Internal symmetries in Kaluza-Klein models

Abstract

The usual approach to Kaluza-Klein considers a spacetime of the form M4 × K and identifies the isometry group of $$ {g}_K^0 $$ g K 0 , the internal vacuum metric, 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 $$ {g}_K^0 $$ 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 $$ {g}_K^0 $$ 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) × SU(3))/ℤ3 down to (SU(3) × SU(2) × U(1))/ℤ6, 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.