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HETEROTIC STRING MODELS IN CURVED SPACETIME ∗

arXiv:hep-th/9207118v1 1 Aug 1992hep-th/9207118USC-92/HEP-B4July 1992HETEROTIC STRING MODELS IN CURVED SPACETIME ∗ITZHAK BARSPhysics DepartmentUniversity of Southern CaliforniaLos Angeles, CA 90089-0484, USAABSTRACTWe explore the possibility of string theories in only four spacetime dimensions withoutany additional compactified dimensions. We show that, provided the theory is defined incurved spacetime that has a cosmological interpration, it is possible to construct consistentheterotic string theories based on a few non-compact current algebra cosets.

We classifythese models. The gauge groups that emerge fall within a remarkably narrow range andinclude the desirable low energy flavor symmetry of SU(3)×SU(2)×U(1).

The quark andlepton states, which come in color triplets and SU(2) doublets, are expected to emerge inseveral families.∗Research supported in part by DOE, under Grant No. DE-FG03-84ER-401681

1. IntroductionDuring the past year there has been considerable interest in strings propagating incurved spacetime backgrounds.This was spurred by the fact that some such modelscan be formulated as conformally exact current algebra coset models, or equivalently asgauged WZW models, based on non-compact groups [1], and that their geometry describesgravitational singularities of both black hole [2] and cosmological types.

Before we beginthe main technical part of this paper it is appropriate to make some remarks on why it isinteresting to further study such models. Of course, they provide a setting for investigatingthe very interesting problem of gravitational singularities, but is there more?From experience with string theory we have learned that a conformal field theory thatis a candidate for a “classical” string vacuum may be expected to describe the physicsat Planck scales.

It has been popular to make the assumption that the string vacuum isflat in four Minkowski dimensions and that there are additional compactified “internal”dimensions. In the past all low energy model building efforts have been based on thisunjustified assumption.

Although initially it appeared very promising, the later discoveryof hundreds of thousands of “vacuua” have diminished the confidence of model builders.It must be noticed that the multitude of string vacuua occur in the extra dimensions.Of course, it was not necessary to assume that the first four dimensions are flat.Instead, one could imagine a cosmological scenario in which the four dimensions evolvetoward flat spacetime as a function of time. Furthermore, it was not necessary to assumethat there are more than four dimensions.

Recall that extra dimensions appeared his-torically because the mathematical consistency of flat strings required it. But in curvedspacetime, conformal invariance can perfectly well be satisfied in any dimension, as thenon-compact coset models have demonstrated (even just two dimensions is mathematicallyconsistent).

Therefore, it is conceivable that there are no extra dimensions at all.String theory is needed to describe physics at very early times or very short distancesnear the Planck length. Let us consider a sigma model formalism which provides a glimpseof the geometry at short distances.

Which features of this geometry can be extrapolated tolarger distances? Since there are a few phase transitions that must be taken into accountit is important to distinguish the features that are likely to be different at large distancesafter the phase transitions.

First, there is the dilaton which starts out massless. Since itshould not spoil the long range gravitational forces, it must get a mass near the Planckscale through a phase transition.

So far very little effort has been put into this issue, and2

it remains as one of the challenges for string theory. Perhaps this requires understandingnon-perturbative effects.

Unfortunately, the present state of affairs allows us to hide manyproblems behind this unresolved point. Next, from experience with grand unified theoriesone also knows that phase transitions associated with gauge forces, through the mechanismof inflation, can explain why the universe is homogeneous and isotropic.So, the universe (as described by the sigma model metric) need not start out homo-geneous and isotropic or flat in four dimensions.

It would be sufficient to start out witha part of the universe which is expanding in four dimensions, and that by the time itssize reaches a few Planck scales it approaches a flat universe. If this part of the universeundergoes inflation it may describe our observed universe.

The background geometriesprovided by the non-compact coset models include such geometries in 3d and 4d (see e.g. [3] [4]).

In addition, heterotic string models with such backgrounds predict gauge fieldsand a spectrum of matter that provides candidates for the low energy quarks and leptons.We know that the forces associated with gauge fields and self couplings of matter couldexplain the mechanism for mass generation. So, we may defer the mass generation problemto energies well below the Planck scale.Therefore we may consider a scenario in which there are only three space and one timedimensions.

Then the conformal string theory must be in curved spacetime and is designedto satisfy the conditions of exact conformal invariance. The geometry at the Planck lengthis not necessarily homogeneous or isotropic.At least some bundle of geodesics (thatrepresent the early evolution of part of the universe that gets later inflated) migrate toregions of flat spacetime within a few units of Planck time, perhaps exponentially (as inthe d = 3, 4 non-compact models [3][4] ).

The gauge and matter fields of the heterotictheory can survive to low energies through the mechanism of gauge symmetries and chiralsymmetries. Some of this “low energy matter” will become all of the matter in the inflateduniverse.

Therefore, such a heterotic string theory can be used to at least classify theparticles in multiplets of the symmetry group and compare to the known low energyclassification of quarks and leptons.The initiation of such a program is one of the purposes of the present paper. Wewill classify the heterotic string models in just four dimensions that can be constructedas exact conformal theories based on non-compact groups.

We find that the list of suchmodels is rather short. We will be able to extract the gauge symmetry content of thesemodels and show that the possible gauge groups fall within a remarkably narrow range,and always include the desirable low energy symmetries.

This approach does not explain3

why we live in four dimensions, and of course the program can be carried out also withadditional compact dimensions. But it seems very interesting to find out what kinds ofresults emerge if there are in fact only four dimensions.There exists by now a few models of strings propagating in curved spacetime thatare in principle solvable due to the fact that they are formulated as non-compact currentalgebra cosets based on non-compact groups.

The classification of the cosets G/H thatyield a single time coordinate [1] is known [5]. The cosets that lead to models in four curvedspacetime dimensions (D = 4) always include SO(d −1, 2)/SO(d −1, 1) for d ≤4.

In thispaper we will assume that there are no more than D = 4 dimensions and therefore use onlySO(d −1, 2) for d = 2, 3, 4. For D = d = 4 there are no other bosonic coordinates.

Whend ≤3, then D −d = 4 −d additional bosonic coordinates are supplied by taking directproducts with other groups (including space-like U(1) or IR factors) and then gauging anappropriate subgroup. Furthermore, we include in our list the possibility of a time-likebosonic coordinate and denote it by a factor of T instead of IR.

All possibilities are listedin Table-1 in the column labelled “right movers”.#left movers with N=1 SUSYright movers1SO(3, 2)−k × SO(3, 1)1/SO(3, 1)−k+1SO(3, 2)−k/SO(3, 1)−k2SL(2,IR)−k1×SL(2,IR)−k2×SO(3,1)1SL(2,IR)−k1−k2+2× IRSL(2,IR)−k1×SL(2,IR)−k2SL(2,IR)−k1−k2× IR3SO(2, 2)−k × SO(3, 1)1/SO(2, 1)−k+2× IRSO(2, 2)−k/SO(2, 1)−k× IR4SL(2, IR)−k × SO(3, 1)1 × IRSL(2, IR)−k × IR5SL(2,IR)−k1×SL(2,IR)−k2×SO(3,1)1T ×IRSL(2, IR)−k1 × SL(2, IR)−k2/(T × IR)6SL(2, IR)−k1 × SU(2)k2 × SO(3, 1)1/IR2SL(2, IR)−k1 × SU(2)k2/IR27(SL(2, IR)−k × IR2 × SO(3, 1)1)/IR(SL(2, IR)−k × IR2)/IR8T × IR3 × SO(3, 1)1T × IR39T ×SU(2)k1×SU(2)k2×SO(3,1)1SU(2)k1+k2+2T ×SU(2)k1×SU(2)k2SU(2)k1+k210T × SO(4)k × SO(3, 1)1/SU(2)k+2T × SO(4)k/SU(2)k11T × SU(2)k × SO(3, 1)1T × SU(2)k12(T × IR × SU(2)k × SO(3, 1)1)/IR(T × IR × SU(2)k)/IR13(T × IR × SL(2, IR)−k × SO(3, 1)1)/T(T × IR × SL(2, R)−k)/TTable-1. Current algebraic description of left movers and right movers.For brevity we used IR where we could have used either IR or U(1).

Case 3 is obtainedfrom case 2 in the limit k1 = k2 = k, while case 4 is the k1 = k, k2 = ∞limit of either case4

2 or 5. Similarly, cases 10,11 are limits of case 9.

Furthermore, case 8 may be consideredthe large k limit of case 11. This last case has unique properties in that its geometry isflat, homogeneous and isotropic (modulo boundary conditions on the IR3 factor).

One mayalso notice that cases 9-13 are analytic continuations of cases 3-7. We have listed all theselimits or analytic continuations separately because they lead to different gauge groups aswill be seen in Table-2 below.For the numerator factor T in cases 8-13 we allow a background charge Q0.

Thebackground charge for the time-like coordinate contributes cT = (1+12Q20) to the Virasorocentral charge, and this quantitity is always larger than one. Similarly, every space-likecoordinate associated with the factors of IR in the numerators may be allowed to have a non-trivial background charge Q.

This contributes cIR = (1−12Q2) for a space-like coordinate,and is always less than one and positive. In the following, to keep our expressions simple,we will assume that Q = 0.

A non-zero Q makes no difference for the discussion below,but we will indicate separately the changes that occur at intermediate steps.The cases 5,6,7,12,13 which contain a T or IR factor in the denominator may furtherbe generalized by multiplying both numerator and denominator by a factor IRn. Whatthis implies is that there are many possible ways of gauging the IR and/or T factors bytaking linear combinations.This may lead to models that are different, however thisgeneralization does not change the results given in Table-2 at all.The heterotic string will have a supersymmetric left-moving sector and a non-supersymmetric right-moving sector.The cosets above describe the four dimensionalspace-time part of the right-moving sector.This contributes cR(4D) toward the Vira-soro central charge.

After we analyse the central charge of the supersymmetric left moversand fix it to be cL = 15 in only four dimensions, we will see that cR(4D) will be fixedto some value less than 26. Therefore, for the mathematical consistency of the theory,we must require that the right moving sector contains an additional “internal” part whichmakes up for the difference, i.e.

cR(int)+cR(4D) = 26. One of the aims of this paper is tocompute cR(int) in each model and then find gauge symmetry groups that precisely givethis value.

This procedure will allow us to discover the gauge symmetries that are possiblein these curved spacetime string models.To construct a heterotic string we introduce four left moving coset fermions ψµ thatare classified under H as G/H and form a N = 1 supermultiplet together with the fourbosons. The construction of the action that pocesses the superconformal symmetry is donealong the lines of [6].

The left moving fermions ψµ are coupled to the gauge bosons in5

H. In the Hamiltonian language, the left moving stress tensor is expressed in the form ofcurrent algebra cosets [7] [8] as listed in Table-1, where SO(3, 1)1 represents the fermions.This algebraic formulation allows an easy computation of the Virasoro central chargesfor left movers cL as well as the right movers cR(4D). For a consistent theory we mustset cL = 15.

This condition puts restrictions on the various central extensions k and/orbackground charges Q0, Q, as listed in Table-2. After inserting these in cR(4D) we findthe deficit from the critical value of 26, i.e.

cR(int) = 26 −cR(4D). As seen in the table,the resulting values for cR(int) fall within a narrow range.

For case 2 or 3 it is possibleto change the central charge within the range 11 12 < cR(int) < 13 by varying k1 + k2.Similarly, the corresponding range for cases 9,10 is 12 12 < cR(int) < 13. For the remainingcases it is not possible to change cR(int) by using the remaining freedom with the k′s.#conditions for cL = 15cR(int) gauge group, right movers1k = 511(E7)1 × SU(5)12k1 −2 = k2−22(−1 +q3k23k2−8)13 −δδ =12(k1+k2−4)(k1+k2−2)3k = 311 12(E7)1 × SU(3)1 × SU(2)2 × U(1)14k = 8/313(E8)1 × SO(10)15k1 = 8k2−203k2−8 ,k1, k2 > 8313(E8)1 × SO(10)16k1 = 8k2+203k2+8 , k2 = 1, 2, 3, · · ·13(E8)1 × SO(10)17k = 8/313(E8)1 × SO(10)18Q20 = 3413(E8)1 × SO(10)19Q20 = 34 + 12(1k1+2 +1k2+2 −1k1+k2+4) 13 −δδ =12(k1+k2+4)(k1+k2+2)10Q20 = 3(k+3)4(k+2), (e.g.

k = 1)12 12(E8)1 × SU(3)1 × SU(2)2 × U(1)111Q20 =3k+84(k+2), k = 1, 2, 3, · · ·13(E8)1 × SO(10)112Q20 =3k+84(k+2), k = 1, 2, 3, · · ·13(E8)1 × SO(10)113Q20 =3k−84(k−2)13(E8)1 × SO(10)1Table-2. Conditions for cL = 15 and examples of symmetries that give cR = 26.At this point we mention the effect of a non-zero background charge Q for the space-like factor IR in cases 2,3,4 and 7.

The formula for cR(int) that is listed in the table forcase 2 remains the same, but the conditions on the k′s change slightly. The new conditionstake the same form as cases 9,10,11 and 12 as listed in the table respectively, except for theanalytic continuations Q20 →−Q2 and ki →−ki.

However, since 0 < Q2 <112, the valuesof the new k′s in cases 2,5 must remain within a narrow range of those already fixed in6

Table-2. Only for cases 2,3,4 this has an effect on cR(int).

For example, for case 3 we get2.9 ≤k ≤3 (instead of k = 3) and 11.24 < cR(int) < 11 12 (instead of 11 12). The presenceof a non-zero Q does not change the discussion that follows.The value of cR(int) = 13 that occurs for most of the cases is the same as the deficitfor the popular heterotic string models that have four flat dimensions plus compactifieddimensions described by a c = 9, N = 2 superconformal theory (i.e.

4 + 9 + 13 = 26).Hence, for these cases, the appearance of (E8)1 ×SO(10)1 as the gauge group has preciselythe same explanation as the usual approach. For the remaining cases we give an exampleof a gauge symmetry that will make up the deficit, as listed in Table- 2.

Other gaugegroups are clearly possible just on the basis of cR(int).For example, for case 1 onecan have SO(22)1,(E8)1 × SU(4)1,(E7)1 × SU(5)1,(E7)1 × SU(3)1 × SU(2)1 ×U(1)1,(E6)1 × SO(10)1, etc., as given in [6].The gauge symmetry is associated with a conformal theory of right movers.Thisadditional part of the action may be constructed from right moving free fermions withappropriate boundary conditions, or by using other devices that are quite familiar. Wecan think of this part as another current algebra associated with the gauge group, andwith the central extensions that are given in Table-2.

This final step completes the actionfor the model.For the complete action for case 1, see [6].Further discussion of themodel is required to determine the symmetries consistent with modular invariance. Atthis stage it is encouraging to note that the desirable low energy symmetries, includingSU(3) × SU(2) × U(1), are contained in these curved space string models that have onlyfour dimensions.The special property of the models constructed in this paper is that they can befurther investigated by using current algebra techniques.

The simplest model is case 8,since it is essentially flat, its quantum theory reduces to the manipulation of harmonicoscillators. For the remaining models the spectrum of low energy particles is obtainedby computing the quadratic Casimir operators of the non-compact groups that define themodel.

The computation of the spectrum will be reported in a future publication. Sincethe flavor groups such as SU(3) × SU(2) × U(1) or SU(5), SO(10), etc.

appear at level 1,it is already evident that the quark and lepton type of matter will appear in triplets anddoublets respectively.An interesting question is how the repetition of the families will come about?Inthe traditional approach that includes compactified dimensions, the number of families isrelated to indices, such as Betti numbers, of the compactified space. In the present case,7

the four dimensional geometry has many new and interesting properties, such as duality,different topological sectors, etc. as seen in the global analysis of [3].

Therefore, we mayexpect that the repetition of families may have something to do with these properties. Therepetition will show up in the algebraic approach by the number of distinct ways that itis possible to satisfy the on-mass-shell conditions for the same quark or lepton quantumnumber (e.g.

different representations of the non-compact group). When this criterion isapplied to the flat case 8, we see that the presence of the background charge Q0 allowstwo distict states to be associated with the same conformal dimension, thus leading to twofamilies.

Therefore, it is quite possible that the repetition of families may arise from thefour dimensional geometry alone as described above.We want to point out another possible source of family replication. It may be feasibleto interpret part of the gauge group as a “family group”, as it was done in the days beforestring theory.

If this latter alternative is utilised for family replication, then one mightquit the idea of “hidden sectors” attributable to groups such as E8, and instead adopt aversion of the gauge group which has complex representations. For example, in case 1,one of the possibilities that give cR(int) = 11 was (E6)1 × SO(10)1 , which has complexrepresentations.During the past year there has been many investigations [2-17]that exploredthe geometry of the sigma model-like action associated with some of these models.

Whilethe geometry for d = 2 is interpreted as a black hole the singularity structure for d = 3, 4is considerably more involved and interesting. For example, for d = 3 a global analysisof the manifold shows that there are two topologically distinct sectors that can be pic-tured as the “pinched double trousers” or the “double saddle” [3].

Furthermore, the timedependent backgrounds that emerge allow for cosmological interpretations. These resultswere initially obtained at the semi-classical level using the lagrangian method (in patchesof the geometry).

More recently, fully quantum mechanical results were obtained by usingconformally exact current algebra methods in a Hamiltonian formalism [4]. The algebraicmethod simultaneously yields the full global geometry, as was illustrated for d = 2, 3, 4.Furthermore, the heterotic and type-II supersymmetric versions of these models were in-vestigated and their conformally exact global geometry determined.

By now the globalgeometry of all the above models have essentially been completed in [3][4][17]. The casesnot covered directly in these references can be obtained with analytic continuation tech-niques.8

Are there additional models beyond the ones listed in the tables? Undoubtedly thereare more, but they may not have the virtue of being solvable like the present ones thathave a current algebra formulation.It is, of course, possible to imagine perturbationsof the present models that may be formulated in the current algebra language and yieldsolvable cases.

Such perturbations will tend to change the formulas for the central chargesand it would be interesting to investigate how stable is cR(int) against these perturbationsand how the gauge group is affected by them.On the basis of Table-2 it seems thatone cannot wander too far away from cR(int) = 13. Therefore the desirable low energyflavor symmetries are likely to remain.

These are interesting questions that should beinvestigated.We have argued that it is interesting to consider the possibility of heterotic stringtheories in only four spacetime dimensions and no additional compactified dimensions.This is possible only in curved spacetime, and such a string can be imagined to describethe very small distances or very early times in the Universe.9

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