Flag Superspaces and Supersymmetric Yang-Mills Equations

Abstract

The self-dual Yang-Mills and Einstein equations have a simple geometric meaning, since they imply the vanishing of a part of the curvature tensor of a connection. This connection, the physicists’ gauge potential, is given either on an external vector bundle (the Yang-Mills case) or on the spinorial bundle (the Einstein case) over space-time. After a suitable base change, the relevant part of the curvature becomes the total curvature of the lifted connection along the leaves of a foliation. At least locally (with respect to the initial base manifold), this foliation is a libration and the self-dual field in question can be represented by the vector bundle of horizontal sections along the leaves on the base space of the foliation (Yang-Mills) or by the base space itself (Einstein). This representation is called the Penrose transform. The idea is closely related to the classical Radon transform. One of Penrose’s discoveries was the possibility of using the rigidity of the holomorphic geometry to effectively construct the solutions of the differential equations by geometric means. A mathematician may profitably consult M. F. Atiyah [1] and the references cited therein.