Supersymmetric self-dual Yang-Mills and supergravity as background of the Green-Schwarz superstring

We present the Green-Schwarz σ-model coupled to the N=1 supersymmetric Yang-Mills and supergravity in a four-dimensional space-time with the indefinite signature (+, +, -, -). We first confirm the κ-invariance of the Green-Schwarz action, and show that all the β-functions for the backgrounds vanish consistently after the use of their superfield equations. Subsequently, we inspect the supersymmetric 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 supersymmetric self-duality conditions is consistent with the couplings of the Green-Schwarz superstring. Such a Green-Schwarz superstring is supposed to be the underlying theory of the supersymmetric self-dual Yang-Mills theory, which is conjectured to generate all exactly soluble supersymmetric systems in lower dimensions.