1 - Geometrical structures in (Q)FT

Summary

Part I of this book is introductory in nature. Its purpose is to motivate our geometric approach to supersymmetric field theory. We show how geometric structures arise in classical and quantum field theories on quite general grounds. In Chapter 1 we consider the basic geometric structures which hold independently of supersymmetry. In Chapter 2 we specialize to the supersymmetric case (rigid and local) where more elegant structures emerge. Not being part of the technical body of the book, these chapters are rather elementary and sketchy. However, we show how dualities, modularity, and other stringy patterns are universal features of field theory.

Throughout this book, by a field theory we shall mean a Lagrangian field theory, that is, a classical or quantum system whose dynamics is described by a Lagrangian L with no more than two derivatives of the fields.

(Gauged) σ–models

Most quantum field theories (QFTs) have scalar fields. Usually we can understand a lot about the dynamics of a field theory just by studying its scalar sector. This is a fortiori true if the theory has (enough) supersymmetries, since in this case all other sectors are related to the scalar one by a symmetry. The understanding of the scalars' geometry is relevant even for theories, like quantum chromodynamics (QCD), that do not have fundamental scalar fields in their microscopic formulation. At low energy, QCD is well described by an effective scalar model whose fields represent pions (the lightest particles in the hadronic spectrum). Historically, this effective theory was the original σ–model. It encodes all current algebra of QCD, and its phenomenological predictions are quite a success [303, 304, 305]. Our first goal is to generalize this model. We begin by considering a theory with only scalar fields. In the next section we will add fields in arbitrary (finite) representations of the Lorentz group.

1.1.1 The target space M

We consider a general field theory in D space–time dimensions whose Lagrangian description contains only scalar fields which we denote as øⁱ, with i = 1, 2, …, n.