Integrability in supersymmetric gauge theories. I

This is the first part of a review devoted to the exact, nonperturbative solutions of supersymmetric gauge theories and their formulation in terms of integrable systems. The general phenomenon of integrability as it appears in the formulation of effective actions for various models of topological, low-dimensional string theories and almost realistic supersymmetric gauge field theories is discussed. First the basic features of the string-theory path integral are discussed in order to understand better the nonperturbative properties of the theory. Then a formulation of the exact effective actions based on systems of nonlinear differential equations is proposed. It is demonstrated that the resulting nonlinear differential equations belong to the class of integrable models of the Kadomtsev-Petviashvili and Toda type. Particular models of this class are discussed, with special focus on the integrable systems appearing in the context of multidimensional supersymmetric gauge theories. Their Lax representations and spectral curves are studied in detail, and a classification of the exact solutions of N=2 supersymmetric gauge theories along these lines is proposed.