9 - Matter-coupled N = 1, d = 5 supergravity

Summary

So far we have studied four-dimensional supergravities. There are several reasons why higher-dimensional supergravities are also interesting, but, from our point of view, the most important reasons are that they naturally arise in superstring or M-theory compactifications and that their own compactification gives rise to the four-dimensional supergravities that we have studied. This will allow us to rewrite many theories and their classical solutions in a higher-dimensional language, geometrizing many quantities and objects. In turn, this will allow us to generate new four-dimensional or higher-dimensional solutions using the solution-generation techniques explained in Chapter 15.

In this chapter we are going to study the simplest example of higher-dimensional supergravity: the minimal five-dimensional supergravity (the one with the minimal spinors) and some of its couplings to matter multiplets. Since the supersymmetry parameter, the gravitino, and other fermions are given by just one minimal spinor (eight real components), these theories deserve to be called N = 1, d = 5 supergravities. This is what we will do here. However, in the literature they are very often called N = 2, d = 5 supergravities, the reason being their close relation to the N = 2, d = 4 theories: the compactification of all these theories in a circle gives rise to an N = 2, d = 4 supergravity. Not all the four-dimensional N = 2 theories can be obtained in this way, but for the many that can (most of them very interesting from several points of view) there is an alternative and most useful rewriting in five-dimensional language. In five dimensions the scalars are real and life is usually much simpler. All this, together with the fact that many of them can be obtained from compactifications of 11-dimensional supergravity on Calabi–Yau 3-folds [274], makes the study of these theories very attractive.