Symmetry-Enriched Topological Phases and Their Gauging: A String-Net Model Realization

We present a systematic framework for constructing exactly-solvable lattice models of symmetry-enriched topological (SET) phases based on an enlarged version of the string-net model. We also gauge the global symmetries of our SET models to obtain string-net models of pure topological phases. Without invoking externally imposed onsite symmetry actions, our approach promotes the string-net model of a pure topological order, specified by an input unitary fusion category $\mathscr{F}$, to an SET model, specified by a multifusion category together with a set of isomorphisms. Two complementary construction strategies are developed in the main text: (i) promotion via outer automorphisms of $\mathscr{F}$ and (ii) promotion via the Frobenius algebras of $\mathscr{F}$. The global symmetries derived via these two strategies are intrinsic to topological phases and are thus termed blood symmetries, as opposed to adopted symmetries, which can be arbitrarily imposed on topological phases. We propose the concept of symmetry-gauging family of topological phases, which are related by gauging their blood symmetries. With our approach, we construct the first explicit lattice realization of a nonabelian-symmetry-enriched topological phase -- the $S_3$ symmetry-enriched $\mathbb{Z}_2 \times \mathbb{Z}_2$ quantum-double phase. The approach further reveals the role of local excitations in SET phases and establishes their symmetry constraints.