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 Nkij and spins si. 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.
For example, we find that, up to invertible
p+ip fermionic topological orders, there
are only four fermionic topological orders with one non-trivial topological
excitation: (1) the K=(−1002)
fractional quantum Hall state, (2) a Fibonacci bosonic topological order
214/5B stacking with a fermionic product state, (3) the time-reversal
conjugate of the previous one, (4) a primitive fermionic topological order that
has a chiral central charge c=41, whose only topological excitation has
a non-abelian statistics with a spin s=41 and a quantum dimension
d=1+2. We also proposed a categorical way to classify 2+1D invertible
fermionic topological orders using modular extensions.Comment: 23 pages, 8 table