Seiberg-Witten Theory and the Integrable Systems

First, the book reviews what is known about the physical problem: the construction of low-energy effective actions for the N=2 Yang -- Mills theories from the traditional viewpoint of quantum field theory. Then the necessary background information from the theory of integrable systems is presented. In particular the author considers the definition of the algebro-geometric solutions to integrable systems in terms of complex curves or Riemann surfaces and the generating meromorphic l-form. These definitions are illustrated in detail on the basic example of the periodic Toda chain. Several "toy-model" examples of string theory solutions where the structures of integrable systems appear axe briefly discussed. Then the author proceeds to the Seiberg -- Witten solutions and show that they are indeed defined by the same data as finite-gap solutions to integrable systems. The complete formulation requires the introduction of certain deformations of the finite-gap solutions described in terms of quasiclassical or Whitham hierarchies. The explicit differential equations and direct computations of the prepotential of the effective theory are presented and compared when possible with the well-known computations from supersymmetric quantum gauge theories. Finally, the book discusses the properties of the exact solutions to Susy Yang -- Mills theories and their relation to integrable systems in the general context of the modern approach to nonperturbative string or M-theory.