Edge theories of 2D fermionic symmetry protected topological phases protected by unitary Abelian symmetries.

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. Comparing to the bosonic SPT, fermionic SPT with Abelian total symmetry $G_f$ has two new features: (1) it may support gapless majorana fermion edge fields, (2) some nontrivial bosonic SPT may be trivialized if there is a nontrivial $Z_2^f$ extention. By studying many examples, we show enough technical details to achieve the edge theories of almost all fermionic SPT phases with unitary Abelian $G_f$ except the so-called intrinsic interacting fermionic SPT phases. 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.