On Supersymmetric Constraint Equations

Abstract

It is known that the self-dual Yang-Mills equations are completely integrable in a sense of the existence of a related linear problem, infinity of conservation laws and a possibility of generating solutions. WITTEN [1] and ISENBERG, YASSKIN and GREEN [2], following Ward’s approach to self-dual fields, proposed a. construction, which in principle should yield also non self-dual solutions of the Yang-Mills equations (however no application of this scheme exists). Witten also generalized this method to the case of the supersymmetric Yang-Mills equations with N = 3,4. It was possible because the Susy YM equations are equivalent to the so-called supersymmetric constraint equations [3] which resemble selfduality conditions. Thus the constraint equations can provide a key to understanding the structure of the Susy YM equations and in particular the ordinary Yang-Mills equations. In this seminar I discuss recent approaches to the problem of the integrability of the constraint equations (see [4] for details and references).