Abstract:
We argue that supersymmetry breaking by gaugino condensation in the strongly coupled heterotic string can be described by an analogue of Scherk-Schwarz compactification on the eleventh dimension in M-theory. The M-theory scale is identified with the gauge coupling unification mass, whereas the radius of the eleventh dimension $\rho$ is at an intermediate scale $\rho^{-1}\sim 10^{12}$ GeV. At the lowest order, supersymmetry is broken only in the gravitational and moduli sector at a scale $m_{3/2}\sim\rho^{-1}$, while it is mediated by gravitational interactions to the observable world. Computation of the mass splittings yields in general a hierarchy of soft masses at the TeV scale $(\sim\rho^{-2}/M_p)$ with matter scalars much heavier than gauginos.

Abstract:
The status of gaugino condensation in low-energy string theory is reviewed. Emphasis is given to the determination of the efective action below condensation scale in terms of the 2PI and Wilson actions. We illustrate how the different perturbative duality symmetries survive this simple nonperturbative phenomenon, providing evidence for the believe that these are exact nonperturbative symmetries of string theory. Consistency with T duality lifts the moduli degeneracy. The $B_{\mu\nu}-axion$ duality also survives in a nontrivial way in which the degree of freedom corresponding to $B_{\mu\nu}$ is replaced by a massive $H_{\mu\nu\rho}$ field but duality is preserved. S duality may also be implemented in this process. Some general problems of this mechanism are mentioned and the possible nonperturbative scenarios for supersymmetry breaking in string theory are discussed.

Abstract:
The weakly-coupled heterotic string is known to have problems of dilaton/moduli stabilization, supersymmetry breaking (by hidden-sector gaugino condensation), gauge coupling unification (or the Newton's constant), QCD axion, as well as cosmological problems. We study these problems by adopting the viewpoint that they arise mostly due to our limited calculational power, little knowledge of the vacuum structure, and an inappropriate treatment of gaugino condensation. It turns out that these problems can be solved or are much less severe after a more consistent and complete treatment. There are two kinds of non-perturbative effects in the construction of effective field theory: the field-theoretical non-perturbative effects of gaugino condensation (with a constraint ignored in the past) and the stringy non-perturbative effects conjectured by Shenker, which are best described using the linear multiplet formalism. Stringy non-perturbative corrections to the K\"ahler potential are invoked to stabilize the dilaton at a value compatible with a weak coupling regime. Modular invariance is ensured through the Green-Schwarz counterterm and string threshold corrections which, together with hidden matter condensation, lead to moduli stabilization at the self-dual point where the vev's of moduli's F-components vanish. In the vacuum, supersymmetry is broken at a realistic scale with vanishing cosmological constant. As for soft supersymmetry breaking, our model always leads to a dilaton-dominated scenario. For the strong CP problem, the model-independent axion has the right properties to be the QCD axion. Furthermore, there is a natural mass hierarchy between the dilaton/moduli and the gravitino, which could solve the cosmological moduli problem and the cosmological problem of the model-independent axion.

Abstract:
The status of the gaugino condensation as the source of supersymmetry breaking is reexamined. It is argued that one cannot have stable minima with broken supersymmetry in models where the dilaton is coupled only linearly to the gaugino condensate. We show that the problems of the gaugino condensate mechanism can be solved by considering nonstandard gauge kinetic functions, created by nonperturbative effects. As an example we use the principle of S-duality to modify the coupling of the gaugino condensate to effective supergravity (superstring) Lagrangians. We show that such an approach can solve the problem of the runaway dilaton and lead to satisfactory supersymmetry breaking in models with a {\em single} gaugino condensate. We exhibit a general property of theories containing a symmetry acting on the dilaton and also shed some light on the question whether it is generically the auxiliary field of the modulus $T$, which dominates supersymmetry breaking.

Abstract:
We study gaugino condensation in the context of superstring effective theories using the linear multiplet formulation for the dilaton superfield. Including nonperturbative corrections to the K\"ahler potential for the dilaton may naturally achieve dilaton stabilization, with supersymmetry breaking and gaugino condensation; these three issues are interrelated in a very simple way. In a toy model with a single static condensate, a dilaton $vev$ is found within a phenomenologically interesting range. The effective theory differs significantly from condensate models studied previously in the chiral formulation.

Abstract:
The principle of S-duality is used to incorporate gaugino condensates into effective supergravity (superstring) Lagrangians. We discuss two implementations of S-duality which differ in the way the coupling constant is transformed. Both solve the problem of the runaway dilaton and lead to satisfactory supersymmetry breaking in models with a {\em single} gaugino condensate. The breakdown of supersymmetry is intimately related to a nontrivial transformation of the condensate under T-duality.

Abstract:
We investigate the formation of dynamical gaugino condensates and supersymmetry breaking in the compactifications of Horava-Witten theory with perturbative nonstandard embeddings. Specific models are considered where the underlying massless charged states of the condensing sector are determined by the spectra of $Z_2 \times Z_2 $ and $Z_4 $ orbifolds with nonstandard embeddings. We find among them viable examples where gaugino condensation is triggered on the wall with the weakest gauge coupling at $M_{GUT} $. In all these cases the magnitude of the condensate formed is below the energy scales at which extra dimensions are resolved, and so justifies the analysis of condensation in an effective 4-dimensional framework. We make some comments concerning the size of the largest extra dimension in the models considered. We discuss racetrack scenarios in the framework of perturbative nonstandard embeddings.

Abstract:
We study dynamical gaugino condensation in superstring effective theories using the linear multiplet representation for the dilaton superfield. An interesting necessary condition for the dilaton to be stabilized, which was first derived in generic models of static gaugino condensation, is shown to hold for generic models of dynamical gaugino condensation. We also point out that it is stringy non-perturbative effects that stabilize the dilaton and allow dynamical supersymmetry breaking via the field-theoretical non-perturbative effect of gaugino condensation. As a typical example, a toy S-dual model of a dynamical E_8 condensate is constructed and the dilaton is explicitly shown to be stabilized with broken supersymmetry and (fine-tuned) vanishing cosmological constant.

Abstract:
We investigate gaugino condensation in the framework of the strongly coupled heterotic $E_8 \times E_8$ string (M--theory). Supersymmetry is broken in a hidden sector and gravitational interactions induce soft breaking parameters in the observable sector. The resulting soft masses are of order of the gravitino mass. The situation is similar to that in the weakly coupled $E_8 \times E_8$ theory with one important difference: we avoid the problem of small gaugino masses which are now comparable to the gravitino mass.

Abstract:
We study gaugino condensation in presence of an anomalous $U(1)$ gauge group and find that global supersymmetry is dynamically broken. An example of particular interest is provided by effective string models with 4-dimensional Green-Schwarz anomaly cancellation mechanism. The structure of the hidden sector is constrained by the anomaly cancellation conditions and the scale of gaugino condensation is shifted compared with the usual case. We explicitly compute the resulting soft supersymmetry breaking terms.