices. They form even numbers if the entries of each elements of P are added. Because we consider E 8 × E ′ 8 , the product of E 8 and E ′ 8 parts must be even. They need not be even separately. But, there is a distinction in even × even lattice and odd × odd lattice. In the former case, E 8 and E ′ 8 gauge quatum numbers do not change, but in the latter case we change the signs of the quantum numbers. In the table captions, we take into account this fact. In the multiplicity calculation in Θ 0 , there is a factor 1 2 between the lattice shifts by Wilson lines. Even though the Wilson lines cannot distinguish the fixed points, we consider V + and V − also as if Wilson lines distinguish fixed points.

Chiralitys
−s · φs −pvec · φs, P · V0 (3/2)φ 2 • Two index vector form for V 3 + : For a vector form of P , Massless states are shown in Table V, which are exactly the same as those of Table IV.
The chiral spectrum we obtained for the two index tensors in T 3 is These make up three chiral families together with Ψ [ABC] R from U . The number in (43) is the same as the one if we treat T 3 with multiplicity 2. This multiplicity is because 3V is a Z 4 twist which has two fixed points in a two-dimensional torus. Since there is no Wilson line, we could have treated only 3V with multiplicity 2 of Z 4 . The multiplicity 2 is accounted by T 0 3 and T + 3 . But, T − 3 produce additional vectorlike pairs, which must be fictitious. We may consider the spectra in Table VI are fictitious. In the remainder of the paper, we will not consider V 3 + and V 3 − . We consider only V 3 0 and take into account the multiplicity 2 of T 3 . Now let us proceed to consider one index tensors in T 3 . In the final result, we will multiply the overall multiplicity 2 as commented above.
• One index spinor form for V 3 0 : which make up 9 8 . The oscillator contributions of 2 3 12 are needed to satisfy the masslessness condition. Chiralities and multiplicities are tabulated in Table VII. • One index vector form for V 3 : vector forms do not give massless states because (P + 3V − ) · a 3 = 0.
Chiralitys −s · φs −pvec · φs, P · V0 (6/2)φ 2  • One index vector form for V 6 0 : vector type forms cannot satisfy the masslessness condition.   The remaining SU(7) non-singlet massless particles together with SU(4) ′ nonsinglets are presented in Appendix. The SU(7) and SU(4) ′ indices are represented by A and α ′ , respectively. Therefore, twisted sectors T 3 , T 6 , and T 9 may be guessed that they are not affected by Wilson lines. However, T 3 and T 9 are affected by Wilson lines because in the calculation of the phase ∆ 0 k there is an additional factor k 2 (viz. Eq. (19)). Indeed, the inclusion of this factor k 2 correctly produces a combination of an anomaly free set. Let us comment on the multiplicities in T 3 and T 6 . In T 3 , the multiplicity is 2 as mentioned in subsubsection T 3 . In T 6 , we note that it is a Z 2 shift which is in fact an untwisted sector. The multiplicity of Z 2 untwisted sector is 2, but it must include antiparticles also. Thus, the multiplicity of Z 2 untwisted sector is 1 [25]. It is taken into account in the twisted sector Z 6 .
The SU (7) and SU(4) ′ non-singlet massless states are summarized as R-handed fields in Table IX. The matter fields are denoted as Ψ and vectorlike representations are represented by Φ. Some of Φ fields develop VEVs. The Φ fields can be removed at the GUT scale if correct combinations of sectors and oscillators are satisfied. The chiral fields of Table IX are which do not have the SU(7) nonabelian anomaly. In the untwisted sector, there is no [1 ]. Thus, the family Ψ [ABC] from U has more suppressed Q em = 2 3 quark Yukawa coupling and Ψ [ABC] is interpreted to include the 1st family members. All Ψ [A]R 's appear in twisted sectors. Two chiral fields, (e) and one combination from (c), form a vectorlike pair and removed at a high energy scale. The field in (d) and the remaining 7 fields from (c) are the needed 8 fields for i.e. from T + 5 , is interpreted as u c because it can lead to the smallest Yukawa coupling among Q em = 2 3 quarks. The other 7 fields Ψ [A]R from T 3 have the same fate. Then, note that t c and c c are located in T 3 . Since both Ψ [AB] and t c (from Ψ [A] ) arise at T 3 , the cubic Yukawa coupling is possible from the BEH boson from T 6 . There are many possibilities for assigning H u and H d of the MSSM in T 3 and T 6 . We will choose a specific one in Subsec. IV D.
C. U(1) charges and anomalous U (1) We use the normalization that the index ℓ for fundamental representation N of SU(N ) is 1. Then, the indices of some representations are [26], where [2] where the redefined primed U(1) combinations give the identical sum for the SU (7) and SU(4) ′ anomalies. Note that Q ′ 6 itself is anomaly free. In terms of Q ′ 2 , · · · , Q ′ 5 , we can redefine anomaly free combinations. Six nonabeliananomaly-free U(1) combinations are denoted with tilde, , The remaining U(1) must carry anomaly, which can be represented as Parameters a, b and c are determined by how one breaks the SU(7) anti2 , which defines the electroweak hypercharges or the electromagnetic charges of the SM particles. Note that SU(5) GG subgroup of SO (10) (7) anti2 can be fixed only after SU(7) anti2 is broken down to the SM gauge group. The U(1) X of SU(7) anti2 is given by which is anomaly-free. The orthogonalities of Q a with Q X and the above two SU(7) anti2 directions determine three parameters of (50).

D. Yukawa couplings
For Yukawa couplings, we must satisfy all the symmetries of low energy effective fields and the selection rules in the orbifold compactification. For the fields from twisted sectors, the Yukawa coupling structure is simpler than those involving the untwisted sector fields. Consider for example a vectorlike set from T 6 in Table VIII. For the coupling, Φ [A] R · Φ [A]R , we must satisfy the selection rules for the right-mover and for the left-mover conditions. For the rightmover condition, 36 times p vec · φ s is 0 mod. integer. It is satisfied for the coupling Φ For the left-mover condition, 36 times P · V is 0 mod. integer. It is also satisfied for the coupling Φ As commented above, t, c, t c and c c quarks are located T 3 . On the other hand, u is located in U and u c is located in T + 5 . Order 1 Yukawa coupling of the form 21( is not removed at the GUT scale. This requires a hierarchy of scales, where M 3 is a vacuum expectation value of a singlet 1 in T 3 , At the SU(7) anti2 level, still we have eight light 7's. Thus, the BEH fields are located at T 6 . Depending on θ, the BEH fields contain small components from T 3 , viz. Eq. (55). We interpret this angle as the ratio m c /m t = tan θ.
The t-quark Yukawa coupling in SU(5) anti2 is The BEH fields giving mass to the b-quark are located in T 3 . There are 40 Φ [A] fields in (f ) of Table IX. Because of the mass democracy, there can remain some light fields. Most of them will be removed when SU(7) anti2 is broken, but we need one Φ [A] for the Q em = − 1 3 quark masses. For the b-quark mass, we need the coupling Thus, the b-quark mass is expected to be much smaller than the t-quark mass, O( T 21 3,BEH T 7 3,BEH /M s T 7 6,BEH ), where T 21 3,BEH is the SU(5) splitting VEV Φ [67] . Thus, we expect m b /m t ∼ Φ [67] Ms tan β . Even if tan β = O(1), we can fit m b /m t to the observed value by appropriately tuning Φ [67] . A similar suppression occurs for the second family members.
For the 1st family members, the story is different. This is because d c appears in 35 of SU(7) anti2 , appearing in U . The d-quark mass may arise from Let us check whether this coupling is present. p spin · φ s = 10 12 for 35 U1 35 U1 . From Table VII, we note p vec · φ s = −9 12 for 7 T,BEH . We need the remaining singlet combinations to provide −1 12 . Since we do not list singlets here, it cannot be shown at this stage, but there are numerous singlets and we assume that it is possible. For the left-movewr conditions which are the gauge invariance conditions, the above coupling satisfies the condition. For the u-quark mass, we must consider a higher dimensional operator than Eq. (59), It is because u c appears in T + 5 , requiring another field carrying another Wilson shift − to remove the Wilsone shift +. Thus, there exists a possibility that m u < m d . It is a new mechanism for the inverted 1st family quark mass structure.

E. Missing partner mechanism
At the SU(7) anti2 level, we assume that one pair Φ [A] = 7 (giving mass to the t quark) and Φ [A] = 7 (giving mass to the b quark) are survibing down to low energy. The missing partner mechanism is discussed in this setup.
In a sense, the absence of 7 BEH · 7 BEH is not guaranteed at field theory level. In the MSSM, it is related to the µ problem, "Why there does not exists H u H d at the GUT scale" [27]. Some interesting solutions with hidden-sector quarks exist [28,29]. These solutions are based on the Peccei-Quinn (PQ) symmetry with the very light axion [30]. In the effective SUSY framework language, the superpotential W should not allow µH u H d by assigning a nonvanishing PQ quantum number to the combination H u H d . But, the global PQ symmetry is spoiled by gravity [31]. We may resort to some discrete subgroup, e.g. matter parity [32], of a U(1) gauge symmetry [33]. Suppose assigning the mother gauge charges of H u and H d as Q(H u ) = Q(H d ) = 1 such that the matter parity forbids H u H d at the GUT scale. But we must allow the t-quark mass at the cubic order. It means, t and t c carry the mother gauge charge, Q(t) = Q(t c ) = − 1 2 for example. In string compactification, we do not worry the gravity spoil of global symmetries. Just string selection rules are enough to consider the coupling. It has been noted that some string compactifications do not lead to quadratic term in W as in Z 3 [29], but in non-prime orbifolds the absence of 7 · 7 must be studied case by case. In our example discussed above, the coupling 7 BEH · 7 BEH is not allowed because 7 BEH is located in T 3 , and 7 BEH is located in T 6 . But, GUTs need the doublet-triplet splitting that in the same GUT scale BEH multiplet the colored fields are superheavy while H u and H d remain light. In the absence of the coupling 7 BEH · 7 BEH , the missing partner mechanism of SU(7) anti2 is realized. Consider the coupling, , and/or where Φ [1] is Q em = − 1 3 colored boson whose partner is Φ [23] . The color-weak separating VEV Φ [45] ≈ M GUT ∼ M s is the key making the colored scalar heavy. In the same multiplet Φ [A] , the BEH doublet H d is present at Φ [4] and Φ [5] . But, the indices 4 and 5 are already used for the GUT scale VEV, hence H d does not find a partner in Φ [AB] . This is the missing partner mechanism we realize in SU(7) anti2 .

V. CONCLUSION
In this paper, we proposed unification of families in string compactification. Here, we suggested anti-SU(N ) scheme [6] where the adjoint representation of SU(N ) is not needed for breaking the GUT group down to SU(3) c ×U(1) em . It is pointed out that the anti-SU(N ) scheme has a merit in string compactification and can even save the SO(32) heterotic string theory for many phenomenological purposes. The minimal model for UGUTF is SU (7) GUT with the representation [ 3 ] + 2 [ 2 ] + 8 [1 ]. We show it explicitly that this representation is realized in the Z 12−I orbifold compactification of E 8 × E ′ 8 heterotic string. The large top quark mass is possible in SU(7) anti2 , where t c is in 7 = [1 ]. In the example discussed, 7's in T 3 contain t c and c c . The cubic coupling Ψ

[AB]
R,T3 Ψ [A]R,T3 Φ [B]R,T6,BEH gives a dimension-3 superpotential for the t mass. This cubic coupling is the only possible dimension 3 superpotential in our model and hence only m t is expected to be of order the electroweak scale. Other fermion masses are much smaller than m t . We also presented an argument why there is an inverted mass ratio in the u-quark family. It is because u c is located in T + 5 which requires another Wilson line shifted singlet field. Finally, we presented the missing partner mechanism in string compactification. The key assumption, the absence of the coupling 7 BEH · 7 BEH in the superpotential, is achieved here by locating 7 BEH and 7 BEH separately in T 3 and T 6 . Then, it is shown that the missing partner mechanism works for H u and H d in 7 BEH and 7 BEH . The colored particles in 7 BEH and 7 BEH find their partners in 21 BEH and 21 BEH and obtain superheavy masses.
Here, we neglected the details of singlet vacuum expectation values, toward removing vectorlike representations, though the singlet VEVs have been widely used in other string compactification papers toward the MSSM [35]. In this sense, the SU(7) anti2 presented in this paper may be an aethetic choice toward a desirable UGUTF. Other physics implications such as the quark and lepton mass textures, dark matter, and very light axions, including SU(7) singlet representations, will be presented elsewhere. • One index spinor form for V 5 + :