Formal power series solutions of supersymmetric (N=3) Yang–Mills equations

Abstract

Formal power series solutions of the linear system with one spectral parameter associated to the constraint equations for Yang–Mills superconnections on N-extended super-Minkowski space are considered. For N=3 the integrability equations reduce to the supersymmetric field equations. The method of approach is identical to that used by Takasaki for the self-dual equations, based upon formal power series in a spectral parameter and in spatial variables. The problem is reduced to a linear system of equations for a superfield with values in an ∞-dimensional Grassmann manifold. The formal solution is expressed in terms of data on a (3‖2N)-dimensional superhypersurface. However, a difficulty arises with respect to the Cauchy problem, which becomes formally solvable only for an extended system, breaking the relativistic invariance through introduction of additional superfields.