Extended supersymmetry and self-duality in 2 + 2 dimensions

The N = 2 supersymmetric self-dual Yang-Mills theory and the N = 4 and N = 2self-dual supergravities in 2 + 2 space-time dimensions are formulated for the first time. These formulations utilize solutions of the Bianchi identities subject to the super-Yang-Mills or supergravity constraints in the relevant N-extended superspace with the space-time signature (2,2). The self-duality condition on the Yang-Mills field strength or on the Riemann tensor is not modified, but is accompanied by the field equations of other superpartners. These self-dual systems are conjectured to generate supersymmetric exactly soluble models in lower dimensions.