Monopoles, affine algebras and the gluino condensate

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 a 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. For gauge groups SU (N), SO (N) and USp (2N) our results for the gluino condensate are in precise agreement with the “weak coupling instanton” expressions (and not with the “strong coupling instanton” calculations). For the exceptional gauge groups we calculate the values of the gluino condensate for the first time.