Dual models of gauge unification in various dimensions

We construct a compactiﬁcation of the heterotic string on an orbifold T 6 / Z 6 leading to the standard model spectrum plus vector-like matter. The standard model gauge group is obtained as an intersection of three SO ( 10 ) subgroups of E 8 . Three families of SO ( 10 ) 16 -plets are localized at three equivalent ﬁxed points. Gauge coupling uniﬁcation favours existence of an intermediate GUT which can have any dimension between ﬁve and ten. Various GUT gauge groups occur. For example, in six dimensions one can have E 6 × SU ( 3 ) , SU ( 4 ) × SU ( 4 ) × U ( 1 ) 2 or SO ( 8 ) × SO ( 8 ) , depending on which of the compact dimensions are large. The different higher-dimensional GUTs are ‘dual’ to each other. They represent different points in moduli space, with the same massless spectrum and ultraviolet completion. 

handed quarks and leptons can be grouped in three SU(5) multiplets [1], 10 = (q L , u c R , e c R ), 5 = (d c R , L ) and 1 = ν c R . Here we have added right-handed neutrinos which are suggested by the evidence for neutrino masses. All quarks and leptons of one generation can be unified in a single multiplet of the GUT group SO(10) [2], (1) 16 = 10 + 5 + 1.
It is a remarkable property of the standard model that the matter fields form complete SO(10) multiplets whereas the gauge and Higgs fields are 'split multiplets'. They have to be combined with other split multiplets, not contained in the standard model, in order to obtain a complete unified theory. It is also well known [5] that exceptional groups play an exceptional role in grand unification, and the embedding (2) SO(10) ⊂ E 6 ⊂ E 8 appears, in particular, in compactifications of the heterotic string [6] on Calabi-Yau manifolds [7].
As we shall see, complete SO(10) matter multiplets together with split gauge and Higgs multiplets arise naturally in orbifold compactifications of higher-dimensional unified theories. Orbifold compactifications have first been considered in string theory [8,9] and subsequently in effective higher-dimensional field theories [10,11]. They provide a simple and elegant way to break GUT symmetries, while avoiding the notorious doublet-triplet splitting problem. More recently, it has been shown how orbifold GUTs can occur in orbifold string compactifications [12][13][14].
In the following we shall first search for a scheme of Z N twists which allows to break E 8 , a common ingredient of string models, to the standard model group. A Z N twist is an element of the gauge group G, with (3) P = exp (−2πiV N · H ), P N = 1.
Here the generators H i form the (Abelian) Cartan subalgebra of G, and V N is a real vector. The twist P acts on the Cartan and step generators E α as follows: where α is a root associated with E α . Clearly, P breaks G to a subgroup containing all step generators which commute with P , i.e., [P , E α ] = 0.
The symmetry breaking is conveniently expressed in terms of the Dynkin diagrams. This technique has been employed to classify possible symmetry breaking patterns in string models [15][16][17] and, more recently, in orbifold GUTs [18,19]. Starting with the extended Dynkin diagram which contains the most negative root in addition to the simple roots, regular subgroups of a given group are obtained by crossing out some of the roots of the Dynkin diagram. In particular, the action of the Z N orbifold twist essentially amounts to crossing out a root with the (Coxeter) label N , or more generally, roots whose labels sum up to N .
As an example, consider the breaking of SO(10), displayed in Fig. 1. For each simple root the Coxeter label is listed, which determines the order of the twist required for the corresponding symmetry breaking. Crossing out one of the nodes with label 2 breaks SO (10) to the semi-simple subgroup G PS , while crossing out one of the roots with label 1 together with the most negative root θ breaks SO(10) to G GG . The intersection of the two groups gives the standard model with an additional U(1) factor [11], where '∼' means 'modulo U(1) factors'. Under the Z 2 twisting, the group generators divide into those with positive and negative parities P with respect to the twist. Combining the two parities P GG and P PS , one can construct the third Z 2 parity P GG · P PS = P fl which breaks SO(10) to the flipped SU(5), The standard model group G SM can also be obtained as an intersection of the two SU(5) embeddings, G GG and G fl , As another example, consider now E 6 breaking to the standard model group. From the extended Dynkin diagram Fig. 2 it is clear, in analogy with the SO(10) breaking, that three Z 2 twists, can break E 6 to the standard model up to U(1) factors, As in the SO(10) example, one can check that the same breaking can be obtained as an intersection of three different SO(10) embeddings in E 6 which correspond to the twists (10) E 6 such that Let us remark that it is not possible to distinguish the three SO(10) embeddings in E 6 (as well as G GG and G fl embeddings in SO(10)) at the level of Dynkin diagrams. The corresponding subalgebras are related by Weyl reflections within the embedding group. To distinguish them, an explicit analysis of the shift vectors V N is required.
Our final goal is to break E 8 to the standard model gauge group. This can be achieved by combining the above three Z 2 twists with a Z 3 twist which breaks E 8 to E 6 × SU(3) (cf. Fig. 3). The Z 2 twists can then also break the SU(3) factor to SU(2) × U(1). In this way one obtains three Z 6 twists which break E 8 to subgroups containing SO(10), such that the intersection is the standard model group up to U(1) factors, In an orthonormal basis of E 8 roots, three Z 6 shift vectors which realize the described symmetry breaking read explicitly: , 0, 0, 0, 0, 0 , Note that the differences between the Z 6 shift vectors are Z 2 shift vectors, which will play the role of Wilson lines in the next section.
To summarize, in this section we have presented a group-theoretical analysis of E 8 breaking to the standard model with intermediate E 6 and SO(10) GUTs, suggested by the structure of matter multiplets.

Orbifold compactification
Let us now construct an orbifold compactification of the heterotic string, which realizes the symmetry breaking described above. As is clear from the above discussion, we will need a Z 6 or a higher-order orbifold and choose the former for simplicity.
To obtain a four-dimensional theory, 6 dimensions of the 10D heterotic string are compactified on an orbifold. In our case, this is a Z 6 orbifold obtained by modding a 6D torus together with the 16D gauge torus by a Z 6 twist, On the three complex torus coordinates z i , i = 1, 2, 3, the Z 6 twist acts as Here 6v 6 has integer components. The compact string coordinates are described by the complex variables Z i = X 2i−1 +iX 2i , i = 1, . . . , 3. The Z 6 action on the string coordinates reads, up to lattice translations (cf. [16]), where 6V I is an E 8 × E 8 lattice vector. The torus T 6 is spanned by basis vectors e κ , κ = 1, . . . , 6. In general, a torus allows for the presence of Wilson lines, i.e., a translation by a lattice vector n κ e i κ can be accompanied by a shift of the internal string coordinates, Here the discrete Wilson lines W κ are restricted by symmetry and by modular invariance.
The basis vectors e κ are taken to be simple roots of a Lie algebra, whose choice is dictated by the required symmetry of the lattice. In our case the lattice must have a Z 6 symmetry and allow for the existence of 3 independent V 6 shift vectors (14) (or two Wilson lines of order 2). This leaves two possibilities for the Lie lattice [16] We shall base our analysis on the first lattice, which has recently been studied in detail by Kobayashi, Raby and Zhang [12]. These authors have obtained models with the Pati-Salam gauge group in four dimensions, which then has to be broken to the standard model by the Higgs mechanism. The model described in the following differs from those in the choice of Z 6 twists and the pattern of symmetry breaking.
For the G 2 × SU(3) × SO(4) lattice, the action of the Z 6 twist is given by Eq. (18) with z 1 , z 2 and z 3 are the coordinates of the G 2 , SU(3) and SO(4) T 2 -tori, respectively. The Z 6 twist v 6 has two subtwists, An interesting feature of this orbifold is the occurrence of invariant planes. Clearly, the Z 3 twist leaves the SO(4)-plane invariant whereas the Z 2 twist leaves the SU(3)-plane invariant. The corresponding fixed points and invariant planes are shown in Fig. 4. Our construction requires two Wilson lines in the SO(4) plane, W 2 and W 2 , such that there are 3 independent gauge shift vectors (14) acting at different fixed points in this plane. The rules of orbifold compactifications of the heterotic string have recently been reviewed in [13,14]. We are interested in the states whose masses are small compared to the string scale M S . These states are described by fields Here r labels the gauge quantum numbers and is given by where q is an element of the SO(8) weight lattice and v 4 6 = 0. In our convention, the last component of q gives the 4D helicity. For example, 4D vectors correspond to q = (0, 0, 0, ±1), 4D scalars to q = (±1, 0, 0, 0) with all permutations of the first 3 entries, and fermions correspond to q = (± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 ) with an even number of '+' signs. 1 The physical states are invariant under the action of the orbifold symmetry group which consists of twists and translations. In our case only translations in the SO(4) plane have a non-trivial action on the gauge degrees of freedom, due to the presence of Wilson lines. Then the invariance conditions read 2 ( = 1, . . . , 5): where we have included the Wilson lines in the local shift vectors kV 6 . We note that here two sources of symmetry breaking are present: local, due to twisting, and non-local, due to the Wilson lines. In the first case, symmetry breaking is restricted to the fixed points in the compact space. Indeed, since orbifold fixed points are invariant under twisting (up to a lattice vector), the first condition can be satisfied only for certain p, which indicates symmetry breaking at the fixed points. These sets of p's are generally different at different fixed points and only their intersection survives in 4D, since in this case the wave function can be constant in the compactified dimensions leading to a massless state. In the case of Wilson line symmetry breaking, the second and third conditions apply to all points in the G 2 and SU(3) planes and the symmetry breaking is non-local.
To define our string model, it is necessary to specify the action of the twist on the second, 'hidden', E 8 . We find that the desired symmetry breaking pattern and the appearance of three 16-plets at fixed points with unbroken SO(10) lead to (29) in the orthonormal E 8 × E 8 basis. In string theory, these quantities must satisfy certain consistency conditions (see [13] for a recent discussion). First of all, 6V 6 and 2W 2 , 2W 2 must be elements of the E 8 × E 8 root lattice which is required by embedding of the orbifold symmetry group ('space' group) in the gauge degrees of freedom. Second, modular invariance requires (32) 6 (mV 6 + nW 2 + n W 2 ) 2 − m 2 v 2 6 = 0 mod 2, m, n, n = 0, 1. Our choice of the hidden sector components of V 6 , W 2 , W 2 is strongly affected by these conditions.
We note that N = 1 supersymmetry in 4D requires whereas N = 2 would require, in addition, v i 6 = 0 mod 1 for some i. In the former case, there is one gravitino satisfying q · v 6 = 0 mod 1 whereas in the latter case there are two of them.
Finally, massless states in 4D must satisfy the following conditions: for the untwisted sector, and for the kth twisted sector. HereÑ is an oscillator number and c k , c k,Ñ are certain constants (see, e.g., [13]). In our model, all states which transform non-trivially under SU(3) c × SU(2) L haveÑ = 0.
In this section we have described the necessary ingredients of our orbifold model. In the next section we compute the massless spectrum of the model and discuss localization of various states.

Massless spectrum of the model
First let us identify the gauge group in 4D. For N = 1 vector multiplets q · v 6 = 0. Hence, the surviving gauge group in 4D is given by the root vectors satisfying It is straightforward to verify that these roots together with the Cartan generators form the These are precisely the groups discussed in the first section. Their intersection yields the surviving group SU(3) × SU(2) × U(1) 5 .
Let us now consider matter fields. These can be either in the untwisted sector U or in one of the twisted sectors T 1 . . . T 5 . Below we analyze each of them separately. Before we proceed, let us fix the chirality of the matter fields to be positive 3 , i.e., q 4 = +1/2 for their fermionic components.
The states represent bulk matter of the orbifold. By choosing an appropriate right-mover, these massless states can be made invariant under the Z 6 orbifold action and thus are present in the 4D spectrum. Apart from SU(3) c × SU(2) L singlets, 4 the untwisted sector of our model contains (1, 2) in terms of the SU(3) c × SU(2) L quantum numbers. From the field-theory perspective, these fields correspond to the compact space components of the E 8 gauge fields and their superpartners.

T 1 + T 5 sector
These matter fields are located at the 12 orbifold fixed points (Fig. 4) and satisfy Since Wilson lines are present only in the SO(4) plane, only SO(4)-plane projections of the fixed points matter. The G 2 and SU(3) projections do not affect the local twist. They only lead to a multiplicity factor 3 due to the three identical SU(3) fixed points. Any massless state in the T 1 sector survives the orbifold projection, i.e., is invariant under the Z 6 action, and is therefore present in the 4D spectrum. The twisted matter fields located at a given fixed point appear in a representation of the local gauge group at this point. In our case, twisted matter with SU(3) c × SU(2) L quantum numbers is It is convenient to keep the notation (16) of SO(10) even though the unbroken group in 4D is only G SM , since it represents one complete generation of SM fermions including right-handed neutrinos. In terms of SU(3) c × SU(2) L quantum numbers we have (1, 2), where again we have omitted singlets.

T 2 + T 4 sector
These states are localized at the fixed points in the G 2 and SU(3) planes, while being bulk states in the SO(4) plane (Fig. 4). If the T 1 sector corresponds to the string with the boundary condition twisted by Θ = diag(e 2πiv 1 6 , e 2πiv 2 6 , e 2πiv 3 6 ), the T 2 sector corresponds to the strings twisted by Θ 2 . Since Θ 2 has a fixed plane, T 2 states are bulk states in this plane and localized states in the other two planes.
The orbifold action on this sector is Z 3 , and is given by Since there are no Wilson lines in the G 2 and SU(3) planes, all fixed points are equivalent. The massless N = 1 multiplets obey Both the G 2 and the SU(3) lattice have 3 fixed points under Z 3 , so the multiplicity factor is 9. The local gauge groups at the fixed points are determined by At each Z 3 fixed point, the unbroken gauge group and the twisted sector matter fields are (cf. Fig. 6) These states are subject to further projection and not all of them survive. Indeed, by construction they are only invariant under the Z 3 action, but not under the full Z 6 . Furthermore, the Z 3 fixed points in the G 2 -plane are only fixed under Z 3 and the Z 6 action transforms them into one another. Physical states are formed out of their linear combinations which are eigenstates of the Z 6 twist.
The Z 3 invariance of a physical state requires where q + v 3 is the shifted SO(8) momentum and p + V 3 is the shifted E 8 × E 8 momentum. This is satisfied automatically as long as the gauge embedding of the twist and the Wilson lines obey modular invariance (yet it may require shifts by a lattice vector). A non-trivial Z 2 invariance condition is The extra term γ = (0, 0, 1/2) appears due to the 'mixing' of the fixed points [12,22]. There are three combinations of the Z 3 fixed points which are eigenstates of Z 6 with eigenvalues e 2πiγ . An important note is in order. The SO(8) lattice momentum q is found via the masslessness condition for the right-movers, Since v 3 has a fixed plane, there are always two sets of solutions, with opposite chiralities. Both of them survive the projection (48), which leads to N = 2 hypermultiplets. The conditions (49) break the symmetry between the two chiralities and one obtains N = 1 chiral multiplets. As a result, 9 × (27)N = 2 hypermultiplets produce the following N = 1 multiplets with SU(3) c × SU(2) L quantum numbers: (1, 2).

T 3 sector
These states are localized at the Z 2 fixed points in the G 2 and SO(4) planes and are bulk states in the SU(3) plane (Fig. 4). They correspond to strings twisted by Θ 3 . The massless T 3 states satisfy and the local gauge groups at the fixed points are determined by The result for gauge groups and matter multiplets reads (cf. Fig. 7) (n = 0, n = 0): SO (16) As usual we have omitted SU(3) c × SU(2) L singlets and included a multiplicity factor 4 from the G 2 -plane fixed points. These states are located at the Z 2 fixed points which are mixed by the action of the full Z 6 twist. Again, one has to form linear combinations of the states transforming covariantly under Z 6 .
The matter states are, as before, subject to projection conditions. The Z 2 condition for the relevant states is With a proper redefinition of W 2 by a lattice vector shift, this condition is satisfied by all states with both chiralities, i.e., both solutions q of the equation for massless right-movers, Therefore, these states form N = 2 hypermultiplets. The further Z 3 projection reads where now γ = (0, 0, 1/3, −1/3). The four Z 2 fixed points in the G 2 -plane lead to four eigenstates under Z 3 with eigenvalues e 2πiγ . The above condition projects out some of the states. The surviving N = 1 multiplets with SU(3) c × SU(2) L quantum numbers are (59) T 3 = 7 × (3, 1) ⊕ 5 × (3, 1) ⊕ 10 × (1, 2).

Summary of the massless spectrum
Combining all matter multiplets from the untwisted and the five twisted sectors we finally obtain plus SU(3) c × SU(2) L singlets. Note that in addition to three SM generations contained in the three 16-plets we have only vector-like matter. This result is partly dictated by the requirement of anomaly cancellations. Vector-like fields can attain large masses and decouple from the low energy theory. A detailed analysis of this issue, including U(1) factors, will be presented in a subsequent publication.

Intermediate GUTs
So far we have made no assumption on the size of the compact dimensions. These are usually assumed to be given by the string scale, R i ∼ 1/M S . However, this is not necessarily the case and, furthermore, unification of the gauge couplings favours anisotropic compactifications where some of the radii are significantly larger than the others [23,24]. In this case one encounters a higher-dimensional GUT at an intermediate energy scale. Indeed, the Kaluza-Klein modes associated with a large dimension of radius R become light and are excited at energy scales above 1/R M S . At these energy scales we obtain an effective higher-dimensional field theory with enhanced symmetry in the bulk.
In our model there are four independent radii: two are associated with the G 2 and SU(3) planes, respectively, and the other two are associated with the two independent directions in the SO(4)-plane. Any of these radii can in principle be large leading to a distinct GUT model.
The bulk gauge group and the amount of supersymmetry are found via a subset of the invariance conditions (28). Consider a subspace S of the 6D compact space with large compactification radii. This subspace is left invariant under the action of some elements of the orbifold space group, i.e. a subset of twists and translations G. The bulk gauge multiplet in S is a subset of the N = 4 E 8 × E 8 gauge multiplet which is invariant under the action of G, i.e. a subset of conditions (28) restricted to G.
Consider first the case with two large compact dimensions, for instance those associated with the SO(4)-plane. The SO(4)-plane is invariant under the Z 3 subtwist as well as translations by a lattice vector in the G 2 and SU(3) planes. The latter do not lead to non-trivial projection conditions since there are no Wilson lines in these planes, while the former leads to gauge symmetry and supersymmetry breaking. The light gauge states are described by fields which are constant with respect to z 1 and z 2 . Invariance under Z 3 requires (see Eq. (28) with = 2) (61) Ψ p,q (x; z 3 ) = e 2πi(p·V 3 −q·v 3 ) Ψ p,q (x; z 3 ).
Gauge multiplets satisfy q · v 3 = 0 which has two sets of solutions for q corresponding to N = 2 supersymmetry. Then the condition p · V 3 = 0 breaks E 8 to E 6 × SU(3). At the four fixed points of the SO(4)-plane symmetry is broken further to the four subgroups discussed in Section 3. Altogether, we obtain a 6D E 6 × SU(3) orbifold GUT with the distribution of gauge symmetries in the fundamental region of the orbifold shown in Fig. 8. Similarly, untwisted matter satisfies (61) with q · v 3 = ±1/3. We note that all three SM generations live at the origin in Fig. 8.
This breaks N = 4 E 8 to N = 2 SU(4) × SU(4) × U(1) 2 in the bulk (Fig. 9). At the fixed points, the symmetry is broken further by the Z 3 twist leaving only the standard model gauge group (up to U(1)'s). Each of the three fixed points carries one generation of the standard model matter.
A different picture arises when the G 2 -plane compactification radius is large. The G 2plane is not invariant under any of the twists, thus there is no projection condition due to twisting. The only non-trivial projection conditions are due to the Wilson lines, Thus we have N = 4 supersymmetry and the gauge group is SO(8) × SO (8). Three generations of the standard model are localized at the origin where the Z 6 twist breaks the symmetry to the standard model gauge group (Fig. 10).
In principle, there is nothing special about six dimensions, and the same analysis can be carried out for five, seven, eight, nine and ten dimensions. The results are summarized in Table 1. A variety of orbifold GUTs appears, with gauge groups ranging from E 8 to SU(4) × SU(4) × U(1) 2 . These GUTs represent different points in moduli space. Values of the corresponding T-moduli determine the compactification radii. It is remarkable that all these GUT models in various dimensions are consistent with gauge coupling unification. 5 This is true even though in some cases SU(3) c and SU(2) L are contained in different simple factors, i.e. SO(8) × SO (8) or SU(4) × SU(4). The beta functions for both SO(8)'s or SU(4)'s are the same. In the former case this is enforced by N = 4 supersymmetry, while in the latter case the two SU(4)'s have identical bulk matter content, 2 × (6, 1) + 16 × (4, 1) N = 2 multiplets. In all other cases SU(3) c × SU(2) L is contained in a simple factor such that unification of the gauge couplings in the bulk is automatic.
On the other hand, different GUTs differ in the value of the gauge coupling at the unification scale, since the power law running depends on the number of extra dimensions and the bulk gauge group. Realization of some of the GUTs may require non-perturbative string coupling [23,24]. Different models also lead to different Yukawa couplings which depend on the compactification radii. These phenomenological aspects are similar to those of orbifold GUTs [25] and will be discussed elsewhere.

Summary
We have presented a Z 6 heterotic orbifold model leading to the standard model spectrum and additional vector-like matter in four dimensions. Standard model generations appear as 16-plets of SO(10). They are localized at different fixed points in the compact space with local SO(10) × SU(3) × U(1) symmetry.