Superstring Phenomenology after 1995

The recent progress in understanding nonperturbative issues of string theory [10], [54] has necessarily strong impact on the phenomenological questions, we are only starting to explore these implications which can be sumarized as follows.

(i)$\$Unification of theories: We mentioned in the introduction that there are five consistent superstring theories and each has thousands or millions of different vacua. It is now believed that the five string theories are related by strong-weak coupling dualities and furthermore, they appear to be different limits of a single underlying fundamental theory, probably in 11D, the $M$ theory (probably related with membranes or higher dimensional objects such as five-branes), which is yet to be constructed. If this is true it may solve the arbitrariness in the number of fundamental string theories by deriving them from a single theory.

(ii)$\$Unification of vacua (?): Recent work based on comparison of string compactifications with the Seiberg-Witten theory, has lead to the conclusion that many and probably all Calabi-Yau compactifications are connected. Then it seems that not only the five different theories are unified, but also all the vacua of these theories could also be unified: since, if they are all connected, we can foresee a mechanism that lifts the degeneracy and select one point in the web of compactifications, something it could not have been done before because they were thought to be disconnected vacua. These transitions occur in singular points of the corresponding moduli space where a particular state (massless black hole) or even an infinite tower of states (tensionless strings) become massless. They were partially understood for $N>1$ compactifications, but recently extensions to the phenomenologically interesting $N=1$ case have been found [105], implying for instance that models with different number of families would belong to the same moduli space reducing in some sense the discrete degeneracy problem to the level of the continuous degeneracy problem and so we may expect that probably one particular number of families could eventually be selected dynamically.

(iii)Nonperturbative vacua: The fact that the strong coupling regime of a given string theory would simply be the weak coupling regime of another string theory would be very dissapointing since that means that the problems present at weak coupling would remain at strong coupling. Fortunately this is not the case. For instance, the strong coupling limit of the $E_8\times E_8$ string is believed to be given by $M$ theory compactified in the orbifold $S^1/{\bf Z}_2$ which is just a one dimensional interval. $M$ theory contains elementary membranes and their magnetic dual, $5-$branes. The membranes can end at each of the two $10 D$ ends of the interval (fixed points) which are $9$-branes and generate an $E_8$ symmetry at each end. The distance between the two $9$-branes $\rho$ is proportional to the heterotic coupling and when this is very small the two $E_8$'s collapse to a single $10 D$ point which is the heterotic string. For any finite coupling the membrane is a cylinder between the two $9$-branes with heterotic strings at the intersection. This reproduces the standard perturbative spectrum of heterotic strings. The new ingredient comes mostly from the $5$-branes which for the $E_8\times E_8$ case, carry two-index antisymmetric tensors, therefore introducing more than one of these fields in the spectrum after compactifications (in the $SO(32)$ version they may lead to extra vector fields depending on the compactification). In perturbative heterotic string there was a single antisymmetric tensor $B_{\mu\nu}$ that we saw is dual to an axion field. The appearance of several of those fields in the spectrum shows clearly that the corresponding vacuum is nonperturbative and may eventually create more possibilities for using these axion fields for solving the strong CP problem in string theory. There is even a model with zero tensor fields. This may be relevant because $B_{\mu\nu}$ is a supersymmetric partner of the dilaton and having a model without antisymmetric tensors would mean that somehow the dilaton was fixed, lifting the corresponding degeneracy, and acquired a mass (avoiding the cosmological moduli problem)! Furthermore for compact spaces with nontrivial $4$-cycles, the corresponding $5$-branes could wrap around those cycles giving rise to another string (different of course from the one obtained from the membrane). These nonperturbative strings will generically have their own nonperturbative gauge group, therefore enhancing the maximum rank required in perturbation theory [55] (the world record seems to be right now a group of rank of order $10^5$! [106]). The physical relevance of the nonperturbative gauge fields is yet to be explored.

(iv)$\$Scales in M theory: It is interesting to analyze the different scales present in a $4D$ model built from $M$-theory. There are three relevant scales: the $11D$ Planck scale $\kappa$, the length of the interval $\rho$ and the overall volume of the compactified $6D$ space $V$. In the $11D$ theory, the gauge and gravitational couplings can be written as:

$$L=-{1\over 2\kappa^2}\int_{M^{11}} d^{11}x \sqrt g R -$$

$$\sum_i {1\over 8\pi (4\pi \kappa^2)^{2/3}}\int_{M^{10}_i}d^{10}x\sqrt g {\rm tr} F_i^2.$$ (46)

Where $M^{11}$ is the $11D$ space (bulk) and $M_i^{10}$, $i=1,2$ are the two $10 D$ $9$-branes at each end of the interval. We can see that after compactification, the $4D$ Newton constant and gauge couplings are given by $G_N={\kappa^2\over{16 \pi^2 V\rho}}$ and $\alpha_{GUT}={\left(4\pi \kappa^2\right)^{2/3}\over {2V}}$. Notice that now $M_{GUT}^2=V^{-1/3}={\alpha_{GUT}\over 8\pi^2 G_N^{2/3}\rho}$, since we have an extra parameter, $\rho$, we can get $M_{GUT}\sim 10^{16} GeV$ by setting $\rho^{-1}\sim 10^{12-14} Gev$ something we could not have done in perturbative heterotic strings. This has been used by Witten to claim that it may be possible to solve the string unification problem by tunning the extra parameter as in standard GUTs [107]. We then get the following picture: at large distances the universe looks $4D$ at energy scales between $10^{12-14} GeV$ and $10^{16} Gev$ it looks $5D$ and at higher scales (smaller distances) it looks $11D$. This new intermediate scale ($\rho$) may play an interesting role for other phenomenological and cosmological questions. There is a complication that for $\rho^{-1}\leq 10^{15} GeV$ the gauge coupling of one of the gauge groups blows up, this has been argued by Witten that could put a bound on Newton's constant on a generic model. There are some specific models which avoid this problem which makes them more attractive. Also, the process of gaugino condensation can be reanalyzed in this picture [108]. A single condensate in the hidden $E_8$ $9$-brane, does not break supersymmetry in its vecinity nor in the $5D$ bulk but due to a topological obstruction it can break supersymmetry in the observable sector [108]. Note that in this picture the standard model lives at one of the `end of the world' branes while gravity and the moduli fields live in the $5D$ bulk. The possible physical consequences of this new picture are only starting to be explored [111] (see next section).

(v)$\$Nonperturbative superpotential: It is quite remarkable that recently Witten and others have been able to extract information about superpotentials derived from stringy nonperturbative effects [109]. At the moment there have been found three classes of results, depending on the compactification: $W=0$, $W\sim e^{-\Phi}$, $W=$ a modular form. Here $\Phi$ is one of the moduli fields. The first case is interesting because it means there are compactifications for which the nonperturbative superpotential vanishes so the only source of superpotential could be strong coupling infrared effects such as gaugino condensation making the field theoretical discussion above more relevant. The second case gives the standard runaway behaviour of the scalar potential and the third possibility is a realization of the kind of duality invariant potentials proposed in the past [37],[10], in this case there are nontrivial minima and it is yet to be studied in detail whether supersymmetry could be broken, in particular these models seem suitable for a realization of the two steps scenario alluded to before. We hope more progress will be made in this direction which is addressing the main problem of superstring phenomenology from a nonperturbative formulation.

(vi)Stringy $e^{-1/g}$ effects: Some time ago, Shenker proposed that in string theory, there would appear nonperturbative effects of the form $e^{-1/g}$ on top of the standard field theoretical effects of the form $e^{-1/g^2}$. These have been argued to correct the Kähler potential and contribute to the dilaton potential in such a way that the dilaton can be fixed even with a single exponential in $W$ [58]. Recently, these effects were explicitly computed for the heterotic string for a particular compactification [110].

We can see that many of the results from string perturbation theory are modified by the nonperturbative information obtained so far. Some of the other results are expected to be modified or need revision, for instance the nonexistence of global symmetries was proved using CFT techniques which are explicitly perturbative, it is expected that being string theory a theory of gravity, global symmetries will not be allowed (as usually found studying black holes and wormholes), but a general nonpertubative proof is not available yet. Also, the main problems such as supersymmetry breaking, are still open which is a good motivation to work on this field. Now we turn to probably the most striking result of the second string revolution, namely the possibility that we live on a hypersurface inside a higher dimensional spacetime.