Integrable systems and supersymmetric gauge theory

Abstract

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 $\Sigma$. In particular, the integral of a special differential ${\lambda }_{SW}$ over (a subset of) the periods of $\Sigma$ gives the mass formula for BPS-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, $G^{\vee }$, 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 $\rho$ of $G^{\vee }$; however, different choices of $\rho$ lead to equivalent constructions. The Seiberg-Witten differential ${\lambda }_{SW}$ 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 ${\Sigma }_{\gg ,\rho }$ and ${\lambda }_{SW}$.

20 pages, latex, 3 uuencoded figures (needs epsf.tex); minor errors corrected, references added

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