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

Abstract

Self-consistent (non-)Abelian statistics in 2+1 dimensions (2+1D) are classified by modular tensor categories (MTCs). In recent works, a simplified axiomatic approach to MTCs, based on fusion coefficients N[ij over 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 realized 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 (i.e., realize their fusion coefficients N[ij over k] and spins s[subscript 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 over k], s[subscript i]) is a classifying theory at least for simple bosonic 2+1D topological orders (up to invertible topological orders).