On enriching the Levin–Wen model with symmetry

Symmetry protected (SPT) and symmetry enriched topological (SET) phases of matter are of great interest in condensed matter physics due to new materials such as topological insulators. The Levin?Wen (LW) model for spin/boson systems is an important rigorously solvable model for studying 2D topological phases. The input data for the LW model is a unitary fusion category, but the same model also works for unitary multi-fusion categories. In this paper, we provide the details for this extension of the LW model, and show that the extended LW model is a natural playground for the theoretical study of SPT and SET phases of matter.