Flux-Breaking of Gauge and Central Symmetries in the Hosotani Model

The purpose of this work is to examine the flux-breaking mechanism for breaking gauge symmetries in the context of the generalised Hosotani model. This model consists of either fermions or scalars (in one arbitrary representation of the gauge group) minimally coupled to Yang-Mills fields on a background spacetime of the form Rm x S1. In this model, the one loop effective potential can be explicitly calculated in terms of the non-trivial background field components satisfying Fmuv = 0 which exist in the circular dimension and therefore the particular breaking direction for any given case can be determined by minimising this potential. We begin in Chapter 1 by describing the flux-breaking mechanism. The group theory required for this analysis is then reviewed in chapter 2 before we go on to discuss the effective potential and its calculation in the generalised Hosotani model. We show in Chapters 5 and 6 that, for the case of either antiperiodic fermions or periodic scalars, the zero background is preferred as the global minimum of the potential and hence no symmetry breaking occurs. For periodic fermions or antiperiodic scalars, the zero background is destabilised by the matter field contribution to the potential. If such fields are in a representation whose congruency class does not contain the adjoint representation, then there exist non-zero backgrounds which preserve the gauge symmetry and are preferred as the global minima of the potential. Such non-zero backgrounds, however, lead to the matter fields having no zero modes on compactification of the theory to Rm and hence there is an apparent breaking of the central symmetry of the gauge group. If such destabilising fields are included in an adjoint class representation, then the zero background is found to be the only symmetry preserving background and. as this is eventually becomes a local maximum of the potential, the gauge symmetry must break. Examples illustrating this fact are given in numerical and graphical form. It is also found that the critical fermion number required to destabilise the zero background decreases as the spacetime dimension is increased, whereas the corresponding critical scalar number increases. This then leads to the conjecture that fermions will be more conducive to symmetry breaking than scalar fields in higher dimensional theories. An additional feature is found whereby scalars given a phase delta = pi/2 can break the gauge symmetry providing the representation containing them generates a group with Z2 centre. Examples of this type of breaking are also given, along with a discussion of permissible phases for the matter fields. Finally, Lagrangians containing more than one matter representation are examined in Chapter 7 in order to determine if the single representation features persist. It is found that the situation becomes more complicated, but symmetry breaking is again only found if destabilising fields in the appropriate representations are included. This leads to a general conjecture about the possibilities for flux breaking of gauge symmetries in any model, a conjecture which seems to be validated in a brief review of other toy models. An original example of an E6 model is also given, in which a realistic subgroup is obtained by flux breaking, provided that an upper limit is imposed on the number of fundamental representation fermion generations. We conclude by summarising the contributions made to the field in this thesis and we review the work of other authors on flux-breaking toy models