Abstract
Recent developments involving strongly coupled superstrings are discussed from a phenomenological point of view. In particular, strongly coupled is described as an appropriate longwavelength limit of Mtheory, and some generic phenomenological implications are analyzed, including a long sought downward shift of the string unification scale and a novel way to break supersymmetry. A specific scenario is presented that leads to a rather light, and thus presently experimentally testable, sparticle spectrum.
CTPTAMU42/97
ACT15/97
MPhenomenology^{1}^{1}1Based on invited talks delivered during the summer, 1997.
D. V. Nanopoulos
Center for Theoretical Physics, Department of Physics
Texas A&M University, College Station, TX 77843–4242, USA
Astroparticle Physics Group, Houston Advanced Research Center
The Mitchell Campus, The Woodlands, TX 77381, USA
Academy of Athens, Chair of Theoretical Physics, Division of Natural Sciences
28 Panepistimiou Avenue, Athens 10679, Greece
1 Introduction
The standard model (SM) of particle physics, an gauge group with the appropriate field representations, seem to fit all presently available experimental data, including, notably, the LEP high precision electroweak tests[1]. Most remarkably, a supersymmetric extension of the standard model (SSM), while it more than doubles the SM particle content in the mass range (), it does not only escape unscathed from all the LEP severe tests[1], but it provides the first evidence for Superunification[2]. Indeed, renormalization group extrapolation of the three coupling constants (, , ), as measured at LEP, at very high energy show that they do converge at some scale
(1) 
at a common value
(2) 
as theoretically predicted a long time ago[3]. While very suggestive, this value of the Grand Unified scale brings us suspiciously close to the SPlanck scale:
(3) 
implying that gravitational effects may be nonnegligible and should be taken into account.
Supergravity is the local extension of rigid supersymmetry (SUSY), that automatically involves gravity. While Supergravity (SUGRA) cannot provide a finite quantum field theory, and thus a consistent quantum theory of gravity, still it may serve as an effective theory for energy scales below the SPlanck scale (3). Usually, SUGRA models are plagued by a major catastrophe, namely by an unacceptably large value for the cosmological constant, , at the classical level. Interestingly enough, there is a specific class of models consisting of the so called noscale supergravity framework[4], that

provides a naturally vanishing cosmological constant, , at least at the classical level due to the available flat directions () of the scalar potential: [5]

supergravity is spontaneously broken, but the gravitino mass, , is undetermined at the classical level:[5]
(4) 
quantum corrections curve, in principle, the flat directions of the scalar potential , thus creating dynamically a , and provide vev’s to the ’s. In other words, we have not only succeeded to get radiative electroweak breaking (REWB), but we do also have a dynamical determination of the SUGRA breaking scale: . In principle, in the noscale supergravity, as it is suggested by its name, all mass scales are dynamically determined in terms of a single scale (our yardstick), say the SPlanck scale[6, 7, 8].
These rather unique characteristics find their natural habitat in string theory, whose infrared limit is nothing else but noscale supergravity[9]. Sections 2 and 3 describe the weak and strong coupling limits of superstrings, respectively, while section 4 provides a phenomenological profile of strongly coupled viewed as an appropriate limit of Mtheory. Mtheory inspired supersymmetry phenomenology is discussed in section 5 and some conclusions are drawn in section 6
2 Superstrings: Weak Coupling (10D 4D)
Superstrings, onedimensional extended objects, provide a consistent quantum theory of gravity and a natural framework for realistic unification of all fundamental interactions. While superstrings are intrinsically different from pointlike particles in many respects, one may still, at least, initially try to use perturbation theory in order to get out some physics. The initial stages of this (perturbative) programme has yielded rather interesting results

useful gauge groups are available, , , …

useful particle content is present, fitting into highly desirable representations of the above gauge groups, such as, three generations of quarks and leptons, two Higgs doublets, etc.

useful superpotential form, yielding among other things successful Yukawa couplings, e.g.
(5) at the string scale implying a top quark mass[10, 11]
(6) or successful fermion mass relations[12], e.g.,
(7) More generally, the “technology” has been developed to understand the so called, in the late 70’s, generation problem and the fermion mass problem at the most fundamental level of dynamics, i.e. at the string scale. Dynamically calculable “nonrenormalizable” terms[13], may provide the rather intrigued textures needed to explain the observed fermion mass spectrum.

Supersymmetry emerges naturally due to the highly constrained form of string dynamics, in other words supersymmetry is a prediction of string theory. Furthermore, the lowenergy limit of string theory is, generically, described by an effective theory belonging to the noscale supergravity framework[4].
Despite all the above remarkable, indeed, successes of weakly coupled strings, there are enough stumbling blocks in the way towards a realistic superunification, that shed shadows of doubt on the whole picture, such as

it has not delivered yet a unique theory at the string scale. Actually, the problem is not that we are short of “theories”. On the contrary, we seem to have more than it is necessary. Instead of one, we got five theories: type I, type IIA, type IIB, heterotic SO(32), and heterotic . A rather unpleasant proliferation of Theories Of Everything (TOE). Things are getting worse, if we just recall the fact that the above five consistent string theories live in D=10 dimensions, and it is through compactification that we eventual reach D=4 dimensions. Since the compactification procedure, even if it is severely constrained, is rather arbitrary, at least in perturbation theory, we are landing in a D=4 landscape made of myriads and myriads of consistent string vacua. Nonperturbative string effects may help (and do help!), but one wants to make sure that they don’t undo hardearn successes of perturbation theory, e.g. Yukawa couplings, noscale structures and the likes.

while supersymmetry is a highly desirable symmetry, we better make sure that it gets broken. Weakly coupled string theory has not produced a clearcase SUSY breaking mechanism. Even if certain scenaria, e.g. gaugino condensation, has been the usual playground for string phenomenologists, clearly not pinning down the SUSY breaking mechanism implies not pinning down the SUSY particle mass spectrum, in other words no hard string predictions

it suffers from an embarrassing disparity between the observed, apparent scale of (grand) gauge unification and the dynamically calculable string unification scale[14].
(8) i.e. a discrepancy of about a factor of 20. Certain ways out in the present framework (that of weakly coupled strings) have been proposed, including large threshold effects, extra matter multiplets, nonminimal KacMoody levels and the likes[15]. They all suffer from a common inadequacy. They turn a wonderful prediction of the supersymmetric standard model (1) to a mere fitting of parameters. On the other hand, we may have yet another case where nonperturbative string effects may play a major role. But how?
Since the, observed at LEP, superunification of the gauge coupling constants is of such obvious fundamental importance, it is worthwhile unearthing the origin of the discrepancy between the and . The heterotic string, the most relevant for phenomenology, will be used as a working example. It is well known that[15], in the heterotic string both gravitational and gauge interactions are produced from the closed string sector, thus establishing a relation
(9) 
with the 10D string coupling constant, while () is the 4D one. It is pretty clear that, with the values of and as given by (1) and (2) respectively, and assuming weakly coupled strings in 10dimensions (), we overshoot in the “prediction” of Newton’s constant () by about three orders of magnitude! On the other hand, if one “fits in” the observed value of , one gets a value for , much bigger than 1 indicating that we are really probing the strong coupling limit of the string in 10dimensions. By recalling the fact that
(10) 
with , the inverse of the string tension and the compactified 6D volume, one may entertain the hope that there is a suitable strong coupling limit, such that while , and go both to infinity in a suitable manner. Then, one may recover both the standard prediction of the SSM about and the right value of . In such a case, the strong coupling behavior can be deduced from what happens in the strongly coupled 10dimensional theory. It is highly remarkable that the bottomup approach that we have followed until now has lead us to deduce that the “vacuum of the world” seems to be a “strongly coupled heterotic string vacuum in 10D,” which has also been the focus of stunning theoretical developments (topdown approach) the last few years[16]. So let us shift our attentions to
3 Superstrings: Strong Coupling (10D)
The strong coupling limit of quantum systems is usually fairly complicated. Sometimes, we may be lucky and get into the following situation. Consider a quantum system with its fundamental degrees of freedom (d.o.f) denoted as , and let some relevant parameter, say to go to infinity (strongly coupled limit). It may happen that in this limit () some of the fundamental d.o.f, turn into some new d.o.f say , that describe the fundamental d.o.f of another quantum system , but such that the corresponding relevant parameter say is much smaller than one (weakly coupled limit). In such a situation, by using the mapping , we can extract information about the physics of the strongly coupled system, by working in the familiar perturbation regime of relevance to us, of the system ! While all the above may sound and look as a pipedream, that is exactly what is happening in string theory. Intense theoretical work of the last few years[16] have indicated that all five string theories, discussed in the previous section, are interrelated in a similar way that systems and are related above, with referring to the corresponding string coupling constants. The magic property that is responsible for all these correlations is called string duality. This is nothing else but a nontrivial generalization of the electromagnetic duality, observed by Dirac, and which lead him to his famous charge quantization condition[17], in the presence of magnetic monopoles
(11) 
where and refer to electric and magnetic charges respectively. In the modern counterpart of nonAbelian gauge theories, the existence of ’t HooftPolyakov type magnetic monopoles, which are not pointlike, but extended objects of solitonic nature resurrected Dirac’s ideas. Properly modified by Montonen and Olive[18], put to work in supersymmetric YangMills theories. It was shown that in the Eweak limit (), the solitonic monopoles become superheavy () and thus decouple, while in the Mweak limit (), corresponding through (11) to , the solitonic monopoles become massless and provide the new fundamental degree of freedom! Actually, extended supergravities, as mentioned in the introduction, contain naturally noncompact continuous global symmetries, that act as duality symmetries, that is, exchange ordinary particles with solitons corresponding to weakstrong coupling interchange. Since, string theory yields naturally extended supergravities in its longwavelength limit (e.g. in in ), it shouldn’t be that surprising that string duality is present with all its spectacular consequences. While, grosso mondo, the generic analysis above for super YangMills theories hold true, even more exciting tricks are involved in string theory. String duality multiplets may contain, vibrating strings (the basic quanta of string theory), smooth classical objects of solitonic type, singular classical objects (black holes) and Dbranes[19], stringy type of topological defects. It all depends on the specific string theory and strong coupling limit chosen, which of the above objects will become massless and will provide the “new” fundamental degree of freedom. In 10dimensions, the strong coupling limit of type I () string theory is given by the weakly coupled heterotic theory, while the IIB string is self dual. Things become more interesting in the case of IIA strings. IIA strings in 10D contain D0 branes, topological defects with pointlike particle behavior, of mass , which in the weakly coupled limit () are superheavy and leave the vibrating strings to provide the fundamental d.o.f. Witten[20] has shown that in IIA strings we get towers of D0 brane supersymmetric bound states of mass
(12) 
In the strong coupling limit () the D0 bound states form a continuum, recastable as
(13) 
which, of course, is nothing else but the definition of an extra 11thdimension, á la KaluzaKlein! The strongly coupled limit of IIA strings is the weakly coupled limit of some theory in 11dimensions, compactified on a circle of radius , as given in (13). This new, 11D theory is called Mtheory[21]. Actually as Horava and Witten[22] have shown, a different compactification of Mtheory, this time on a semicircle (or segment) of radius , it provides the strong coupling limit of 10D heterotic string, to be discussed in detail later. Thus, we see that through string duality, we have been able to interconnect all five string theories in 10D, and in addition, we lead to the discovery of a new M(ysterious) Theory in 11D. Of course, as (13) indicates, if we insist on the weakly coupled limit () , (or ) are much smaller than the string scale () and thus invisible. Hopefully, all the above analysis will help understand the string duality motto:

The strongly coupled limit of any string theory is the weakly coupled limit of another “string” theory,
with the understanding that “string” theory contains also Mtheory.
Nonperturbative string effects, attainable through string dualities, have lead to a much more satisfactory picture of string theory. Nowadays, all five string theories in 10D and the M(ysterious) theory in 11D are considered as the limits of one theory. In other words, over most of the theory space, , except various specific limits where , corresponding to the above mentioned string theories, and Mtheory. In fact, currently the expression Mtheory is used to describe an unknown, fundamental, 11D theory, that cannot formulated as a traditional quantum theory, due to the lack of a parameter to be utilized in some perturbation expression, and approximates to 11D supergravity at long wavelengths. A further speculative theory may exist in t͡welvedimensions, dubbed as Ftheory[23], which gives upon reduction on a two torus the type IIB theory. Whatever is the name of the new theory, it is clear that it resolves one of the problems of weakly coupled strings, that of the existence of five different theories in 10D, by considering them as different limits of a single 11D or 12D theory. On the other hand, the excursion in 11 or 12dimensions shouldn’t be taken lightheartedly, since D=10 has been advertized as the critical dimension of consistent superstring theories. What’s going on? How these extra dimensions (11 or 12) popped out in string theory?
While the jury is still out in answering the above kind of fundamental questions, let me describe a resolution proposed recently by Ellis, Mavromatos and myself[24] in the framework of the socalled noncritical or Liouville strings[25, 26]. It is well known that string theory contains a degree of freedom, that of the Liouville filed , that decouples both at the classical and quantum levels in the critical D=10 superstrings. It does not decouple, though, at the quantum level, if we go away from D=10. Actually, one may employ this new available d.o.f to construct exact string solutions involving curved spacetimes, such as a RobertsonWalker expanding Universe[27, 28, 29], 2dimensional blackholes[30], or to get new solitonic solutions like NS fivebranes[31] and the likes, that play an important role in nonperturbative string theory[32]. Sometimes these solutions are referred to as the linear dilaton solutions, because the dilaton is proportional to time (cosmological solution) or to some specific space direction (soliton solution). Taking into account the Liouville filed in the worldsheet dynamics, allows for extrapolation between critical points (corresponding to conformal field theories), thus providing a way to get out of D=10 and still obtain a consistent string theory. We have shown[24] that in the Liouville string theory, one may introduce worldsheet topological defects of the Liouville field, like vortices and monopoles, described by a deformed sineGordon model. Dbranes in target space can be described in terms of these worldsheet topological defects, and their connection to black holes become apparent in our framework[25, 26]. The statistical system of vortices and monopoles suffers a KosterlitzThouless phase transition at a dynamically determined critical dimension which turns out to be ! Away from criticality, , since the Liouville field , is not decoupled. It is highly tempting to identify[24] such an emerging target space theory with Mtheory while the corresponding with Ftheory! What such an identification buys us? Well, to start with, even if we cannot provide a traditional quantum field theory in 11 or 12 dimensions, we may have at least a worldsheet renormalizable theory that represents them. As we have stressed, in dissent and for several years[25, 26], we may be in for surprises on the form of quantum field theory that is descenting from string theory. A further indication of the real worth of the Liouville string framework, is provided by contemplating on the well known fact that the maximum number of dimensions in which supersymmetry can exist is , provided that the signature of spacetime is (10,2). The tale of two times! Our proposal[24] for managing the appearance of these two times is to identify the zero modes of the a priori distinct quantum fields ( in critical strings) and , in the neighborhood of each fixed point corresponding to the five string theories, and the 11dimensional theory. In other words, each fixed point in the space of “string” theories has its own timelike coordinate , which is a reversible coordinate, in the Einstein sense, so that a fixed point is Lorents invariant. In addition, in the bulk of the theory space there is a second timelike coordinate, the Liouville field , with respect to which evolution is irreversible in general. This means that the general 12dimensional target space Ftheory is “nonequilibrium” and does not have a simple fieldtheoretic interpretation. Thus it is possible to turn an apparent embarrassment of riches, that of signature (10,2) to our advantage and resolve a hundred year mystery, that of the (microscopic) origin of the arrow of time, while keeping Einstein physics basically intact. Last and not least, our approach reestablishes the singular importance of strings. Maybe, L(iouville) theory will come to be known as the theory formerly known as M/F theory.
4 Strongly coupled Mtheory: A phenomenological profile
The heterotic string, in its weakly coupled form has been the focus of intense phenomelogical studies, due to its rich structure that yields realistic models describing the world at long wavelengths. In the strong coupling limit, emerges with certain unique characteristics, that once more make it a very promising framework to study phenomenology. One may, up front, question the meaning of the whole exercise, by noticing that we don’t know what Mtheory is, so what we are talking about, its compactification and the likes? The idea here is the following: we do know that the infrared limit of Mtheory has to be superqravity, since there is no other consistent quantum field theory available! Furthermore, since supergravity (or supergravity) are fairly wellstudied, we may get quite a lot of information about Mtheory, by studying SUGRA, if it happens that , the 11th (or 5th) compactified dimension is much bigger than the 11th Planck scale (). As we are going to see soon, phenomenological requirements put us in a regime and thus enabling us to study Mphenomenology. The lowenergy consequences of the unknown, at microscopic level (), Mtheory may be unveiled by studying suitably tailored SUGRA (or SUGRA). After all, Fermi’s theory of decay, augmented with parity violation and Cabibbo currents, provided a pretty good picture of low energy weak interactions (), we didn’t have to wait till the ’70s, when a microscopic () theory of electroweak interaction was available, in order to study low energy weak interactions!
As discussed in the previous section, the strongly coupled , is described by Mtheory compactified on a segment (or semicircle) , of dimension [22]. Of course, realistically one considers compactifications of the strongly coupled , which corresponds to a compactification of Mtheory, with denoting an appropriate CalabiYau threefold (= 6 space dimensions). The 11th dimension () has an orbifold structure that is instrumental:

at one end live the observable fields contained in , at the other end live the hidden sector fields contained in , and in the middle (“bulk”) propagate the gravitational fields. The fields that live on the two boundaries, may be considered as comprising the “twisted” sector, while the gravitational fields in the bulk make up the “untwisted” sector.

as such, the fields living at the boundaries, are oblivious to the existence of the 11th dimension (), whatever is its compactification scale! Such a property has farreaching phenomenological consequences:

Since the observable fields live in the ( after compactification) boundaries, chirality is not an issue, in sharp contrast to conventional manifold compactifications of SUGRA where it is fatal.

There are no KaluzaKlein towers of particles, based on the boundary living fields. Thus, a standard severe problem of the “large compactification radius” type models[33], that of the breakdown of perturbation theory at the compactification radius (), is naturally evaded. Gauge unification, and for that matter, Yukawa coupling unification proceeds normally as in the SSM, i.e., independent of the size of !

Furthermore, one may naturally identify the 11th dimensional Planck mass () with , which also provides the characteristic CalabiYau compactification scale. One then finds that (9) is replaced by[34, 35]
(14) 
which for , the compactification radius of the 11th dimension,
(15) 
gives the right value for ! Of course what we have realized here[34] is what we have foresaw in section 2, but with a “twist”. Indeed, the existence of an extra 11th dimension , has allowed us to enlarge suitably the compactified volume (), such that remains much smaller than one, while at the same time the 11th dimension is large enough (see (15)) so that an 11D SUGRA approach is justifiable! At an intuitive level what is really happening is the following. While the three gauge coupling constants evolve dynamically with energy, and meet at , because they are 11th dimension blind, the gravitational constant after we hit , it is replaced by a 5dimensional one, so that for , instead of considering as the dimensionless relevant constant, we ought to work with that increases much faster with energy and enables unification of all interactions at ! The reader may have already noticed that all the above marvelous picture depends on the specific value of as given by (15). In order to claim that we have resolved the GUT scalestring scale disparity problem (see section 2), we need to determine dynamically the value of , and hopefully it will be given still by (15). This issue brings us naturally to our next point.
The scalar potential is independent of , thus it contains a flat direction, at the classical level, that may serve as the basis for implementing the noscale supergravity framework (4)[4]. Indeed, several groups[35, 36, 37, 38, 39, 40] have reached the same conclusion, namely that noscale supergravity is the long wavelength limit of Mtheory in 4dimensions. In particular, we provided the first explicit calculation[38] that supports the above remarks. Upon compactification of Mtheory, in its 11D SUGRA form, on a CalabiYau manifold with Hodge numbers and and boundary , a noscale Kähler potential, superpotential and gauge kinetic function were obtained explicitly[38]. In four dimensions, this result is related to the previous weaklycoupled string noscale supergravity result, obtained by Witten[9], through a field transformation, which means that they are equivalent in 4dimensions. This robust behavior of the noscale supergravity framework[4], all the way from the weakly coupled to strongly coupled heterotic string is rather remarkable, and highly suggestive that (4) may be implemented at the phenomenologically relevant strong coupling limit of heterotic string. After all, noscale supergravity traces its origins[5, 7, 8] in the noncompact continous global symmetries of extended supergravities, whose subgroups, suitably treated, provide string duality!
Supersymmetry breaking may be supplemented at the string level by employing[35, 38, 39, 40] the ScherkSchwarz (SS) mechanism[41] on the 11th dimension. As is well known, the ScherkSchwarz SUSY breaking mechanism makes use of a symmetry of the theory transforming the gravitino nontrivially. In our case, since the 5th dimension is compactified on , the symmetry under discussion must be a rotation on the plane of the 5th dimension and one of the internal CalabiYau coordinates. In a way, such a symmetry acts on the 5D fields as the spacetime parity , i.e., changes sign for the fermions and leave bosons invariant[39]. Thus
(16) 
Notice that, it is only the fermions in the “bulk” (“untwisted” sector) that receive a uniform shift in their momentum (, while both fermions and bosons on the boundaries (“twisted” sector), as living on the semicircle edges have no momenta and thus no supersymmetry breaking contributions. Supersymmetry breaking will, then, be communicated from the “bulk” by gravitational interactions. It is highly amazing, how closely the above scenario resembles the noscale framework[4]. Indeed, the SS mechanism provides a flat potential (nocosmological constant at the classical level) along the 11th (or 5th) dimension , with SUSY breaking at the classical level (see (16)), but with the magnitude of the SUSY breaking (alias gravitino mass) undetermined at the classical level. But this is the noscale SUGRA framework! We then can employ quantum corrections to dynamically determine everything, á la (4), including the magnitude of the compactification scale of the 11th (or 5th) dimension , as promised above. In fact, one may see that what we expect to get out is in the right ballpark, i.e. (15) gets satisfied. Indeed, one expects naively that the noscale mechanism will dynamically fix, as usual, the amount of the observable SUSY breaking scale, relevant for the gauge hierarchy problem, at , which is related to by , with some scale in the () range, and thus fixing dynamically , and thus through (16) to be in the right domain, given by (15)! Work in progress[42] indicates that such an optimistic scenario is not far from reality. All the above results depend critically on the stability of the noscale framework, which in turn depends strongly on the fact, that there are no quadratic divergences in the effective supergravity. Indeed, another, of utmost importance, result of the specific SS SUSY breaking mechanism considered here is the vanishing of Str after supersymmetry breaking[39].
The communication of supersymmetry breaking to the observable sector, attached to one of the boundaries, from the “bulk”, where it was originated, through gravitational interactions, is rather intrigued and maybe not very clear, at the time of writing. Horava[37] has argued that supersymmetry breaking () is not felt immediately in the observable sector because of a topological obstruction (essentially the 11th dimension of length that separates the two sectors). In fact he has argued[37] that there is a hidden 11D supersymmetry, broken only by the global topology of the orbifold dimension (), that explains the “conspiracy” that leads in the weakly coupled heterotic string theory to the noscale structure, with SUSY breaking and vanishing cosmological constant, at the classical level. Supersymmetry breaking becomes apparent only after the renormalization scale is low enough to not reveal the presence of the 11th dimension anymore. In practice, one is to allow for nonvanishing SUSY breaking parameters only for scales [35]. A similar conclusion can be reached by looking in the dual, strongly coupled theory, where as Witten has shown[34], one reaches a strongly coupled , suitable for gaugino condensation, only when gets some critical value, . In fact, , very close, at least numerically, to , as given by (15). Thus, once more in order to “feel” the SUSY breaking in the observable sector. This effect can leave a deep imprint on the low energy sparticle spectrum, which depends quantitatively on the amount of “running” of these parameters[35]. Until now I have tried to present general characteristics of the anticipated Mphenomenology, without resorting to a specific scenario or model. In order to get some experimentally testable predictions, we need now to be more specific in the selection of boundary conditions for the softly broken parameters at [35]. Opinions are divided on this issue, and it is fair to say that things are not yet crystal clear. We have chosen[35] to take , at , as it avoids FCNC problems and make the effect of taking most noticeable. While this choice arises in certain scenaria, cannot be claimed to be indispensible. Anyways, let us see what type of SUSY phenomenology is coming out.
5 Mtheory Inspired Supersymmetry phenomenology
We now proceed to the analysis of the lowenergy sparticle spectrum under the assumptions of and . Thus, the only free parameters are and . We find that the requirement of radiative electroweak symmetry breaking plus two basic phenomenological requirements, allow solutions in the plane for only one sign of and only within a completely bounded region. For the case of , this region is shown in Fig. 1, where to facilitate comparison with experiment we also show the region in the plane. The upper limit on (for a fixed value of ) follows from the requirement that the lightest supersymmetric particle be neutral[43]. Above the upper boundary the righthanded selectron () becomes lighter than the lightest neutralino (). The bottom boundary is obtained by imposing the absolute lower limit on the sneutrino mass from LEP 1 searches (). The area to the right of the rightmost tip of the region is excluded by these two conflicting constraints. The dependence of these constraints may be understood from the Dterm contribution to the and mass formulas
(17) 
where the are some RGEdependent constants and whereas . The dotted line indicates the lower bound on that is consistent with the topquark mass () and perturbative Yukawa couplings up to the unification scale. In practice, the LEP 172 lower bound on the chargino mass ()[44] gives the strongest constraint on the parameter space (dashed line on bottom panel in Fig. 1). Nonetheless, a portion of the parameter space remains allowed, and in fact it is within the reach of future LEP 2 energy upgrades, as we discuss below.
To give a more detailed picture of the lowenergy spectrum, in Fig. 2 we display representative sparticle masses as a function of the chargino mass for and . This choice of allows the widest range of sparticle masses (see Fig. 1). This figure shows that the spectrum “terminates” when approaches from below, as mentioned above in connection with the upper boundary in Fig. 1. It is interesting to note the significant splitting of the topsquark () masses around the average squark () mass.
In the LEP 172 allowed region in Figs. 1 and 2 we find and . Both of these particles appear within the reach of LEP 2. More to the point, one might wonder whether the such light righthanded selectron masses might have already been excluded by LEP 2 searches, as they have been certainly kinematically accessible. We have calculated the cross section at LEP 161, for which explicit limits have been released by the OPAL Collaboration[45]. We find , which in would have yielded a maximum of two events. Indeed, the experimental sensitivity to this mode is at the 0.5 pb level[45]. Thus, past LEP 2 searches in the selectron channels do not restrict the allowed parameter space any further. Moreover, near the upper end of the parameter space the experimental detection efficiency should be greatly reduced because approaches . One should also consider the predictions for trilepton events at the Tevatron. We find for . The leptonic decays of the chargino and neutralino are maximally enhanced because of the lighter righthanded sleptons and sneutrinos, respectively. That is, and , where . Combining these numbers we arrive at a single channel (i.e., any single one of , , , or ) cross section of pb. This result is slightly below the sensitivity reached at the Tevatron in trilepton searches[46], and thus these also do not constrain the allowed parameter space any further.
6 Conclusions
Strongly coupled strings, as studied by the use of string duality, seem to provide one single theory and hold the potential to lead us to a unique string vacuum, hopefully involving . Stringy new Mtheory may cause a paradigmshift in the way we are understanding lowenergy physics. A new way of understanding gaugegravitational unification is suggested and already put to work, a new way of SUSY breaking is emerged, which, while contains seeds of the past, it resolves several severe problems and may lead eventually, into a clearcut SUSY spectrum, among other things, that may provide a smoking gun for physics at the (11th dimension) Planck scale. For example, a deeper understanding of proton stability () in SUSY theories may be needed, and such constraints may reduce considerably the class of acceptable Mcompactifications[47]. On a different wavelength, black hole dynamics may be studied explicitly and quantum mechanics may suffer modifications, similar to the ones occurred to the Newtonian gravity after the innocent looking equivalence principle was implemented correctly by Einstein. Here, duality symmetries are realized only at the quantum level, and while, once more, innocent looking, may carry the seeds to a complete revision of quantum theory[48]. It is not, yet, judgment day, it is only the beginning excellent and fair
Acknowledgments
This work has been supported by DOE grant DEFG0395ER40917.
References
 [1] For reviews see: G. Altarelli. hepph/9611239; J. Ellis. hepph/9707443.
 [2] J. Ellis, S. Kelley, and D. V. Nanopoulos, Phys. Lett. B249, 441 (1990); ibid B260 131 (1991); ibid B287 95 (1992); Nucl. Phys. 373 55 (1992).
 [3] For a review see: S. Dimopoulos, S. A. Raby, and F. Wilczek, Phys. Today Oct, 25 (1991).
 [4] For a review see: A. B. Lahanas and D. V. Nanopoulos, Phys. Rept. 145, 1 (1987).
 [5] E. Cremmer, S. Ferrara, C. Kounnas, and D. V. Nanopoulos, Phys. Lett. 133B, 61 (1983).
 [6] J. Ellis, A. B. Lahanas, D. V. Nanopoulos, and K. Tamvakis, Phys. Lett. 134B, 429 (1984).
 [7] J. Ellis, C. Kounnas, and D. V. Nanopoulos, Nucl. Phys. B241, 406 (1984).
 [8] J. Ellis, C. Kounnas, and D. V. Nanopoulos, Nucl. Phys. B247, 373 (1984).
 [9] E. Witten, Phys. Lett. 155B, 151 (1985).
 [10] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B327, 279 (1994).
 [11] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Rev. D53, 5253 (1996).
 [12] J. L. Lopez and D. V. Nanopoulos, Phys. Lett. B268, 359 (1991).
 [13] S. Kalara, J. L. Lopez, and D. V. Nanopoulos, Nucl. Phys. B353, 650 (1991).
 [14] V. S. Kaplunovsky, Nucl. Phys. B307, 145 (1988); Erratum: B382 436 (1992); I. Antoniadis, J. Ellis, R. Lacaze, and D. V. Nanopoulos, Phys. Lett. B268, 188 (1991).
 [15] For a review see: K. R. Dienes, Phys. Rept. 287, 447 (1997).

[16]
For reviews see:
J. Polchinski, Rev. Mod. Phys. 68, 1245 (1996); hepth/9611050;
M. R. Douglas, hepth/9610041;
C. Vafa, hepth/9602201;
R. Dijkgraaf, hepth/9703136;
S. Ferrara, hepth/9708164;
S. Muhki,
tt hepph/9710470.  [17] P. A. M. Dirac, Proc. Roy. Soc. Lond. A133, 60 (1931).
 [18] C. Montonen and D. Olive, Phys. Lett. 72B, 117 (1977).
 [19] J. Polchinski, Phys. Rev. Lett. 75, 4724 (1995).
 [20] E. Witten, Nucl. Phys. B443, 85 (1995).
 [21] For reviews see: J. H. Schwarz, Nucl. Phys. Proc. Suppl. 55B, 1 (1997); M. J. Duff, Int. J. Mod. Phys. A11 5623 (1996).
 [22] P. Horava and E. Witten, Nucl. Phys. B475, 94 (1996); ibid B460, 506 (1996).
 [23] C. Vafa, Nucl. Phys. B469, 403 (1996).
 [24] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos. hepth/9706125
 [25] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, in International School of Subnuclear Physics, From Supersymmetry to the Origin of SpaceTime, edited by A. Zichichi (World Scientific, 1994), p. 1.
 [26] D. V. Nanopoulos, Riv. Nuovo Cim. 17, 1 (1994).
 [27] I. Antoniadis, C. Bachas, J. Ellis, and D. V. Nanopoulos, Phys. Lett. 211B, 393 (1988).
 [28] I. Antoniadis, C. Bachas, J. Ellis, and D. V. Nanopoulos, Nucl. Phys. B328, 117 (1989).
 [29] I. Antoniadis, C. Bachas, J. Ellis, and D. V. Nanopoulos, Phys. Lett. B257, 278 (1991).
 [30] E. Witten, Phys. Rev. D44, 314 (1991).
 [31] C. G. Callan, J. A. Harvey, and A. Strominger, Nucl. Phys. B359, 611 (1991).
 [32] For a review see: M. J. Duff, R. R. Khuri, and J. X. Lu, Phys. Rept. 259, 213 (1995).
 [33] I. Antoniadis, Phys. Lett. B246, 377 (1990).
 [34] E. Witten, Nucl. Phys. B471, 135 (1996).
 [35] T.J. Li, J. L. Lopez, and D. V. Nanopoulos, hepph/9702237.
 [36] T. Banks and M. Dine, Nucl. Phys. B479, 173 (1996).
 [37] P. Horava, Phys. Rev. D54, 7561 (1996).
 [38] T.J. Li, J. L. Lopez, and D. V. Nanopoulos, Phys. Rev. D56, 2602 (1997).
 [39] I. Antoniadis and M. Quiros, Phys. Lett. B392, 61 (1997); hepth/970537; hepth/9707208; hepth/9709023.
 [40] E. Dudas and C. Grojean. hepth/9704177; E. Dudas, hepth9709043; H. P. Nilles, M. Olechowski, and M. Yamaguchi, hepth/9707143; Z. Lalak, and S. Thomas, hepth9707223.
 [41] J. Scherk and J. H. Schwarz, Nucl. Phys. B153, 61 (1979); and Phys. Lett. 82B, 60 (1979).
 [42] K. Benakli, J. Ellis, and D. V. Nanopoulos, in preparation.
 [43] S. Kelley et al., Nucl. Phys. B398, 3 (1993).
 [44] LEP Experiments Committee Open Session, Nov, 1996, presentations by F. Ragusa (ALEPH), J. Temmermans (DELPHI), D. Stickland (L3), A. Ball (OPAL). http://alephwww.cern.ch/ALPUB/seminar/lepc_nov96
 [45] K. Ackerstaff et al., Phys. Lett. B396, 301 (1997).
 [46] K. W. Merrit, hepex/9710009; T. Kamon, private communication.
 [47] J. Ellis, A. E. Faraggi, and D. V. Nanopoulos. hepth/9709049
 [48] E. Witten, Physics Today May, 28 (1997).