Majorana-Weyl Spinors and Self-Duality Gauge Fields in 2+2 Dimensions

TL;DR

This paper explores the properties of spinors and gauge fields in 2+2 dimensional space-time, introducing Majorana-Weyl spinors, supersymmetry, and self-dual theories, with potential links to integrable systems.

Contribution

It introduces the first formulations of supersymmetric self-dual Yang-Mills and supergravity theories in 2+2 dimensions, highlighting their mathematical structure and relevance.

Findings

Existence of Majorana-Weyl spinors in 2+2 dimensions

Formulation of supersymmetric self-dual Yang-Mills theories

Development of self-dual supergravity models

Abstract

The properties of Dirac gamma matrices in a four-dimensional space-time with the $(2,2)$ signature are studied. The basic spinors are classified, and the existence of Majorana-Weyl spinors is noted. Supersymmetry in $2 + 2$ dimensions is discussed, and the existence of the {\it real chiral} scalar supermultiplet is discovered. Supersymmetric {\it self-dual} Yang-Mills theories and {\it self-dual} supergravity model in $2 + 2$ dimensions, that are apparently relevant to integrable systems, are formulated for the first time.