  We study topological field theory describing gapped phases of gauge theories
where the gauge symmetry is partially Higgsed and partially confined. The TQFT
can be formulated both in the continuum and on the lattice and generalizes
Dijkgraaf-Witten theory by replacing a finite group by a finite 2-group. The
basic field in this TQFT is a 2-connection on a principal 2-bundle. We classify
topological actions for such theories as well as loop and surface observables.
When the topological action is trivial, the TQFT is related to a
Dijkgraaf-Witten theory by electric-magnetic duality, but in general it is
distinct. We propose the existence of new phases of matter protected by higher
symmetry.
