Realistic three-generation models from SO(32) heterotic string theory

We search for realistic supersymmetric standard-like models from SO(32) heterotic string theory on factorizable tori with multiple magnetic fluxes. Three chiral ganerations of quarks and leptons are derived from the adjoint and vector representations of SO(12) gauge groups embedded in SO(32) adjoint representation. Massless spectra of our models also include Higgs fields, which have desired Yukawa couplings to quarks and leptons at the tree-level.


Introduction
Superstring theory is a good candidate for the unified theory of the gauge and gravitational interactions, and quark, lepton and Higgs fields. Indeed, there have been several approaches to derive the realistic string vacua by comparing the theoretical predictions with the data of the cosmological observations as well as the collider experiments which is known as the subjects of string cosmology and phenomenology.
Beginning with the work of Ref. [1], there are much progresses to find the standard-like models from the E 8 × E 8 heterotic string theory instead of the SO(32) heterotic string theory. (See for a review, e.g. [2].) This is because E 8 gauge group involves several candidates of the grand unified groups such as E 6 , SO (10) and SU(5) as the subgroups of E 8 and the E 8 adjoint representation includes matter representations such as 27 of E 6 , 16 of SO(10) and 10 and5 of SU (5). However, in SO(32) heterotic string theory, for example, the 16 spinor representation of SO (10) is not involved in the adjoint representation of SO(32). (In the framework of toroidal Z N orbifold, there are some possibilities to obtain the spinor representation of SO groups as discussed in Ref. [3].) Therefore, as one of the procedures to find the realistic string vacua, we try to derive the (non-)supersymmetric standard-like models from the SO(32) heterotic string theory without going through the grand unified groups. These approach might be useful to search for the realistic standard model, because the standard-like model given through the decomposition of GUT groups have the extra matters which should be decoupled from the low-energy dynamics in terms of some non-trivial mechanisms.
The standard model is a chiral theory. Thus, the keypoint to realize the standard model is how to realize a chiral theory. Toridal compactification is simple, but it can not realize a chiral theory unless introducing additional backgrounds. Orbifold and Calabi-Yau compactifications can lead to a chiral theory. Toroidal compactification with magnetic fluxes can also lead to a chiral theory. Here, we study such a background. That is, our key ingredients are the multiple U(1) magnetic fluxes inserted into SO(32) gauge group. These magnetic fluxes are first discussed in Ref. [1], where the SU(5) grand unified groups can be realized from the SO(32) heterotic string theory. Furthermore, there are much progresses on the resolved toroidal orbifold in [4] and on more general Calabi-Yau manifolds for E 8 × E 8 and/or SO(32) heterotic string theory via the spectral cover construction [5] and the extension of it [6] ( see e.g., Refs. [7]). 1 In this paper, we study SO(32) heterotic string theory on six-dimenional (6D) torus with magnetic fluxes, which is one of the simplest compactifications leading to a chiral theory. Then, we search the models, where the unbroken gauge group includes SU(3) × SU(2) × U(1) Y and massless spectra correspond to three chiral gennerations of quarks and leptons.
The paper is organized as follows. In Sec. 2, we show our set-up and typical theoretical constraints which are required from the consistency of heterotic string theory. For example, in the standard embedding scenario of the Calabi-Yau compacfitication, the internal gauge backgrounds are set to be equal to spin connections of the Calabi-Yau manifolds. On the other hand, in the non-standard embedding scenario, the gauge fields are not always identified as the spin connections of the internal manifold due to the existence of the fluxes. We discuss the consistency conditions for such fluxes on 6D torus in Sec. 2.1. In addition, the U(1) Y gauge boson should be massless, even if the consistent fluxes are inserted into SO(32) gauge groups in order to derive the standard-like model gauge groups. Generically, U(1) gauge bosons appeared in the low-energy effective theory couple to the universal and Kähler axions through the ten-dimensional (10D) Green-Schwarz term [9] which implies that the linear combination of U(1) gauge bosons may absorb these axions by their Stueckelberg couplings and become massive. Thus the axionic couplings of U(1) Y gauge boson should be absent, otherwise U(1) Y gauge boson would become massive as discussed in Sec. 2.2. In addition to the merits of gauge symmetry breaking, the fluxes are important tools to realize the degenerate zero-modes, i.e., three-generation of the elementary particles and the four-dimensional (4D) N = 1 supersymmetry (SUSY) at the same time. In fact, in Sec. 2.3, the chiral theory with degenerate zero-modes can be obtained from the considerations of zero-mode wavefunctions on tori. At the same time, the existence of 4D N = 1 SUSY depends on the ansatz of U(1) fluxes due to the flux-induced Fayet-Illipoulous terms.
In Sec. 3.1, we discuss the concrete embeddings of the standard model gauge groups into SO(32) gauge group in terms of the multiple U(1) fluxes. The correct matter contents of the standard model are then derived from the adjoint and vector representations of SO(12) given by the subgroup of SO(32). Since the number of generations corresponds to the number of U(1) fluxes, we search for the desired matter contents of the standard model satisfying the U(1) Y massless conditions as well as the SUSY conditions as can be seen in Sec. 3.2. Finally, Sec. 4 is devoted to the conclusion. The normalization of SO(32) generators and useful trace identities of SO(32) gauge group are summarized in Appendices A and B, respectively.
2 SO(32) heterotic string theory on tori with U (1) magnetic fluxes 2.1 Low-energy description of SO(32) heterotic string theory We briefly review the SO(32) heterotic string theory on a general complex manifold with multiple U(1) magnetic fluxes. The notation is based on Refs. [10,11,12]. The low-energy effective action of SO(32) heterotic string theory is given by which is the bosonic part of the action at the string frame in the notation of [10]. The gravitational and Yang-Mills coupling is set by 2κ 2 10 = (2π) 7 (α ′ ) 4 and g 2 10 = 2(2π) 7 (α ′ ) 3 and φ 10 denotes the ten-dimensional dilaton. Here the field-strength of SO(32) gauge groups F has the index of vector-representation. In what follows, "tr" and "Tr" represent for the trace in the vector and adjoint representation of the SO(32) gauge group, respectively. In addition, H denotes the heterotic three-form field strength defined by where w YM and w L are the gauge and gravitational Chern-Simons three-forms, respectively. From the action given by Eq. (1), the kinetic term of the B-field is extracted as where we add the Wess-Zumino term which describes the magnetic sources for the Kalb-Ramond field B (6) . Such sources correspond to the non-perturbative objects, i.e., the stacks of N a five-branes which wrap the holomorophic two-cycles Γ a and their tensions are given by Here, δ(Γ a ) denote the Poincáre dual four-form of the two-cycles Γ a . By employing the ten-dimensional Hodge duality, the Kalb-Ramond two-form B (2) and six-form B (6) are related as * dB (2) = e 2φ 10 dB (6) , and then the kinetic term of Kalb-Ramond field and Wess-Zumino term (3) are rewritten as S kin + S WZ = − 1 4κ 2 10 M (10) e 2φ 10 dB (6) ∧ * dB (6) where N a = ±1 represent for the contributions of heterotic and anti-heterotic five-brane, respectively. The equation of motion of B (6) leads to the following tadpole condition of the NS-NS fluxes in the presence of five-branes, in cohomology and whereF stand for the gauge field strengths of the internal gauge fields whose gauge groups are embedded in SO(32). When the extra-dimension is compactified on the flat space such as three 2-tori, (T 2 ) 1 × (T 2 ) 2 × (T 2 ) 3 , the tadpole cancellation requires the following consistency conditions, which should be satisfied on (T 2 ) i × (T 2 ) j with i = j, i, j = 1, 2, 3. Thus if the nonvanishing fluxes are not canceled by themselves, the non-perturbative objects would contribute to the cancellation of anomalies. It suggests that the modular invariance of heterotic string theory is recovered by the existence of these non-perturbative objects [13,14] which can be also realized in the framework of heterotic orbifold [15]. 2

Generalized Green-Schwarz mechanism
In addition to the consistency condition as discussed in Sec. 2.1, it must be ensured that our models do not have gauge and gravitational anomalies. In heterotic string theory, it is known that some gauge and gravitational anomalies are canceled by considering the following one-loop Green-Schwarz term at the string frame [9], whose normalization factor is determined by the S-dual type I theory as shown in Appendix of [18] and the eight-form X 8 reads, Although the gauge and gravitational anomalies for the non-Abelian gauge groups are canceled by the above Green-Schwarz term (8) and the tadpole condition (6) as shown in Ref. [1], the anomalies relevant to the multiple Abelian gauge groups, which appear in low-energy effective theory, can be also canceled by same Green-Schwarz mechanism, for more details see Refs. [11]. In fact, since we derive just the three-generation standard-like models, our phenomenological models do not receive these anomalies. However, as pointed out in Refs. [11], even if the Abelian gauge symmetries are anomaly-free, the Abelian gauge bosons may become massive due to the Green-Schwarz coupling given by Eq. (8). In order to ensure that the hypercharge gauge boson is massless, they should not couple to the axions which is hodge dual to the Kälb-Ramond field.
Before evaluating the mass term of U(1) gauge bosons, for completeness, we show the definition of three 2-tori (T 2 ) i ≃ C/Λ i with i = 1, 2, 3, where the lattices Λ i are generated by two vectors e i = 2πR i and e i = 2πR i τ i . Here, R i and τ i are the radii and complex structure moduli of (T 2 ) i , respectively. The metrics of three 2-tori are then given by where x m are the coordinates of T 2 with m, n = 4, 5, 6, 7, 8, 9, z i = x 2+2i + τ i x 3+2i and the rank 2 diagonal matrices g (i) and h (i) are given by From this expression, we expand the Kalb-Ramond field B (2) and internal U(1) a field strenghsf a , (a = 1, · · · , 13) in the basis of Kähler forms, w i = idz i ∧ dz i /(2 Imτ (i) ) on tori (T 2 ) i derived from the metrics (21), where m (i) a are the integers or half-integers determined by Dirac quantization condition. Since Dirac quantization is satisfied in the adjoint representation of SO(32), the factional numbers of m (i) a can be allowed as pointed out in Ref. [1]. From the Eqs. (14) and (15), we can extract the Stueckelberg couplings, where the trace identities are employed as shown in Appendix B. If the U(1) gauge fields couple to the universal axion b S which is the hodge dual of the tensor field b S , one of the multiple U(1) gauge fields absorbs the universal axion and become massive. In our model, since the hypercharge U(1) Y is identified as the linear combinations of multiple U(1)s, i.e., U(1) Y = 1 6 (U(1) 3 + 3 c U(1) c ) as shall be discussed in Sec. 3 3 , the U(1) Y gauge field becomes massless under the condition 3 m 3 m which means no interaction between U(1) Y and the universal axion b (0) S . Here the following formulas are satisfied Except for the universal axion, there are other axions associated with the internal cycles, that is, Kähler axions which couple to the U(1) gauge bosons originated from the action given by Eq. (5). Along with the Kälb-Ramond field B (2) , we expand the dual field B (6) as whereŵ i are the Hodge dual four-forms of the Kähler forms, which are defined as those satisfying T 2 ×T 2 ×T 2 w i ∧ŵ j = δ ij . After inserting these expressions into the action given by Eq. (5), we can extract the mass terms of the U(1) gauge bosons, In the same way as the case of universal axion, the U(1) Y gauge field should not couple to the Kähler axions, otherwise it becomes massive. Thus the U(1) Y gauge boson is massless under the following condition, with i = 1, 2, 3. As a step to realize the realistic models, the massless conditions for U(1) Y gauge boson given by Eqs. (24) and (28) should be satisfied. It is remarkable that these U(1) fluxes are sensitive to the consistency condition given by Eq. (7) as shown in the Sec. 2.1. When the heterotic five-branes are absent in our system, the following conditions, are required from the consistencies of heterotic string theory, otherwise the NS-NS tadpole could be canceled by the existence of heterotic five-branes.

The chiral fermions and degenerate zero-modes
The heterotic string theory on three 2-tori has N = 4 supersymmetry in the language of 4D supercharges which have to be broken to at least N = 1 supersymmetry in the four-dimension, otherwise the chiral matters do not appear in the low-energy effective theory. Although it is known that there are much progresses in the framework of toroidal orbifold, in this paper, we focus on the realization of chiral fermions by employing the multiple U(1) fluxes as discussed in this section. 4 First we define the 10D Majorana-Weyl spinor λ which satisfies the Majorana-Weyl condition, where Γ is the 10D chirality matrix. The following analysis is based on Ref. [19]. In order to discuss the 4D chirality, we decompose the 10D Majorana-Weyl spinor λ into four 4D Weyl spinors λ 0 and λ i with i = 1, 2, 3 as the representation of SU(4) ≃ SO (6). The 10D chirality matrix Γ is also decomposed into the product of three 2D chirality operators , satisfying the Clifford algebra. Then the 4D chirality is fixed as which lead to the following 4D Weyl spinors, where the subscript indexes denote the eigenvalues of Γ i with i = 1, 2, 3. When we insert the magnetic fluxes on three 2-tori, one of the four 4D Weyl spinors would be chosen. In order to prove the above statements, we show the zero-mode wavefunction of fermions originating from the 10D gaugino field by solving their Dirac equations. The zero-modes of 10D gaugino field λ and gauge field A M are defined through the following decompositions, where M = 0, 1, · · · , 9 and x µ , µ = 0, 1, 2, 3 are the coordinates of the 4D spacetime. The zero-modes of gaugino fields, ψ where hereafter we omit the subscript 0 of the zero-modes, that is, On the other hand, the extra dimensional components of U(1) a gauge backgrounds A (i) a (z i ) (a = 1, 2, · · · , 13) are given by which lead to the magnetic fluxes given by Eq. (22) along the Cartan direction of SO(32).
Then the zero-mode equations of fermions ψ (i) (z i ) with the U(1) a charge q a are given by where the Gamma matrices and covariant derivatives in terms of the complex coordinates, (z i ,z i ) are defined as which can be derived from the Gamma matrices in flat space (31) and the metric of torus (21) and The spin connections are vanishing due to the topology of tori. Thus the Dirac equations on (T 2 ) i are rewritten as Then ψ (i) However, such magnetic fluxes may break all N = 4 SUSY through the D-terms or Fayet-Iliopolous terms in the language of 4D N = 1 SUSY. When N = 1 SUSY is preserved in the system, the vanishing D-terms imply that the hermitian Yang-Mills equations for the U(1) a field strengths should be satisfied at the vacuum, In our set-up, these conditions are equal to where A i = (2πR i ) 2 Im τ i are the areas of tori, (T 2 ) i . Indeed, when these conditions are satisfied, massless scalar fields appear for A M (M = 4, · · · , 9), and they correspond to superpartners of the above massless fermions. At the perturbative level, the D-term conditions receive at most one-loop corrections [20] which have the dilaton dependence. Finally, we comment on the Wilson lines which play a role of breaking the gauge group into its subgroups without changing the rank of gauge groups. In fact, when we introduce the Wilson lines ζ (i) a , along the U(1) a directions, the internal components of U(1) a gauge fields take the following shifts compared to Eq. (36), 3 Three-generation models in the SO(32) heterotic string theory

Matter content
In this section, we show the concrete decomposition of SO(32) gauge group into the standard model gauge groups and then the parts of adjoint representation of SO(32) are identified as the matter contents of the standard model. As the first step to obtain the standard model gauge groups, we consider the decomposition of SO(32) illustrated as where the multiple U(1) fluxes are assumed along the Cartan directions of SO(32).
In order to derive the matter contents of the standard model, we examine whether the adjoint representation of SO(12) involves the candidates of elementary particles or not. When we put three U(1) 1,2,3 fluxes along the Cartan directions of SO(12) gauge group, it is found that SO(12) involves the candidates of SU(3) C and SU(2) L , where the Cartan directions of U(1) 1,2,3 are given by Eq. (11) . Then the adjoint representation of SO(12) is decomposed as which are singlets of SO (20), where the subscript indices denote the U(1) 1,2,3 charge q 1,2,3 . The normalization of U(1) generators are given by Appendix A. Thus when we identify the hypercharge as U(1) Y = U(1) 3 /6, we can extract the candidates of the quarks, charged leptons and/or Higgs, As shown in the above analysis, the adjoint representation of SO(12), 66 does not involve the candidate of right-handed leptons. Therefore, we further decompose the SO(20) gauge group into U(1) 4,5,··· ,13 gauge groups, where the nonvanishing U(1) fluxes along all U(1) 4,··· ,13 directions are inserted shown in Eq. (11). Now SO (2) is identified as U(1). The vector representation and the singlet of SO(12), 12 v and 1 give the suitable matter contents, i.e., right-handed quarks and leptons, charged-leptons and/or Higgs, where the underlines for U(1) 4,5,··· ,13 charge q 4,5,··· ,13 denote the all possible permutations. It is remarkable that the correct U(1) Y charge can be also realized as

Three-generation models
Since the matter contents of the standard model are correctly identified in the previous section, we show the number of generations for each representation in this section. As discussed in Sec. 2.3, the U(1) fluxes generate the degenerate zero-modes if these zeromodes have U(1) charges. It implies that the number of generations for the representations embedded in the adjoint and vector representations of SO(12), 66 and 12 v are determined by the following formulas, and respectively. Now we are ready to search for the realistic three-generation models in the framework of SO(32) heterotic string theory. In the light of U(1) Y massless conditions given by Eqs. (24) and (28), the nonvanishing U(1) 3 fluxes seem to violate these massless conditions. Therefore, in this paper, we restrict ourselves to the case that U(1) 3 fluxes are absent in our system, which lead to no chiral generations of right-handed quarks, u c R and d c R from the adjoint representation of SO(12) as can be seen in Eq. (51). Only left-handed quarks Q and charged-leptons L are then generated from the adjoint representation of SO (12). As for the left-handed quarks, Q, there are two possibilities to reproduce the three-generations of Q,  Table 1 and they lead to the same generation numbers. We omit such possibilities in both tables.  , m e c a R , m n a 2 ) = (0, 0, 0, 0), (a = 6,7,8,9,10,11,12,13). (55) In the case of Type A, only "Case I" is allowed as the realistic three-generation models. The   Table 3: The typical values of U(1) fluxes in the model of type A and "Case I" given by Eqs. (53) and (54). "Case I" and "Case II" are allowed and they are then categorized as the three types of models, BI : "CaseI" in type B, BII : "CaseII" in type B with m n 1 = 0, BIII : "CaseII" in type B with m n 1 = 0. (57) For each model, the typical U(1) fluxes and the number of generations of matters are summarized in Tables 5, 6 , 7, 8, 9 and 10. In the type BI model summarized in Tables 5 and 6, the realistic Yukawa couplings of quarks and leptons are allowed in terms of the renormalizable operators, 3 , L 9 4 , u c 9 R 2 , d c 9 R 3 , e c 9 R 1 , n 9 1 ) (0, 0, 0, 1, 1, 0) (L 10 3 , L 10 4 , u c 10 R 2 , d c 10 R 3 , e c 10 R 1 , n 10 1 ) (−1, 0, 0, 0, 0, 0) (L 11 3 , L 11 4 , u c 11 R 2 , d c 11 R 3 , e c 11 R 1 , n 11 1 ) (−1, 0, 0, 0, 0, 0) (L 12 3 , L 12 4 , u c 12 R 2 , d c 12 R 3 , e c 12 R 1 , n 12 1 ) (0, 1, 0, 0, 0, 0) (L 13 3 , L 13 4 , u c 13 R 2 , d c 13 R 3 , e c 13 R 1 , n 13 1 ) (0, 1, 0, 0, 0, 0)  Table 7: The typical values of U(1) fluxes in the type BII model given by Eq. (57).
(L 9 3 , L 9 4 , u c 9 R 2 , d c 9 R 3 , e c 9 R 1 , n 9 1 ) (7, 0, 0, 0, 0, 0) (L 10 3 , L 10 4 , u c 10 R 2 , d c 10 R 3 , e c 10 R 1 , n 10 1 ) (0, −7, 0, 0, 0, 0) . . . . . . quarks and leptons are allowed in terms of the renormalizable operators, whereL 4 3 , L 5 4 are identified as Higgs doublets. Note that in our models, the consistency conditions given by Eq. (7) do not satisfied without introducing the heterotic five-branes. In this case, we have to take care of the Witten anomaly [16,17] on the heterotic five-branes with Sp(2N) gauge groups which is the case that the number of heterotic five-branes is N. In order to avoid the Witten anomaly, the number of chiral fermions under the fundamental represenations of Sp(2N) are even [16,17]. These fundamental representations of (32, 2N) under SO(32) ⊗ Sp(2N) can be read in the type I string with D5-and D9-brane system which is expected as the S-dual of the SO(32) heterotic string. The generations of the chiral fermions included in (12, 2N) under SO(12) ⊗ Sp(2N) and (20, 2N) a for a = 1, 2, · · · , 13. In our most supersymmetric models, the chiral fermions arise from (32, 2N) under SO(32) ⊗ Sp(2N). Thus we require the non-trivial mechanism to obtain the even number of chiral fermions such as U(1) fluxes on the heterotic five-branes in order to avoid the Witten anomaly.

Conclusion
In this paper, we have shown the realistic standard model gauge groups from the framework of Our models also include Higgs fields, which have Yukawa couplings to quarks and leptons at tree level. Possible configurations of magnetic fluxes are severely constrained by the massless condition of U(1) Y hypercharge gauge boson and the consistency condition of heterotic string theory. It is remarkable that in general, the ten-dimensional Green-Schwarz term induces the Stueckelberg couplings to multiple U(1) gauge bosons which might lead to the mass term of U(1) Y hypercharge gauge boson. In this respect, the numbers of fluxes have been constrained by the massless condition of U(1) Y gauge boson. Since the torus is flat, our models requires the existence of heterotic five-branes in order to satisfy the consistency conditions without introducing the extra Stueckelberg couplings to U(1) gauge boson, in contrast to the E 8 × E 8 heterotic string theory. At that time, the Witten anomaly cancellation constrains the number of U(1) fluxes due to the nature of symplectic groups on the heterotic five-branes. In fact, the chiral fermions under the fundamental representation of symplectic gauge groups do not arise in the parts of our models, whereas the other parts of our models requires the non-trivial mechanisms such as U(1) fluxes on heterotic five-branes to cause even number of these chiral fermions to avoid the Witten anomaly. We listed supersymmetric three-generation standard models with massless U(1) Y gauge bosons and desirable Yukawa couplings of quarks, leptons and Higgs. The detailed phenomenological analysis of our models such as mass matrices would be studied in a separate work and the detail of this paper is applicable in the framework of type I string.
The unbroken gauge sector in our models has N = 4 supersymmetry, that is, three adjoint scalar fields and four types of gaugino fields. However, the existence of (anti-)heterotic fivebranes would lead to the breaking of (all) partial breaking of supersymmetry in our model. Orbifolding would be useful to reduce N = 4 supersymmetry to N = 1. Zero-mode wavefuctions have been also studied on orbifolds with magnetic fluxes [24,25]. Such extensions would be also interesting.