Five-dimensional gauge theories and relativistic integrable systems

We propose a non-perturbative solution of N = 2 supersymmetric gauge theory in five dimensions compactified on circle of a radius R. We consider the cases of the pure gauge theory as well as the theories with matter in the fundamental and in the adjoint representations. The pure theory as well as the one with adjoint hypermultiplet give rise to the known relativistic integrable systems with 1/R playing the rôle of the speed of light. The theory with adjoint hypermultiplet exhibits some interesting finiteness properties.