$SO(32)$ heterotic standard model vacua in general Calabi-Yau compactifications

We study a direct flux breaking scenario in $SO(32)$ heterotic string theory on general Calabi-Yau threefolds. The direct flux breaking, corresponding to hypercharge flux breaking in the F-theory context, allows us to derive the Standard Model in general Calabi-Yau compactifications. We present a general formula leading to the three generations of quarks and leptons and no chiral exotics in a background-independent way. As a concrete example, we show the three-generation model on a complete intersection Calabi-Yau threefold.


Introduction
String theory is an attractive candidate not only for a theory of quantum gravity but for a unified theory of elementary forces. It predicts phenomenologically promising higher-dimensional nonabelian gauge theories which are expected to include the Standard Model (SM) as well as its realistic spectra. Indeed, string theory naturally incorporates non-abelian gauge groups, appearing from stacks of D-branes in type I and II string theories, seven-branes in F-theory and closed string sector in heterotic string theories.
Non-trivial gauge backgrounds such as internal gauge fluxes and Wilson lines play an important role in breaking the higher-rank gauge group down to the SM one [1,2], but Wilson lines are only allowed for restricted non-simply-connected manifolds [2]. As an example, there exist 195 non-simply-connected complete intersection Calabi-Yau (CY) threefolds (CICYs) among 7820 CICYs [3,4,5]. Hence, it is interesting to check whether or not gauge fluxes directly lead to the SM gauge group. This approach allows us to open up a new window for the string model building in more general CY compactifications. In the context of F-theory grand unified theories (GUTs), such a direct flux breaking called hypercharge flux breaking is an attractive scenario to break the SU(5) gauge group [6,7] (also discussed in the dual heterotic string side [8,9,10,11]). In contrast, the authors of Ref. [12] pointed out that the realization of hypercharge flux scenario is generically difficult to achieve in E 8 × E 8 heterotic string theory on smooth CY threefolds due to the large number of index conditions, corresponding to the three generations of quarks and leptons. It therefore motivates us to search for models with a hypercharge flux in other string theories.
In this paper, we systematically study SO(32) heterotic line bundle models as a realization of direct flux breaking. 1 In a way similar to Ref. [12], we search for the three-generation SM against several branchings of SO(32) in a background-independent way. After solving a large number of index conditions for each elementary particle, together with the K-theory condition and hypercharge masslessness conditions, it turns out that the SM-like spectrum can be realized in general CY compactifications. Note that supersymmetric and stability conditions are required to be checked for each CY threefold.
The remainder of this paper is as follows. In Sec. 2, we first show the model-building approach to derive heterotic Standard Models on smooth CY threefolds. After discussing the hypercharge direction for several group decompositions of SO(32) and corresponding spectrum, we next present the general formula satisfying the index conditions, K-theory condition and hypercharge masslessness conditions. The obtained formula is applicable to general CY threefolds. In Sec. 3, the specific MSSM (minimal supersymmetric Standard Model)-like spectrum is shown for a concrete CICY. In Appendix A, we present the algorithm to compute the particle spectrum for all the group decompositions discussed in this paper. where SO(2m) gauge group includes the SM one. Now U(1)s descend from U(m) ⊂ SO(2m) and their number is restricted to be a = 1, 2, · · · m − 3, where 3 is the rank of non-abelian gauge groups in the SM. In particular we focus on line bundles L a each with structure group U(1) a , that is the internal bundle of the form Then, U(1) a gauge fluxes are inserted into the Cartan direction of SO(2m) to realize the SMlike gauge group. We expand U(1) a gauge field strengths tr(F a ) in the basis of Kähler form w i , i = 1, 2, · · · , h 1,1 with h 1,1 being a hodge number of CY, namely tr(F a ) = 2π where T a are U(1) a generators descending from U(m) ⊂ SO(2m).
Here and in what follows, "tr" denotes the trace in the fundamental representation and m (i) a are integers constrained by the Dirac quantization condition. Note that the hypercharge is a linear combination of U(1) a , namely with normalization factors f a . Throughout this paper, we assume the uncorrelated U(1) a line bundles, otherwise the U(1) gauge groups are enhanced to be a non-abelian one. According to the group decomposition (1), the adjoint representation of SO(32), corresponding to the massless mode in the heterotic string, decomposes under SO(2m) × SO(32 − 2m), where the adjoint representation of SO(2m), Adj SO(2m) , is expected to include the SM particles, whereas the vector and adjoint representations of SO(32 − 2m) correspond to the exotic particles. On this line bundle background, the net number of chiral zero-modes with U(1) a charges Y a is counted by the index where we consider internal bundles ⊗ n a=1 L Ya a and for the later purpose, we define Here, d ijk are the intersection numbers of the basis of two-forms w i and the second Chern class of the tangent bundle of CY threefolds is expanded in their Hodge dual four-formsŵ i , namely c 2 (T M) = i c 2,iŵ i . Variables {X abc , Z a } are written in terms of the internal gauge fluxes m (i) a along U(1) a with generators T a descending from U(m) ⊂ SO(2m). Since X abc are totally symmetric tensors with respect to a, b, c from the fact that d ijk are totally symmetric tensors with respect to i, j, k, we note that variables {X abc , Z a } consist of m−3 C 3 +2( m−3 C 2 )+ m−3 C 1 = m−1 C 3 and m − 3 degrees of freedom, totally m−1 C 3 + m − 3, determined by the values of gauge fluxes and topological data of CY. The aim of this work is to search for variables {X abc , Z a } leading to the three-generation SM without specifying the topological data of CY. An advantage of this approach is the possibility to perform the systematic search on general CY manifolds. Before going to the detailed analysis, we remark three consistency conditions in the four-dimensional effective action of heterotic string. (For more details, see, e.g., Refs. [8,9].) First one is the "K-theory condition" to admit the spinorial representation in the first excited mode [16,17] where c 1 (W ) is the first Chern class of the total internal bundle W and κ (i) denote integers. Second one is the stability condition of our discussing four-dimensional effective action. Stability of the effective action requires a positive number of heterotic five-branes, constraining the background curvatures through the anomaly cancellation condition, where ch 2 (W ) is the second Chern character of the internal bundle and N i is the number of heterotic five-branes wrapping the internal holomorphic two-cycles on the CY threefold. Note that, in the perturbative heterotic string vacua (N i = 0), by multiplying the stability condition (9) by m for all a = 1, 2, · · · , m − 3.
In addition, internal bundles have to satisfy the zero-slope poly-stability conditions, namely D-term conditions associated to U(1) a gauge symmetries, where φ 10 is the ten-dimensional dilaton and the Kähler form is now expanded as J = l 2 s i t i w i with t i being the Kähler moduli in string units l s = 2π √ α ′ = 1. 2 Here, we include the one-loop correction to D-terms [8,9].
The last one is the hypercharge masslessness conditions. On the non-trivial gauge background, the internal gauge fluxes induce the Stückelberg couplings between string axions and the hypercharge gauge boson through the Green-Schwarz terms [18,19]. It is known that some of U(1)s are anomalous due to the internal gauge fluxes and their number is counted by the rank of U(1) mass matrix in string units l s = 2π √ α ′ = 1 [8,9], where the first and second lines are coming from Stückelberg couplings of Kähler axions and dilaton axion, respectively. To ensure the masslessness of the hypercharge direction U(1) Y = a f a U(1) a , we impose two constraints originating from Kähler axions and dilaton axion, where we note that a tr(T 2 a )f a Z a = 0 is satisfied under the constraint (13). Let us take a closer look at the K-theory condition (8) and hypercharge masslessness conditions (13) which are summarized as 2 Note that we now use a different notation for the Kähler moduli, i ) compared with Ref. [9]. Here, b where Here and in what follows, two generators T 1,2 are chosen such that tr(T 1 )tr(T 2 2 )f 2 − tr(T 2 )tr(T 2 1 ) f 1 = 0. It then allows us to rewrite variables {X αBC , X αβC , X αβγ , Z α } in terms of others, namely where As a result, independent variables are Thus, the number of variables reduces to m(m − 4)(m − 5)/6 + 2(m − 4) from m−1 C 3 + m − 3. We now also use the totally symmetric properties of d ijk .
In addition to the theoretical requirements such as K-theory condition (8) and hypercharge masslessness conditions (13), variables {X ABC , X ′ AB , X ′′ A , X ′′′ , Z A , Z ′ } are further constrained by phenomenological requirements. To realize phenomenologically consistent models, we impose the three generations of quarks and charged leptons 3 and no chiral exotics, namely for all 1 ≤ p ≤ p ex , where the number of chiral exotics p ex depends on the branching of SO(32) but is at least p ex ≥ m − 3 appearing from exotic states (2m, 32 − 2m) in Eq. (5). We have taken into account the order of freely-acting discrete symmetry group of CY threefolds |Γ| in order to be applicable to the model building on non-simply-connected CY threefolds. The above phenomenological requirements constrain variables {X ABC , X ′ AB , X ′′ A , X ′′′ , Z A , Z ′ } through Eq. (6). It turns out that in the case with m = 6, i.e., SO(32) → SO (12) (20), total six variables are not enough to satisfy at least eight index conditions (20) and remaining hypercharge masslessness condition (14) in general. Here, we have not considered the other stability conditions (9) and (11) which depend on the number of five-branes and values of moduli fields.
To simplify our analysis, we focus on m = 8 case, namely (16) ′ , including the m = 7 case. Note that such a visible SO(16) gauge group can be also embedded into the T-dual E 8 × E 8 and non-supersymmetric SO(16) × SO(16) heterotic string theories, taking into account the spinor representation of SO(16) as a massless mode. The following analysis is then applicable to other heterotic string theories.
To obtain the SM-like spectra without chiral exotics, we require and employing the index theorem on CY threefolds. The above index formulae are applicable to both simply-and non-simply-connected CY threefolds with the order of freely-acting discrete symmetry group |Γ| and here we have not counted the number of Higgs/Higgsino fields. 5 It is possible to consider other hypercharge directions: where the sign is taken in the random order. The algorithm to determine the hypercharge direction is discussed in Appendix A.
Although we have focused on the particular branching of SO(32) with specific hypercharge direction, we systematically investigate several gauge embeddings with possible hypercharge directions listed in Table 1 of Appendix A. In the next section 2.3, we solve the eleven index conditions (27), (28) and remaining hypercharge masslessness condition (14) in terms of 24 variables {X ABC , X ′ AB , X ′′ A , X ′′′ , Z A , Z ′ } against several branchings of SO(32).

General formula
We are now ready to search for 24 variables , satisfying the eleven index conditions (27), (28) and the remaining hypercharge masslessness condition (14). Since the stability conditions depend on the detail of CY data, for the time being, we focus on only K-theory condition and hypercharge masslessness conditions. We remark that the stability conditions have to be checked for each CY threefold. The details of the algorithm computing the particle spectrum is shown in Appendix A.
On simply-connected CY threefolds, our systematic search reveals that a direct flux breaking scenario is available for the following decompositions: In the case with the first branch of Eq. (29) corresponding to the model of Sec. 2.2, the following 24 variables satisfy all the requirements: where p 1 , · · · , p 16 are integers and we now consider the specific hypercharge direction (24). For other gauge branchings, see, Table 2 of Appendix A. As mentioned before, some of U(1)s are anomalous due to the internal gauge fluxes and their number is counted by the rank of U(1) mass matrix (12). Remarkably, for all the gauge decompositions of Table 2, the dilaton-axion induced mass term in Eq. (12) turns out to be due to the correlation between X abc and Z a . Let us consider in more detail the κ (i) = 0 case which means that the right-handed side of K-theory condition (8) vanishes. In such a case, independent variables are {X ABC , Z A }, since other variables are written by from which two U(1) gauge boson mass terms are provided by the other one, i.e., M αi = A K αA M Ai . Thus, the maximal rank of U(1) gauge boson matrix is 3. One of the massless U(1) directions will be identified with U(1) B−L in addition to U(1) Y . On the other hand, in the κ (i) = 0 case, the rank of U(1) gauge boson matrix is generically 4 when h 1,1 ≥ 3 and the remaining gauge symmetry consists of Finally, we comment on the direct flux breaking scenario on non-simply-connected CY threefolds. In contrast to the simply-connected CY cases, some special non-simply-connected CY threefolds, specifically |Γ| = 5Z >0 , allow intermediate GUT-like models without chiral exotics. The detail of phenomenologically acceptable CYs is shown in Table 1 of Appendix A.

Phenomenological aspects of three-generation models
The obtained general formula in Sec. 2.3 is applicable to general CY threefolds. In this section, we discuss three-generation models on a specific CICY.

Three-generation models on simply-connected CY threefolds
We consider simply-connected CY threefolds, namely |Γ| = 1. Among known 7890 CICYs, we consider the following CICY called # 7247 in the list of [20], where the above configuration matrix characterizes how to embed the CY manifold in four projective spaces P 2 . The superscript and subscript indices denote the hodge number of CY (h 1,1 , h 2,1 ) = (4, 40) and the Euler number of CY −72, respectively. (See for details of CICYs, e.g., Ref. [21].) The topological data of the above CICY is calculated as in the basis of dual four-formsŵ i , respectively. Note that this CY has Z 3 and Z 3 × Z 3 discrete symmetries, but we concentrate on the case with |Γ| = 1 in what follows. For concreteness, we study compactifications of the SO(32) heterotic string on the above CICY with line bundles leading to the group decomposition (21). Although the model-building approach is classified into two cases: κ (i) = 0 and κ (i) = 0 appearing in the K-theory condition (8), in the following analysis, we focus on the case with κ (i) = 0, providing just the SM gauge group in the visible sector.
We search for the internal U(1) gauge fluxes within the range −1 ≤ m  27) and (28), the stability conditions (9) and (11) and hypercharge masslessness condition (14). Note that other U(1) 1,2 fluxes are determined by Eq. (16). It turns out that some of the fluxes satisfy all the requirements.
is positive and at the same time, the D-term conditions (11) are satisfied at the physical domain of Kähler moduli and dilaton, namely t i , s > 1 in string units, where the ten-dimensional dilaton field in Eq. (11) is now written by s = e −2φ 10 V/(2π) with CY volume V = i,j,k d ijk t i t j t k /6 in string units. The above fluxes result in the following chiral zero-modes: and singlets: where the particles n c 1 or n c 2 could be identified with right-handed neutrinos. Remarkably, we have phenomenologically interesting perturbative Yukawa couplings, and interestingly, the higher-dimensional proton decay operators are prohibited by the massive U(1) B−L gauge symmetry. However, we require the mechanism generating the mass terms for extra Higgs doublets, which will be the subject of future work.

Gauge coupling unification
So far, we have focused on the number of chiral generations in SO(32) heterotic compactification. In this section, we discuss the unification of gauge couplings at the string scale. The four-dimensional gauge coupling is extracted from the ten-dimensional SO(32) Yang-Mills action, where g 2 10 = 4π(l s ) 6 is the ten-dimensional gauge coupling. After expanding the ten-dimensional gauge field strength as the four-dimensional part F and internal partF , namely F → F +F , the kinetic terms of the four-dimensional gauge bosons become where s is the dilaton field with CY volume V in string units.
Since the generator of hypercharge direction listed in all the gauge decompositions of Table 1 is of the form where the sign is taken in the random order, the four-dimensional gauge field strength of the SM gauge group is expanded as using the Gell-Mann matrices λ A and Pauli matrices σ α . In our case, the non-abelian part is normalized as tr SO(32) (T A SU (3) C T B SU (3) C ) = 2δ AB , tr SO(32) (T α SU (2) L T β SU (2) L ) = 2δ αβ due to the fact that the level of Kac-Moody algebra is one. The trace of gauge field strength then reads Note that off-diagonal U(1) gauge couplings are absent in contrast to the E 8 × E 8 heterotic string case [9]. Taking into account the normalization of U(1) Y generator, the gauge couplings satisfy at tree level. Thus, we cannot arrive at phenomenologically interesting so-called GUT-relation. However, the unification of gauge couplings actually depends on the number of Higgs doublets, radiative corrections to the gauge couplings and supersymmetry-breaking scale. Indeed, the radiative corrections to the non-abelian gauge groups are non-universal in contrast to the E 8 ×E 8 heterotic case [8,9], which allows us to obtain the realistic values of gauge couplings at the string scale as demonstrated on toroidal background [22]. To evaluate their precise values, we have to discuss the stabilization mechanism of moduli fields and supersymmetry-breaking sector. We will postpone the concrete model building for a future analysis.

Conclusion
We have searched for SO(32) heterotic SM vacua directly with the SM gauge group from smooth CY threefolds. The non-trivial internal gauge background allows us to directly embed the SM gauge group G SM into SO(32) one. In this paper, we focus on the branching G SM ⊂ SO(16) ⊂ SO(32) as listed in Table 1. Against several branchings of SO(16), we have derived a general formula leading to the three generations of quarks and leptons and no chiral exotics, taking into account the K-theory condition and the hypercharge masslessness conditions. Since the obtained formula is independent of the topological data of CY, it is applicable in general CY compactifications in contrast to models using Wilson lines. Such a direct flux breaking is attractive scenario not only in F-theory GUTs [6,7], but also in heterotic string theory.
For concreteness, we have discussed phenomenological aspects of direct flux breaking scenario on a specific CICY, where the spectrum consists of MSSM particles, singlets, extra Higgs doublets and vector-like particles whose number cannot be captured by the index theorem. In our setup, the normalization of hypercharge direction is different from conventional SU(5) GUT normalization, which also be stated in E 8 × E 8 heterotic compactification with a hypercharge flux [12]. However, one-loop threshold corrections to gauge couplings are non-universal for non-abelian gauge groups compared with E 8 × E 8 heterotic models. It thus opens up the way for phenomenologically acceptable models.
It is interesting to check whether or not one can achieve direct flux breaking (i.e., hypercharge flux breaking) in the dual global F-theory compactifications.

A Group decompositions
In this appendix, we present the algorithm to compute the particle spectrum for all the group decompositions of SO(16) ⊂ SO(32) treated in this paper. First we decide which weights correspond to particles or anti-particles without determining the hypercharge direction. In our analysis, the matter weights belonging to SO(16) adjoint representation are chosen as Q : , u c , d c : , e c , singlets : To extract the SM particles from Eq. (49) including particles and anti-particles, we define the following maps: Next, we searched for all 2 6 · 2 6 · 4 7 ≈ 6.71 × 10 7 possibilities of {q L (i), q u c (i), q e c (i)} against each branching of SO(16) listed in Table 1. Then we determine the coefficients of hypercharge direction f a to realize the correct hypercharge of quarks and leptons. It is remarkable that, in our analysis, f a (if exist) are determined uniquely against each set of (27) and (28)) to obtain the three generation of quarks and leptons without chiral exotics, taking into account the hypercharge masslessness conditions and the K-theory condition. In Table 1, we show the possible order of freely-acting discrete symmetry group Γ. In particular, on simply-connected CY threefolds (|Γ| = 1), the list of 24 variables {X ABC , X ′ AB , X ′′ A , X ′′′ , Z A , Z ′ } in Table 2 lead to the MSSM-like models, where we choose a specific hypercharge direction for simplicity.
In the search of concrete models in Sec. 3, we use the brute force attack to solve Eqs.