lattice, the gauge theory has a perimeter law for Wilson loops and linearly confines monopoles (open flux lines), but deconfines static charges of the gauge field [@Savit]. The non-trivial case is $J_2 > J_c$ and $J_1=0$ and has the same confining bulk physics only in the composite gauge variable $\tilde{A}$. Notably the transformation $\tilde{A} = A A_{\sigma}$ can be viewed as acting on the flux degrees of freedom by multiplying them with vorticity lines. Since vorticity lines consist of closed polygons, this transformation leaves the monopole configuration unchanged. Consequently the non-trivial phase also confines monopoles. See figure \[fig:conf\] for a representation of the non-trivial phase.

The above CTP is a robust phase of matter. As in the $2D$ model, the non-local transformation $\tilde{A}=A A_{\sigma}$ maps local symmetric and gauge symmetry respecting operators, into local ones, and leaves the free energy invariant. Respecting these symmetries, both the monopole confining phases of $\tilde{A}$ and $A$ are well defined phases [@Fradkin1979]. In addition, we found that breaking the gauge symmetry on an interface or boundary does not destroy the surface physics (see below) suggesting that gauge symmetry is not crucial here.

Surface theory
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To establish the distinction between trivial and non–trivial phases 