We present the Green-Schwarz $\s\-$model coupled to the $N=1$ { \it { supersymmetric}} Yang-Mills and supergravity in a four-dimensional space-time with the indefinite signature $(+,+,-,-)$. We first confirm the $\k\-$invariance of the Green-Schwarz action, and show that all the $\b\-$functions for the backgrounds vanish consistently after the use of their superfield equations. Subsequently, we inspect the supersymmetric { \it self-duality} conditions, that have been developed in our previous paper on the Yang-Mills and supergravity backgrounds. Remarkably, the Majorana-Weyl spinor dictating the { \it supersymmetric self-duality} conditions is consistent with the couplings of Green-Schwarz superstring. Such Green-Schwarz superstring is supposed to be the underlying theory of the { \it supersymmetric self-dual} Yang-Mills theory, which is conjectured to generate { \it all} exactly soluble supersymmetric systems in lower dimensions.
Categories: hep-th