Simple-current algebra constructions of 2+1D topological orders

Abstract

Self-consistent (non-)abelian statistics in 2+1D are classified by modular tensor categories (MTC). In recent works, a simplified axiomatic approach to MTCs, based on fusion coefficients $N^{ij}_k$ and spins $s_i$, was proposed. A numerical search based on these axioms led to a list of possible (non-)abelian statistics, with rank up to $N=7$. However, there is no guarantee that all solutions to the simplified axioms are consistent and can be realised by bosonic physical systems. In this paper, we use simple-current algebra to address this issue. We explicitly construct many-body wave functions, aiming to realize the entries in the list (\ie realize their fusion coefficients $N^{ij}_k$ and spins $s_i$). We find that all entries can be constructed by simple-current algebra plus conjugation under time reversal symmetry. This supports the conjecture that simple-current algebra is a general approach that allows us to construct all (non-)abelian statistics in 2+1D. It also suggests that the simplified theory based on $(N^{ij}_k,s_i)$ is a classifying theory at least for simple bosonic 2+1D topological orders (up to invertible topological orders).