On integrable systems and supersymmetric gauge theories

We discuss the properties ofN=2 supersymmetric gauge theories underlying the Seiberg-Witten hypothesis. We consider the main points of the theory that describes the finite-gap solutions to integrable equations in terms of complex curves and generating differentials. We clarify the invariant meaning of these definitions. This formalism is applied to the exact nonperturbative solutions found recently in theN=2 supersymmetric non-Abelian gauge theories. In the known cases, we compare this formalism with the results that can be obtained using standard quantum field-theory methods. The paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 3–46, July, 1997.