String-Inspired Special Grand Unification

We discuss a grand unified theory (GUT) based on an $SO(32)$ GUT gauge group broken to its subgroups including a special subgroup. In the $SO(32)$ GUT on six-dimensional (6D) orbifold space $M^4\times T^2/\mathbb{Z}_2$, one generation of the SM fermions can be embedded into a 6D bulk Weyl fermion in the $SO(32)$ vector representation. We show that for a three generation model, all the 6D and 4D gauge anomalies in the bulk and on the fixed points are canceled out without exotic chiral fermions at low energies.

Recently, a new-type GUT has been proposed by the author in Ref. [23]. In usual GUTs, GUT gauge groups are broken to their regular subgroups; e.g., E 6 ⊃ SO(10) × U (1) ⊃ SU (5) × U (1) × U (1) ⊃ G SM × U (1) × U (1). In the new GUT called a special GUT, GUT gauge groups are broken down to special subgroups. (For Lie groups and their regular and special subgroups, see e.g., Refs. [3,[24][25][26].) In Ref. [23], the author proposed an SU (16) special GUT whose gauge group SU (16) is broken to a special subgroup SO (10). The results are summarized as follows. In a 4D SU (16) special GUT, one generation of quarks and leptons can be embedded into a 4D SU (16) 16 Weyl fermion; the 4D SU (16) gauge anomaly restricts the minimal number of generations. Unfortunately, the minimal number is 12 in 4D framework. In a 6D SU (16) special GUT on 6D orbifold space M 4 × T 2 /Z 2 , one generation of quarks and leptons can be embedded into a 6D SU (16) 16 Weyl fermion; the 6D SU (16) gauge anomaly and the 4D SU (16) gauge anomaly on the fixed points restricts the minimal number of generations; three generation of quarks and leptons is allowed without 4D exotic chiral fermions.
Superstring theory [27,28] has been considered as a candidate for unified theory to describe all the interaction including gravity. There are a lot of attempts to construct the SM from string theories. One of its trials is based on E 8 ×E 8 and SO(32) heterotic string theories [29][30][31][32][33][34][35][36][37][38][39][40]. Usually, the E 8 ×E 8 heterotic string model building is much more popular than SO(32) one. One of the biggest reason seems that when we only consider regular embeddings, for the branching rules of SO(32) ⊃ SO(10)(×U (1) 11 ), the SO(32) vector and adjoint representations 32 and 496 do not contain SO(10) spinor representations 16 and 16, while for the branching rules of E 8 ⊃ SO(10)(×U (1) 3 ), the E 8 adjoint representation 248 contain SO(10) spinor representations 16 and 16. However, for a special embedding, on the other hand, the branching rules of SO(32)(⊃ SU (16) × U (1) Z ) ⊃ SO(10) × U (1) Z for SO (32) vector and adjoint representations 32 and 496 are given by where we follow the convention of their U (1) normalization in Ref. [3]. Obviously, an SO(32) vector representation is decomposed into SO(10) spinor representations, and an SO(32) adjoint representation contains an SO(10) bi-spinor representation 210. When we take into account the special embedding SU (16) ⊃ SO(10), SO(32) gauge theories contain SO(10) spinors, easily. In the following discussion, we will not consider how to realize models from string theories. There are several good features of special GUTs pointed out in Ref. [23]. First, almost all unnecessary U (1)s can be eliminated; e.g., SO(32) ⊃ G SM × U (1) 12 by using only regular embeddings, while SO(32)(⊃ SU (16) × U (1) Z ⊃ SO(10) × U (1) Z ) ⊃ G SM × U (1) 2 by using regular and special embeddings. Second, by using only regular embeddings, the SM fermions cannot be embedded into an SO(32) vector representation 32, while by using regular and special embeddings SO (32) It is known in e.g., Refs. [2,3] that any 4D SO(32) gauge theory is a vectorlike theory since an SO(32) group has only real representations. To realize the SM, i.e., a 4D chiral gauge theory, we take orbifold space construction [41,42]. It allows us to realize 4D Weyl fermions from 5D Dirac fermions, 6D Weyl fermions, etc. In the 6D SU (16) special GUT [23], the nonvanishing VEV of a 5D SU (16) 5440 brane scalar is responsible to break the SU (16) GUT gauge group to its special subgroup SO(10) via the Higgs mechanism [43,44]. For SO(32) special GUTs, the SO(32) GUT gauge group can be broken to SO(10) by using the nonvanishing VEV of a scalar in an appropriate representation of SO(32); the lowest dimensional representation is 86768. (The spontaneous symmetry breaking of SU (n) to its special subgroups has been discussed in e.g., Refs. [45,46].) In this paper, we will discuss an SO(32) special GUT on 6D orbifold spacetime M 4 × T 2 /Z 2 . As in 6D SU (16) special GUTs, we need to take into account 6D and 4D gauge anomalies. As the same as the 6D SU (16) gauge anomaly in the 6D SU (16) special GUT [23], the 6D SO(32) gauge anomaly can be canceled out by introducing 6D positive and negative Weyl fermions in the same representation of SO(32) gauge group. Unlike an SU (16) gauge group, an SO(32) gauge group itself has no 4D gauge anomaly for any fermion in any representation of SO(32), but there can be 4D gauge anomalies for its subgroups. We will see it in Sec. 3 in detail.
The main purpose of this paper is to show that in a 6D SO(32) special GUT on M 4 × T 2 /Z 2 we can realize three generations of the 4D SM Weyl fermions from six 6D SO(32) 32 bulk Weyl fermions without 4D exotic chiral fermions at low energies, and without any 6D and 4D gauge anomaly.
This paper is organized as follows. In Sec. 2, before we discuss a special GUT based on an SO(32) gauge group, we quickly review basic properties of SO(32) and its subgroups shown in Ref. [3]. In Sec. 3, we construct a 6D SO(32) special GUT on M 4 × T 2 /Z 2 . Section 4 is devoted to a summary and discussion.
2 Basics for SO(32) and its subgroups Since the SO(32) vector representation 32 is real, a 4D Weyl fermion in SO(32) 32 representation includes not only 4D SM Weyl fermions but also their conjugate fermions. To realize chiral fermions, we take orbifold symmetry breaking mechanism [41,42]. After taking into account orbifold effects, we can regard the zero modes of an SO(32) 32 fermion as one generation of the SM fermions plus a right-handed neutrino. Note that there is no 4D pure SO(32) gauge anomalies of any representation of SO(32) gauge group, while there can be 4D SU (16) and U (1) anomalies generated by 4D Weyl fermions in complex representations of SU (16) and U (1), respectively. Then, after orbifolding, a maximal regular subgroup SU (16) × U (1) Z of SO(32) may be anomalous. We will discuss how to cancel out 4D pure SU (16) We consider a symmetry breaking pattern from SO(32) to G SM . One way of achieving it is to use orbifold symmetry breaking boundary conditions (BCs) and several GUT breaking Higgses. One example is to choose orbifold BCs breaking SO(32) to SU (16) × U (1) and to introduce three SO(32) 86768, 496, 32 scalar fields, where we assume their proper components get non-vanishing VEVs. First, the following orbifold BC for the SO(32) vector representation 32 breaks SO(32) to SU (16) × U (1) Z : where the projection matrix P 32 is proportional to the U (1) Z generator and satisfies (P 32 ) 2 = I 32 .
(The matrix form of P 32 depends on basis.) Next, the non-vanishing VEV of the SO(32) 86768 scalar field is responsible for breaking

Summary and discussion
In this paper, we constructed an SO(32) special GUT by using a special breaking SU (16) to SO (10). In this framework, the zero modes of the 6D SO(32) 32 Weyl fermion can be identified with one generation of quarks and leptons; the 6D SO(32) and the 4D SU (16) × U (1) gauge anomalies on the fixed points allow a three generation model of quarks and leptons in 6D framework; as in the SU (16) special GUT [23], exotic chiral fermions do not exist due to a special feature of the SU (16) complex representation 120 once we take into account the symmetry breaking of SO(32) to SO (10).