Dynamical mechanisms for Kaluza–Klein theories

Abstract

We present variational formulations of gauge theories and Einstein–Yang–Mills equations in the spirit of Kaluza–Klein theories. For gauge theories, only a topological fibration is assumed. For gravitation coupled with gauge fields, no fibration is assumed: Fields are defined on a ‘space-time’ \({\mathcal {Y}}\) of dimension \(4 + r\) without any structure a priori, where r is the dimension of the structure group. If the latter is compact and simply connected, classical solutions allow to construct a manifold \({\mathcal {X}}\) of dimension 4 to be the physical space-time, in such a way that \({\mathcal {Y}}\) acquires the structure of a principal bundle over \({\mathcal {X}}\) and leads to solutions of the Einstein–Yang–Mills systems. The special case of the Einstein–Maxwell system is also discussed: It suffices that at least one fiber closes in on a circle to deduce that the five-dimensional space-time has a fiber bundle structure.