Doublet-Triplet Splitting in Fertile Left-Right Symmetric Heterotic String Vacua

Classification of Left-Right Symmetric (LRS) heterotic-string vacua in the free fermionic formulation, using random generation of generalised GSO (GGSO) projection coefficients, produced phenomenologically viable models with probability $4\times 10^{-11}$. Extracting substantial number of phenomenologically viable models requires modification of the classification method. This is achieved by identifying phenomenologically amenable conditions on the Generalised GSO projection coefficients that are randomly generated at the $SO(10)$ level. Around each of these fertile cores we perform a complete LRS classification, generating viable models with probabilility $1.4\times 10^{-2}$, hence increasing the probability of generating phenomenologically viable models by nine orders of magnitude, and producing some $1.4\times 10^5$ such models. In the process we identify a doublet-triplet selection mechanism that operates in twisted sectors of the string models that break the $SO(10)$ symmetry to the Pati-Salam subgroup. This mechanism therefore operates as well in free fermionic models with Pati-Salam and Standard-like Model $SO(10)$ subgroups.


Introduction
The Standard Model of particle physics agrees with all observational data to date.The discovery of a scalar resonance, compatible with the Standard Model Higgs particle, lends further support to the possibility that the Standard Model provides viable parameterisation of all experimental observables up to the GUT or Planck scales.Further elucidation of the Standard Model parameters can therefore only be obtained by fusing it with gravity, i.e. in a theory of quantum gravity.String theory is the leading contemporary framework that enables the pursuit of this synthesis, as its consistency conditions mandate the existence of the gauge and matter structures that form the bedrock of the Standard Model.This necessitates the construction of string models that are compatible with the Standard Model data [1].
An appealing feature of the Standard Model is the embedding of its matter states in chiral SO (10) spinorial 16 representations.This characteristic is reproduced in the heterotic E 8 × E 8 string theory [2] that gives rise to chiral 16 SO (10) representations in the perturbative spectrum.The construction of phenomenological string models proceeds by studying compactifications of the heterotic-string to four dimensions.Among the string models that reproduce a large number of phenomenological three generation models with SO (10) embedding of the chiral spectrum are the heterotic-string models in the free fermionic formulation that correspond to Z 2 × Z 2 orbifold compactifications at special points in the moduli space with discrete Wilson lines [3].
The case of PS models utilise solely RNS boundary conditions [11], whereas the three other cases utilise RNS and complex boundary conditions.The FSU5 models utilise a single basis vector that breaks the SO (10) gauge symmetry [12,13], whereas the SLM models necessarily include two such basis vectors [14].The SLM models therefore contain a proliferation of sectors that include an SO (10) breaking vector and produce exotic states.The result is that the frequency of viable three generation models in the total space of models is reduced, making the random based classification method inefficient.To circumvent this problem fertile conditions have been identified that facilitate the extraction of three generations SLM vacua with varying phenomenological characteristics [14].We remark that the genetic algorithm developed in ref. [21] provides an alternative method to extract vacua with phenomenological characteristics, albeit not to classify large classes of them.We further note that employing fertility condition analysis of string vacua is also adopted in analysis of other classes of string vacua [22].
The situation in the case of the LRS models is similar to that of the SLM models, with the added complexity that the LRS models do not admit the E 6 embedding of the charges in the extension of SO(10) × U( 1) to E 6 [7].While the LRS models can be constructed with a single SO(10) basis vector α, the vector 2α breaks the SO(10) symmetry as well [15].Thus, exotic states producing sectors arise in the LRS string models from basis vector combinations with the vectors α and 2α resulting again in proliferation of exotic producing sectors and diminishing the frequency of viable three generation vacua.A remedy to this situation is provided by identifying a set of fertile conditions in the space of LRS free fermionic heterotic-string vacua.
In this paper we undertake this task.In the process we uncover a doublettriplet mechanism in the twisted sectors of the heterotic-string vacua.At the SO(10) level vectorial 10 representations arise from the untwisted and twisted sectors.These decompose as 5 + 5 under SU(5) and as (3,1,1) In the case of the untwisted states a doublet-triplet splitting mechanism has been identified in PS, SLM and LRS string vacua that utilise asymmetric boundary conditions [23].However, the free fermionic systematic classification method utilises symmetric boundary conditions.In PS, SLM and LRS string vacua with symmetric boundary conditions the untwisted sector produces three pairs of colour triplets rather than electroweak doublets.In this paper we identify a doublet-triplet splitting mechanism in terms of the discrete torsions that appear in the one-loop partition function of the models and that operates in the twisted sectors of the LRS models.The core of our fertility conditions revolve about the doublet-triplet splitting mechanism in the twisted sectors, thus increasing the frequency of models that contain heavy and light Higgs representations.
As in the case of the SLM models, the classification is performed in two stages.The fertility conditions include GGSO phases that involve only basis vectors that do not break the SO(10) GUT symmetry.Thus, the fertility conditions are implemented by a random search for SO (10) vacua that satisfy these conditions, resulting in 19374 fertile cores.To these cores we add the SO( 10) breaking basis vector and generate a complete classification of LRS string vacua, generating some 9.92 × 10 6 models from which 1.4 × 10 5 satisfy all our phenomenological criteria.This result exceeds the random classification method of [15] by four orders of magnitude in about 1/10 computational time on a computer platform of similar power.
Our paper is organised as follows: in section 2 we discuss the general structure of the free fermionic LRS models; section 3 summarises the fertility conditions employed in the analysis; in section 4 we discuss the results of the analysis and in section 5 we introduce the doublet-triplet splitting mechanism that operates in the twisted sectors of the LRS models; in section 6 we analyse an exemplary model in some more detail; section 7 concludes our paper.

Left Right Symmetric Free Fermionic Models
This paper utilises the free fermionic formulation [24] of the heterotic string to explore the space of string vacua which possess the Left-Right Symmetric (LRS) subgroup of SO (10).The classification of such vacua was performed in [15].The models are constructed by defining a set of basis vectors and the Generalised Gliozzi-Scherk-Olive (GGSO) projection coefficients of the one-loop partition function.An overview is outlined in the following section but more details of the LRS classification can be found in [15].
In order to obtain LRS vacua, the SO(10) GUT symmetry is broken directly at the string scale and the unbroken LRS subgroup of SO (10) in the low energy effective field theory is SU(3) C ×U(1) C ×SU(2) L ×SU(2) R .Resulting models obey N = 1 spacetime supersymmetry and preserve the SO(10) embedding of the weak hypercharge.Fermionic matter representations of the Standard Model are found in the spinorial 16 representation of SO (10) decomposed under the unbroken SO (10) subgroup.Similarly, vectorial representations, including the Standard Model Light Higgs, derive from the 10 representation of SO (10).

The Free Fermionic Formulation
In this section, a brief overview of the the free fermionic formulation will be outlined.We will also draw attention to key features relevant for the discussion of fertile regions and doublet-triplet splitting in the LRS models.
A free fermionic string model is defined through boundary condition basis vectors that specify the transformational properties of the free fermions as they propagate around the two non-contractible loops of the one-loop partition function.These basis vectors are 64-dimensional and are of the form: where the boundary condition of a fermion, α(f ), is defined through: so that α(f ) = 0, 1 correspond to real boundary conditions and α(f ) = 1 2 corresponds to a complex boundary condition.
A model is constructed with two ingredients.First, is a set of basis vectors v i=1,...,k , which span a space Ξ of all linear combinations, α, which we call sectors.Second, is a set of distinct GGSO projection coefficients C v i v j , where i > j due to modular invariance consistency conditions leaving 2 N (N −1)/2 independent coefficients.
With these two ingredients, we can construct the modular invariant Hilbert space H of states |S α of the model through the one-loop GGSO projection such that: where F α is the fermion number operator and δ α = 1, −1 is the spin-statistics index.
A key role is played by the vectors b 1 and b 2 in these models as they define the SO (10) gauge symmetry and correspond to Z 2 × Z 2 orbifold twists which break the N = 4 supersymmetry, obeyed by the other 10 vectors, to N = 1.The third twisted sector is given by the linear combination b 3 = b 1 + b 2 + x, where the x vector is the combination: This vector plays an important role in these models as a map from spinorial 16 sectors of SO (10) to vectorial 10 sectors.
In order to break the SO(10) models down to the LRS subgroup we add a single breaking basis vector: which will leave the unbroken SO (10) subgroup SU(3) × SU(2) 2 × U(1).

GGSO Projections
The next ingredient of the free fermionic models are the GGSO projection coefficients C v i v j .Since we have 13 basis vectors, our GGSO coefficients span a 13 × 13 matrix.Due to modular invariance constraints, the lower triangle of the matrix containing 78 coefficients are fixed by the upper triangle.Modular Invariance constraints also lead to the following demands on the leading diagonal phases: The matrix entries are further constrained through imposing N = 1 supersymmetry.This can be done by requiring: where i = 1, . . ., 6 and k = 1, 2. All these constraints leave us with 66 independent coefficients and therefore 2 66 ≈ 7.38 × 10 19 distinct LRS string vacua.
This is too large a space to explore with a computer program and so in [15] a sample of 10 11 was explored and found to produce phenomenologically viable vacua with probability 4 × 10 −11 .This tiny probability is the key motivation for modifying this classification procedure through the use of imposing phenomenological constraints in the smaller space of SO( 10) models.This corresponds to constraining the GGSO coefficients relating to the first 12 basis vectors, which form a 12 × 12 matrix.

Properties of the String Spectrum
The sectors in the model can be characterised according to the left and right moving vacuum separately.Physical states must however satisfy the Virasoro matching condition: where N L and N R are sums over left and right moving oscillators, respectively.In our models, sectors which have the products ξ L • ξ L = 0 and ξ R • ξ R = 0, 4, 6, 8 can produce spacetime vector bosons, which determine the gauge symmetry in a given vacuum.We note that only massless states are phenomenologically interesting as massive states will be at scales comparable to the Plank mass.From the untwisted sector vector bosons we obtain a full gauge group of: 1) 1,2,3 Hidden : where the weak hypercharge is given by: such that U(1 In order to obtain the charge Q associated to a U( 1) current generated by a fermion f we use: where α(f ) is the boundary condition of the fermion in the sector and F (f ) is the fermion number given by: for fermionic oscillators and their complex conjugates, whereas for degenerate Ramond vacua it is: where |+ R = |0 is a degenerated vacuum with no oscillator and |− R = f † 0 |0 is the degenerated vacua with one zero mode oscillator.

Fertility Conditions
In order to narrow the search of the 2 66 vacua on phenomenologically promising regions, we examine the GGSO coefficients at the SO(10) level, which means a 12 ×12 matrix with 55 independent coefficients.The aim of this section is to apply further constraints that we call 'fertility conditions' on the SO(10) models.The models satisfying the fertility conditions we call 'fertile cores'.The conditions are chosen so as to increase the likelihood of finding phenomenologically viable vacua at the LRS level.
After obtaining fertile cores we perform a comprehensive classification of all models resulting from the cores by iterating over all α projection coefficients values.This methodology was used with great success in [14] where phenomenologically viable standard-like vacua were found in great abundance through the use of fertility conditions.

Observable Spinorial Sectors
The choice of basis vectors in equation ( 2) means that sectors giving rise to states of a particular representation of the gauge group can be written compactly as a function of p, q, r, s = 0, 1.These 16 possibilities correspond to the 16 fixed points of each twisted plane of the Z 2 × Z 2 orbifold.For example, the observable SO (10) spinorial sectors are: where p, q, r, s = 0, 1 and b 3 = b 1 + b 2 + x.These 48 sectors contain the 16 and 16 spinorial representations of the SO (10).
With this information we can begin classifying the spinorial/antispinorials, 16/16, of SO (10).The spinorials/antispinorial can be determined to give rise to either left or right chirality states, leaving 4 classification numbers: N L , N L , N R , N R .To determine whether a sector gives rise to a spinorial or antispinorial, we inspect the projectors on B A , A = 1, 2, 3: and the chirality phases: which together let us define N 16 , N 16 as: p,q,r,s=0,1 p,q,r,s=0,1 The number of spinorials/anti-spinorials alone is not sufficient to describe the phenomenological properties of the models under consideration as we need to consider what happens as the SO( 10) GUT is broken.Recall that the basis vector v 13 = α induces SO( 10) gauge symmetry breaking.The spinorial representations of SO (10), are decomposed under the residual ) Only one of the spinorial components The same is true for the anti-spinorials.That is, in order to accommodate the fields of one fermion generation we need at least four SO(10) spinorials and properly adjusted projections.This poses a challenge to any computer-based model scan.A lot of computer time is allocated in examining unacceptable incomplete generation models.
Remarkably, there is a way of partially overcoming this important problem.It turns out that the GGSO projection of the vector 2α + x when acting on spinorials differentiates between left and right states.In addition, as dictated by properties, this projection does not act on the GGSO phases associated to the α vector.Indeed, the GGSO projection of the vector 2α + x = { ψ45 , φ1,...,6 } gives: where we have used that 2α + x ∩ B (A) pqrs = { ψ45 } and the notation ch ψ4,5 stands for the SO(4) ∼ SU(2) L × SU(2) R chirality.In other words, the 2α + x projection selects between left and right states.Adopting the convention: we can write analytic formulas for the number of left spinorials, N L , right spinorials, N R , as well as the left and right anti-spinorials N L , N R respectively: p,q,r,s=0,1 p,q,r,s=0,1 p,q,r,s=0,1 N R = 1 4 A=1,2,3 p,q,r,s=0,1 An eventually complete generation SO(10) configuration should satisfy where n g stands for the number of generations.The factor of two in the last equation is necessary in order to compensate for the additional truncation imposed by the α vector projection.Furthermore, a consistent low energy model should include Higgs fields to break the SU(3 2) R gauge symmetry to that of the Standard Model.The necessary states, referred to as heavy Higgs fields transform as righthanded doublets and lie in an additional pair of spinorial/anti-spinorial (16/16) SO( 10) representations.This leads to the additional constraint We will refer to ( 23), (25) as fertility conditions regarding spinorials.SO (10) configurations enjoying this property are most likely to end up in phenomenologically viable Left-Right Symmetric Models when the α vector projection is also applied.

Observable Vectorial Sectors and Doublet-Triplet Splitting
Vectorials of SO (10) gauge symmetry are of great importance to phenomenology for they accommodate the light Standard Model Higgs doublets.In the class of models under consideration, massless SO( 10) vectorial states arise from the sectors which contain four periodic right-moving complex fermions and consequently they admit one Neveu-Schwarz right-moving fermionic oscillator The vectorial representation of SO( 10) where the colored triplets are generated by the ψ and the bi-doublet is generated by ψ 4,5 1/2 /ψ * 4,5 1/2 oscillators.At the SO(10) level, i.e. taking into account GGSO projectors associated to v 1 , . . ., v 12 vectors, the total number of surviving vectorials, N 10 , is given by N 10 = A=1,2,3 p,q,r,s=0,1 R A pqrs (28) where However, the full GGSO projections, include also the gauge symmetry breaking α vector projections associated to C α v i , i = 1, . . ., 12 phases.These projections act differently on the three states in (27).As a result, only one of the vectorial segments (triplet, anti-triplet or bi-doublet) survives.Depending on the phase configuration, the α related projections can eliminate all Standard Model doublets leading to unacceptable phenomenology.The mere existence of SO(10) vectorials does not guarantee the presence of Higgs doublets in the low energy massless spectrum.One has to assure the appropriate action of the α projections takes place, which is a time-consuming task from the point of view of model search.
There exists an elegant solution to the above problem that is related to a stringy doublet-triplet splitting mechanism.Moreover, it turns out that the relevant information, whether one of the triplets or the bi-doublet will survive, is encoded in each SO (10) model; it does not depend on the GGSO projectors associated to the SO(10) breaking α vector.In order to prove this we consider the action of the 2α + x GGSO projection on the SO (10) In other words, only SO (10) vectorials originating from sectors with C V A pqrs x = −1 could give rise to Higgs doublets.We call these states fertile vectorials.Their number, N f 10 , is given by In general N f 10 ≤ N 10 .As a minimal requirement a viable SO( 10) configuration should possess Nevertheless, the 2α + x projection considered above is not completely equivalent to the α projection.The latter can in principle completely project out the bidoublet even in the case where C V A pqrs x = −1, so this fertility condition should be considered as necessary but not sufficient.
The advantage of this stringy doublet-triplet mechanism lies in the fact that it not only preserves the Higgs doublet pair but it also guarantees the absence of the associated triplet pair.We should note that the above mentioned triplet representations are colour triplets, usually referred to as leptoquarks in the literature, which mediate proton decay via dimension five operators.Therefore, these states must be either sufficiently heavy so as to agree with the current proton lifetime of ≥ 10 33 years [26] or must be projected out of the string spectrum by the GGSO projections.The elegance of the string doublet-triplet mechanism has been previously noted, for example, in [23], which works with NAHE-set based [27] free fermionic models.In the NAHE models the doublet-triplet splitting occurs only in the untwisted sector, whereas here it can be applied to any twisted sector SO( 10) vectorial.

Top Quark Mass Coupling
In the class of models under consideration the top mass term stems from a superpotential coupling of the form where the left/right quarks and Higgs fields Q L , Q R , h were defined in ( 16), (27).
The conditions that assert the presence of this coupling at the tri-level superpotential were derived in [28].The advantage of the formalism described in [28] is that it also fixes some of the degeneracy that appears in the free fermionic formulation (e.g.orbifold plane interchange).Without loss of generality we can choose that Q L arises from the sector B 1 0000 = S + b 1 , Q R comes from the sector B 2 0000 = S + b 2 , and h comes from the sector V 3 0000 = S + b 3 + x = S + b 1 + b 2 .In order for these states to survive the GGSO projections associated to v 1 , . . ., v 12 vectors, the following conditions must be met In addition, the states that participate in (33) are subject to the 2α + x GGSO projection.As explained in sections 3.1,3.2this projection is related to the SU(2) L × SU(2) R symmetry representations.Assuring the correct L/R transformation properties for Q L , Q R translates to the additional constraints Only two of these constraints are independent and can be used to fix two additional GGSO coefficients, e.g.Furthermore, the states that give rise to top quark mass coupling are subject to the GGSO projections related to the SO(10) gauge symmetry breaking vector v 13 = α.Their survival is assured only in the case that two additional constraints are met From the technical point of view the above results have the advantage that they are explicit and consequently can be utilised to reduce the scanned parameter space.
2. Constraints on SO (10) spinorial states related to the presence of complete fermion families and SU(2) R × U(1) C symmetry breaking Higgs fields.For n g ≥ 3 these read 3. Constraints related to the presence of the Standard Model breaking Higgs fields The above constraints do not guarantee the existence of phenomenologically promising LRS models, however, they result in a high likelihood that such models will arise after employing the full GGSO projections on the massless string spectrum.We call SO( 10) models that comply with the above constraints "fertile SO (10) cores".As explained, the first class of conditions can be expressed explicitly in terms of GGSO phases that define our parameter space.However, the second and third class of constraints cannot be explicitly solved in terms of GGSO phases.A scan of the related parameter space is required in order to extract SO (10) models that satisfy these criteria.A comprehensive scan of the full parameter space, numbering 4.40 × 10 12 models, albeit straightforward, requires considerable computer resources and computing time.It turns out that a random scan of the parameter space is quite efficient in capturing the salient phenomenological characteristics of these fertile cores.To this end we examine a sample of 10 9 randomly selected configurations which corresponds to analysing one in one thousand models.A number of approximately 42000 fertile cores is collected through this procedure.
As part of our methodology here we decided to incorporate an analysis of enhancements arising at the SO( 10) level and filtered out fertile cores which contained gauge group enhancements to the observable sector, whilst keeping those with no enhancements or enhancements affecting only the hidden gauge group factors.This procedure is described in the following section.

Hidden Enhancements
In the previous LRS classification [15], it was noted that approximately 29.1% of LRS models contain additional gauge bosons but only the non-enhanced models were classified.In general, additional space-time vector bosons enhancing the gauge factors of ( 8) may arise from the following 26 sectors: where x is defined in equation ( 3).However, in the current work, we are interested in exploring models with no enhancements or solely enhanced hidden sector gauge factors.Such hidden enhancements may arise from the sectors: z 1 , z 2 and z 1 + z 2 .Such enhancements can be tested for at the SO( 10) level as they do not concern α GGSO phases and therefore fertile cores containing them can be found and included alongside non-enhanced cores in the analysis.In particular, after obtaining the approximately 42000 fertile cores from our scan of 10 9 SO( 10) configurations, we then tested these cores for SO (10) enhancements and filtered out those with observable enhancements and included cores within our sample with hidden enhancements.The hidden enhancement cases are presented in following tables.Note that in the tables we choose to use the arguments of the GGSOs: • z 1 + z 2 = { φ12345678 } gives rise solely to spinorial hidden enhancementss.

Enhancement Condition Resulting Enhancement
• z 1 = { φ1234 } which gives rise to massless states of the form: {ȳ i , wi , ψ12345 , η123 , φ5678 } | φ1234 .In the following table however, only the cases that result in enhancements to the hidden gauge group only are analysed, in particular the states: ψ µ 1 2 {ȳ i , wi , φ5678 } |z 1 are analysed.

Enhancement Condition
Resulting Enhancement (z • z 2 = { φ5678 } which gives rise to massless states of the form: {ȳ i , wi , ψ12345 , η123 , φ1234 } |z 2 .In the following table, the cases that result in enhancements to the hidden gauge group only are analysed, which are states of the form:

Enhancement Condition
Resulting Enhancement (z Having filtered out observably enhanced cores, we were left with 19374 fertile cores free from observable SO( 10) enhancements.It's interesting to note that this means 53.9% contain observable enhancements and so there's a higher correlation between these enhancements for fertile cores compared with a randomly generated SO (10) cores.The origin of this correlation can be motivated by noting the appearance of constraints on the (b j |z k ), j, k = 1, 2, GGSO coefficients from the top quark mass coupling in equation ( 36) coinciding with enhancement conditions.The main characteristics of our remaining 19374 fertile cores, including the number of generations (n g ), the number of spinorials/anti-spinorials that give rise to left and right states (N L /N R and NL / NR ) as well as the number of vectorials that give rise to SM doublets, are presented in Table 1.

Results and Analysis
The use of fertile cores in our analysis means we are splitting the parameter space of LRS models into two components: Π = Π 1 × Π 2 .Where Π 1 is the space of SO( 10) models in which we select our fertile cores using our fertility conditions and the Π 2 subspace includes the GGSO phases related to the SO(10) breaking vector α.
As mentioned in section 3.5, using a code written in Python we performed a scan over a random sample of 10 9 vacua in the space Π 1 which consists of 4.4×10 12  independent SO(10) cores once the constraints from Section 3 are implemented.Cores satisfying the fertility conditions and containing no observable enhancements at the SO(10) level are collected and in our sample 19374 fertile SO(10) cores were found.
These 19374 cores are now to be explored in the LRS subspace Π 2 by iterating over its possible α GGSO coefficients.Considering equations (5,39) there are in fact 9 independent α coefficients so each core results in 2 9 = 512 LRS models which can be analysed and classified very quickly using our Python code.
Before analysing the results at the LRS level, we first present Figure 1  The next step is the analysis of the LRS statistics resulting from our cores, which we obtain through a comprehensive scan of Π 2 .The results are shown in Table 2.These results ought to be compared and contrasted to the results of the classification using the random classification method of [15], which are shown in a corresponding table on page 24 of ref. [15] for a sample of 10 11 LRS vacua.
The methodology just described is analogous to that used in the Standard-Like Model classification of [14] except that here our comprehensive scan of Π 2 includes an analysis of the Enhanced, Hidden and Exotic sectors too.As mentioned in section 3.5, an important feature of our analysis was the inclusion of fertile cores which admitted an enhancement to the hidden sector gauge group.As can be seen in table 2 compared with Table 4 of [15], the probability of finding a model meeting all listed phenomenological criteria has increased from 4 × 10 −11 to 1.4 × 10 −2 due to our application of fertility conditions.This increase in probability by 9 orders of magnitude exhibits the power of the fertility methodology.In order to explore the LRS statistics more closely, we can break down the results with respect to the important quantum numbers coming from the observable 16 representation and those coming from the vectorial 10 telling us the number of triplets and Higgs doublets at the LRS level.These statistics are presented in Table 3 for three generation models in which the number of triplets, n 3 , and anti-triplets, n3, are matched.
We note here that the doublet-triplet splitting mechanism, discussed in section 3.2, involves the projection of the sector (2α + x), on the observable vectorial states arising in the vectorial sectors in eq. ( 26).The (2α + x) projection breaks the underlying SO (10) GUT symmetry to the Pati-Salam subgroup.Hence, a doublettriplet splitting mechanism, similar to the one that we discussed here, is operational in the Pati-Salam, as well as the Standard-like, heterotic-string models, that employ a Pati-Salam symmetry breaking basis vector in the construction.
The absence of triplet-free three generation models in our sample may indicate that they are very rare, and hence are not generated in our statistical sampling, or may result from a deeper reason in the structure of the LRS heterotic-string models.Three generation triplet-free models may also exist in the PS and SLM heterotic-string models and can be searched for, by employing a similar analysis to that of section 3.2.
The fact that there are some four generation, triplet-free models though is somewhat reminiscent of the result for from the classification of FSU5 models in [12] that exophobic models exist only for even generation models.
It is also worth reiterating here that all the models obtained by using the free fermion classification method of refs [9,10,11,12,15] contain three pairs of vectorlike triplet from the untwisted Neveu-Schwarz sector, due to the fact that all these       ,6 .Projecting out the untwisted colour triplets and retaining the electroweak Higgs bidoublets, requires assignment of asymmetric boundary conditions for this set of worldsheet fermions, in the basis vector that breaks the SO(10) symmetry to the Pati-Salam subgroup [23].Implementation of the untwisted doublet-triplet splitting mechanism requires therefore extension of the classification method to free fermion models with asymmetric boundary conditions.Models which are free of both untwisted and twisted additional vector-like colour triplets may exist, but such models have not been generated to date.
Since the extra triplets appear in vector-like representation, mass terms can be generated from cubic level and higher order nonrenormalisable terms in the superpotential.In that case their masses may be intermediate, rather than at the Planck scale.
This situation is similar to that of the exotic fractionally charged states that are endemic in the heterotic-string models [29].A phenomenological requirement on such states is that they appear in vector-like representations and are sufficiently massive or sufficiently rare to satisfy observational bounds.Models in which fractionally charged states appear only in the massive string spectrum, but not among the massless physical states, were dubbed as exophobic string models.Exophobic 3 generation models were found in the case of the PS heterotic-string models [11], but not in the cases of the FSU5 [12], the SLM [14], or the LRS [15], models.We may anticipate a similar situation with respect to the extra vector-like colour triplets that appear in these constructions.

Analysis of one Exemplary Model
It is interesting to examine in detail one of our 141568 exemplary models.As already mentioned, some of these models contain hidden enhancements but it's preferable to select a minimal model with no enhancement.We also choose a model with a minimal number of exotics states.It would have been preferable to have found a model with none of the vectorial triplets/antitriplets but, as already mentioned, no good models were found to derive from fertile cores with no triplet/anti-triplets.

Conclusion
The left-right symmetric models represent an appealing extension of the Standard Model [30] restoring the left-right symmetry in its spectrum, and attributing its violation to spontaneous symmetry breaking.Furthermore, it mandates the existence of right-handed neutrinos and has a natural embedding in SO (10).From the point of view of heterotic-string model building they also represent an interesting case, as they do not follow from the more common SO( 12) × E 8 × E 8 route, but rather from the pattern SO( 16) × E 7 × E 7 [7].Resulting in models in which all U(1) symmetries are anomaly free [7], and in particular TrU(1) 1,2,3 = 0.In this paper, we presented a model with TrU(1) 1 = TrU(1) 2 = 0, in which case the contribution to the anomalies arises from exotic states producing sectors.
In terms of the fermionic Z 2 ×Z 2 classification program the LRS models present challenges that are similar to the SLM classification.In both cases there is a proliferation of exotic states producing sectors, lowering the frequency of viable three generation models in the total space of models.For that purpose, one identifies fertile conditions at the SO(10) level and selects cores that are amenable to producing three generation configurations.Around these fertile cores a complete classification of the SO (10) breaking phases is performed.In ref. [15] a classification using the random generation method was performed producing a small number of three generation models.Adopting the two stage classification method in this paper, the number of viable models is increased by four orders of magnitude.Furthermore, we showed that the fertility conditions are associated with a novel doublet-triplet splitting mechanism that operates in the twisted sectors of the LRS vacua.While a doublet-triplet splitting was demonstrated in the past for untwisted states [23], the doublet-triplet splitting mechanism identified herein operates in the twisted sectors, and may be employed in SLM and PS heterotic-string models as well.The stage is now ripe for adopting novel computational methods in the classification program [31] to identify patterns in the GGSO coefficient space that are conducive for producing viable phenomenological characteristics.
last two conditions are the doublet-triplet splitting constraints for the vectorials stemming from the sector S + b 3 + x = S + b 1 + b 2 .

Figure 1 :
Figure 1: Number of three generation fertile cores versus number of twisted fertile vectorial representations from our set of 19374 fertile cores S e 1 e 2 e 3 e 4 e 5 e 6 b 1 b 2 L L s = 1 = nL L s and n L R s = 1 = n RL s , whilst the vectorial exotic numbers are n 3v = 1 = n3 v and n 1v = 5 = n1 v .Additionally, the Pati-Salam exotic numbers are n L L e = 4 and n L R e = 10.Another feature of this model is that it has an anomaly under the U(1) 2 and U(1) 3 gauge group factors since: Tr U(1) 2 = 12 and Tr U(1) 3 = 12 (48) which results in an anomalous U(1) combination of U(1) A = U(1) 2 + U(1) 3 .

Table 1 :
(10) characteristics of fertile SO(10)cores with no observable enhancements for 19374 distinct models derived through a scan over 10 9 randomly selected configurations (in a total of 4.4 × 10 12 possibilities)

Table 2 :
Statistics for the LRS models derived from fertile cores

Table 3 :
LRS quantum number statistics for three generation models.