Asymmetric Gepner Models II. Heterotic Weight Lifting
NNIKHEF/2010-30IFF-FM-2010/02September 2010
Asymmetric Gepner Models
II. Heterotic Weight Lifting
B. Gato-Rivera a,b, and A.N. Schellekens a,b,ca NIKHEF Theory Group, Kruislaan 409,1098 SJ Amsterdam, The Netherlands b Instituto de F´ısica Fundamental, CSIC,Serrano 123, Madrid 28006, Spain c IMAPP, Radboud Universiteit, Nijmegen, The Netherlands
A systematic study of “lifted” Gepner models is presented. Lifted Gepner models are ob-tained from standard Gepner models by replacing one of the N=2 building blocks and the E factor by a modular isomorphic N = 0 model on the bosonic side of the heterotic string. Themain result is that after this change three family models occur abundantly, in sharp contrastto ordinary Gepner models. In particular, more than 250 new and unrelated moduli spaces ofthree family models are identified. We discuss the occurrence of fractionally charged particlesin these spectra. Also known as B. Gato a r X i v : . [ h e p - t h ] S e p Introduction
In a previous paper [1] an analysis of spectra of simple current invariants of Gepner models[2] was presented, completing work that was initiated in 1989 in [3]. The motivationfor this new, extended analysis were the statistical results on the number of families inorientifold models. The observed number of families, three, turned out to occur two tothree orders of magnitude less frequently [4, 5] than the number one, two or four. Wewanted to verify if the same suppression also occurred in heterotic models.This return to the heterotic string forced us to face the problem which led one of usto essentially abandon the subject after 1989: the appearance of particles with fractionalelectric charge in the spectrum, an undesirable feature first observed in a class of (2 , ,
2) com-pactifications with the standard SU (3) × SU (2) × U (1) embedding [8] based on exactCFT constructions.The results of [1] can be summarized as follows. No substantial progress was made onthe issue of family numbers, despite a huge enlargement of the class considered: as before,three families only occur for the exceptional model usually denoted as (1 , ∗ , ∗ , ∗ ) [9],and for any other exceptional or non-exceptional combination the number of families isalways even (see also [10]). Hence it appeared that also in the heterotic case, threefamilies is a rare feature. On fractional charges a somewhat more optimistic conclusionwas reached, namely that they are fairly often vector-like. Within the context of RCFT(rational conformal field theory) that is about the best one can reasonably hope for.However, this analysis led to a new idea, presented in [11], and called “Heterotic WeightLifting”. Essentially this enables us to go one step further away from the lamppost ofheterotic (2 ,
2) models, and get closer towards genuine (0 ,
2) models. In [3] this was doneby not imposing the equivalent of space-time supersymmetry in the bosonic sector of theheterotic string; in [1] the world-sheet supersymmetry constraints between CFT buildingblocks were relaxed in the bosonic sector, and furthermore the SO (10) group was broken(see also [12]). Heterotic weight lifting allows us to replace a minimal N=2 model plus the E gauge group in the bosonic sector by a new CFT that has exactly the same modular S and T matrices. This implies that modular invariance is manifestly preserved. However,the spectrum is drastically changed: the conformal weight of all ground states is shifted byintegers, and the dimension of these ground states is changed as well. Because the weightof most ground states is moved up, we called this procedure “weight lifting”. However,there are also some ground states that move down. Therefore, in order to be able tocompute massless spectra of lifted Gepner models it is essential to be able to computethe exact spectra of standard Gepner models. This is indeed possible in the constructionsconsidered here. In [11] a few examples of lifted Gepner models were presented, whichindicated in particular that the chances for getting three families looked better. Herewe want to present a complete analysis of all 435 cases of heterotic weight lifting oneobtains by applying the list of lifts shown in [11] to the 168 Gepner models. We willfocus on the same two issues considered in [1]: the number of families and the presenceof fractional charges. On the first issue our conclusion is much more positive than the2ne in our previous paper, whereas on the second issue the conclusion remains essentiallyunchanged.This paper is organized as follows. In the next section we explain our motivations.Readers who are already sufficiently motivated may skip this section, but our perspectiveof string phenomenology is rather different from that of others working in this area, and sowe believe some explanation may be useful. In section three we summarize the heteroticweight lifting procedure. In section four we present our results on charge quantizationand in section five the results on the number of families. Section six contains results ondistributions of vector-like particles. In section seven we investigate whether matter inthis class of models generically couples to the extra non-abelian groups provided by theheterotic weight lifting procedure, instead of the familiar E factor. Finally, in sectioneight we formulate our conclusions. This paper is about RCFT, and by their nature such constructions are more suited forstudying the structure of the standard model (gauge groups and representations) thanfor parameters and moduli. Obviously the first question to ask regarding the standardmodel structure is: can one build exact string theories that have a gauge group SU (3) × SU (2) × U (1), three families, and no fractionally charged exotics? It has been amplydemonstrated that the answer to this question is positive, both in the case of orientifoldsand heterotic strings (see our previous paper [1] for references). The common attitude instring phenomenology seems to be that this is all we need to know about the standardmodel structure, and that we should move on to issues like parameter values and modulistabilization. Clearly that is indeed important. But there are other questions regardingthe standard model structure we should not ignore.A couple of decades ago, it was hoped that string theory would determine the standardmodel uniquely. Already very soon that turned out to be unrealistic, and it was replacedby the hope that some mysterious missing selection principle would uniquely select thestandard model. Furthermore it was hoped that while waiting for the discovery of thatprinciple, one could at least try to make a complete enumeration of all candidate vacua,and find the one that agrees with all available data, in particular the standard model andall of its parameters. Having identified it, a plethora of predictions would be available, andthe correctness of the theory would be proved unambiguously by testing those predictions.This point of view – although rarely expressed so explicitly – could still be maintaineduntil the beginning of this century, but it seems now unlikely that we will be in that situa-tion in the foreseeable future, because the amount of string vacua is likely to vastly exceedthe currently available amount of experimental data (since the latter increases monoton-ically, this balance may tip in the future). One may still hope that there is somethingfundamentally wrong with our understanding of moduli stabilization, supersymmetry orthe cosmological constant, and that we eventually will end up with a much smaller set.One may also hope that the large numbers are somehow irrelevant to the problem at hand3 i.e. getting the standard model from string theory), and that they can be cleanly fac-tored out from the problem, contributing only to the tuning of the cosmological constant.Even if that were true, we are still facing the problem that the methods we use to explorethe string theory landscape are primitive and limited. On the exact CFT side they aremostly limited to non-interacting CFT’s (free bosons, free fermions and orbifolds), a veryspecial case, whereas geometric approaches to heterotic strings are constrained to theneighbourhood of (2 ,
2) models. In other words, at present the chances of finding “thestandard model” in string theory and make exact predictions based on it seem very small.A possibly achievable goal is to prove that the landscape is dense enough in all relevantdirections to contain the standard model (and anything discovered beyond it at a giventime), and perhaps this is all that can be done in the foreseeable future.But one may still hope to be able to make generic predictions for classes of models.Possible examples are extra U (1)’s, rank-2 tensors that are ubiquitous in orientifold mod-els, mirror fermions or particles with fractional electric charge. Such predictions based ongeneric features of certain classes will never be absolutely falsifiable, but if such particlesare found they will nevertheless boost the confidence in the theory. Conversely, if it canbe shown that certain particles are generically present in string theory, but are neverfound, one should start having serious doubts about either string theory in general, orspecific classes of string theories. However, before making such predictions, it would beprudent to check if similar ones would have worked in the past. For example, had therenot already been strong limits on fractionally charged particles a few decades ago, theearly results of heterotic strings might have led to predict their existence. This predic-tion, which is actually a postdiction, would have failed. One should be highly skepticalof pre dictions if the post dictions are wrong. In other words, one should not only demandthat the standard model is among the string vacua in a class, but also that its observedfeatures are not excessively rare in that class. If indeed fractional charges are abundantin most heterotic string vacua that otherwise produce the standard model correctly, thenthis should be viewed as a serious problem.Most of string phenomenology is focusing on getting the observed features out whileignoring the neighbouring landscape. We would argue that scanning the landscape in theneighbourhood of the standard model is essential in order to have any confidence in thecorrectness of a certain approach. If certain features systematically come out wrong, andcan be gotten right only in very special and rare conditions, then we must assume thatthere is something missing in our understanding. Such failures may be due to a number ofcauses. Perhaps one is looking at the wrong kind of string theory (orientifolds, heteroticstrings, F-theory) or one is seeing artifacts of a computation method ( e.g. free fields orRCFT), or one uses a too severe approximation to a true string vacuum (for exampleunbroken supersymmetry or unstabilized moduli). It is also possible that certain rarefeatures are needed for the existence of (intelligent) life, and hence it is not surprising thatwe observe them. If all this fails one can always claim that apparently we ended up in auniverse that has some exceptionally rare features for reasons that we cannot figure out.In our opinion this is the least attractive option. In any case it is important to confrontthese problems, rather than ignoring them by focusing solely on string vacua that fit the4bservations.Anthropic arguments, in the sense defined above, are unavoidable and perfectly ac-ceptable in the context of a landscape [13, 14, 15, 16], if indeed it can be convincinglydemonstrated that a certain feature is or is not important for the existence of life. Butusually such arguments are just too complicated to be used reliably. What one shoulddo instead, before worrying about a rare feature, is to try to be as certain as possiblethat at least anthropic arguments are not likely to make a difference. In [1] we discussedthis issue for the number of families and the presence of fractionally charged particles:although it cannot be demonstrated rigorously, it is hard to believe that the existence oflife requires three families (as opposed to two or four) or is inhibited by the mere exis-tence of very rare, and heavy, fractionally charged particles (on the other hand, if they areeither light or abundant the argument is far too complicated to arrive at decisive generalconclusions). We would be perfectly happy to be convinced otherwise, but we take as ourworking hypothesis that anthropic arguments do not play a rˆole here.Any study of distributions of standard model properties is plagued by bias and measureproblems (see for example [17]). Some features of distributions may be artifacts of theclass considered or the method used. Furthermore distributions can be affected by factorsbeyond current knowledge, such as cosmological selection or moduli stabilization. Biaseffects can partly be avoided by investigating the same issues using different methods orin different regions of the landscape. For example, the paucity of three family modelsin orientifold constructions was observed both in Gepner orientifolds and in Z × Z orientifolds; some questions raised in the present paper have also been examined in thecontext of free fermion models; and the present paper itself examines if earlier results onGepner models hold also in a different class of heterotic models. Any investigation ofdistributions makes the implicit assumption that unknown factors are smoothly behavedin the relevant region of parameter space. For example, there is no reason to believethat cosmological selection or moduli stabilization favors three families, so it is difficultto believe that such effects will lift the large dip in orientifold distributions. As long as nosuch argument is found, the problem should be taken seriously, and not simply dismissedas a statistical fluke.It is a matter of taste what one defines as “rare” or “unnatural”. In the literature,undesirable features at the 1% level are either ignored or stressed, depending on the pointof view one wishes to express. Most people would perhaps not worry about a dip of afactor of hundred in the family distribution of orientifold models, precisely at the valuethree. But on the other hand, most people would say that a class of models predictsa certain property if 99% of the set had that property. Likewise, convergence of gaugecouplings with a precision at the 1% level is considered strong evidence by some, andignored by others. The discrepancy of the GUT scale and the string scale by two ordersof magnitude, on the other hand, is considered small by adepts of the heterotic string,and taken to infinity in discussions on F-theory GUTs.5 .1 Remarks on vector-like particles Essentially all string spectra that produce the known particles contain large numbers ofadditional particles: singlets, mirror quarks and leptons, higher rank tensors (in orientifoldmodels) and in some cases even more exotic ones such as fractionally charged particles.It is reasonable to require that all additional particles should be vector-like, and that allmatter that is chiral with respect to SU (3) × SU (2) × U (1) has already been observed.String theory provides robust chiral spectra modulo non-robust vector-like states. Toidentify potential string vacua that match our world, we have to be able to match at leastthe robust part of the spectra.However, it has become standard practice in string phenomenology to go much furtherand also match the non-robust part of the spectrum, as if we are already certain that thecurrently known spectrum (plus the Higgs and all superpartners) is all that exists. In otherwords, one assumes the absence of any vector-like particles other than those mentioned.This is a remarkable assumption given that, on the one hand, some vector-like particles areexpected to exist, but on the other hand we do not (yet) have experimental informationabout any of them.Exact RCFT constructions are more seriously affected by the problem of masslessvector-like particles because they cannot be continuously deformed in any practical way.However, usually one gets a large number of discrete points in what appears to be the samemoduli space. This results in a discrete scan over possible masses of vector-like particles,which manifests itself in the form of discrete distributions of the number of masslessvector-like particles. This phenomenon has been observed both in orientifold models andheterotic models. If one is lucky enough the set includes points where certain vector-like particles are completely absent. However, it is unlikely that the problem is solvedentirely within the context of RCFT. This is simply not the right tool for solving thisproblem. Several steps are needed to move from the exact RCFT to a phenomenologicallyacceptable string vacuum, and masses for vector-like particles may be generated at anyof these steps. They may acquire masses from simply moving away from the specialRCFT point into the full moduli space, from moduli stabilization or from supersymmetrybreaking. What is known about this is mainly folklore, based on common beliefs andsporadic examples (see e.g. [18, 19, 20, 21] for discussions of decoupling of exotic statesin various models).Let us assume, for simplicity, a scenario where the physics of moduli stabilizationis cleanly separated from the physics of supersymmetry breaking, and where the latterscale is low, in the LHC range. It is quite generally accepted that all vector-like particleswill acquire a mass after supersymmetry breaking (as well as the breaking of any othersymmetries beyond SU (3) × SU (2) × U (1)). It is less clear what a generic string spec-trum looks like when all moduli have been stabilized but supersymmetry is unbroken. Ifgenerically such a spectrum contains massless vector-like states beyond the MSSM, thenthis implies a generic prediction of the existence of such vector-like states with masses With a slight abuse of terminology (since we apply the term also to bosons) we will call all particlesthat can get a mass without breaking SU (3) × SU (2) × U (1) “vector-like”. The observationof low energy couplings roughly in agreement with the SUSY-GUT scenario does indeedeliminate many candidate heterotic string spectra, but none of these arguments completelyforbids the existence of light vector-like particles, and they might even have the virtue ofhelping to reduce the notorious GUT-scale/string scale gap.However, light ( e.g. near the TeV-scale) fractionally charged particles are almostcertainly a phenomenological disaster, since they would be stable. The upper limit ontheir abundance on Earth is far less than 10 − per nucleon [22], and stable light particleswould be produced far too copiously in the early universe to be consistent with thatlimit [7]. Hence if we allow fractionally charged particles as vector-like massless particlesin exact RCFT, we are making the implicit assumption that in the full theory theyacquire essentially Planck scale, or at least GUT scale masses (the lower limit depends onvarious assumptions about the cosmological evolution of the universe). Provided this iswhat generically happens, vector-like fractional charges are acceptable in an exact RCFTspectrum. At least one can give them the benefit of the doubt.If, on the other hand, generically some vector-like states remain light, this producesan interesting dilemma for heterotic strings. First of all one would be forced to look forexamples with a rare feature, i.e. complete absence of fractional charges in the exactRCFT massless spectrum, with no argument why we find ourselves in such a specialuniverse. Secondly it would undermine the usual assumption that the low-energy spectrumshould be that of the MSSM, and nothing more. Other vector-like particles do not satisfythe same strict limits as stable, fractionally charged ones, and without having such limits,the logical conclusion would be to predict their discovery at the LHC, rather than bend Note however that even for fractionally charged exotics it is possible to find multiplets that have noimpact on gauge coupling unification. An example is a standard model family with the opposite sign forall Y -charges, which gives rise to third-integral charges. Just like an extra family, this affects only thevalue of the unified coupling, but not the convergence itself. The starting point of the construction of heterotic strings using interacting rational confor-mal field theory is a diagonal modular invariant partition function with identical buildingblocks in the bosonic and the fermionic sectors. In the fermionic sector the choice ofbuilding blocks is tightly constraint by the requirement of world-sheet supersymmetry.But on the bosonic side the only constraint is conformal invariance, and hence a muchlarger set of building blocks is in principle available. However modular invariance makesit nearly impossible to combine distinct building blocks in the left and right sectors.Heterotic weight lifting [11] is a special case of a presumably very large set of modifi-cations of the heterotic string, based on the idea of replacing some building blocks in thebosonic sector by isomorphic ones in the sense of the modular group. The difficulty is thento find building blocks that are isomorphic, but not identical, and have the same centralcharge. Some examples are known, such as the affine algebras of the E × E and SO (32)heterotic strings or the meromorphic c = 24 CFTs [23]. All of these are CFTs with justa single primary field. But what we need here are building blocks that are isomorphic to N = 2 minimal models.It is an interesting question whether there exist CFTs with the same S and T matricesand central charge as a given N = 2 minimal model, without being identical to it. Wedo not know of any example. However, we can remove the superfluous E factor that ispresent in the bosonic sector of a (2 ,
2) heterotic string and allow the central charge todiffer by 8. There is a well-known example of such an isomorphism: SO ( N ) × E versus SO ( N + 16) .In a nutshell, the method works as follows. We start with the level-k minimal modeltimes E . The minimal model can be realized as a coset CFT: SU (2) k × O (2) U k +2) The precise construction of this CFT involves “field identification”, which can be viewedas a formal extension of the chiral algebra with a spin-0 current of order 2. Now we“deconstruct” this coset construction by formally removing the field identification, thenwe embed the denominator algebra U k +2) in E instead of SU (2) k × O (2), and then were-establish the field identification as a standard simple current extension.The resulting CFT has a chiral algebra SU (2) k × O (2) × X , where X is the remain-der of E after dividing out U k +2) . In other words, the tensor product U k +2) × X can be chirally extended to E . The factor X has central charge 7, and its modulartransformation matrices S and T are the complex conjugates of those of U k +2) . In [11]8 list of 33 such combinations was given, which was not claimed to be complete. We willcall the isomorphic N = 0 CFT the “lift” of the corresponding N = 2 CFT. For somevalues of k no such lift was found, and for some values there are two. Meanwhile we haveattempted to find more examples, but so far without success. Nevertheless, we feel thatthe list published in [11] is just the “low-hanging fruit”. These examples were easy toget, because they can be realized in terms of tensor products of affine Lie algebras. Thepresence of these affine Lie algebras is important in order to reproduce the large simplecurrent groups of the minimal models, but there may well be other ways of achieving that.Another possibility considered in [11] is to replace two minimal model factors and the E factor by an isomorphic CFT. One example was found, but it will not be consideredhere.A Gepner model is characterized by a set of values ( k , . . . , k M ) for the levels of theminimal model factors. In principle each of the values can be lifted, provided a lift exists.We will denote a lifted Gepner model by ( k , . . . , (cid:98) k i , . . . , k M ) if the i th factor is lifted. If k i has two distinct lifts, we use the notation ˜ k i for the second one. There are 168 standardGepner models. Applying all 33 single lifts listed in [11], we end up with 435 lifted Gepnermodels.Note that we may combine lifts with any modular invariant partition function of thestandard Gepner models, including exceptional ones. Here we will not consider the latter.Apart from that, we will consider exactly the same set of MIPFs as in our previouspaper [1], allowing in particular the breaking of SO (10) to any of the eight subgroupslisted there. We also allow breaking of world-sheet and space-time supersymmetry inthe bosonic sector. Of course this is already the case as soon as we replace one of theminimal models by its lift. However, in addition to that we also allow the space-timesupersymmetry current and the world-sheet supersymmetry alignment currents of thefermionic sector to be mapped to any isomorphic current in the bosonic sector. In [1] we presented a detailed discussion of SO (10) breaking and the associated fractionalcharges (by which we always mean electric charges of color singlet particles). In thepresent work, SO (10) breaking is treated in exactly the same way, and therefore a priori the possibilities are exactly the same. We just summarize the main points for convenience.The heterotic string revolutionized string theory in 1984 [24] partly because it becameclear almost immediately [25] that chiral four-dimensional spectra could be obtained witha remarkable resemblance to the observed particle spectrum. It is not an exaggeration tosay that by merely imposing chirality and four uncompactified space-time dimensions oneis almost inescapably led to spectra consisting of a number of families of (16)’s of SO (10)(or in the special cases discovered earliest [25], (27)’s of E ). This is especially easy todemonstrate by using the “bosonic string map” [26].Unfortunately, this does not work so beautifully anymore if one tries to break SO (10)to SU (3) × SU (2) × U (1). This cannot be done with a field-theoretic Higgs mechanism, be-9ause the required Higgs representations cannot occur as massless states for affine SO (10)at level 1. Instead one can do the breaking directly in string theory. In our language,the breaking amounts to writing the theory in terms of SU (3) × SU (2) × U (1) affineLie algebras and not allowing the SU (5) roots to appear in the massless spectrum. Here U (1) N denotes a free boson compactified on a circle such that there are N primaryfields with conformal weights q / N , for q = − N + 1 , . . . , N . Applying the standardgroup theory breaking SU (5) ⊃ SU (3) × SU (2) × U (1) to affine SU (5) level 1 yields SU (3) × SU (2) × U (1) (we will omit the level indices on SU (3) and SU (2) henceforth).However, the only way to eliminate the SU (5) roots while respecting modular invarianceis by allowing some of the fractionally charged representations to appear. They must ap-pear in the full theory, although not necessarily in the massless sector. In other contexts(orbifolds, Calabi-Yau constructions) the breaking is often achieved by using backgroundgauge fields on Wilson lines, but no matter how it is done, as long as the result can bedescribed in terms of SU (3) × SU (2) × U (1) as above, this description applies.Rather than breaking SO (10) we start from its sub-algebra SU (3) × SU (2) × U (1) × U (1) , which we extend by a simple current of order 30 to SO (10) in the fermionic sector.In the bosonic sector we allow some powers of this current to be replaced by other currentsof the same order and relative monodromies. This allows in total eight different subgroupsof SO (10), listed in Table 1. Which of these groups can be realized for a given tensorproduct depends on the values of the levels k i exactly as described in [1], and independentof the lifting. Each of these groups can still be further extended by means of currentswith components outside SO (10).In this setup the first three factors, SU (3) × SU (2) × U (1) , are destined to becomethe standard model gauge group. Only a limited set of representations of this group hasconformal weight less than or equal to one, and can occur in the massless spectrum. Theseare the standard quark and lepton multiplets Q, U, D, L, E and their conjugates, singletsand the SU (3) × SU (2) × U (1) representations (3 , , ± ). The latter have weight one, andoccur as extended roots in SU (5) GUT models (yielding the familiar X and Y bosons).These representations do not occur as matter particles. All other representations that canoccur in the massless spectrum violate the standard model charge integrality sum rule t + s + Y ∈ Z (where t is SU (3) triality and s is the SU (2) spin, modulo 2) and hencelead to color singlets with fractional electric charge. In the following we reserve the name“exotics” to particles that violate the charge integrality sum rule.The SU (3) × SU (2) × U (1) embedding we consider here is the standard one which wouldbe obtained if one breaks SO (10) to SU (5) and then to SU (3) × SU (2) × U (1) Y , with U (1) Y completely embedded in SU (5). It is singled out by the requirement that the couplingsconverge in the canonical SU (5) way. This embedding offers the best opportunity tounderstand the two “GUT miracles” (coupling convergence and family structure) in anatural way in heterotic strings, and perhaps even in string theory in general. It followsfrom the structure of the CFT that in this context the only fractional electric charges thatcan occur are , and . There are many other ways of embedding the standard modelin the heterotic string. Examples are flipped SU (5) models, which provide a different wayto embed the standard model in SO (10). Furthermore, in (2 ,
2) models one may consider10eneral embeddings of U (1) Y in E , as was done for example in [6] and [7]. In thesepapers other fractional charges, like , are mentioned. This must be due to a differentchoice for the U (1) embedding. Furthermore one may consider affine Lie algebras SU (3)and SU (2) at several higher levels, and distribute them freely over the available c = 22internal CFT. In the rest of this paper we will often use the term “heterotic strings” torefer to the specific SU (3) × SU (2) × U (1) class described above. If the algebra SU (3) × SU (2) × U (1) is extended, the set of allowed fractional chargescan be reduced, but it can only be reduced to integers if the extension includes SU (5).For each group type, the allowed fractional charges are listed in the last two columns ofTable 1. The first of these columns lists the fractional charges one would expect on thebasis of the gauge group (given the fact that by construction the quantization of the U (1)factor is in units of ), and the second column lists the true quantization in string theory,as explained in [1].Of these eight groups, the SU (5) GUT and the SO (10) GUT are unacceptable, since,as explained above, they come without Higgs bosons to break the GUT gauge symmetry(we do not consider the possibility that such Higgses are generated dynamically as boundstates of other massless matter). Higgs bosons to break the Pati-Salam group or theleft-right symmetric groups to the standard model can exist in the massless spectrum,and hence these options will be considered viable. The most attractive possibility is nr.2, called “SM, Q=1/2”, because it has a gauge group closest to the standard model, anda charge quantum closest to 1. Within the SO (10) sector of the theory it has a gaugegroup which is exactly SU (3) × SU (2) × U (1), times a superfluous extra U (1). In somecases this extra U (1) is B-L, but there are other possibilities.In the figures we will distinguish these group types by means of the color codes shownin Fig. 1. In the text we will usually refer to them by the names in the second column ofTable 1. In heterotic strings of the type considered here, anomalies cancel by means of the standardGreen-Schwarz mechanism involving the B µν field. Note that string theories obtained byheterotic weight lifting do not have a known geometric interpretation, and cannot bederived in any known way by compactifying ten-dimensional heterotic strings. Hence theonly way to derive anomaly factorization here is by computing the “elliptic genus” of theCFT, and using its modular properties, as explained in [27]. This computation appliesin this case, because it is valid in complete generality for all (0 ,
2) CFT constructions (infact even for (0 ,
1) constructions). The color codes used here are slightly different than those used in [1] in order to make them moreeasily distinguishable in back and white printouts.
SM, Q=1/6 SU (3) × SU (2) × U (1) × U (1)
16 16 SM, Q=1/3 SU (3) × SU (2) × U (1) × U (1)
16 13 SM, Q=1/2 SU (3) × SU (2) × U (1) × U (1)
16 12 LR, Q=1/6 SU (3) × SU (2) L × SU (2) R × U (1)
16 16 SU(5) GUT SU (5) × U (1) LR, Q=1/3 SU (3) × SU (2) L × SU (2) R × U (1)
16 13 Pati-Salam SU (4) × SU (2) L × SU (2) R
12 12 SO(10) GUT SO (10) List of all standard model extensions within SO (10) and the resulting group theoryand CFT charge quantization (last two columns). We refer to these subgroups either by thelabel in column 1 or by the name in column 2, where “LR” stands for left-right symmetric. This result implies that the anomaly factorizes in the standard way in a factor linear in F and R , times a factor Tr F − Tr R , where F is evaluated in the vector representationof an orthogonal group. Furthermore it follows that Tr F receives contributions fromevery factor in the gauge group. If the E factor in the gauge group is unbroken, itcannot produce a term of the form Tr F Tr F (cid:48) , where F is an E two-form and F (cid:48) a U (1)two-form. This is because the only non-trivial representation of E level 1 that can bemassless is the (248), and since this exactly saturates the conformal weight, there is noroom for a U (1) charge. Hence there are no massless representations charged under both E and any U (1). If E cannot contribute to Tr F , it follows that there cannot be aTr F − Tr R factor at all, and hence the anomaly must be identically 0. Since heteroticweight lifting breaks the E factor, it is no longer present in the gauge group. Thereforein general there can be a non-trivial Green-Schwarz mechanism in these models. Thisalso implies that U (1) factors can be anomalous. This may happen in particular to thestandard model U (1) factor Y or the U (1) corresponding to B-L. The corresponding U (1)gauge boson will then become massive because of the Stueckelberg mechanism, absorbingthe B µν field. Note that for this to happen, the U (1) factor must really be anomalous.This is different from orientifold models: there even a non-anomalous U (1) can acquirea mass from axion mixing. It follows that in a heterotic model with a certain number ofstandard model families, but no chiral exotics, U (1) Y must remain unbroken, because itis anomaly free. However, the extra U (1) that commutes with the standard model gaugegroup within SO (10) can be anomalous. This is because we are not requiring this chargeto coincide with B-L for all quarks and leptons.In the absence of fractionally charged matter, anomaly cancellation would force the12
10 20 30020000400006000080000100000120000 SM,Q=1/6SM,Q=1/3SM,Q=1/2LR,Q=1/6SU(5)LR,Q=1/3Pati-SalamSO(10)
Figure 1: Color codes for group types.spectrum into a certain number of standard model families. One question we will beinterested in is to which extent the existence of fractionally charged matter changes that.
First we will examine how often we find each group type if we simply select MIPFs atrandom. MIPF selection requires choosing a subgroup of the full set of simple currents,plus a discrete torsion matrix defined on that subgroup [28]. We have randomized thischoice by first choosing N random elements of the full simple current group, with equalweight for each element. Then we compute the subgroup generated by these N elements.Finally we either enumerate all allowed discrete torsions defined on that subgroup if thereare fewer than a hundred, or otherwise we take a random sample. The amount of timeneeded to compute spectra increases rapidly with N , and in practice values of N largerthan 4 are difficult to deal with. Our data are based on nearly saturated samples for N = 0 , N , but we have checked that the gross features are notstrongly dependent on thoseIn principle there is an exact correspondence between MIPFs of standard and liftedGepner models: the same MIPF gives rise to the same group type. Hence if one randomlyselect tensor products and MIPFs one would get the same distribution for these differentgroup types (though the fact that different standard Gepner models may have differentnumbers of lifts, as well as differences in randomization introduce some discrepancies).This distribution is shown for the lifted Gepner models in Fig. 2. It is based on 9 . × MIPFs, each counted once, before comparing spectra and identifying identical ones.This distribution contains all spectra, including those with chiral fractionally chargedparticles. If we require absence of chiral exotics, the distribution is the one shown inFig. 3. This figure is based on 6 . × MIPFs. Note that requiring absence of chiralexotics reduces the number of spectra by only about 30%. The reduction is strongest forthe spectra with third-integer charges and surprisingly small for those with sixth-integer13 .9%4.9%11.3%8.1%19.4% 8.2% 14.4% 32.8%
SO(10) Pati-Salam LR, Q=1/3 SU(5)LR, Q=1/6 SM, Q=1/2 SM, Q=1/3 SM, Q=1/6
Wednesday, 18 August 2010
Figure 2: Overall distribution of group types.charges. The two half-integer charged types, Pati-Salam and SM, Q= behave ratherdifferently: the former is only reduced by less than 10%, whereas the latter is reduced bya factor 10. In the vast majority of these spectra, fractionally charged exotics are presentin the massless spectrum, but they are vector-like.In most cases, the absence of chiral exotics is simply due to the fact that the entirespectrum is non-chiral. So as a final step we consider only those cases that have at leastone chiral family. Then we get Fig. 4. Now there are 1 . × spectra left. The Q = spectra are reduced to just a few hundred, and are not visible. The Q = spectra nrs.1 and 5 occur respectively about 100.000 and 24.000 times, and are also invisible. Theonly ones that remain with a substantial frequency are Pati-Salam, SM, Q= , and thetwo GUT models SU (5) and SO (10). Interestingly, the SO (10) spectra are reduced muchmore than the SU (5) models, i.e. the latter are more often chiral. Often the appearance of fractionally charged particles in broken heterotic GUTs is hand-waved away. Indeed, they may be massive or confined by additional forces, but theelegance of the original GUTs ( SU (5) or SO (10)) is lost in either case. More seriously,we loose a deeper understanding of why the standard model matter comes in the formof the chiral families we observe. Without the availability of fractionally charged repre-sentations, anomaly cancellation necessarily imposes the observed family structure on thechiral spectrum. If in addition one makes the plausible assumption that only the chiralspectrum survives at low energies, the observed family structure is nicely explained. This14 SO(10) Pati-Salam LR, Q=1/3 SU(5)LR, Q=1/6 SM, Q=1/2 SM, Q=1/3 SM, Q=1/6
Wednesday, 18 August 2010
Figure 3: Overall distribution of group types for spectra without chiral exotics.
SO(10) Pati-Salam LR, Q=1/3 SU(5)LR, Q=1/6 SM, Q=1/2 SM, Q=1/3 SM, Q=1/6
Wednesday, 18 August 2010
Figure 4: Overall distribution of group types without chiral exotics and at least one family.15s indeed true in SU (5) field theory models, but not in heterotic GUTs of the type con-sidered here (it would be true in heterotic GUTs based on higher level affine Lie algebras,but in that case the set of allowed massless representations is not naturally limited to theobserved ones). As soon as fractionally charged matter is available in principle, there areseveral other solutions to anomaly cancellation. Therefore the observed chiral structure ofa standard model family is not understood in this class of heterotic strings, even thoughit seemed that with (16)’s of SO (10) we were on the right track. If the initial emergenceof (16)’s of SO (10) is considered a success, by the same standards the appearance offractionally charged particles must be seen as a failure.It is well-known [29] that in models of this type (heterotic strings with SO (10) brokento SU (3) × SU (2) × U (1)) quarks and leptons are not really unified into a single (16) of SO (10). They typically originate from several different (16)’s in the original manifold.However, if indeed there were no fractional charges available, one could legitimately arguethat in this class of heterotic strings the group-theoretical fact that a family of quarks andleptons fits in a (16) is understood, even if they do not really belong to the same multiplet.As soon as fractionally charged anomaly-free multiplets are available in principle , the bestone can still hope for is that models with the observed family structure dominate thelandscape statistically. Observe that in field-theoretical SO (10) models, upon breaking SO (10), a representation (16) automatically decomposes into standard quarks and leptons,whereas in string-theoretic SO (10) models in the heterotic class considered here, this isnot automatic. Of course one can impose the bias that only standard families are allowedin the chiral spectrum, but then on cannot argue anymore that string theory fully explains the family structure. In RCFT constructions like the one we are discussing here, SU (3) × SU (2) × U (1) spectraare sampled without any further bias towards the standard model family structure (ofcourse the gauge group choice itself is a bias). So in this context it becomes natural toask if, despite the existence of other options, the standard model family structure stilldominates in a statistical sense.The “other options” are models with chiral multiplets that are not of the form of astandard model family. We merely count these spectra before rejecting them, in order tosee to what extent one can still argue that the structure of a standard model family isunderstood in this class of heterotic strings.As remarked above, about 71% of randomly generated spectra turned out to be free ofchiral exotics, whereas for the standard Gepner models this ratio is about 63%. One mightbe tempted to conclude that fractional charges are generically vector-like in these models.But this is a slightly misleading statement for two reasons: on the one hand the set isdominated by SO (10) and SU (5) models, for which this is automatically true, and on theother hand it is dominated by non-chiral spectra, for which it is also automatically true.Another way to ask the question is: which percentage of the chiral spectra with a brokenGUT group ( i.e. all types except SU (5) and SO (10)) and a non-anomalous Y -charge have16 standard family structure with at least one family? For standard Gepner models, theanswer is about 20%. For lifted Gepner models it is about 4 . SU (3) × SU (2) × U (1) that emerges from these heterotic strings. Indeed, some of them arelacking some or all standard quarks and leptons. However, analyzing this is unfortunatelyfar beyond anyone’s capabilities.It would obviously be very interesting to have information about this issue in otherconstructions (such as orbifolds, Calabi-Yau compactifications) but usually spectra withchiral exotics are not even considered in these approaches. In [30, 31, 32] however, a similaranalysis has been done for Pati-Salam models based on free fermions. Their conclusionsappear to agree with ours qualitatively: spectra with chiral exotics dominate those withonly standard families by about a factor of five. In most of the aforementioned 4 .
6% of chiral models without chiral exotics, the fractionallycharged exotics exist as vector-like particles. Here we mean vector-like with respect to SU (3) × SU (2) × U (1). We did not examine chirality with respect to other factors inthe gauge group. Spectra of this kind are potentially acceptable, provided they acquire avery large mass in a proper string vacuum, as discussed in section 2. We also found examples where the fractional charges are completely absent from themassless spectrum. In [1] we found only very few examples, all of them with zero families;for lifted Gepner models we also have examples with a non-vanishing, though always even(2, 4, 6, 8, 12 or 24) number of families. These were found only for the Pati-Salam andSM, Q= group types; in other words, only if the CFT charge quantum is . Theyoccur very rarely indeed. To get an idea, let us see the precise numbers for the mostcommon one, the Pati-Salam model. In total, we found about 864.000 distinct Pati-Salam type models without chiral exotics, of which about 100.000 were chiral, i.e. theyhave a non-vanishing number of standard model families. About 6000 Pati-Salam modelshad no massless exotics at all, but of these 6000 only 143 had a non-zero number of chiralfamilies. For type SM, Q= these numbers are respectively 75.000, 36.000, 330 and 36.In both cases, the chance that a chiral model has no fractionally charged exotics at all Note that here each distinct spectrum is counted only once, whereas the foregoing figures were basedon total occurrences
17s about 1 in 1000. It seems unreasonable to claim that this is the reason we observe nofractional charges experimentally.This same issue has been investigated recently for free fermionic constructions in[30, 31, 32]. These authors only constructed Pati-Salam models and found a somewhatsmaller rate for the absence of fractional charges, about 10 − , but did find examples withthree families. This demonstrates that there does not exist a no-go theorem against suchspectra in the general class of (0 ,
2) models. Given the fact that the total number ofexamples without massless fractional charges we found is of order 100, there is a chancethat three family examples occur in our set as well at a rate of 10 − , but are still hiddenin the noise.The most important conclusion here is that non-exotic models are rare. As explainedbefore, in our opinion in RCFT one should aim for absence of chiral exotics, but requiringcomplete absence of vector-like exotics (fractionally charged or not) is simply too muchto ask for. To put it differently: if one were to reject models because they have somevector-like exotics at the level of exact RCFT, one risks loosing many examples that wouldsurvive perfectly if one perturbs the theory.It is known that massless fractional charges can be avoided using Wilson line symmetrybreaking in combination with freely acting discrete symmetries [6, 7]; for recent examplessee e.g. [33, 34]. Let us emphasize that this is not in disagreement with the statementabove. The question we are trying to answer is how common such non-exotic models arein the full class of examples that have chiral families. This question is not answered byfocusing only on the case where massless fractional charges are absent by construction.It should be noted that most models in the “heterotic mini landscape” studied prior to[34] (see e.g. [20]) did have exact spectra with massless fractional charges, whose massescould be lifted by certain terms in the superpotential. It would be tremendous progressif a non-phenomenological argument could be found to explain why freely acting discretesymmetries should be favored over other ways to get the standard model from the heteroticstring.Nevertheless, it would be interesting to find out if the examples we found, or those of[30], can be understood in terms of some freely acting discrete symmetry (which is notmanifest in our formalism), or if they are merely statistical fluctuations. The distributionshown in Fig. 13, and which will be discussed in section 6, suggests the latter, since thesmall peak at zero is not inconsistent with the rest of the distribution (for comparison,the peaks at zero in the first two plots of Fig. 16, to be discussed later, are clearly notstatistical fluctuations). For comparison we show in Figs. 5 and 6 the distribution of the number of familiesfor standard Gepner models (168 minimal model combinations, plus 59 with exceptionalinvariants of SU (2)) and lifted Gepner models (435 combinations). On the vertical axis weshow the number of MIPFs. We distinguish MIPFs on the basis of the numbers of standard18odel representations Q, U, D, L, E and their conjugates, the total number of standardmodel singlets, the total number of fractionally charged particles (not counting their coloror weak isospin degrees of freedom, but counting their dimension in any group that is notpart of the standard model), the total number of gauge bosons, the observable part ofthe gauge group (gauge bosons that are neutral with respect to SU (3) × SU (2) × U (1)are merely counted, but their mutual interactions are ignored), and any deviations fromthe expected charge quantization in the massless spectrum (for example, if on the basisof the CFT particles of color singlet charge are allowed, but in the observed spectrumthese particles all turn out be massive).Of course this is not strictly the same as distinguishing MIPFs, since two distinctMIPFs might still produce the same spectrum, either by coincidence, or by symmetry-related degeneracies (for example the interchange of identical factors in the tensor prod-uct). If there are such degeneracies, it would be preferable to remove them anyway, so noharm is done by identifying such spectra. Furthermore, such symmetry-related spectrawill be identical in all other respects as well. This still leaves the possibility of coincidentalmatches of only the massless spectrum. We have reasons to believe these are rare (moreabout this later), so that what we are plotting in Figs. 5 and 6 is indeed very close to thenumber of truly distinct MIPFs.Obviously these plots are not true landscape distributions, i.e. distributions of non-super-symmetric moduli-stabilized string vacua. Furthermore one may question the rel-ative weights for different MIPFs, and especially the relative weights given to differenttensor products. However, the main point we want to stress in Fig. 5 is that in standardGepner models the number three is hard to get, and in particular very much disfavouredin comparison to two and four, confirming the experience of two decades ago. This re-sult remains true regardless of the measure one chooses, unless one can find an argumentexplaining why the (1 , ∗ , ∗ , ∗ ) (giving rise to the only 3-family models in this set)should be enhanced by a huge factor over anything else.The main result of the present paper is shown in Fig. 6. We observe that in the liftedGepner model landscape an approximately exponential family distribution is obtained,where the number three is not especially disfavoured with respect to two and is evenfavoured with respect to four. Indeed, the impression one gets from this plot is thatfamilies with a factor of two and/or three are enhanced with respect to the average. Notefor example the peaks at 6, 9 and 12 families. Also for these models the same caveatsregarding true landscape distributions apply. However in this case three family spectrado not just come from a single tensor product, but from 260 of the 435 combinations,so the issue of their relative weight is less important. Of course it is likely that modulistabilization and supersymmetry breaking affect the shape of this distribution, but we arenot aware of any effect that would single out the number three, either by means of a dip ora peak. It is much more plausible that such effects might change the slope, and indeed itis still possible that this suppresses three families with respect to two by a huge amount,but the optimistic interpretation of our results is that in the heterotic landscape, as soonas one moves away from the (2 ,
2) models, the number three is not especially difficult toobtain. 19urther evidence for the special role of the numbers two and three comes from studyingthe greatest common divisor (denoted ∆) of the number of families for all the MIPFs ofeach lifted N = 2 tensor product. The results are listed in Table 2. Most lifted Gepnermodels have a value ∆ = 1, but 0, 2, 3, 4, and 6 also occur. By contrast, for standardGepner models, the values of ∆ we found were 0 , , , . . . families from the ∆ = 1 models; the first number thatwe did not encounter at all is 31.The distribution of families was also studied recently in the context of heterotic stringsusing free fermions [32]. These authors also find a roughly exponential fall-off with thenumber of families, but with a steeper slope of about a factor three per family. Eventhough in that case three is more suppressed, one could still conclude that three familyspectra are not especially rare. An interesting difference with our results is the completeabsence of spectra with seven families (or any larger odd number). This must perhaps beviewed as an artifact of free fermionic constructions.∆ Number0 941 1992 563 614 85 06 17Table 2: The number of lifted Gepner models with a given family quantum ∆.Our conclusion is quite different from what was observed for orientifolds in [4] and[5]. In this case the number of families drops off much faster, and there is a clear addi-tional suppression of odd numbers with respect to even ones, which unfortunately is notunderstood. Consequently, three is suppressed with respect to two and four by two tothree orders of magnitude. It is impossible to tell whether the heterotic results or theorientifolds results are more typical for the string landscape as a whole, but at least thereis a better chance now that the difficulties with obtaining three families can be attributedto artifacts of the chosen method.Note that Fig. 6 is dominated by just four group types, the phenomenologicallyuseless (because unbroken) GUT groups SU (5) and SO (10), the Pati-Salam model, and20he unextended standard model with charge quantization in half-integer units. The otherfour types are also present, as was already discussed in the previous section, but there aretoo few of them to be visible. To make them appear we can add the data for zero familymodels. This is shown for standard Gepner models in Fig. 7 and for lifted Gepner modelsin Fig. 8. Especially in the latter case this yields a huge bar at zero that dominateseverything. Universes with zero families thus seem to dominate the ensemble, but this isnot a problem, since they are anthropically unacceptable.In comparison with the results of [32] we get a much larger peak at zero families. Thiscan be explained, at least partly, as follows. Our plots are in terms of the group typesintroduced in the previous section. These group types define the subgroup of SO (10) thatis realized, but we do not put any constraints on the extension of these groups into theinternal sector of the theory, whereas the authors of [32] require that there be no extensionbeyond the group they consider, the Pati-Salam model. Some of those extensions willrender the gauge group non-chiral, and therefore can only contribute to the zero-familypeak in the distribution, producing an abnormal enhancement of that peak.Note that in Figs. 5 . . . SU (5) and SO (10) cases tendto come out far more often, and hence they dominate the set. The reason for using totaloccurrence frequency rather than the total number of distinct MIPFs in the foregoingsection was explained in [1]: we do not distinguish MIPFs for spectra with chiral exotics.In Table 3 we show all lifted Gepner models with a non-vanishing value for ∆. In thefirst column the combination of levels is shown, with a hat indicating which factor is lifted.For some values of the level k two lifts are possible. The second one is indicated by atilde instead of a hat. Column 2 shows the value of ∆ and column 3 the maximal numberof families for that model. Column 4 lists the number of distinct 3-family spectra weobtained, and column 5 lists the total number of distinct N family spectra, with N ≥ i.e. a spectrum and itsmirror pair is just counted once.As explained in [1] the full set of spectra we obtain is mirror symmetric. This is astatement about RCFT, and does not necessarily imply anything about an underlyinggeometry. Typically, in order to encounter a mirror partner of a given spectrum it isnecessary to add one simple current twist. Therefore, if we go to arbitrary numbers ofgenerators of the simple current subgroup, we will eventually find all mirror pairs. Forpractical reasons, we have limited this number to 4. By checking if the set of spectra isclosed under mirror symmetry we can get a rough idea about the completeness of the set.For this reason we have indicated in the last column which percentage of the total setsof MIPFs is lacking a mirror partner. As expected, this number tends to be largest fortensor combinations with many factors, because here the limitation to four generators ismost restrictive. 21able 3: Results for lifted Gepner models model ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors( (cid:98) , , , , , (cid:98) , , , (cid:98) ,
82) 1 8 10 69 0.00%( (cid:98) , , ,
58) 1 18 189 563 0.71%(1 , , (cid:98) ,
58) 1 64 11 295 0.34%(1 , , , (cid:98)
58) 1 42 7 103 0.00%( (cid:98) , , ,
46) 3 12 126 172 8.14%(1 , , (cid:98) ,
46) 1 14 96 648 12.50%( (cid:98) , , ,
28) 1 17 126 560 6.79%( (cid:98) , , ,
26) 3 6 8 12 0.00%(1 , , (cid:98) ,
26) 3 15 2 10 0.00%(1 , , , (cid:98)
26) 1 12 1 9 0.00%( (cid:98) , , ,
22) 3 24 117 215 24.19%( (cid:98) , , , , ,
10) 3 30 757 1194 31.49%( (cid:98) , , , , , , , ,
1) 3 33 304 584 40.07%(1 , , , , , , , (cid:98)
4) 2 24 0 437 33.64%( (cid:98) , , , , , , ,
4) 3 33 359 670 27.76%(1 , , , , , (cid:98) ,
10) 1 24 26 226 15.04%( (cid:98) , , , , , ,
10) 3 30 98 343 26.24%(1 , , , , , (cid:98) ,
4) 1 24 344 2986 37.51%( (cid:98) , , , , , ,
4) 3 33 1827 3054 27.96%( (cid:98) , , ,
76) 6 6 0 2 0.00%(1 , (cid:98) , ,
76) 3 6 4 7 0.00%(1 , , (cid:98) ,
76) 2 8 0 6 0.00%(1 , , , , , , (cid:98)
4) 1 16 0 100 31.00%( (cid:98) , , , , , ,
4) 3 24 81 186 20.43%(1 , , , , (cid:98) , ,
4) 1 24 13 175 29.14%( (cid:98) , , , , ,
40) 3 18 12 25 36.00%(1 , , , , (cid:98) ,
40) 2 8 0 21 23.81%( (cid:98) , , , , ,
22) 3 24 64 143 14.69%(1 , , , , (cid:98) ,
22) 1 16 0 81 13.58%( (cid:98) , , , , ,
16) 3 27 303 523 29.45%Continued on next page22 able 3 – continued from previous pagemodel ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors( (cid:98) , , , , ,
13) 3 18 7 37 8.11%(1 , , , , (cid:98) ,
13) 2 16 0 19 0.00%( (cid:98) , , , , ,
10) 3 30 706 1377 27.09%(1 , , , (cid:98) , ,
10) 1 24 315 1542 25.42%(1 , , , , (cid:98) ,
10) 1 24 114 1245 22.25%( (cid:98) , , , , ,
4) 3 33 4117 6732 28.58%(1 , , , (cid:98) , ,
4) 1 32 1044 9903 22.56%(1 , (cid:98) , ,
40) 1 18 21 123 0.00%( (cid:98) , , , ,
82) 3 18 28 55 0.00%(1 , , (cid:98) , ,
82) 1 14 16 70 0.00%(1 , , , (cid:98) ,
82) 2 8 0 15 0.00%( (cid:98) , , , ,
58) 1 18 42 156 4.49%(1 , , (cid:98) , ,
58) 1 14 22 160 3.12%(1 , , , (cid:98) ,
58) 1 18 1 60 1.67%(1 , , , , (cid:98)
58) 2 32 0 19 5.26%( (cid:98) , , , ,
46) 3 15 144 211 6.16%(1 , , (cid:98) , ,
46) 1 12 42 174 0.00%(1 , , , (cid:98) ,
46) 1 12 9 195 2.05%( (cid:98) , , , ,
34) 3 21 368 734 19.75%(1 , , (cid:98) , ,
34) 1 18 73 816 12.50%( (cid:98) , , , ,
28) 1 12 98 247 1.62%(1 , , (cid:98) , ,
28) 1 12 42 204 1.96%( (cid:98) , , ,
19) 3 6 14 18 0.00%(1 , (cid:98) , ,
19) 1 14 0 9 0.00%(1 , , (cid:98) ,
19) 6 6 0 1 0.00%( (cid:98) , , , , ,
10) 3 24 219 458 5.68%(1 , , (cid:98) , , ,
10) 1 36 263 1963 21.40%( (cid:98) , , , ,
22) 3 30 200 481 13.31%(1 , , (cid:98) , ,
22) 1 24 43 471 11.46%( (cid:98) , , , , ,
4) 3 24 285 771 13.62%(1 , , (cid:98) , , ,
4) 1 24 24 854 17.80%(1 , , , , (cid:98) ,
4) 1 32 81 1555 20.19%( (cid:98) , , , ,
13) 1 24 220 569 20.74%Continued on next page23 able 3 – continued from previous pagemodel ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors(1 , , (cid:98) , ,
13) 1 48 28 580 5.52%(1 , , , (cid:98) ,
13) 1 36 47 475 6.53%( (cid:98) , , ,
28) 1 34 335 1093 5.67%(1 , (cid:98) , ,
28) 1 80 145 1184 2.20%( (cid:98) , , ,
18) 1 24 37 240 0.00%(1 , (cid:98) , ,
18) 1 48 10 137 0.00%( (cid:98) , , , ,
43) 1 18 51 94 9.57%(1 , , (cid:98) , ,
43) 1 24 6 114 8.77%( (cid:98) , , , ,
28) 1 22 251 922 7.70%(1 , , (cid:98) , ,
28) 1 48 145 1279 15.72%(1 , , , (cid:98) ,
28) 1 32 42 334 4.49%( (cid:98) , , , ,
10) 3 30 2203 3775 11.42%(1 , , (cid:98) , ,
10) 1 24 269 2842 36.17%( (cid:98) , , ,
22) 3 24 34 79 3.80%(1 , (cid:98) , ,
22) 1 12 53 379 15.57%(1 , , (cid:98) ,
22) 1 13 60 305 20.00%( (cid:98) , , , ,
40) 3 18 18 48 0.00%(1 , , (cid:98) , ,
40) 2 16 0 35 20.00%(1 , , , (cid:98) ,
40) 2 12 0 22 4.55%( (cid:98) , , , ,
22) 3 24 504 1016 9.55%(1 , , (cid:98) , ,
22) 1 24 78 1009 12.88%(1 , , , (cid:98) ,
22) 1 48 31 837 9.44%( (cid:98) , , , ,
16) 3 27 900 1476 21.00%(1 , , (cid:98) , ,
16) 1 20 196 1596 10.78%( (cid:98) , , , ,
13) 1 18 43 125 2.40%(1 , , (cid:98) , ,
13) 1 16 8 107 0.93%(1 , , , (cid:98) ,
13) 2 16 0 31 0.00%(1 , , , , (cid:98)
13) 1 18 3 20 0.00%( (cid:98) , , , ,
19) 3 12 5 12 0.00%(1 , , (cid:98) , ,
19) 1 8 7 9 0.00%(1 , , , (cid:98) ,
19) 3 6 7 8 0.00%( (cid:98) , , , ,
10) 3 6 48 71 0.00%(1 , , (cid:98) , ,
10) 1 8 31 121 0.00%Continued on next page24 able 3 – continued from previous pagemodel ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors( (cid:98) , , , ,
7) 3 27 246 315 1.90%( (cid:98) , , , ,
10) 3 24 620 1060 5.38%(1 , (cid:98) , , ,
10) 1 25 764 5365 25.55%(1 , , , , , (cid:98)
4) 2 32 0 293 15.02%( (cid:98) , , , , ,
4) 6 24 0 116 18.97%(1 , (cid:98) , , , ,
4) 2 32 0 1499 11.41%( (cid:98) , , , ,
22) 3 24 136 423 9.46%(1 , (cid:98) , , ,
22) 1 18 204 1717 21.26%(1 , , , (cid:98) ,
22) 1 48 75 1010 16.93%( (cid:98) , , , ,
16) 3 12 3 17 0.00%(1 , (cid:98) , , ,
16) 1 12 0 61 6.56%( (cid:98) , , , ,
58) 1 12 57 203 2.96%(1 , (cid:98) , , ,
58) 1 16 31 245 2.04%(1 , , (cid:98) , ,
58) 1 12 54 493 10.34%(1 , , , , (cid:98)
58) 1 24 0 52 0.00%( (cid:98) , , , ,
18) 2 12 0 24 0.00%(1 , (cid:98) , , ,
18) 2 24 0 76 0.00%(1 , , (cid:98) , ,
18) 2 24 0 103 3.88%(1 , , , (cid:98) ,
18) 2 16 0 51 0.00%(1 , (cid:98) , , ,
10) 1 24 419 3016 15.02%(1 , , (cid:98) , ,
10) 1 20 451 4097 22.19%( (cid:98) , , , ,
6) 6 12 0 18 11.11%(1 , (cid:98) , , ,
6) 2 24 0 172 8.14%(1 , , (cid:98) , ,
6) 4 16 0 35 8.57%(1 , , , (cid:98) ,
6) 2 48 0 305 14.75%( (cid:98) , , , ,
13) 1 18 125 692 3.18%(1 , (cid:98) , , ,
13) 1 36 444 3383 12.33%(1 , , , , (cid:98)
13) 1 36 41 410 0.73%( (cid:98) , , , ,
8) 2 14 0 63 0.00%(1 , (cid:98) , , ,
8) 1 24 18 411 12.65%(1 , , , (cid:98) ,
8) 2 18 0 122 0.82%(1 , , , , (cid:98)
8) 1 18 3 51 1.96%(1 , (cid:98) , , ,
4) 1 24 947 8018 12.91%Continued on next page25 able 3 – continued from previous pagemodel ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors( (cid:98) , , , , (cid:98) , , , , (cid:98) , (cid:98) , , , , (cid:98) , , (cid:98) , , , , (cid:98) , , (cid:98) , , , , (cid:98) , , , , (cid:98) , (cid:98) , , , , (cid:98) , , (cid:98) , , , , (cid:98) , , , , (cid:98) , (cid:98) , , , , (cid:98) , , , , (cid:98) , (cid:98) , , , , (cid:98) , , (cid:98) , , , , (cid:98) , , (cid:98) , , ,
82) 3 48 6 53 0.00%(1 , (cid:98) , ,
82) 1 56 22 81 1.23%( (cid:98) , , , , (cid:98) , , , , (cid:98) , (cid:98) , , , , (cid:98) , , (cid:98) , , , , (cid:98) , , (cid:98) , , , , (cid:98) , , able 3 – continued from previous pagemodel ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors(1 , , (cid:98) , (cid:98) , , , , (cid:98) , , (cid:98) , , ,
94) 3 12 88 168 26.19%(1 , (cid:98) , ,
94) 1 40 43 483 19.46%(1 , , (cid:98) ,
94) 1 32 49 615 15.93%(1 , (cid:98) , ,
86) 2 4 0 5 0.00%(1 , , , (cid:98)
86) 6 12 0 5 0.00%( (cid:98) , , ,
70) 3 9 24 40 10.00%(1 , (cid:98) , ,
70) 1 48 35 471 6.16%( (cid:98) , , ,
58) 1 11 32 243 4.12%(1 , (cid:98) , ,
58) 1 24 59 438 7.31%(1 , , , (cid:98)
58) 2 42 0 79 11.39%(1 , (cid:98) , ,
54) 2 32 0 101 3.96%(1 , , , (cid:98)
54) 2 24 0 28 0.00%( (cid:98) , , ,
46) 3 12 101 163 21.47%(1 , (cid:98) , ,
46) 1 64 69 867 21.34%( (cid:98) , , , , , (cid:98) , (cid:98) , , , (cid:98) , , , , , (cid:98) , (cid:98) , , ,
97) 3 6 4 5 0.00%(1 , , (cid:98) ,
97) 4 8 0 3 0.00%( (cid:98) , , ,
70) 3 15 121 207 18.84%( (cid:98) , , ,
52) 3 18 50 80 2.50%( (cid:98) , , ,
43) 1 10 60 142 2.11%( (cid:98) , , ,
34) 3 24 123 209 1.91%( (cid:98) , , , , (cid:98) , , , , (cid:98) , , , , (cid:100) (cid:98) , , ,
88) 1 19 64 530 12.64%Continued on next page27 able 3 – continued from previous pagemodel ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors(1 , (cid:98) , ,
88) 1 28 43 477 5.87%( (cid:98) , , ,
58) 1 23 726 3734 6.86%(1 , (cid:98) , ,
58) 1 32 333 2234 5.73%(1 , , , (cid:98)
58) 1 84 97 1248 7.05%( (cid:98) , , ,
38) 1 12 12 152 0.66%(1 , (cid:98) , ,
38) 2 10 0 38 0.00%( (cid:98) , , ,
28) 1 36 1530 7738 17.19%(1 , (cid:98) , ,
28) 1 56 584 4618 9.44%( (cid:98) , , ,
31) 3 6 8 10 0.00%(1 , (cid:98) , ,
31) 1 9 1 3 0.00%(1 , , (cid:98) ,
31) 1 8 2 5 0.00%( (cid:98) , , ,
10) 1 12 28 518 0.00%( (cid:98) , , , , ,
2) 1 48 940 14131 8.90%( (cid:98) , , , ,
18) 1 24 75 1029 13.70%(2 , , , (cid:98) ,
18) 2 30 0 286 11.19%( (cid:98) , , , ,
10) 1 24 185 3216 13.50%(2 , , , (cid:98) ,
10) 2 24 0 534 17.42%( (cid:98) , , , ,
6) 1 24 383 9788 5.85%(2 , , , (cid:98) ,
6) 1 48 178 3288 6.60%( (cid:98) , , , ,
8) 1 16 23 443 1.35%(2 , , (cid:98) , ,
8) 1 24 55 1112 2.34%(2 , , , , (cid:98)
8) 1 32 33 360 0.56%(2 , , (cid:98) , ,
4) 2 24 0 471 5.73%( (cid:98) , , , , (cid:98) , , , , (cid:98) , (cid:98) , , , , (cid:98) , , (cid:98) , , ,
98) 1 12 32 243 0.00%(2 , (cid:98) , ,
98) 1 54 30 372 0.81%(2 , , (cid:98) ,
98) 1 32 24 260 0.38%( (cid:98) , , ,
68) 1 12 0 32 6.25%(2 , (cid:98) , ,
68) 2 24 0 79 2.53%Continued on next page28 able 3 – continued from previous pagemodel ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors( (cid:98) , , ,
58) 1 18 169 1340 3.96%(2 , (cid:98) , ,
58) 1 78 308 3150 10.63%(2 , , , (cid:98)
58) 1 40 13 137 8.03%( (cid:98) , , ,
38) 1 24 125 1038 14.84%(2 , (cid:98) , ,
38) 1 108 166 2053 10.42%( (cid:98) , , , , (cid:98) , , (cid:98) , , ,
58) 1 8 62 228 4.39%(2 , (cid:98) , ,
58) 1 12 27 306 12.42%(2 , , (cid:98) ,
58) 2 30 0 95 2.11%(2 , , , (cid:98)
58) 1 32 0 19 15.79%( (cid:98) , , ,
46) 1 12 0 57 8.77%(2 , (cid:98) , ,
46) 1 12 2 29 10.34%(2 , , (cid:98) ,
46) 2 12 0 68 0.00%(2 , (cid:98) , ,
34) 1 8 8 248 9.27%( (cid:98) , , ,
28) 1 12 18 161 0.00%(2 , (cid:98) , ,
28) 1 10 0 79 2.53%( (cid:98) , , ,
22) 1 12 0 61 0.00%(2 , (cid:98) , ,
22) 1 8 2 24 0.00%( (cid:98) , , ,
40) 3 12 4 19 0.00%( (cid:98) , , ,
26) 1 12 22 141 0.71%(2 , , (cid:98) ,
26) 1 5 10 40 0.00%(2 , , , (cid:98)
26) 1 15 4 20 10.00%( (cid:98) , , , , (cid:98) , , , , (cid:98) , (cid:98) , , ,
22) 1 15 19 426 2.11%(2 , (cid:98) , ,
22) 1 12 24 248 0.40%( (cid:98) , , ,
14) 1 20 0 138 2.17%(2 , (cid:98) , ,
14) 1 32 32 224 3.12%(2 , , (cid:98) ,
14) 1 18 67 350 5.71%(2 , , , (cid:98)
14) 1 18 55 348 2.87%( (cid:98) , , ,
70) 1 6 0 20 10.00%Continued on next page29 able 3 – continued from previous pagemodel ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors(2 , (cid:98) , ,
70) 2 2 0 11 9.09%( (cid:98) , , ,
38) 1 16 48 387 2.58%(2 , (cid:98) , ,
38) 1 22 18 320 3.12%(2 , , (cid:98) ,
38) 1 18 1 140 0.00%( (cid:98) , , ,
16) 1 10 14 114 1.75%( (cid:98) , , ,
34) 1 6 2 8 0.00%( (cid:98) , , ,
13) 1 8 2 61 4.92%(2 , (cid:98) , ,
13) 4 8 0 6 0.00%(2 , , , (cid:98)
13) 2 22 0 28 0.00%( (cid:98) , , ,
18) 1 24 311 2310 1.90%(2 , (cid:98) , ,
18) 1 24 395 3115 1.28%( (cid:98) , , ,
58) 1 30 80 486 0.00%(3 , , , (cid:98)
58) 1 16 5 49 0.00%( (cid:98) , , ,
33) 1 16 2 89 0.00%(3 , , (cid:98) ,
33) 2 12 0 9 0.00%( (cid:98) , , ,
28) 1 54 1034 6388 5.42%(3 , , (cid:98) ,
28) 1 48 145 999 0.20%( (cid:98) , , ,
18) 1 68 1802 12434 1.17%( (cid:98) , , , ,
3) 1 50 3189 25714 20.91%( (cid:98) , , , , , (cid:98) , (cid:98) , , ,
18) 2 18 0 76 0.00%(3 , (cid:98) , ,
18) 2 8 0 20 0.00%( (cid:98) , , ,
13) 1 48 11 138 0.00%(3 , (cid:98) , ,
13) 2 12 0 83 0.00%(3 , , (cid:98) ,
13) 1 30 16 79 0.00%( (cid:98) , , , , (cid:98) , , , , (cid:98) , (cid:98) , , ,
43) 1 24 4 16 0.00%(3 , (cid:98) , ,
43) 2 4 0 5 0.00%( (cid:98) , , ,
28) 1 48 340 2638 4.66%(3 , (cid:98) , ,
28) 1 20 71 1116 1.43%Continued on next page30 able 3 – continued from previous pagemodel ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors(3 , , (cid:98) ,
28) 1 32 86 644 1.24%( (cid:98) , , ,
68) 1 24 0 27 0.00%(3 , (cid:98) , ,
68) 1 8 9 13 0.00%(3 , , (cid:98) ,
68) 1 6 9 17 0.00%( (cid:98) , , ,
18) 2 28 0 158 0.63%(3 , (cid:98) , ,
18) 1 8 18 99 0.00%( (cid:98) , , ,
8) 1 72 3046 23592 4.80%(3 , (cid:98) , ,
8) 1 32 3137 21945 3.20%( (cid:98) , , ,
10) 1 20 45 420 2.86%( (cid:98) , , ,
40) 1 12 0 64 3.12%( (cid:98) , , ,
22) 1 14 4 260 5.38%(4 , , (cid:98) ,
22) 4 8 0 26 3.85%( (cid:98) , , ,
16) 1 18 104 491 3.87%( (cid:98) , , ,
13) 1 12 5 166 1.81%(4 , , (cid:98) ,
13) 4 16 0 23 0.00%(4 , , , (cid:98)
13) 2 28 0 25 0.00%( (cid:98) , , ,
19) 4 8 0 2 0.00%(4 , (cid:98) , ,
19) 1 8 1 2 0.00%(4 , , (cid:98) ,
19) 3 3 1 1 0.00%( (cid:98) , , ,
10) 2 2 0 2 0.00%(4 , (cid:98) , ,
10) 4 8 0 9 0.00%( (cid:98) , , ,
7) 2 6 0 14 0.00%( (cid:98) , , ,
12) 1 8 13 82 0.00%(5 , (cid:98) , ,
12) 3 6 29 40 0.00%(5 , , , (cid:98)
12) 1 8 6 22 0.00%( (cid:98) , , ,
6) 1 16 174 1869 0.27%(1 , , (cid:101) , , , (cid:101) ,
46) 1 14 83 599 11.85%(1 , , , , (cid:101) ,
40) 6 12 0 5 20.00%(1 , , , (cid:101) ,
46) 1 12 20 195 0.00%(1 , , , (cid:101) ,
40) 6 12 0 13 0.00%(1 , (cid:101) , , , (cid:101) , , able 3 – continued from previous pagemodel ∆ Max. 3 family N fam. ( N >
0) Missing Mirrors(1 , (cid:101) , , , (cid:101) , , , (cid:101) , , , (cid:101) , , , (cid:101) , , , (cid:101) , , , (cid:101) , , , (cid:101) , ,
82) 3 48 39 74 1.35%(1 , , (cid:101) , , , (cid:101) , , , (cid:101) ,
46) 2 24 0 72 1.39%(2 , (cid:101) , , As already emphasized before, we regard the chiral data (the number of families and theabsence of chiral exotics) as the main area of study for these methods. However, we didalso collect non-chiral data such as the number of mirrors, singlets and fractionally chargedvector-like matter. The rationale for doing that is to determine how many distinct pointsin a given moduli space we obtain. Ideally, one could do that by comparing MIPFs. Butthis is hard to do. In principle any distinct simple current subgroup and/or any differentchoice of discrete torsion parameters defines a different MIPF, but there are usually largedegeneracies, which are not all understood, and are difficult to “mod out” even if theyare. So the simpler question we consider is: how many distinct massless spectra do weget (with “distinct” defined as in section 4). Obviously two identical massless spectramay still belong to different MIPFs, but when we made finer distinctions (as discussed inthe next section) we found that in most cases a given spectrum yielded only one distinctrefined spectrum. Hence it seems that our results give a pretty good estimate for the truenumber of distinct MIPFs.Having gathered all this data we can also use them for a different purpose, namely toplot distributions of non-chiral quantities. The most important lesson that may be drawnfrom this procedure is how easy it would be to find exact RCFT spectra with certaindesired properties, such as absence of certain vector-like exotics. Although this problemshould really be addressed in a continuous geometric approach, as we have emphasizedbefore, at least the existence of an exact RCFT example demonstrates that there are nofundamental obstacles for finding such solutions.32n all of these plots we display on the vertical axis the number of distinct MIPFs,counting only models without chiral exotics, and with at least one family. We treatmirror symmetric MIPFs as distinct. Furthermore, all plots are stacked histograms: thecontributions of the 8 different group types are stacked on top of each other for bettervisibility. In Figs. 9 and 10 we show respectively the distribution of Q and L-type mirrors.It is noteworthy that both distributions have shifted towards zero in comparison to thestandard Gepner model case [1]. For Pati-Salami models, the Q mirror distribution peaksat 4 for standard Gepner models, and at 0 for lifted ones. For SM, Q= models the Qmirrors peak at 3 (standard) and 0 (lifted). So in both cases the peak is at the MSSMvalue. For L mirror pair the MSSM value is one (the Higgs pair H , H ); the peaks doindeed shift in that direction, from a value of around 20 to values of around 10.The analogous plots for U and E mirrors are practically indistinguishable from those ofQ mirrors, and the D-mirror plot is nearly identical to the L-mirror plot. This is becausein SU (5) GUTs, Q, U and E all come from the representation (10), while D, L both comefrom the (¯5). Hence for the dominant contribution to the plots these distributions areidentical. But even for the broken GUTs these quantities are not all unrelated: the spin-2current (3 , , ,
0) of SU (3) × SU (2) × U (1) × U (1) , that is always present in the half-integer fractional charge models, relates U and E. Since the half-integer fractional chargemodels (Pati-Salam and SM, Q= ) dominate the statistics for the non-GUT models,there are only two differences that can still be non-trivial: D-L and Q-E. We show thedistributions for these differences in Figs. 11 and 12 respectively. The main conclusion isthat even in broken GUT models there is a tendency for mirror particles to be in completeGUT multiplets. This has the interesting consequence that mirror particles tend not toaffect the convergence of couplings. The effect is somewhat weaker for members of the(¯5), but it is still no entirely unreasonable to assume that the effect of mirrors on couplingunification could be small.In Fig. 13 we show the distribution of fractionally charged particles. What is countedhere along the x-axis are half the number of SU (3) × SU (2) × U (1) representations. TheSM-dimensions (color triplets, weak doublets) of those representations are not taken intoaccount, but the dimensions with respect to any additional gauge factor are counted asmultiplicities. Most vector-like fractionally charged exotics come in pairs, but the half-integer charged representation (1 , ,
0) is real by itself (this is the only available SU (3) × SU (2) × U (1) representation, apart from the singlet, with that property). However, ittoo must appear with even multiplicities, since otherwise there would be an SU (2) globalanomaly (note that this anomaly cancel within each chiral family, and among the twomembers of any other vector-like pair). For this reason we have divided the number offractionals by two, so that essentially what is shown is the number of vector-like pairs.There is an interesting structure visible in these plots. There are considerably more SM,Q= models where the number of vector-like pairs is even. For the Pati-Salam modelsthis goes even further, with a successive enhancement if the number of pairs is even, amultiple of four and a multiple of eight. The origin of this phenomenon is not clear.The singlet distribution is shown in Fig. 14. The positions of the peaks of thesedistributions are not very different from those of the standard Gepner model case [1], but33he distributions are much broader.One might have hoped that weight lifting, since it increases more conformal weightsthan it decreases, would lower the number of singlets (as indeed it seems to do for themirrors). However, one should keep in mind that not only the conformal weight changes,but that also the ground state dimensions in general increase, because the ground statesare often in non-trivial representations of the extra non-abelian gauge groups of the liftedCFT. When counting states we did take these dimensions into account. Therefore theactual number of singlet particle representations is considerably smaller. This effect ismore important for standard model singlets than for mirrors, because the latter get acontribution to their conformal weight from the standard model representation they carry,so that there is less “space” left for a non-trivial representation of a non-abelian factor:the total conformal weight has to add up to 1. In orientifold models, matter that is charged under additional gauge group factors butnot under the standard model is usually not counted as a singlet, but as hidden sectormatter. One can perform a similar count here. In Fig. 15 we plot the distribution ofstandard model matter singlets (all singlets except the dilaton multiplet) that are in thetrivial representation of the non-abelian factors of the “hidden sector”. Here we are onlyconsidering non-abelian groups originating from the lifts. In general there will also be afew cases where the chiral algebra of the internal sector of the theory (that is, the part ofthe CFT outside SO (10)) is extended to a non-abelian group, but these extensions werenot taken into account. As expected, the distributions move towards zero in comparisonto Fig. 14, and there are even some cases where zero is reached, so that all standard modelmatter singlets are in a non-trivial representation of some non-abelian hidden sector group.The latter are examples of chiral heterotic spectra where the only gauge singlets arethe dilaton and the axion, B µν (and of course the graviton). However, this does not meanthat there are no moduli, apart from the dilaton multiplet. One can observe the samephenomenon in enhanced symmetry points in the moduli space of Narain compactifica-tions. In that case the moduli are in non-trivial representations of non-abelian groups,and giving them a v.e.v. breaks the gauge symmetry. It is likely that the same happenshere and that these models correspond to enhanced symmetry points in a larger modulispace. On the other hand, if most or all of the moduli are in non-trivial non-abeliangauge group representations, this opens the way to dynamical mechanisms to lift themwhen these gauge interactions become strong. Note that even if standard model singletsare also singlets with respect to all non-abelian groups originating from the lift, they willusually still be charged under abelian factors in the internal sector. Indeed, in standardGepner models most singlets (and in particular most moduli) are charged. In this case weknow from comparison with the geometric case that moving in moduli space will breaksome or all of the U (1)’s. For this reason we ignored such U (1) charges when countingsinglets. 34nalogously, one may ask if the fractionally charged matter could be mostly in non-abelian gauge representations. Indeed, one way around the problem of the fractionallycharged particles would be that all of them are confined by some additional gauge forceinto integer charged particles. Although this looks like an ugly and far-fetched solution(and probably is) to a problem that is beautifully solved by field-theoretic SU (5) GUTs,remarkably the first example of heterotic weight lifting we worked out in detail in [11]had precisely that feature. In that example, occurring in the tensor product (3 , (cid:98) , , A , ( SU (2) level 8), and in the massless spectrum allhalf-integer charged particles turned out to be in half-integer spin representations of this SU (2), whereas all integer-charged particles are in integer spin representations (this wasonly checked for the massless spectrum).In the present work we have merely checked if in general the fractionally chargedmatter is in non-trivial representations of non-abelian groups. The answer turns out tobe mostly negative (even if it had been positive, this would still not guarantee that thegauge group confines them). The result is shown in Fig. 16. These distributions show,for all six types of broken GUTs, the number of MIPFs (y-axis) with a certain number ofvector-like pairs of fractionally charged particles that are singlets with respect to the extranon-abelian factors. Clearly in the majority of cases there are at least some fractionallycharged particles that are singlets of all non-abelian hidden sector groups, and hencecannot be confined. Interestingly, for the half-integer charge models there are large peaksat zero, which are completely absent for the third- and sixth integer charge models (ofwhich there are very few; please note that the scales of these plots are very different).For the models contributing to these peaks all fractionally charged particles couple tonon-abelian groups. However, as already remarked above, this still does not mean thatthey are confined, and anyhow this is only a small part of the total surface area.Finally, one may ask if the standard model families couple to the additional non-abelian groups. If they do, and if those non-abelian groups remain unbroken, this willalmost certainly give rise to serious phenomenological problems. For example, in theaforementioned model presented in [11] some of the standard model matter is in tripletsof SU (2) , in such a way that some Yukawa couplings are forbidden. Here we have simplychecked which fraction of the total amount of matter (families and mirror pairs) is fullyabelian in the extra gauge groups. The result is shown in Fig. 17. There is a huge peakat 100%, representing spectra for which all matter is in the singlet representation of theextra non-abelian groups. Many of the other spectra are not yet ruled out though: therewill also be examples where only the mirror fermions couple to the extra gauge groups.This is not a problem and might even be a benefit.However, we have not investigated such more detailed questions. The most importantissue is whether the extra non-abelian groups introduced by heterotic weight lifting giverise to such serious obstacles that the whole class must be rejected. This seems not to bethe case. 35 Conclusions
We have studied a new region of the heterotic landscape that is a bit more genuinely(0 ,
2) than standard Gepner models. This immediately solves an annoying problem ofthe latter: the difficulty to get three families. Other classes we have studied (and whichwill be published in future work) seem to confirm this. If indeed the distributions we getfor these models are typical for heterotic strings, then the observed number of families isnot an issue of concern. The family distribution is fairly smooth around three familiesand does not have the sharp dip observed in orientifold models. Nevertheless, from thesetwo different results in two accessible corners of the string landscape no conclusion canbe drawn about such distributions on the full landscape.What does remain an issue of concern is the occurrence of fractional charges. Unlessin generic heterotic vacua vector-like particles are lifted to scales far above the weak scale,we have not found any convincing reason for their absence or extremely small abundancein our environment. In the class we studied here, as well as those studied in [1] and [32],fractionally charged particles are always present in the massless spectrum, with very fewexceptions, and usually not all of them are coupled to other non-abelian interactions toconfine them. The fact that examples can be found where they are absent does not changethe fact that string theory seems to predict them.Our results demonstrate the existence of hundreds of novel heterotic “mini-landscapes”where the standard model might be located, as shown in Table 3. Further study ofthe feasibility of this class require methods beyond RCFT. In particular, a geometricunderstanding of this class would be most welcome.Presumably all this is just the tip of a huge iceberg. Along the same lines, one couldstill study weight lifting for exceptional N = 2 MIPFs. Indeed, combining these two ideasis completely straightforward, but we have not done this yet. One can also constructinternal sectors of heterotic strings partly out of free fermions and partly out of liftedGepner models. This also yields new examples of three family models [35]. Anotherpossibility is to use double lifts, as explained in [11] (so far a partial investigation did notyield examples with three families).But the most important reason to think that there may be a large number of otherpossibilities is that the list of lifted Gepner models is not known to be complete, andpresumably only contains the simplest possibilities, those that can be obtained by fairlystraightforward manipulations of affine Lie algebras. Acknowledgements:
This work has been partially supported by funding of the Spanish Ministerio de Cien-cia e Innovaci´on, Research Project FPA2008-02968, and by the Project CONSOLIDER-INGENIO 2010, Programme CPAN (CSD2007-00042). The work of A.N.S. has been36erformed as part of the program FP 57 of Dutch Foundation for Fundamental Researchof Matter (FOM).
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10 20 30020000400006000080000100000120000 SO(10)Pati-SalamLR,Q=1/3SU(5)LR,Q=1/6SM,Q=1/2SM,Q=1/3SM,Q=1/6
Figure 5: Family distribution for standard Gepner models.
Figure 6: Family distribution for lifted Gepner models.39
10 20 300100000200000300000400000500000 SO(10)Pati-SalamLR,Q=1/3SU(5)LR,Q=1/6SM,Q=1/2SM,Q=1/3SM,Q=1/6
Figure 7: Family distribution for standard Gepner models, including zero families x x x x SO(10)Pati-SalamLR,Q=1/3SU(5)LR,Q=1/6SM,Q=1/2SM,Q=1/3SM,Q=1/6
Figure 8: Family distribution for lifted Gepner models, including zero families.40
Figure 9: Q-Mirror distribution.
Figure 10: L-Mirror distribution.41
10 -5 0 505000100001500020000 Pati-SalamLR,Q=1/3LR,Q=1/6SM,Q=1/2SM,Q=1/3SM,Q=1/6
Figure 11: Difference between the D and L distributions. -10 -8 -6 -4 -2 0 2 4 6 8010000200003000040000500006000070000 Pati-SalamLR,Q=1/3LR,Q=1/6SM,Q=1/2SM,Q=1/3SM,Q=1/6
Figure 12: Difference between the Q and E distributions.42
50 100 150 200010002000300040005000 Pati-SalamLR,Q=1/3LR,Q=1/6SM,Q=1/2SM,Q=1/3SM,Q=1/6
Figure 13: The number of MIPFs with a certain number of vector-like pairs of fractionallycharged particles.
Figure 14: Singlet distribution.43
100 200 300 400 500 6000500100015002000250030003500 SO(10)Pati-SalamLR,Q=1/3SU(5)LR,Q=1/6SM,Q=1/2SM,Q=1/3SM,Q=1/6
Figure 15: Abelian singlet distribution: singlets that do not couple to non-abelian hiddensector groups. 44
20 40 60 8002004006008001000120014001600 Pati-Salam
Figure 16: Abelian fractional charge distributions.45
20 40 60 80 10001000020000300004000050000 SO(10)Pati-SalamLR,Q=1/3SU(5)LR,Q=1/6SM,Q=1/2SM,Q=1/3SM,Q=1/620 40 60 80 10001000020000300004000050000 SO(10)Pati-SalamLR,Q=1/3SU(5)LR,Q=1/6SM,Q=1/2SM,Q=1/3SM,Q=1/6