INTEGRABLE SYSTEMS AND SUPERSYMMETRIC GAUGE-THEORY

After the work of Seiberg and Witten, it has been seen that the dynami cs of N = 2 Yang-Mills theory is governed by a Riemann surface Sigma. In particular, the integral of a special differential lambda(SW) over (a subset of) the periods of Sigma gives the mass formula for BPS-satu rated states. We show that, for each simple group G, the Riemann surfa ce is a spectral curve of the periodic Toda lattice for the dual group , G(V), whose affine Dynkin diagram is the dual of that of G. This cur ve is not unique, rather it depends on the choice of a representation rho of G(V); however, different choices of rho lead to equivalent cons tructions. The Seiberg-Witten differential lambda(SW) is naturally exp ressed in Toda variables, and the N = 2 Yang-Mills pre-potential is th e free energy of a topological field theory defined by the data Sigma( g,rho) and lambda(SW).