An exactly solvable model for twisted symmetry-enriched phases

Topological phases in 2D have a long history of exotic behaviour, producing anyons and protected edge states. This trend continues when we impose an extra symmetry $G$, producing a symmetry-enriched topological (SET) phase. While the ground state will remain invariant under $G$, the set of anyons $A$ may transform non-trivially. The different ways of implementing the symmetry are classified by the elements of the group cohomology $H^2_{\rho}(G,A)$, where $\rho$ describes the action of $G$ on the set of anyons. Previously constructed models fix $\rho$ to be the identity, meaning that $G$ can only modify anyons by a phase, whereas we could easily envision a case where $G$ permutes anyon types, which we call twisted SETs. In this talk, we will propose a modified string-net model which allows $G$ to act on the anyons in exactly that manner, for any choice of $\rho$. We will also introduce a constructive method of gauging the global symmetry, which allows us to verify that the obtained twisted SETs are distinct by showing that discrete gauge theories produced by gauging $G$ are distinct.