SUPERSYMMETRIC SELF-DUAL YANG-MILLS AND SUPERGRAVITY AS BACKGROUND OFTHE GREEN-SCHWARZ SUPERSTRING

We present the Green-Schwarz sigma-model coupled to the N=1 supersymme tric Yang-Mills and supergravity in a four-dimensional space-time with the indefinite signature (+, +, -, - ). We first confirm the kappa-in variance of the Green-Schwarz action, and show that all the beta-funct ions for the backgrounds vanish consistently after the use of their su perfield equations. Subsequently, we inspect the supersymmetric self-d uality conditions, that have been developed in our previous paper on t he Yang-Mills and supergravity backgrounds. Remarkably, the Majorana-W eyl spinor dictating the supersymmetric self-duality conditions is con sistent with the couplings of the Green-Schwarz superstring. Such a Gr een-Schwarz superstring is supposed to be the underlying theory of the supersymmetric self-dual Yang-Mills theory, which is conjectured to g enerate all exactly soluble supersymmetric systems in lower dimensions .