Supersymmetric Self-Dual Yang-Mills and Supergravity as Background of Green-Schwarz Superstring

TL;DR

This paper demonstrates the coupling of the Green-Schwarz superstring to supersymmetric Yang-Mills and supergravity backgrounds, confirming their quantum consistency and exploring supersymmetric self-duality conditions that relate to integrable systems.

Contribution

It establishes the Green-Schwarz superstring in supersymmetric self-dual backgrounds and confirms their quantum consistency through vanishing beta functions.

Findings

Green-Schwarz action is k-invariant and beta functions vanish

Supersymmetric self-duality conditions are compatible with superstring couplings

Proposes supersymmetric self-dual Yang-Mills as a foundation for integrable systems

Abstract

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.