Classification of Flipped SU(5) Heterotic-String Vacua

We extend the classification of the free fermionic heterotic-string vacua to models in which the SO(10) GUT symmetry at the string scale is broken to the flipped SU(5) subgroup. In our classification method, the set of basis vectors defined by the boundary conditions which are assigned to the free fermions is fixed and the enumeration of the string vacua is obtained in terms of the Generalised GSO (GGSO) projection coefficients entering the one-loop partition function. We derive algebraic expressions for the GGSO projections for all the physical states appearing in the sectors generated by the set of basis vectors. This enables the analysis of the entire string spectrum to be programmed in to a computer code therefore, we performed a statistical sampling in the space of 2^{44} (approximately 10^{13}) flipped $SU(5)$ vacua and scanned up to 10^{12} GGSO configurations. For that purpose, two independent codes were developed based on JAVA and FORTRAN95. All the results presented here are confirmed by the two independent routines. Contrary to the corresponding Pati-Salam classification, we do not find exophobic flipped SU(5) vacua with an odd number of generations. We study the structure of exotic states appearing in the three generation models that additionally contain a viable Higgs spectrum. Moreover, we demonstrate the existence of models in which all the exotic states are confined by a hidden sector non-Abelian gauge symmetry as well as models that may admit the racetrack mechanism.


Introduction
The LHC discovery of a Higgs-like resonance [1] lends further support to the viability of the Standard Model as the effective parameterisation of all observational subatomic data up to the GUT or Planck scales. This hypothesis is further supported by the proton lifetime, the neutrino mass suppression and the logarithmic evolution of the Standard Model parameters in the gauge and heavy generation matter sectors. The logarithmic evolution in the scalar sector is spoiled by radiative corrections from the cutoff scale. Restoration of the logarithmic running in the scalar sector suggests the existence of a new symmetry, with supersymmetry being a concrete example of contemporary interest.
Despite its enormous success in accounting for observational subatomic data, the Standard Model is unsatisfactory. It contains too many ad hoc parameters. The gauge symmetries and representations are not selected by a fundamental principle. The Standard Model * requires at least twenty-six additional parameters to account for the available data. The Standard Model gauge and flavour parameters can only be determined in a theory that unifies the gauge interactions with gravity. String theory provides a framework to study how the elementary particle's attributes may arise from a consistent theory of gauge-gravity unification. This necessitates the construction of quasi-realistic string models and the investigation of their phenomenological properties. The tools assembled for this purpose include target-space and worldsheet constructions [2].
The early quasi-realistic free fermionic models consisted of a few examples that shared an underlying NAHE-based structure [9]. Contemporary research in string model building focuses on explorations of large classes of string vacua. Over the last decade, tools for the classification of the symmetric Z 2 × Z 2 free fermionic orbifolds were derived in [10] for type II superstring and extended in [11,12]. Classification of the heterotic-string vacua with unbroken SO (10) and E 6 GUT groups revealed the existence of a symmetry in the space of Z 2 and Z 2 × Z 2 string models under the exchange of spinorial plus anti-spinorial and vectorial representations of SO(10) [12,13], which resembles mirror symmetry [14]. The classification was extended to the string vacua in which the SO (10) symmetry is broken to the Pati-Salam subgroup in [15]. It revealed the existence of exophobic string vacua, in which fractionally charged states (exotics) appear in the massive spectrum, but do not exist among the massless states. A concrete three generation exophobic model was studied in [16] and was shown to accommodate qualitatively viable phenomenology. The classification method in [18], was used to fish out an exophobic model in which the E 6 symmetry is broken to the maximal SU(6) × SU (2) subgroup, which admits an additional family universal and anomaly free U(1) symmetry beyond the U(1) generators of the SO (10) GUT group [19].
The classification methodology developed in [10,11,12,15], provides a useful tool to explore the properties of large classes of string vacua. In this paper, we extend the classification to models in which the SO(10) symmetry is broken to the flipped SU (5) subgroup [20]. The novel aspect in the classification of this class of string models, is that the basis vectors that generate these contain boundary conditions that give rise to complex phases, whilst the models in the previous studies contained only periodic and antiperiodic boundary conditions. Extension of the classification method to the flipped SU(5) case is also a necessary step towards the classification of Standard-like string vacua, which utilises both the Pati-Salam and flipped SU (5) generating basis vectors. A question of particular interest in the classification is the existence of quasi-realistic exophobic three generation flipped SU(5) models. We find that such a model does not exist in the space of the order of 10 12 that we explore. Our scan shows that the exophobic flipped SU(5) models exist in the string vacua with an even number of generations, but not in those with an odd number.

Flipped SU (5) Free Fermionic Models
The quasi-realistic free fermionic models correspond to Z 2 × Z 2 orbifold compactifications with N = (2, 0) super-conformal worldsheet symmetry. The free fermionic formulation [3] is set at an extended symmetry point in the moduli space, where the compactified directions are represented in terms of two dimensional fermions propagating on the string worldsheet [21,22]. Exactly marginal deformations from the free fermionic point are obtained by incorporating worldsheet Thirring interactions among the worldsheet fermions [23]. The free fermionic formulation provides a set of rules that enables straightforward derivation of the physical states and interactions plus is suited to explore the phenomenological properties of the string vacua. The matter states in the free fermionic models arise from the spinorial 16 representations of SO (10), whilst the Higgs states arise from the vectorial 10 representation. The Z 2 × Z 2 free fermionic orbifold models therefore preserve the Standard Model spectrum embedded in the SO(10) GUT group. The SO (10) symmetry is broken at the string scale, leading to the gauge symmetry in the low energy effective field theory being a subgroup of the SO (10).
In this paper, we extend the classification method to the case of the flipped SU (5) subgroup. The distinctive feature of these models is the utilisation of rational boundary conditions, whereas the SO (10) and SO(6)×SO(4) that were classified previously only used periodic and anti-periodic boundary conditions. Our classification method, entails that the GGSO projections for all the states that arise in the twisted sectors are expressed in terms of algebraic equations. The equations are incorporated in a computer program that facilitates scanning a large number of models.
The basis vectors generate a space Ξ that consists of 2 N +1 sectors which produce the string spectrum. Each sector is given by a linear combination of the basis vectors where N j · v j = 0 mod 2. The basis vectors induce the GGSO projections, with action on a given string state |S ξ given by where δ ξ = ±1 is the space-time spin statistics index and F ξ is the fermion number operator. Different sets of GGSO projection coefficients c ξ v i = ±1; ±i, consistent with modular invariance produce different models. To summarize, a model is defined by a set of basis vectors v 1 , . . . , v N and a set of 2 N (N −1)/2 independent GGSO projection coefficients C v i v j , i > j defining the string spectrum.

SO(10) Models
The flipped SU(5) models we consider here are generated by a set of 13 basis vectors. The first 12, consist of the same basis vectors that were used in the classification of the SO(10) vacua [12], which are given by v 1 = 1 = {ψ µ , χ 1,...,6 , y 1,...,6 , ω 1,...,6 | y 1,...,6 , ω 1,...,6 , η 1,2,3 , ψ 1,...,5 , φ 1,...,8 }, The basis vectors 1 and S, generate a model with the SO(44) gauge symmetry and N = 4 space-time supersymmetry. The vectors e 1 ,. . . ,e 6 , corresponding to all the possible symmetric shifts of the six internal coordinates, break the SO(44) gauge group to SO(32) × U(1) 6 and preserve the N = 4 space-time supersymmetry. The vectors b 1 and b 2 correspond to the Z 2 × Z 2 orbifold twists, which break N = 4 to N = 1 supersymmetry and reduces the rank of the group as the U(1) 6 is broken, leaving the SO(32) symmetry to decompose to the SO(10) × U(1) 3 × SO(16) gauge group. Furthermore, the SO(10) × U(1) 3 group corresponds to our observable and the SO(16) group to our hidden gauge group. The remaining fermions that are not affected by the action of the previous vectors are φ 1,...,8 , which correspond to the SO(16) gauge group. The vectors z 1 and z 2 reduce the untwisted hidden gauge group from SO (16) to SO(8) × SO (8). This choice of basis is the most general set of basis vectors with symmetric shifts for the internal fermions compatible with a SO(10) GUT group. The untwisted vector bosons consistent with the GGSO projections induced by the choice of basis vectors in (2.2), generate the adjoint representation of an SO(10) × U(1) 3 × SO(8) 2 gauge group.

Flipped SU(5) Construction
The SO(10) GUT models generated by (2.2) are broken to the flipped SU(5) subgroup, by the rational boundary condition assignment of the complex right-moving fermions ψ 1,...,5 = ± 1 2 . This is achieved with the addition of the basis vector v 13 = α. The case of the SO(6) × SO(4) models, which were classified in [15], utilize solely periodic and anti-periodic boundary conditions. In this case, the choice of α compatible with the set (2.2) is unique and is given by α = {ψ 4,5 , φ 1,2 }. All other possible assignments that reduce the SO(10) symmetry to the SO(6) × SO(4) are equivalent. As in the cases of other free fermionic flipped SU(5) models constructed to date [4,5], we restrict the assignment of ψ 1,...,5 to the case of positive 1/2 boundary conditions. Furthermore, unlike the case of the SO(6) × SO(4) models, the choice of the basis vector α that breaks the SO(10) symmetry to SU(5) × U(1) is not unique. Also, the assignment of the three complex worldsheet fermions η 1,2,3 = 1/2 is fixed by the modular invariance constraint b j · α = 0 mod 1. Consequently, it follows that the assignment of the boundary conditions of the eight worldsheet complex fermions ψ 1,...,5 , η 1,2,3 is unique and the variation is in the boundary conditions of the worldsheet fermions φ 1,...,8 . Modular invariance constraints, restrict the possibilities to assigning 1/2 boundary conditions of φ 1,...,8 worldsheet fermions to 0, 4 or 8. The null case been given by is automatically excluded because the sector x = 2α = {ψ 1,...,5 , η 1,2,3 } enhances the SU(5) × U(1) gauge group back to the SO(10) symmetry. The condition z 1,2 · α = 0 mod 1, imposes the assignment of 1/2 boundary conditions to 0, 2 or 4 of each of the groups of worldsheet fermions φ 1,...,4 and φ 5,..., 8 . The possible choices of v 13 are then given by The α's above require that the sets of basis vectors are linearly independent. This does not hold for the cases with α 1 and α 2 , since in these cases we obtain In order to keep the set of basis vectors in (2.2), in addition to α 1 or α 2 being linearly independent, we choose to remove the basis vector 1 leaving the set of 12 vectors {S, e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , b 1 , b 2 , z 1 , z 2 , α i }, where i = 1 or 2. In the case with α 3 , the set in (2.2) is linearly independent giving us the set {1, S, e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , b 1 , b 2 , z 1 , z 2 , α 3 }.
The plan for the remainder of the paper, is to give a comprehensive view of the methodology and an insight into the classification of the SU(5) × U(1) models with the inclusion of α 1 in the basis. The classification was carried out by using two independent codes, the first being the JAVA and the second being the FORTRAN95 code. It was then also carried out in the case of α 2 . The details of the formulae needed for this classification can be obtained from the authors and will be published in a separate publication [24]. The classification using α 3 will be reported in future work.

GGSO projections
In order to define the string vacua, the GGSO projection coefficients appearing in the one-loop partition function c v i v j need to be specified. Taking the coefficients to span a 12 × 12 matrix, only the elements i ≥ j are independent. Modular invariance dictates that the 66 lower triangle elements of the matrix are fixed by the corresponding 66 upper triangle elements. Adding the remaining 12 diagonal terms, we are left with 78 independent coefficients corresponding to 2 78 ≈ 3 × 10 23 different string vacua. Moreover, requiring that the models possess N = 1 space-time supersymmetry, we fix eleven of the coefficients. Without loss of generality we set the associated GGSO projection coefficients Modular invariance imposes additional constraints on the diagonal terms. In our case, where the vector 1 is composite, they are given by where C z 2 z 2 is independent of any term. Further analysis of the GGSO projections of interest, shows that there are additional phases that do not effect the properties of the string spectrum. As a result, the following coefficients are fixed in the ensuing analysis (2.6) where i = 1, ..., 6. Taking the equations (2.4), (2.5) and (2.6) we are left with 44 independent coefficients which can take two discrete values ±1, except in the cases C α b 1 , C α b 2 and C α z 2 , where they take the values ±i since α · b 1 = −3 (odd), α · b 2 = −3 (odd) and α · z 2 = −1 (odd). Furthermore, a simple counting gives 2 44 ≈ 1.76 × 10 13 vacua in this class of superstring models. We note that there may still exist some degeneracies in this space of vacua with regard to the characteristics of the low energy effective field theory, and in particular with respect to the observable massless states. For instance, the three twisted sectors of Z 2 × Z 2 toroidal orbifolds possess a cyclic permutation symmetry. Nevertheless, some of the vacua that may seem identical in the low energy effective field theory limit of the observable sector, differ by other properties, such as the massive spectrum, superpotential couplings, hidden sector matter states and are therefore distinct.

String Spectrum
The vector bosons from the untwisted sector generate the (6) gauge symmetry. Depending on the choices of the GGSO projection coefficients, extra space-time vector bosons may be obtained from the following twelve sectors The projections on the sectors 3α, z 1 + 3α, z 2 + 3α, z 1 + z 2 + 3α can be inferred from the projections on the sectors α, z 1 + α, z 2 + α, z 1 + z 2 + α respectively. Therefore, we will not discuss them in detail. The gauge bosons that are obtained from the sectors in (3.1) enhance the untwisted gauge symmetry. We impose the restriction that the only gauge bosons that remain in the spectrum are those that are obtained from the untwisted sector. The gauge groups in these models are therefore Observable : The NS sector matter spectrum is common in these models and consists of three pairs of 5 and 5 representations of the observable SU(5) × U(1) 5 gauge group and twelve that are singlets under the non-Abelian gauge symmetries.

Exotic Matter Spectrum
In the string spectrum, additional sectors exist which produce fractionally charged states under the SU(5) × U(1) symmetry. These sectors arise when we have massless states, which are produced from a linear combination of basis vectors that include the vector α, resulting in the breaking of SO(10) symmetry. Moreover, these sectors produce states that do not fall into representations of the underlying SO(10) GUT symmetry. Specifically, they possess fractionally charged assignments with respect to the U(1) symmetry in the decomposition SO(10) −→ SU(5) × U(1). Consequently, provided that the weak hypercharge has the canonical SO(10) GUT embedding and the canonical GUT prediction sin 2 θ w = 3/8, these sectors produce states that carry fractional electric charges. This is a generic feature of string compactifications [25,26], that may have interesting phenomenological implications [27], as electric charge conservation implies that the lightest of those exotic states is necessarily stable. Many experimental searches for fractionally charged matter have been conducted [28]. However, no reported observation of any such particles has ever been confirmed and there are strong upper bounds on their abundance [28]. This implies that such exotic states in string models should be either confined into integrally charged states [4], or be sufficiently heavy and diluted in the cosmological evolution of the universe [27]. The first of these solutions is problematic, due to the effect of the charged states on the renormalisation group running of the weak-hypercharge and gauge coupling unification. The preferred solution is therefore for the fractionally charged states to become sufficiently massive, i.e. with a mass which is larger than the GUT scale. In this case the fractionally charged states can be diluted by the inflationary evolution of the universe. Due to their heavy mass they will not be reproduced during reheating and the experimental constraints can be evaded. Three generation Pati-Salam heterotic-string models in which the fractionally charged states arise in the massive string spectrum but not as massless states, that were constructed in [15], which are dubbed as the quasi-realistic exophobic Pati-Salam string models. A particular question of interest in the current work is the existence of quasi-realistic flipped SU(5) heterotic-string models. Also, it should be noted that the sectors appearing in (3.4) and (3.5) contain the combination 2α and do not break the SO(10) symmetry. Therefore, these sectors do not produce exotic states under the SU(5) × U(1) gauge symmetry.
In the free fermionic construction, we classify the sectors that produce exotic states according to the product ξ R · ξ R = 4, 6, or 8. In the first case, massless states are obtained by acting on the vacuum with a Neveu-Schwarz fermion or with two oscillators with 1/4 frequencies. In the second case, oscillators with 1/4 frequency are needed to produce massless states, whereas in the third case no oscillators are used to produce massless states. Furthermore, in the third case with no oscillators, we have the following 96 sectors These produce states that are singlets under the observable SU(5) but are charged under the U(1) 5 and are given by 1, − 5 4 and 1, + 5 4 . We now move on to the second case that consists of oscillators with one 1/4 frequency giving rise to additional massless vector-like states given by the following 48 sectors As an example, the sectors in B (19) pqrs produce the following states: pqrs , where |R (19) pqrs is the degenerate Ramond vacuum of the B (19) pqrs sector. These states transform as vector-like representations under the U(1) 1 .
Similarly the sectors in B (20) pqrs and B (21) pqrs produce the states above. What is more, similar states appear in the following 48 sectors The only difference between the sectors in (3.8) and (3.9) is the sign of the 1/2 boundary condition of the worldsheet fermion φ 1,...,4 . This changes some of the U (1) charges arising in (3.8) compared to those arising in (3.9), but the structure and type of states are similar to those listed above. Finally, the first case of exotic states arise in the sectors α and z 1 + α. These exotic states can be eliminated by the same conditions that eliminate the space-time vector bosons arising in these sectors which will be discussed in section 5.

Twisted matter spectrum
The counting of spinorial and vector-like representations in the given string vacua is realised by utilising the so called projectors. Each sector B i pqrs , corresponds to a projector, P i pqrs = 0, 1, which is expressed in terms of GGSO coefficients and determines whether a given sector survives the GGSO projections. It is noted with the basis vectors given in (2.2), each fixed point of the Z 2 × Z 2 orbifold corresponds to a distinct sector ξ in the additive group. In this method, the states arising from each fixed point are, therefore, controlled individually. Furthermore, the computational analysis is facilitated by rewriting the projectors in an analytic form. These are written as algebraic conditions, for the individual states arising in the string spectrum, in terms of the GGSO phases of the basis vectors. The algebraic expressions are inserted into the computer code, which enables the scan of the large space of models spanned by the basis GGSO phases.

Observable spinorial states
In order to get the particle content for the representations for the sectors in (3.2), we used the following normalisations for the hypercharge and the electromagnetic charge: Where the Q i charges of a state, arise due to ψ i for i = 1, ..., 5. The following table summarises the charges of the colour SU (3) and electroweak SU(2) × U(1) Cartan generators, of the states which form the SU (5) Here " + ", and " − ", label the contribution of an oscillator with fermion number corresponds to a part of the Ramond vacuum formed by two oscillators with fermion number F = 0 and one oscillator with fermion numbers F = −1. These states correspond to particles of the Standard Model. More precisely we can decompose these representations under SU ( where L is the lepton-doublet; Q is the quark-doublet; d c , u c , e c and ν c are the quark and lepton singlets. Because of the α-projection, which projects on incomplete 16 and 16 representations, complete families and anti-families are formed by combining states from different sectors.

Chirality Operators
A phenomenologically viable model consists of 3 families of chiral 16 representations of SO(10) decomposed under SU(5) × U(1). Therefore, we have to count the number of 16s and 16s. The choice of GGSO coefficients determine the model we consider and therefore the number of families. In order to be able to distinguish between 16 and 16, one has to define operators that determine the representations in which the states of each observable sector fall into. The operators X (1,2,3) SO(10) pqrs = ±1, defines the SO(10) chirality (16 or 16) for B 1 pqrs , B 2 pqrs and B 3 pqrs , which are given by This is in contrast to the case in the Pati-Salam heterotic-string models, where one needs to determine the chirality of the SO (6) and SO (4) representations separately [15]. Additionally, we determine which components in the 16 and 16 survive the α projection, which breaks SO (10) to SU(5) × U(1). In this respect, we note that the α projection operates identically on the 1 ≡ (1, +5/2) and 5 ≡ (5, −3/2) states and similarly on the conjugate representations 1 ≡ (1, −5/2) and 5 ≡ (5, +3/2).
The surviving components are determined by defining the operators X (5) (10). These conditions are expressed as

Projectors
The states in the sectors in B (1) pqrs , as given in (3.2), can be projected in or out of the string spectrum depending on the GGSO projections of the vectors e 1 , e 2 , z 1 and z 2 . Likewise for B (2) pqrs and B (3) pqrs , we define a projector P such that the states survive when P = 1 and are projected out when P = 0, which are given as: These projectors can be expressed as a system of linear equations with p, q, r and s as unknowns. The solutions of such a system of equations yield the different combinations of p, q, r and s for which sectors survive the GGSO projections. The analytic expressions for each of the different projectors P 1,2,3 pqrs are given in a matrix Here the GGSO phases are defined as where v i and v j refer to the basis vectors and the GGSO projections are defined as in (2.1). The corresponding algebraic expressions for the states from the remaining sectors above are enumerated in the appendix, as well as the states in the hidden sector which are not classified here, may play a substantial role in the string phenomenology, such as in the case of the SUSY breaking. Furthermore, the projectors presented in the appendix determine the number of surviving observable, hidden and exotic states in each model.

Gauge Group Enhancements
The SU(5) × U(1) gauge symmetry generated by the untwisted space-time vector bosons, may be enhanced by the vector bosons that arise from the sectors listed in (3.1). We impose that all the additional space-time vector bosons are projected out. The gauge symmetry is therefore identical in all the models that we scan, though the occurrence of models with enhancements is approximately about 23.8% of the total models. The string models in our classification differ by the string spectrum that arises from the twisted sectors. In our classification method, we encode the GGSO projections coefficients in terms of algebraic equations, which are applied to all the sectors listed in section 3.
The gauge bosons of any given sector in (3.1) transform under a subgroup of the Neveu-Schwarz gauge group. If they survive the GGSO projections, then the NS gauge group is enhanced. We restrict our classification here to the cases without enhancement, by identifying when the gauge bosons survive the GGSO projections and generalize the formulae to eliminate them. We remark that models with enhanced gauge symmetry in the observable or hidden sectors may be of interest for various phenomenological reasons, as, for example, the SU(6) × SU(2) string models presented in [18]. Below, we present the different types of enhancements that can occur within the string spectrum from the sectors given in (3.1). In addition, we assume that only one set of conditions is satisfied from any one given sector in (3.1).

Observable gauge group enhancement
There is one sector contributing only to the enhancement of the observable gauge group i.e. SU (5) (1) 3 . This is the sector z 1 + 2α, given by the conditions: where i = 1, . . . , 6 and k = 1, 2.

Hidden gauge group enhancement
The vector bosons arising from the untwisted sector produce the hidden gauge symmetry, which is given as SU (4) (1) hid . Similar to the observable sector, there is one sector that enhances only the untwisted hidden sector gauge symmetry and is given by the sector z 1 + z 2 , where the conditions are given by: (1) where i = 1, . . . , 6 and k = 1, 2.

Mixed gauge group enhancements
The additional sectors in (3.1), produce vector bosons coming from the mixture of the observable and hidden sector gauge groups. The mixed gauge group enhancements are formed from the untwisted symmetries of the observable and hidden gauge group. These are given from the sectors z 1 , z 2 , α, z 1 + α, z 2 + α and z 1 + z 2 + α. The conditions are as follows: where i = 1, . . . , 6 and k = 1, 2. (8) where i, j = 1, . . . , 6, i = j and k = 1, 2.
Finally, we remark that as noted in section 3.3, the sectors α, z 1 + α, z 2 + α and z 1 + z 2 + α may also give rise to exotic states, when the left-moving ψ µ oscillator is replaced by a left-moving χ i oscillator. Moreover, we note that the GGSO projections of the basis vectors e 1,...,6 , z 1,2 and α do not distinguish between ψ µ and χ i , which can therefore be used to project both the enhancements, as well as the exotic states arising from the sectors α, z 1 + α, z 2 + α and z 1 + z 2 + α.

Results
By use of the algebraic expressions given in the sections previously, as well as in the appendix, we analyse the entire massless spectrum for a given choice of configuration of GGSO projection coefficients. These expressions can be seen as matrix equations which we programmed into a JAVA and FORTRAN95 code independently, that are used to scan the space of the string vacua. The number of possible configurations is 2 44 ≈ 10 13 , which is very large, for a classification of the entire string vacua. For this purpose, a random generation algorithm is used † and the characteristics of the models for each set of random GGSO projection coefficients are extracted. From the generated sample, a model with the desired phenomenological criteria can be fished out. This procedure was followed in [15,16], which produced a three generation Pati-Salam heterotic-string models that do not contain any exotic massless states with fractional electric charge. In this paper, we use this methodology to classify the flipped SU(5) free fermionic string models with respect to some phenomenological criteria. For example, a question of interest is the existence of viable three generation exophobic flipped SU(5) vacua. The observable sector of a heterotic-string flipped SU(5) model is characterised by 15 integers (n 1 , n 1 , n 5s , n 5s , n 10 , n 10 , n g , n 10H , n 5v , n 5v , n 5h , n 1e , n 1e , n 5e , n 5e ), which are given by: n 1 = # of (1, + 5 2 ), n 1 = # of (1, − 5 2 ), n 5s = # of (5, + 3 2 ), n 5s = # of (5, − 3 2 ), n 10 = # of (10, + 1 2 ), n 10 = # of (10, − 1 2 ), n g = n 10 − n 10 = n 5 − n 5 = # of generations, n 10H = n 10 + n 10 = # of non chiral heavy Higgs pairs, n 5v = # of (5, +1), n 5v = # of (5, −1), n 5h = n 5v + n 5v = # of non chiral light Higgs pairs, n 1e = # of (1, − 5 4 ) (exotic), n 1e = # of (1, + 5 4 ) (exotic), n 5e = # of (5, − 1 4 ) (exotic), n 5e = # of (5, + 1 4 ) (exotic). These numbers above are all relevant for the classification of the string vacua. As noted in section 4.2, the α projection dictates that n 1 = n 5s and n 1 = n 5s . Therefore, the counting of n 5 and n 5 is sufficient for the number generations as shown above. Moreover, we note the distinction between the 5 and 5 representations that arise from the spinorial 16 representation of SO(10) decomposed under SU(5) × U(1), denoted by n 5s , n 5s and the 5 and 5 that arise from its vectorial 10 representation, denoted by n 5v , n 5v . While the former gives rise to the Standard Model up-type quark electroweak singlet and lepton-doublet, the later accommodates the light electroweak Higgs doublets. In the flipped SU(5) models they are distinguished by their charges under the U(1) 5 symmetry. Using the methodology outlined in section 4, we obtained analytic formulas for all these quantities. In order to extract a string spectrum from the phenomenologically viable models of the flipped SU (5), we must have: Three light chiral of generations, n 10H ≥ 1 At least one heavy Higgs pair to break the SU(5) × U(1) symmetry, n 5h ≥ 1 At least one pair of light Minimal SM Higgs doublets, n 1e = n 1e ≥ 0 Heavy mass can be generated for vector-like exotics, n 5e = n 5e ≥ 0 Heavy mass can be generated for vector-like exotics.
Here, we imposed the constraints n 5h = n 5h , n 1e = n 1e and n 5e = n 5e , in order to sustain anomaly free flipped SU(5) models.

Minimal exophilic models
Compared to the case of the Pati-Salam classification [15], which yielded 3 generation models that are completely free of massless exotic states, no such models were found in our scan of the flipped SU(5) models. We emphasise that this does not indicate that exophobic free fermionic flipped SU(5) vacua do not exist, but merely that they do not exist in the space of vacua that we explored. Nevertheless, it does show that large spaces of vacua may not contain exophobic models, which is in line with related searches [30]. A model with a minimal number of exotic states that we find in our scan has n g = 3, n 5s = 3, n 5s = 0, n 10 = 4, n 10 = 1, n 10H = 1, n 5h = 4, n 1e = 2 and n 5e = 0. We note that this minimal model still contains exotics which has 4 states. The minimal model is then given by the following GGSO coefficients matrix: S e 1 e 2 e 3 e 4 e 5 e 6 b 1 b 2 z 1 z 2 α S 1 1 1 1 1 1 1 1 1 1 In section 6.3 we elaborate further on the structure of the exotic states in the flipped SU(5) models.

Classification
Next, we elaborate on the classification of the space of the free fermionic flipped SU(5) string vacua by performing a statistical sampling, in a space of 10 12 models out of the 2 44 possibilities. For this purpose, we developed two independent computer codes. One being a FORTRAN95 computer program running on a single node of the Theoretical Physics Division of University of Ioannina, HPC cluster. The other being a JAVA code running on 10 nodes of the University of Liverpool, Department of Physics ULGQCD cluster that runs on AMD Opteron 6128 2GhZ CPUs. Additionally, assistance from five servers of the University of Liverpool, Department of Mathematics were also used, which totalled to 200 CPUs that ran for about 2 weeks in order to scan 10 12 random models. Some of the results are presented in figures 1 -3 and table 1 -3.
In figure 1, the number of models against the number of generations is displayed. This is in agreement with the results of [11,12,15], where the number of models peaks for 0 generation and decreases as the number of generations increases. Also in this figure, we note the absence of models with 7, 9, 11 and greater than 12 generations. These result can be understood in light of the corresponding results in the SO(10) classification [12]. We recall that the α projection which breaks the SO(10) symmetry to the flipped SU (5), truncates the number of generations by two. Examining the corresponding figure in the SO(10) classification, we observe the absence of the models with double the number of generations, i.e. with 14, 18, 22 and more than 24 generations. We remark that this result is applicable to the case in which all gauge group enhancements are projected out, as discussed in section 5. Therefore, models with the excluded number of generations may occur when the hidden gauge group is enhanced. However, we compare figure 1 to the corresponding figure in [15] and notice the existence of models with 16 generations, which seems to contradict our argument. This apparent contradiction is resolved by noting that the α projection in the Pati-Salam case projects out some of the gauge group enhancements, whereas from section 5 we note that this is not the case for the basis vector α 1 we used here. Therefore, some of these models descend from SO (10) n 5h /n 10H   0  1  2  3  0  281477  28518  0  0  1  3626622 275967 8197 651  2  630727  61910 2092  0  3  23924485  63774 5901  0  4  78959  67900  0  0  5 139642 12380 0 0 Table 1: Number of three generation models as a function of the flipped SU (5) breaking Higgs pairs (n 10H ) and SM breaking Higgs pairs (n 5h ) in a random sample of 10 10 models. Models with n 10H = 0 or n 5h = 0 are not SM compatible, the former because SU(5) cannot be broken to the SM via the Higgs mechanism and the latter due to the lack of SM breaking Higgs scalars.
GUTs with enhanced gauge group, which do not arise in the case of the flipped SU(5) models studied here.
In table 1, we display the number of three generation models against the number of pairs of light and heavy Higgs representations appearing in the models, with the light and heavy pairs being 5 + 5 and 10 + 10 representations of SU (5), respectively. Clearly, the null cases are not viable phenomenologically and the minimal cases are models with one pair of each. In models with a larger number of light Higgs pairs it may be easier to accommodate the Standard Model fermion mass textures, whereas models with a larger number of heavy Higgs pairs may facilitate gauge coupling unification at the string scale [5,33].
As seen in section 3.3, some of the exotic matter states in the models transform in vector-like representations of the hidden sector non-Abelian group factors. They carry fractional electric charge and must be sufficiently massive or confined. These exotic states may nevertheless have interesting phenomenological implications. In table 2, we explore the structure of the exotic states arising in the models, which are labelled by four integers, (n e 5 , n e 1 , n e 4 , n e 4 ′ ), where n e 5 = n 5e + n 5e is the number of exotic states that transform as 5 and 5 of the observable SU (5); n e 1 = n 1e + n 1e is the number of exotic states that transform as singlets of all non-Abelian group factors; n e 4 = n 4e + n 4e is the number of exotic states that transform as 4 and 4 of the hidden SU (4); n e 4 ′ = n 4 ′ e + n 4 ′ e is the number of exotic states that transform as 4 and 4 of the hidden SO(6) gauge group.
In figure 2, we display the number of exophobic models against the number of generations. The striking feature in this figure is the absence of models with three chiral generations. This is in contrast to the case of the Pati-Salam models that yielded numerous three generation exophobic models. Figure 2 also reveals the absence of any exophobic models with 1,3,5,7,9,11 and any exophobic models with more than 12 generation, whereas exophobic models arise for even number of generations, up to 12. As a result, exophobic models in this class arise in configurations with even number of generations but not in models with an odd number of generations. We emphasize that these results hold in the space of models that we explore here and does indicate absence of three generation exophobic flipped SU(5) models. In figure  3, we display the number of three generation models against the number of exotic multiplets. We note from the figure that the minimal number of exotic multiplets is 4.

The structure of the exotic states
One of the main highlights of our classification method in the case of the Pati-Salam heterotic-string models has been the discovery of the exophobic heteroticstring models, in which all exotic states are limited to the massive spectrum and do not appear among the massless states. As shown in figures 2 and 3, in the class of 10 12 flipped SU(5) models that we analysed here, there are no exophobic 3 generation vacua with statistical a frequency larger than 1 : 10 12 . The structure of the exotic states arising in the models is analysed further in table 2. All the models given in the table 2 contain three chiral generations of which at least one-pair is the light Higgs states and at least one pair is the heavy Higgs states. Thus, in all these models the gauge symmetry can be broken to the Standard Model in the effective field theory limit and contain all the fields required for viable Standard Model phenomenology.

Number of Multiplets
Number of Models Figure 3: Logarithm of the number 3 generation models against the number of exotic multiplets (n e = n 1e + n 1e + n 5e + n 5e ) in a random sample of 10 12 flipped SU (5) configurations.
From table 2, we note the occurrence of models in which all exotic states transform in representations of an hidden non-Abelian gauge group. In this case, the exotic states are confined into integrally charged states and produce the so-called cryptons [31]. We further note from this table, the existence of models with equal number of 4 and 4 ′ states. This suggests the possible existence of the free fermionic models that admit the race-track mechanism to stabilise the vacuum expectation value of the dilaton field [32]. Moreover, table 2 reveals interesting observations and directions for future research. The first eleven models in the table contain only states that transform in non-trivial representations of an hidden non-Abelian gauge group. Thus, this class of models may give rise to the so-called crypton states that are confined into integrally charged states. We see that there is an abundance of such models. There are also numerous models with small number of crypton states that may remain asymptotically free and therefore, confined at some scale. A well known example of a model that gives rise only to crypton like states is given in [4]. The table shows the existence of a large space of models with similar characteristics. One notable difference between the vacua in this table and the one of [4], is the fact that the model in [4] uses asymmetric internal shifts, whereas the models in this table only use symmetric internal shifts. The models in the six and twelfth rows of the table, with n 4 = n 4 ′ = 2 are interesting to study for implementation of the racetrack mechanism [32].
Turning to the other types of exotic states. The non-Abelian singlet states that are counted in the second column are fractionally charged and must decouple from the light spectrum or be sufficiently diluted. The fields counted in the first column transform as 5 and 5 of the observable SU(5) and carry 1/2 of the hypercharge compared to the standard flipped SU (5) states. Such states do not arise in the flipped SU(5) model studied in [4], but their colour triplet and electroweak doublet components arise generically in the standard-like heterotic-string models [7,33]. These fields may be instrumental as intermediate matter states to resolve the conflict between heterotic-string scale unification and the low scale gauge coupling experimental data [33]. The models appearing in the 13 th and 25 rd rows in table 2, are interesting examples of flipped SU(5) models admitting such states. The models in the 25 rd row with n 5 = 2, n 1 = 6 , n 4 = 2 and n 4 ′ = 2 may accommodate both the intermediate matter thresholds and the racetrack mechanism, therefore be of particular interest.

An illustrative example
In this section, we analyse the model given in (6.2) as an illustrative of the exotic spectrum appearing in the flipped SU(5) models. The twisted sectors of the model given here produce three chiral generations; one pair of heavy Higgs states; one pair of light Higgs representations. Therefore, this model may yield viable Standard Model phenomenology.
Additionally, the model contains the following states that transform under the hidden SU(4) gauge group: six non-exotic pairs of (4 + 4); one non-exotic state transforming in the vectorial 6 representation; one pair of exotic states transforming as (4 + 4). The model contains the following states that transform under the hidden SU(4) ′ gauge group: four non-exotic pairs of (4 + 4); one non-exotic state transforming in the vectorial 6 representation; one pair of exotic states transforming as (4 + 4). Thus, the β-functions of the SU(4) and SU(4) ′ hidden gauge groups are β 4 = −4 and β 4 ′ = −6, respectively. Depending on the mass scales for the hidden sector matter states, this model may therefore provide workable example for implementing the racetrack mechanism. The model also contains one pair of exotic (5 + 5) states of the observable flipped SU(5) group that can be used to mitigate the gauge coupling unification problem.  Table 2: Number of three generation models with (n 10H ≥ 1, n 5h ≥ 1) against fractional charge state multiplicities in a sample of 10 10 randomly selected models. Here n 5 is the number of 5 + 5 SU(5) pairs, n 1 is the number of fractional SU(5) singlet pairs, n 4 is the number of 4 + 4 pairs transforming under hidden SU(4) and n ′ 4 is the number of 4 + 4 pairs transforming under hidden SU(4) ′ .

Conclusion
String theory continues to provide the only viable contemporary framework to explore the unification of gravity with the gauge interactions. For this reason, models with three generations must be constructed phenomenologically. Whilst examples of such fully realistic models may be a long way into the future, string theory provides an abundance of these concrete quasi-realistic examples that can be explored as toy models on the way to achieving the ultimate goal.
Here, we have continued to develop the methodology for the classification of the free fermionic heterotic-string models. The free fermionic construction [3], gave rise to some of the most realistic string models constructed to date [4,6,7,8,15,16]. These models correspond to Z 2 ×Z 2 orbifold compactifications at special points in the moduli space with discrete Wilson lines [21,22]. The classification methodology was developed in [10] for the type II superstrings and then adopted for the classification of free fermionic heterotic-string models in [11,12,15]. In this method, the set of basis vectors are fixed incorporating all the possible symmetric Z 2 -shifts along the internal compactified directions, in the form of the six basis vectors e i with adequate boundary conditions. The enumeration of the models is then obtained in terms of the GGSO projection coefficients and enables the scanning of large number of models with the aid of computers. The initial classification in [11] was with respect to chiral 16 and 16 representations of an unbroken SO(10) GUT group. Inclusion of the xmap [17] in the methodology facilitated the classification with respect to the vectorial 10 representations, which led to the discovery of spinor-vector duality [12,13]. The classification methodology relies on writing the GGSO projections in algebraic forms, which makes the enumeration of the massless spectrum for a given configuration of GGSO phases straightforward. This classification was also extended to models in which the SO(10) symmetry was broken to the Pati-Salam subgroup leading to the discovery of exophobic string vacua [15,16].
In this paper, we extended the development of the classification methodology to the class of the free fermionic heterotic-string vacua in which the SO(10) symmetry is broken to the flipped SU (5) subgroup. This case presents several complications with respect to the previous ones. Whilst the earlier cases use only periodic and anti-periodic boundary conditions, the flipped SU(5) class of vacua requires the use of rational boundary conditions. Another, is the variation of the basis vectors that are used to break the SO(10) symmetry. With these modifications, while adaptation of the algebraic expressions is readily available, the computerised classification is substantially complicated. For this purpose, we developed two completely independent software routines, one based on JAVA and the second one being FORTRAN95 and all results presented in this paper are crossed checked using them. Our classification is limited to 1/2 rational boundary conditions, which is the case in all the quasi-realistic free fermionic models to date.
In table 3, we tabulate the number of models with sequential imposition of phe-  Table 3: Statistics for the flipped SU(5) models with respect to phenomenological constraints. Here we note that the results of 4a and 4b have no effect on each other and this also holds for 6a and 6b.
nomenological constraints. The total number of models in the sample is 10 12 . We first impose that there is no enhancement of the four dimensional gauge symmetry and there are approximately 76.2% of the models that satisfy this criteria. Next we impose, that the flipped SU(5) models are anomaly free with respect to the U(1) 5 group factor and about 14% of the total models satisfy this criterion. A further reduction, by three orders of magnitude, results from the restriction to the three chiral generations. Next, imposing the existence of both the heavy and light Higgs states to break the flipped SU(5) gauge symmetry to the Standard Model gauge group and the electroweak breaking respectively, leads to a further reduction i.e. one order of magnitude. Finally, imposing the minimal number of massless exotic states results in the reduction of the number of models by a further order of magnitude. Therefore, the number of string vacua in the space of models scanned, reduces from 10 12 to 10 6 which satisfy all the constraints that were imposed. This leaves a substantial number to accommodate further phenomenological constraints. In conclusion, we comment on the results obtained by using α 2 and α 3 in (2.3) to break the SO(10) symmetry. In both cases a JAVA code was used to classify the models. The results are not that substantially different compared to the classification with α 1 and we also do not find any three generation exophobic vacua in these cases.

Acknowledgements
We would like to thank Laura Bernard and Ivan Glasser for collaboration in the initial stages of this project. JR would like to thank the University of Liverpool and CERN, AEF would like to thank CERN, the University of Ioannina and Oxford University for their hospitality, and HS would like to thank Johar Ashfaque for the many useful discussions. AEF is supported in part by STFC under contract ST/J000493/1. JR's work has been supported in part by the ITN network "UNILHC" (PITN-GA-2009-237920). HS is supported by the STFC studentship award. This research has been co-financed by the European Union (European Social Fund-ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning", of the National Strategic Reference Framework (NSRF) (Thales Research Funding Program) investing in knowledge society through the European Social Fund.

The sectors
B (19,20,21) pqrs = B (1,2,3) pqrs + α produce massless states that are obtained by acting on the vacuum with a fermionic oscillator. Below we list the type of states that are produced and the projectors that act on them.