On integrable systems and supersymmetric gauge theories

The properties of the $\mathcal N=2$ SUSY gauge theories underlying the Seiberg–Witten hypothesis are discussed. The main ingredients of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential are presented, the invariant sense of these definitions is illustrated. Recently found exact nonperturbative solutions to $\mathcal N=2$ SUSY gauge theories are formulated using the methods of the theory of integrable systems and where it is possible the parallels between standard quantum field theory results and solutions to integrable systems are discussed.