Non-linear σ-models are a common element of many of the actions considered in the main text that contain scalar fields: in the N = 1, d = 4 matter-coupled supergravities studied in Chapter 6 we find σ-models which correspond to Kähler–Hodge manifolds; in the N = 2, d = 4 theories studied in Chapter 7 we find special Kähler and quaternionic-Kähler manifolds, and the latter and real special manifolds naturally arise in the σ-models of the N = 1, d = 5 matter-coupled supergravities studied in Chapter 9. There are many other examples in the text, and that is why generic non-linear σ-models have been included in the generic actions Eqs. (2.147) and (2.178).

In many cases the metrics of these σ-models have isometries and the σ-model action is invariant under an associated global symmetry. Furthermore, there are many instances in which we want to gauge one or several of those symmetries, deforming the action so that it becomes invariant under the local (gauge) version of those symmetries. For example, gauging this kind of symmetry in the above-mentioned supergravities one obtains theories with non-Abelian Yang–Mills fields and symmetries and a scalar potential with many potential uses. Gauged σ-models also arise in the construction of KK-brane effective actions studied on p. 706.

Due to the couplings of the scalars to other fields, the symmetries of a σ-model are not automatically symmetries of the full theory. For general kinds of couplings to certain kinds of bosonic fields (differential forms of various ranks) the situation has been studied in Section 2.6.2. In the above-mentioned supergravity theories the coupling to supergravity (hence, of the σ-model scalars to vectors and spinors) requires the geometries of their σ-models to be Kähler–Hodge, special Kähler, quaternionic-Kähler, or real special. There, the Riemannian structure (the metric) is not the only structure that needs to be preserved for a transformation to be called a symmetry. These geometries and their symmetries have been reviewed in the previous appendices, and here we want to study the gauging of all these symmetries in order of increasing complexity and with an (essentially) homogeneous notation that is used in Chapters 6 and 7 to describe the complete theories.