  We study whether broken dual gauge symmetry, as detected by a monopole order
parameter introduced by the Pisa group, is necessarily associated with the
confinement phase of a lattice gauge theory. We find a number of examples,
including SU(2) gauge-Higgs theory, mixed fundamental-adjoint SU(2) gauge
theory, and pure SU(5) gauge theory, which appear to indicate a dual gauge
symmetry transition in the absence of a transition to or from a confined phase.
While these results are not necessarily fatal to the dual superconductor
hypothesis, they may pose some problems of interpretation for the present
formulation of the Pisa monopole criterion.
