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, 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; 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 . Dedicated to the memory of Claude Itzykson, recalling his gift for combining elegant physics and beautiful mathematics