Integrable systems and supersymmetric gauge theory

After the work of Seiberg and Witten, it has been seen that the dynamics of N=2 Yang-Mills theory is governed by a Riemann surface Σ. In particular, the integral of a special differential λSW over (a subset of) the periods of Σ gives the mass formula for BPS-saturated states. We show that, for each simple group G, the Riemann surface is a spectral curve of the periodic Toda lattice for the dual group, G or, whose affine Dynkin diagram is the dual of that of G. This curve is not unique, rather it depends on the choice of a representation ρ of G or; however, different choices of ρ lead to equivalent constructions. The Seiberg-Witten differential λSW is naturally expressed in Toda variables, and the N=2 Yang-Mills pre-potential is the free energy of a topological field theory defined by the data Σg,ρ and λSW. (orig.)