Flux-breaking in space-times with toroidal compactification

The flux-breaking of gauge symmetries is examined in a Yang-Mills system minimally coupled to massless Dirac fermions on a space-time manifold of the form Rm x Td in which the residual symmetries are determined by minimising the one-loop effective potential with respect to the non-vanishing toroidal components of the classical background gauge field. Previous analysis for the one-torus case is reviewed (where the one-loop potential is expressible in terms of the polylogarithm function), and it is recalled that breaking of the gauge algebra can only occur if there exist fermions obeying periodic boundary conditions and transforming as a faithful, single-valued representation of the adjoint group. On extending to higher-dimensional tori, it is found that the potentials cannot be identified with known standard functions, but it is demonstrated explicitly that the same criterion for symmetry breaking still applies for the two-torus, and it is argued that this criterion should apply in all toroidal compactifications. Finally, the effects of replacing the fermions by complex scalars are discussed. (orig.)