The Minkowski and Conformal Superspaces

The following sections are included:

• Preface
• Introduction
• Contents

Supergeometry represents a generalization of the theory of manifolds and algebraic varieties. In classical geometry, geometric objects such as manifolds are best understood as a pair consisting of a topological space and an assignment of the appropriate functions on each open set of the topological space. We consider differentiable functions when we speak of differentiable manifolds, algebraic functions when we speak of algebraic varieties and so on. In algebraic geometry we allow ‘functions’ with a nilpotent part, so strictly speaking, they are not functions on the topological space, but sections of a certain sheaf called the structural sheaf of the manifold, variety or scheme. In supergeometry, a further generalization is made by making the local rings ℤ₂ = ℤ/2ℤ-graded and supercommutative. This generalization is thus ultimately based on the category of super vector spaces…

In this chapter we want to introduce the ordinary Minkowski and the conformal spacetimes as homogeneous spaces for the actions of their symmetry groups, namely the Poincaré and conformal group respectively…

This chapter is devoted to understanding the meaning and the role played by supersymmetry in physics. From a mathematical point of view, supersymmetry can be presented just as a generalization of the ordinary Lie groups that are spacetime symmetries of physical theories, the Poincaré and the conformal groups. This generalization implies the substitution of the category of commutative algebras by the larger category of commutative superalgebras. Commutative superalgebras contain even and odd generators: the even ones commute among themselves while the odd ones, which are nilpotent generators, anticommute among themselves. In spite of the non commutativity of the odd generators, this generalization has nothing to do with quantum physics: it is a classical generalization…

In this chapter we want to describe a generalization of spacetime that appears in supersymmetric field theories. It is the superspace or super spacetime in two versions: the Minkowski and the conformal superspaces…

In this chapter we construct a quantum deformation of the Minkowski and conformal spaces, together with the natural actions of their groups of symmetries, the Poincaré and conformal groups. The deformation is also constructed for the chiral Minkowski and conformal superspaces, according to the philosophy stated in the Introduction…

The following sections are included:

• Appendix A: Categories
• Appendix B: Representability Criterion
• Appendix C: Lie Superalgebras and Lie Supergroups of Classical Type
• Appendix D: Super Harish-Chandra Pairs
• Appendix E: Quantum Supergroups
• Bibliography
• Index