We examine supersymmetric SU(N) gauge theories on R 3 × S 1 with a circle of circumference β. These theories interpolate between four-dimensional N = 1 pure gauge theory for β = ∞ and three-dimensional N = 2 gauge theory for β = 0. The dominant field configurations of the R 3 × S 1 SU(N) theories in the semi-classical regime arise from N varieties of monopole. Periodic instanton configurations correspond to mixed configurations of N single monopoles of the N different types. We semi-classically evaluate the non-perturbatively generated superpotential of the R 3 × S 1 theory and hence determine its vacuum structure. We then calculate monopole contributions to the gluino condensate in these theories and take the decompactification limit β = ∞. In this way we obtain a value for the gluino condensate in the four-dimensional N = 1 supersymmetric SU(N) Yang-Mills theory, which agrees with the previously known 'weak coupling' expression but not with the 'strong coupling' expression derived in the early literature solely from instanton considerations. Moreover, we discover that the superpotential gives a mass to the dual (magnetic) photon, which implies confinement of the original electric photon and disappearance of all the massless modes.

We examine the low-energy dynamics of four-dimensional supersymmetric gauge theories and calculate the values of the gluino condensate for all simple gauge groups. By initially compactifying the theory on a cylinder we are able to perform calculations in a controlled weakly-coupled way for small radius. The dominant contributions to the path integral on the cylinder arise from magnetic monopoles which play the role of instanton constituents. We find that the semi-classically generated superpotential of the theory is the affine Toda potential for an associated twisted affine algebra. We determine the supersymmetric vacua and calculate the values of the gluino condensate. The number of supersymmetric vacua is equal to c 2 , the dual Coxeter number, and in each vacuum the monopoles carry a fraction 1/c 2 of topological charge. As the results are independent of the radius of the circle, they are also valid in the strong coupling regime where the theory becomes decompactified. In this way we obtain values for the gluino condensate which for the classical gauge groups agree with previously known "weak coupling instanton" expressions (but not with the "strong coupling instanton" calculations). This detailed agreement provides further evidence in favour of the recently advocated resolution of the the gluino condensate puzzle. We also make explicit predictions for the gluino condensate for the exceptional groups.

Certain exact results in supersymmetric gauge theories are generated by non-perturbative effects different from instantons. In supersymmetric QCD with N colours and N F fundamental flavours we examine the Affleck-Dine-Seiberg (ADS) superpotential using controlled semi-classical analysis. We show how for N F < N −1 the ADS superpotential arises from monopole contributions to the path integral of the supersymmetric gauge theory compactified on R 3 × S 1 . These are the monopole effects leading to gaugino condensation and confinement of the low-energy SU(N −N F ) supersymmetric gauge theory.