Internal symmetries in Kaluza-Klein models

The usual approach to Kaluza-Klein considers a spacetime of the form $M_4 \times K$ and identifies the isometry group of the internal vacuum metric, $g_K^0$, with the gauge group in four dimensions. In these notes we discuss a variant approach where part of the gauge group does not come from full isometries of $g_K^0$, but instead comes from weaker internal symmetries that only preserve the Einstein-Hilbert action on $K$. Then the weaker symmetries are spontaneously broken by the choice of vacuum metric and generate massive gauge bosons within the Kaluza-Klein framework, with no need to introduce ad hoc Higgs fields. Using the language of Riemannian submersions, the classical mass of a gauge boson is calculated in terms of the Lie derivatives of $g_K^0$. These massive bosons can be arbitrarily light and seem able to evade the standard no-go arguments against chiral fermionic interactions in Kaluza-Klein. As a second main theme, we also question the traditional assumption of a Kaluza-Klein vacuum represented by a product Einstein metric. This should not be true when that metric is unstable. In fact, we argue that the unravelling of the Einstein metric along certain instabilities is a desirable feature of the model, since it generates inflation and allows some metric components to change length scale. In the case of the Lie group $K = SU(3)$, the unravelling of the bi-invariant metric along an unstable perturbation also breaks the isometry group from $( SU(3) \times SU(3)) / Z_3$ down to $( SU(3) \times SU(2) \times U(1) )/ Z_6$, the gauge group of the Standard Model. We briefly discuss possible ways to stabilize the internal metric after that first symmetry breaking and produce an electroweak symmetry breaking at a different mass scale.