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We analyze the phases of theories which only have a microscopic$\mathbb{Z}_N$ 1-form symmetry, starting with a topological BF theory anddeforming it in accordance with microscopic symmetry. These theories have awell-defined notion of confinement. Prototypical examples are pure $SU(N)$gauge theories and $\mathbb{Z}_N$ lattice gauge theories. Our analysis showsthat in spacetime dimensions $d=2$ only the confined phase is genericallypossible. In $d=3,4$ the confined phase and a topological BF phase are bothgeneric, while in $d=4$ a phase with a massless photon is also possible. Weconstruct a $\mathbb{Z}_N$ lattice gauge theory with a deformation which,surprisingly, produces up to $(N-1)$ photons. We give an interpretation ofthese findings in terms of two competing pictures of confinement --proliferation of monopoles and proliferation of center vortices -- and concludethat the proliferation of center vortices is a necessary but not sufficientcondition for confinement, while that of monopoles is both necessary andsufficient.

Phases of theories with $\mathbb{Z}_N$ 1-form symmetry and the roles of center vortices and magnetic monopoles