A study of the mathematics of supersymmetry theories

Abstract

This Thesis consists of three parts. In the first part a theory of integration is constructed for supermanifolds and supergroups. With this theory expressions for the invariant integral on several Lie supergroups are obtained including the super Poincaré group and supers pace. The unitary irreducible representations of the super Poincaré group are examined by considering the unitary irreducible representations of a certain set of Lie groups equivalent to the super Poincaré group. These irreducible representations contain, at most, particles of a single spin.

In the second part a detailed examination of the massive representations of the super Poincaré algebra is undertaken. Supermultiplets of second quantized fields are constructed for each of the. massive representations, which allows an understanding of the auxiliary fields of supersymmetry theories.

In the third part super Poincaré invariant superfields on superspace are constructed from the supermultiplets of the second part. This enables a connection between the representations of part one and those of part two to be established. An examination of action integrals on superspace is made enabling the relationship between the integration theory constructed in part one and the Berezin integral to be established.