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Exactly Solvable (0,2) Supersymmetric String Vacua With GUT Gauge Groups

We present a construction of modular invariant partition functions for heterotic (0,2) supersymmetric classical string vacua. This generalization of Gepner's construction yields GUT gauge groups E_6, SO(10), SU(5) and SU(3) x SU(2) x U(1)^r, respectively. By calculating the massless spectrum of some of these models we find strong indications that they correspond to (0,2) string vacua discussed recently in the context of CYM/LG phases.


Introduction
Due to the lack of a nonperturbative formulation of string theory we are still restricted to a perturbative search for reasonable string vacua. In the last years classical solutions with N = 1 space-time supersymmetry have been studied intensively. A necessary condition is that the non flat space-time directions are compactified on a Calabi-Yau manifold (CYM) [5] or that the internal conformal field theory (CFT) has (0, 2) world sheet supersymmetry [2], respectively. Besides early indications that generic (0, 2) string models might be destabilized by world sheet instantons [6], the symmetric (2, 2) models are much easier to handle, so that most effort focused only on their investigation. The implied restriction for the nonlinear σ model is that the spin connection is identified with the gauge connection breaking one of the E 8 factors down to E 6 . However, ever since the work of Witten [19] in 1986 as well as Distler and Greene [7] in 1988, it has been known that weakening the latter identification leads to more realistic GUT gauge groups like SO (10) or SU (5). At the same time CFT models with different kinds of gauge groups have been constructed using free bosons, free fermions or orbifold techniques [1]. In modern terminology those examples with N = 1 space-time supersymmetry are of type (0, 2). For the (0, 2) CY models of Witten, Distler and Greene the left moving fermions of the σ model are not any longer sections of the tangent bundle of the CYM, but of a more general stable holomorphic vector bundle of rank four or five, respectively. Since there were those already mentioned reasonable doubts about the consistency at all, these models lost their attraction very fast.
A revival of these models was initiated by Witten's work [20] on the correspondence between nonlinear σ models on CYMs and orbifolds of Landau-Ginzburg (LG) models with isolated singularities. Applying his techniques also to the (0, 2) case yielded a LG description which allowed one to obtain more detailed information about the properties of a possible conformal fixed point [8]. In [9] it was shown that at least for marginal deformations of (2, 2) models by gauge singlets one gets a bona fide CFT. Recently, Silverstein and Witten [18] argued that even for all (0, 2) models described by linear σ models, the CFTs exist. Nevertheless, the explicit construction of such CFTs was still unclear. Fortunately, some aspects of the structure of these CFTs can already be explored in the LG framework [8]. Due to the left moving R-invariance there exists a left moving U (1) current, the spectral flow operator of which extends SO (8) to SO (10) or SO (6) to SU (5), respectively. Furthermore, the central charges for the left and right moving sector can be calculated correctly; for SO (10) they are (c, c) = (10,9) and for SU (5) one obtains (c, c) = (11,9). Thus, it seems to be quite a tough problem to systematically construct modular invariant partition functions for this class of purely heterotic CFTs. In this paper we present a class of CFTs which satisfies all the conditions mentioned above and which exhibits net numbers of generations which can be reproduced using the CYM/LG framework. One well known way of building new modular invariant partition functions is the simple current technique developed by Schellekens and Yankielowicz [15,16]. In the spirit of an idea mentioned in [17] we show that it is also suitable for the construction of the desired (0, 2) string vacua. Using light cone gauge, for SO(10) we start with a diagonal invariant partition function of a (c, c) = (24, 24) CFT, which contains the fourdimensional space-time part, an internal (c, c) = (9, 9) N = 2 supersymmetric part written as a non supersymmetric CFT, a U (1) 2 part and the Kac-Moody algebra SO(8) × E 8 of level one. Then we use simple current projections on the right moving side to extend firstly SO(8) × U (1) 2 to SO (10). This allows to apply the bosonic string map to yield a right moving superstring with c = 12. The idea of starting only with a subalgebra of SO(10)×E 8 and extending it has already been mentioned in [17] but not carried out further. The new feature in our construction is essentially the U (1) 2 factor. Afterwards, we project onto N S-N S and R-R couplings guaranteeing that we choose the 'supersymmetric tensor product' on the r.h.s. The last operation to be carried out on the right is the GSO projection onto even overall U (1) charges. If we would stop at this stage we would get nothing else but the usual Gepner models with E 6 gauge group [11]. However, because of the new U (1) 2 factor there occur new possibilities of preventing all the right moving operations to act also on the left. Thus, we divide out the most complicated simple current one can think of containing both pieces of N S sectors and R sectors. In general, this breaks the left moving N = 2 supersymmetry and the E 6 gauge group. The last step is to perform the left moving GSO projection extending SO(8) × U (1) c=9 × U (1) 2 to SO(10) × U (1). Note that this extension is different from the one carried out on the r.h.s. Finally, we arrive at a modular invariant partition function with gauge group SO(10) and a (c, c) = (10,9) CFT in the internal sector. Since there occur new combinations of left moving excitations which are massless, in general the spectrum of the string changes drastically. Comparing these spectra to those of the Distler/Kachru models we found indications that we have really constructed CFTs, describing certain points in the moduli space of the latter models. The SU (5) case is quite analogous, instead of one U (1) 2 CFT one uses two such factors. As expected for consistent string vacua, the gauge anomaly cancellation comes out automatically [14]. Of course, the above series can be extended further, three U (1) 2 factors generically yield the non GUT gauge group SU (3) × SU (2). In general, in four dimensions our construction allows all E r gauge groups with 3 ≤ r ≤ 6, which are defined by removing successively one simple root of a long leg of the E 6 Dynkin diagram. In six and eight space-time dimensions the usual Gepner construction yields the remaining exceptional gauge groups E 7 and E 8 , respectively. This paper is organized as follows. In section 2 we review some basic facts about the simple current technique. Then we present our construction of (0, 2) modular invariant partition functions. The SO(10) case is discussed in detail, whereas SU (5) and SU (3) × SU (2) are dealt with rather briefly. A discussion of the general massless spectra follows in section 6. In section 7 we present the results of a computer calculation for some exemplary models like the quintic and compare them to the results gained by CYM/LG techniques.

Review of the simple current technique
This chapter contains only a very short review of the work of Schellekens and Yankielowicz about generating new modular invariant partitions using simple currents. For a more detailed discussion we refer the reader to the original literature [15,16,17]. Suppose there is given a rational conformal field theory (RCFT) with at least one modular invariant partition function, e.g. the diagonal one. If this RCFT contains a simple current J, i.e. J × Φ i = Φ j for every highest weight representation Φ i of the chiral algebra in the model, then one can obtain a new modular invariant in the following way: First, define the index N of the simple current J to be the smallest integer so that J N = 1l. Furthermore, the monodromy parameter r is determined by the conformal dimension of J: so that r is defined modulo N for N odd and modulo 2N for N even. Next, one defines the (monodromy) charge of a primary field Φ: which takes values t N , t ∈ Z Z. By the action of the simple current all primaries of the RCFT are arranged in orbits Φ, J × Φ, . . . , J d × Φ, where d is a divisor of N . The charges of the fields occurring in an orbit are t+rn N mod 1. If one can choose r to be even, one can form a new modular invariant partition function with the matrix M determined by the orbits and the charges of the fields with respect to the simple current J: where δ 1 (x) = 1 for x ∈ Z Z and zero otherwise. The slightly modified chargeQ is defined on each orbit byQ Two different kinds of invariants occur. On the one hand, those that result from simple currents of integer conformal dimension. These can be regarded as diagonal invariants for a subset of orbits having integer monodromy charge. Thus, some of the original representations are really projected out. On the other hand, simple currents of non integer dimension lead to invariants corresponding to automorphisms of the fusion algebra which in particular means that only the pairing of the left and right moving sector changes. Obviously, the product of two matrices (2.4) also defines a modular invariant partition function which can be divided consistently by an integer in order to guarantee the vacuum to appear only once. Thus, in general one is allowed to form partition functions like The method of simple currents provides one with a powerful laboratory for the construction of modular invariant partition functions. In [17] it has extensively been used for the construction of four-dimensional, N = 1 space-time supersymmetric string vacua with an internal (c, c) = (9,9) CFT. There, it already appeared that in general one gets only (0, 2) world sheet supersymmetry. However, unlike to our models the gauge group SO(10) is linearly realized and not a result of a left moving GSO projection implying the internal left moving central charge to be also c = 9 and not c = 10. In the following sections we investigate whether this large laboratory can also provide us with models of the Distler/Kachru type [8].
3. String models with (0,2) supersymmetry and gauge group SO (10) In this section we make use of the simple current technique to find modular invariant partition functions which satisfy all the properties known for the conformal fixed points of the (0, 2) string vacua. First, we concentrate on the SO(10) case resulting from choosing a stable vector bundle of rank four. In [8] the following information about the CFT has been extracted from an LG analysis: (a) The left and right conformal anomalies of the internal CFT are (c, c) = (10,9). (b) Besides the right moving U (1) current which is part of the right moving N = 2 Virasoro algebra there exists a left moving U (1) current satisfying the following operator product expansion (OPE): of the gauge group is linearly realized, the remaining roots are generated by taking orbits with respect to the spectral flow of conformal dimension (h, q) = ( 1 2 , 2). Furthermore, we know that there is still a CYM in the model. As we have learned from Gepner's work on (2, 2) models [11], some of them correspond to tensor products of unitary N = 2 models. Using light cone gauge we start with the diagonal partition function for the (c, c) = (24, 24) CFT model shown in Table 3 The remarkable change compared to Gepner's models is the appearance of a free boson compactified on a circle of radius R = 2 denoted as U (1) 2 . The diagonal partition function for this part can easily be expressed in terms of Θ-functions: Note, that this is nothing else but the partition function of a Dirac fermion. The fusion rules are quite simple: The current is j U(1) 2 = i∂φ and satisfies the following OPE: Furthermore, even though U (1) 2 is surely not N = 2 supersymmetric, there exists a spectral flow between the sector of even index m and odd index m or between the N S sector and the R sector of the Dirac fermion, respectively. The spectral flow operator is exp( iφ(z) 2 ) and has conformal dimension and charge (h, q) = ( 1 8 , 1 2 ). Now, it becomes obvious why we have chosen this special c = 1 theory. Combining it with the left moving c = 9 theory offers the possibility to define an overall U (1) current J which satisfies the conditions (b) and (c). The sum of the N = 2 current j c=9 = i √ 3∂Φ(z) and the U (1) 2 current satisfies the OPE in (b) and the left moving spectral flow operator is given by Later on we will see that taking orbits with respect to this spectral flow operator really extends SO(8) × U (1) to SO(10). Now we proceed by discussing the right moving sector. To this end let us recall some facts about the representations of SO(2n) Kac-Moody algebras at level k = 1. All we need for the following discussion is summarized in Table 3.2. character h q mod 2 degeneracy The charge q is taken with respect to the sum of all Cartan elements of the Lie algebra SO(2n) and ϑ i denotes the Jacobi ϑ-functions. The fusion rules for the representations are different for n even and n odd as one can read off from Table 3.3. In order to apply the bosonic string map for SO(10) (10). This can be done by using the simple current which generates the following orbits: Now we can proceed on the right moving side in the same way as in Gepner's construction.
Here we have split the characters in the usual way into two non supersymmetric pieces: Then, in order to ensure that we are actually dealing with an N = 2 supersymmetric model, we have to impose further projections where G i means the supercurrent in the i-th factor of the tensor product. These projections allow only couplings between same kinds of sectors. The last step to be carried out on the r.h.s. is the right moving GSO projection onto states with even overall charge. The necessary simple current is where Σ c=9 denotes the spectral flow operator of dimension (h, q) = ( 3 8 , 3 2 ) of the internal c = 9 CFT. In a concrete model Σ c=9 simply contains one Φ 0 1,1 primary field for each factor. So far, the partition function looks like which produces exactly the ordinary Gepner models, for all projections act also on the left. Thus, in order to get something new we have to prevent this by introducing more simple currents from the left which do not commute with the simple currents in (3.13). Which simple currents are suitable depends on the concrete model one is dealing with. However, on account of the new U (1) 2 factor there occur simple currents which are not present in the Gepner case. In general we are interested in simple currents which both break the left moving N = 2 supersymmetry and the E 6 gauge group resulting from the J GSO R projection. Suppose now we have found such fields Υ l . What remains is only the left moving GSO projection which is performed by the simple current which is actually the same as for the right moving GSO projection. However, since the simple currents J i and J (1×8→10) do not any longer act on the left, it does not yield an extension of the gauge group to E 6 but only to SO(10). On the level of characters this can be seen by using a general theorem about orbits of spectral flows of chiral dimension (h, q) = ( k 2 , k) [13]. Since in the N S sector all orbits contain only states with integral charge, Hermite's lemma † tells us that every orbit can be expanded into a finite number of z dependent functions where the coefficients depend only on the variable q. In our case the chiral spectral flow is twice the flow Σ c=10 and therefore has dimension (h, q) = (2, 4). Consequently, there are only four invariant functions which can also be written in terms of Θ functions: Note, that f 0,4 , f 2,4 have even charge and f −1,4 , f 1,4 odd charge. Since Σ c=10 acts on the invariant functions by every orbit under J GSO L with even charge can be expanded in the following way: Reordering yields After some algebra neglecting the z dependence this can be written as showing explicitly the extension of the gauge group to SO (10). Summarizing, the entire model has the form and by construction exhibits all the properties required at the beginning of this section. The remaining question is whether one can really find simple currents Υ l which break both the left moving supersymmetry and the E 6 gauge group. An explicit computer calculation shows that generically this is not difficult. Apparently, at least for the moment we have no other criteria to decide what the influence of a set of simple currents Υ l is than to perform the explicit calculation. Of course, those which act trivially on the U (1) 2 should correspond to ordinary orbifold constructions of the CYM, whereas others reflect the choice of a different vector bundle for the left moving σ model fermions and thus reducing the rank of the gauge group. In section 7 we present some first results of an explicit calculation showing what kinds of spectra one can expect from the models in (3.21).
4. String models with (0,2) supersymmetry and gauge group SU (5) The generalization of the above construction to SU (5) is straightforward, so that it will be presented more briefly. The CYM/LG analysis reveals the following information about the conformal fixed point:    (6) to SO(10) can be achieved by the following two simple currents: The projections ensuring N = 2 supersymmetry on the right moving side are still given by and the GSO projection leading to N = 1 space-time supersymmetry is So far, the model looks like The left moving U (1) current is the sum of the N = 2 current and the two U (1) 2 currents and in particular satisfies the OPE (4.1). The associated spectral flow operator of dimension (h, q) = ( 5 In order to perform the left moving GSO projection we use the simple current again. In the N S sector there exist five series invariant with respect to the square of the flow Σ c=11 Similar to the Gepner case [3,10] they are not of definite charge parity, so that we have to use the decomposition into Θ functions f Q,5 (q, z) = 1 η(q) (Θ 2Q,10 (q, z) + Θ 2Q+10,10 (q, z)) mod 20. (4.9) Taking into account the action of Σ c=11 on the Θ functions: Σ c=11 : Θ i,10 → Θ i+5,10 mod 20, (4.10) every orbit of even overall charge can be expanded in the following way: Reordering and the fact that the characters of SU (5) can be written as sums over products of those of SU (4) and Θ functions at level ten yields This shows very nicely the extension of the gauge group to SU (5). Unlike E 6 and SO(10) there exist two different representations for chiral space-time fermions, the 10 and the 5 which together contains one generation of the standard model. As we will see in the following sections the number of total generations in 10 and 5 in general are not the same, whereas the number of net generations are. Thus, the whole partition function is Apparently, the whole construction can be extended further starting with three copies of U (1) 2 and the group SO(4)×E 8 which extends to SU (3)×SU (2)×E 8 . Since at our exactly solvable points there are a lot U (1) factors around, one might get the supersymmetric standard model. However, analogously to the (2, 2) case these U (1) factors are believed to exist only at this special point of the moduli space of the CYM and would be broken by a generic marginal deformation. Thus, very sensitive fine tuning is necessary to choose such a special string vacuum.

String models with (0,2) supersymmetry and gauge group SU (3) × SU (2)
The next kind of models are those which exhibit gauge group E 3 = SU (3) × SU (2). At certain points of the moduli space E 3 may be extended by at least some U (1) factors, so that the gauge group contains the standard model. The ansatz is shown in Table 5 gauge group SO(4) × E 8 10 10  (4) to SO(10) can be achieved by the following three simple currents: The GSO projection is (4) s .

(5.2)
This simple current is also used to perform the left moving GSO projection. In the N S sector there exist six series invariant with respect to the square of the flow Σ c=12 : Thus, every orbit of the left moving GSO projection can be expanded in the following way: which can be rewritten in terms of E 3 characters Thus, there appear four chiral representations. The whole partition function is Remarkably, the above generalized Gepner type construction yields all gauge groups E 6 , E 5 = SO(10), E 4 = SU (5) and E 3 = SU (3) × SU (2). The extension of the gauge group SO(2n) × U (1) to E n+1 is schematically represented by the extension of the Dynkin diagrams shown in Figure 5.1. In order to extract more concrete information about these models we have to calculate some physical quantities. As a first step we concentrate on the massless spectrum, especially the number of generations which can also be calculated in the CYM/LG scheme.   (10) Since the right moving sector is the same as in ordinary (2, 2) models, the same combinations of primaries from the different factor models contribute to the right moving part of the massless states. Thus, in a concrete calculation one starts with these combinations on the r.h.s. of the chain in (3.21) and follows their way through all the simple currents to determine to which states they couple on the l.h.s. The appearance of spin one particles in the spinor representations indicates an extension of the gauge group, in the SO(10) case usually to E 6 . However, in general this does not mean that also the N = 2 supersymmetry is restored in the left moving sector. In the case of SU (5) only the singlet and the four spinor representations occur. The quantum numbers in the internal (c, c) = (11,9) CFT are listed in Table 6.2.

The massless spectrum
Massless spectrum for SU (5) In this case further gluons can extend the gauge symmetry to SO(10), E 6 and also to SU (6). Like in all string models where only Kac-Moody algebras at level one occur, one has to use a mechanism for breaking the GUT down to the standard model different from the spontaneous symmetry breaking by attaching a nonzero vacuum expectation value to some Higgs fields in the adjoint representation. For instance, if the fundamental group of the CYM is nontrivial one has the possibility to use Wilson lines. For the gauge group E 3 the massless spectrum has the internal (c, c) = (12, 9) quantum numbers listed in Table 6.3. rank, usually U (1) factors. In most cases these extensions are not written down explicitly in the following.
• Gauge group SO(10) We start with a model like in Table 3.1 where we choose (3) 5 as the internal c = 9 part. Besides all the projections in (3.21) we only include one further simple current (for simplicity we will not write down the '⊗' in the following) 0). Note, that (7.1) contains both factors from the N S and the R sector. The resulting massless spectrum is given in Table 7.1.  Since no further gluons appear, the gauge group is SO(10) by construction. Looking in more detail where the 80 generations are coming from, one realizes that 60 of them are ordinary N = 2 states from the r.h.s. surviving all projections. However, the remaining 20 states arise in some orbits of Υ and contain both a nontrivial contribution of the U (1) 2 part and a mixing of N S and R states. The numbers of particles in 10, 16 and 16 agree with a model discussed in [8] which was defined on a complete intersection in the weighted projective space W P 5 (1,1,1,1,2,2) . Therefore our model could lie in the moduli space of that model which is based on a totally different approach. That would imply that our model, although built up on the (3) 5 tensor product, does not live on the quintic CYM. Like in ordinary Gepner models there exists an isomorphic CFT, for which generations and antigenerations are interchanged. This mirror model can be obtained from the original one simply by including the following simple currents in front of the right moving states χ(τ ) in (3.21): The generalization to the general case is straightforward. Thus, the class of models investigated in this paper exhibits mirror symmetry. For the class of linear σ models in [7] mirror symmetry has not been established so far.
It is easy to calculate at least net numbers of generations for the Distler/Kachru models. Due to [7,17] the defining data of stable, holomorphic vector bundles V = V 1 , V 2 are given by the exact sequence Here r = 3, 4, 5, 6 yields gauge group E 9−r , and the n a and m i are positive integers chosen such that c 1 (V 1 ) = c 1 (V 2 ) = 0 and the gauge anomaly vanishes, i.e. c 2 (V 1 )+c 2 (V 2 ) = c 2 (T ) with T denoting the tangent bundle. The net numbers of generations are given by an index theorem By stochastically calculating some of these numbers for the quintic one realizes that only multiples of 5 occur and 80 really appears. Of course, much more work has to be done to definitely identify this model with a concrete σ model. There are at least two candidates, for the model could be based on the quintic CYM or on the weighted projective space mentioned above.
• Gauge group E 6 Even though the construction yields gauge group SO(10) it may happen that further extended gauge groups occur. For instance, without any further simple current Υ one gets the ordinary (2, 2) Gepner model with gauge group E 6 . However, including the simple current destroys the left moving N = 2 supersymmetry but the E 6 remains unbroken. This can be read off explicitly from the SO(10) massless spectrum listed in Table 7.2.
which is only slightly different from the current (7.6) one obtains a totally different spectrum as given in Table 7.3.  Table 7.3 E 6 model N gen = 20 also appears in the list of Distler/Kachru spectra.
• Gauge group SO (12) It is known that the Gepner model (1) 3 (2)(6) 2 describes a compactification on K 3 × T 2 and yields both an extension of the gauge group to E 7 and an enhanced N = 2 space-time supersymmetry. In [12] it has been argued that such models with at least (0, 4) world sheet supersymmetry have dual realizations as type II compactifications on CYMs. Using the simple current in the (1) 3 (2)(6) 2 model one obtains the spectrum in Table 7.4.  This model has still N = 2 space-time supersymmetry which requires (0, 4) world sheet supersymmetry. In addition, the two vectors in 10 extend the gauge group to SO (12). Thus, there are three hypermultiplets in the two spinor representations 32 and 32 of SO (12). Nevertheless, the 14 singlets may even form a further extension by a nonabelian factor G. In principle, starting with a general CYM different from K 3 × T 2 it can also happen that (0, 4) models occur. Now, let us come to gauge groups of smaller rank.
• Gauge group SU (6) Starting with a model like in Table 4.1 the gauge group is at least SU (5). Analogously to the former case there occur extensions to larger gauge groups, in particular to SO(10) and E 6 . However, it can also happen that the gauge group is SU (6). Including the simple current in (4.13) one obtains the massless SU (5) spectrum shown in Table 7.5. • Gauge group SU (5) A model with SU (5) gauge group can be achieved by choosing the following two simple currents: The massless spectrum is listed in Table 7.6.  The gauge anomaly cancellation condition for SU (5) requires the same number of chiral fermions in the 10 and 5 representation. This condition is satisfied in our example yielding a net number of N gen = 64 generations which is not divisible by 5. However, there exists a model with N gen = 320, so that we expect our model to be an orbifold of that model. It is a general problem of our construction that we do not have control over the geometric interpretation of dividing out a certain set of simple currents. It can correspond either to an orbifold construction or to a new vector bundle.
There is another example which shows that the number of generations in 10 does not need to be zero. Therefore, we consider the tensor product of minimal models (1)(4) 4 also adding up to c = 9. This model has already turned out to yield a lot of different massless spectra [15]. For instance, choosing the two simple currents one obtains the spectrum presented in Table 7.7.  It is clear, that the smaller the rank of the generic gauge group the larger the set of possible extensions of the gauge group. For SU (3) × SU (2) models we find also extensions to the semisimple groups SU (6) × SU (2) and SU (4) × SU (2).
• Gauge group SU (6) × SU (2) For the quintic (3) 5 the simple current yields the E 3 spectrum in Table 7.8. • Gauge group SU (4) × SU (2) Analogously to the SU (5) case one can obtain smaller gauge groups by including more simple currents. For instance, gives the spectrum in Table 7.9.  In order to reduce the gauge group to E 3 one has to include three simple currents.
gives the model with 50 net generations listed in Table 7.10. What can also happen is the occurrence of gauge groups of rank higher than six.
• Gauge group E 7 For this extended group we again start with the model (1)(4) 4 and select only the simple current Due to the decomposition of the adjoint representation of E 7 into irreducible representations of SO (10) 133 = 45 + 10 + 10 + 16 + 16 + 16 + 16 + 1 + 1 + 1 + 1 (7.17) the massless spectrum in Table 7.11 yields 36 generations in the 56 representation of E 7 .  Table 7.11 E 7 model N gen = 36 can also be reproduced using (7.4). Furthermore, one can even produce the gauge group SU (7).
• Gauge group SU (7) Again we choose (1)(4) 4 and include the following two simple currents: The extension of SU (5) to SU (7) can be seen from Table 7.12 taking into account the decomposition 49 = 24 + 5 + 5 + 5 + 5 + 1 + 1 + 1 + 1 (7.19)  Table 7.13. This model is likely an orbifold of a Distler/Kachru model.  All these examples show that one can really find suitable simple currents Υ breaking both the left moving N = 2 supersymmetry and the gauge group E 6 . Furthermore, the massless spectra obtained can all be found in the context of (0, 2) (non)linear σ models encouraging further investigations about their relationship.

Conclusion
In this paper we have presented a method to construct modular invariant partition functions of four space-time dimensional heterotic string vacua with (0, 2) world sheet supersymmetry and generic gauge groups E r with 3 ≤ r ≤ 6. This constructively proves the existence of bona fide CFTs with all the properties known for the conformal fixed points of (0, 2) supersymmetric CYM/LG models. In particular, these vacua are not suffering from destabilizing instanton corrections. Clearly, our construction is not unique and there might exist different construction schemes of (0, 2) CFTs, especially those starting with a truly heterotic modular invariant partition function. However, (2, 2) models have taught us that due to the GSO projection the spectra obtained are highly degenerate. Thus, we hope that more than only a very small subset of all exactly solvable (0, 2) vacua can be realized by the simple current method.
Since both orbifolds of the CYM and the choice of a more general stable vector bundle for the left moving σ model fermions are encoded in the same manner in this class of CFTs, a direct correspondence between simple currents and the defining data of the latter bundles might be hardly to reveal. One ansatz to construct such a map could be the coincidence of the model in Table 7.1 and a complete intersection in a weighted projective space. If such a one to one identification could be achieved for at least a few models, it would be possible to calculate and compare further properties of physical importance. For instance, one could address questions like the exactness of a first order calculation of some of the Yukawa couplings in the CYM/LG framework. Furthermore, since the class of exactly solvable models exhibits exact mirror symmetry, one expects such an isomorphism for the CYM/LG formulation, as well. Thus, in order to learn more about the degeneracy of the spectra and the appearing net numbers of generations one has to extend further the set of explicitly known models [4].