The edge theories of 2D fermionic symmetry protected topological phases*

Abelian Chern-Simons theory, labeled by the so-called $K$ matrices, have been quite successful in characterizing and classifying Abelian fractional quantum hall effect(FQHE) as well as symmetry protected topological(SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fermionic SPT phases protected by Abelian total symmetry $G_f$ that is a central extension of bosonic symmetry $G_b$ by fermion parity symmetry $Z_2^f$.
In particular, we study the edge theories with the proper anomalous symmetry action on the edge fields for various examples.
Moreover, we also discuss the construction of edge theories with central charge $n-1$ for Type-III bosonic SPT phases protected by $(Z_n)^3$ symmetry.