Kaluza [615] and Nordström's [728] original idea/observation that electromagnetism could be seen as part of five-dimensional gravity, combined with Klein's curling up of the fifth dimension in a tiny circle [626], constitutes one of the most fascinating and recurring themes of modern physics. Kaluza–Klein theories are interesting both in their own right (in spite of their failure to produce realistic four-dimensional theories [960], at least when the internal space is a manifold) and because of the usefulness of the techniques of dimensional reduction for treating problems in which the dynamics in one or several directions is irrelevant. We saw an example in Chapter 9, when we related four-dimensional instantons to monopoles.

On the other hand, the effective-field theories of some superstring theories (which are supergravity theories) can be obtained by dimensional reduction of 11-dimensional supergravity, which is the low-energy effective-field theory of (there is no real consensus on this point) M theory or one of its dual versions. In turn, string theory needs to be “compactified” to take a four-dimensional form and, to obtain the four-dimensional low-energy effective actions, one can apply the dimensional-reduction techniques.

Here we want to give a simple overview of the physics of compact dimensions and the techniques used to deal with them (dimensional reduction etc.) in a non-stringy context. We will deal only with the compactification of pure gravity and vector fields, leaving aside compactification in the presence of more general matter fields (including fermions) until Part III. We will also leave aside many subjects such as spontaneous compactification and the issue of constructing realistic Kaluza–Klein theories, which are covered elsewhere [342, 957].