Symmetry breaking by Wilson loops in gauge field theory

An analysis is presented of the gauge symmetry breaking caused by Wilson loops on a space-time whose spatial section is ${\mathrm{openR}}^d$×$S^3$/Γ, for all those fundamental groups Γ that give a homogeneous space. We concentrate on pure SU(3) and SU(5) gauge field theories and find that symmetry breaking can occur when d=0, for all Γ. If d=3, the extra minimal scalars prevent any breaking and one must include other fields to achieve this. Explicit forms for the vacuum energies are exhibited in the case of lens and prism spaces, the former for SU(n). For Γ=$Z_m$, when m and the radius of the sphere become infinite, we recover the results on the space-time ${\mathrm{openR}}^{d+3\mathrm{}}$×$S^1$.