Sector Theory of Levin-Wen Models II : Fusion and Braiding

Abstract

This is the continuation of our study of the Levin-Wen model based on an arbitrary unitary fusion category $\mathcal{C}$ on the infinite plane. The ground state of the Levin-Wen model hosts anyonic excitations whose fusion and braiding properties are captured by the associated braided ${\mathrm{C}}^{\ast }$-tensor category of superselection sectors $\mathsf{SSS}$. By constructing explicit isomorphisms between the fusion spaces of $\mathsf{SSS}$ and those of the Drinfeld center $Z(\mathcal{C})$, we show that these two categories have isomorphic $F$- and $R$-symbols. It follows that the full subcategory of finite sectors is unitarily braided monoidally equivalent to the Drinfeld center, $${\mathsf{SSS}}_f\simeq Z(\mathcal{C}).$$ This provides the first complete characterisation of the category of superselection sectors for a class of two-dimensional lattice models supporting anyons with non-integer quantum dimensions.

49 pages, lots of figures