A theory of 2+1D fermionic topological orders and fermionic/bosonic topological orders with symmetries

Understand

We propose that, up to invertible topological orders, 2+1D fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry $G$ are classified by non-degenerate unitary braided fusion categories (UBFC) over a symmetric fusion category (SFC); the SFC describes a fermionic product state without symmetry or a fermionic/bosonic product state with symmetry $G$, and the UBFC has a modular extension.

• We developed a simplified theory of non-degenerate UBFC over a SFC based on the fusion coefficients $N^{ij}_k$ and spins $s_i$.
• This allows us to obtain a list that contains all 2+1D fermionic topological orders (without symmetry).
• We find explicit realizations for all the fermionic topological orders in the table.