Geometric Structures of Supersymmetry and Gravitation

Abstract

In this chapter we present several basic constructions in the theory of fields which employ the idea of a superspace. In § 1 the Minkowski superspace M is described. If one begins with twistors, then the compact complex version of M is naturally realized as a flag space. The geometry of the flat case corresponding to the minimal number (N = 1) of odd coordinates, after a suitable twist, turns into the geometry of simple gravity according to Ogievetskii-Sokachev, to which § 7 is devoted. In § 2, we explain for the simplest example of scalar superfields several fundamental ideas of supersymmetric field theory which are used in the physical literature. In § 3 connections and dynamical equations for Yang-Mills superfields are studied on the basis of an idea of Witten to treat them as equations of integrability along light supergeodesics. In § 4 we present a method for constructing solutions of supersymmetric Yang-Mills equations using the Penrose transform in supergeometry; the coordinate computations relating to this are carried out in § 5. In § 6 we classify the other flag superspaces whose underlying space is the Penrose model. They have several exotic properties, but they can be useful for understanding such constructions as the Fayet-Sohnius multiplet and other questions of extended supersymmetry and supergravity.