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

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 E to describe the symmetry. Here, E=sRep(Z₂ᶠ) describing particles in a fermionic product state without symmetry, or E=sRep(Gᶠ) [E=Rep(G)] describing particles in a fermionic (bosonic) product state with symmetry G. Then, topological orders with symmetry E are classified by nondegenerate unitary braided fusion categories over E, 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+1.0pt0exi1.0pt0exp fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the K=(arraycc -1& 0\\ 0& 2array) fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge c=1/4, whose only topological excitation has non-Abelian statistics with spin s=1/4 and quantum dimension d=1+√2.