Self-dual supersymmetry and supergravity in Atiyah-Ward space-time

Abstract

We study supersymmetry and self-duality in a four-dimensional space-time with the signature (2,2), that we call the Atiyah-Ward space-time. Dirac matrices and spinors, in particular Majorana-Weyl spinors, are investigated in detail. We formulate N ⩾ 1 supersymmetric self-dual Yang-Mills theories and self-dual supergravities. An N = 1 “self-dual” tensor multiplet is constructed and a possible ten-dimensional theory that gives rise to the four-dimensional self-dual supersymmetric theories is found. Instanton solutions are given as the zero modes in the N = 2 self-dual Yang-Mills theory. The N = 2 superstrings are conjectured to have no possible counter-terms at quantum level to all orders. These self-dual supersymmetric theories are to generate exactly soluble supersymmetric in lower dimensions.