we mean that one deforms the energy functional gradually and maintains the local constraint. We establish the existence of inequivalent phases by providing concrete examples of models between which any continuous deformation must involve a phase transition. Notably, given that such distinct phases exist, by definition their distinction does not involve a local order parameter or confinement-deconfinement transitions. Their difference is of topological origin. This is manifested on interfaces between distinct phases, where either long range correlated or quasi long range correlated phases emerge.

In the next sections we will explore these ideas for the choice $G=G'=Z_N$, considering models in both $2D$ and $3D$ where we find many distinct topological phases with the accompanying exotic boundary phenomena. The latter include a “forbidden" [@VanHove1950] symmetry breaking order along $1D$ boundary and an unusual $2D$ critical phase corresponding to a theory of a compact boson in which the basic $\pm 2\pi$ vortices are linearly confined. Just as group cohomology was shown to be the basis for quantum SPTs phases, we will show that tools from cellular cohomology [@Cohomology1991] give a powerful mathematical framework for writing down models of CTPs and analyzing them. The models thus produced are compact, isotropic and, to a 