In the previous chapter we have seen how scalar fields coupled to gravity arise naturally in KK compactification. In Part III we are also going to see that scalar fields are also present, even before compactification, in some higher-dimensional supergravity theories that are the low-energy effective-field theories of certain superstring theories. In all these examples the scalar fields couple in a characteristic way to vector (or p-form in higher dimensions) field strengths. In this chapter we are going to study first, in Section 12.1, a simple model that synthesizes the main features of those theories.

The a-model describes a real scalar coupled to gravity and to a vector-field strength. The coupling is exponential and depends on a parameter a (hence the name “a-model” that we are giving it here). Since the scalar can be identified in some cases with the string dilaton (or with the KK scalar, which is called also the dilaton sometimes), these models are also generically referred to as dilaton gravity. We will be able to obtain BH-type solutions for general values of a and in any dimension d ≥ 4; however, only a handful of values of a actually occur in the theories of interest, although they occur in many different ways (embeddings [620]).

After studying the main properties of these dilaton BHs, we are going to study in Section 20.1 a more complex (four-dimensional) model that involves several scalar and vector fields. We are going to obtain extreme BH solutions that can be understood as composite BHs.