Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries

摘要

We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry G. The key is to use the so-called symmetric fusion category epsilon to describe the symmetry. Here, epsilon = sRep(Z(2)(f)) describing particles in a fermionic product state without symmetry, or epsilon = sRep(G(f)) [epsilon = Rep(G)] describing particles in a fermionic (bosonic) product state with symmetry G. Then, topological orders with symmetry epsilon are classified by nondegenerate unitary braided fusion categories over epsilon, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to p + i p fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the K =