Symmetry enriched string-nets: Exactly solvable models for SET phases

We construct exactly solvable models for a wide class of symmetry enriched topological (SET) phases. Our construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group G and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. In particular, our construction realizes onsite the ${\mathbb Z}_2$ exchange symmetry of the charge and flux excitations in a model with toric code topological order. More generally, our construction makes use of a correspondence between so-called G-extensions of a fusion category $C$ and the braided G-crossed extensions of its quantum double.