The boundary of 2+1D fermionic topological orders

We propose a systematic approach to investigate the boundary of 2+1D abelian fermionic topological orders, in which S and T matrices are not well-defined. The trick is to realize the fermionic system "on the top" of the Z2 topological order, motivated by the hierarchical construction of fractional quantum Hall states. We explicitly show the construction of the K matrix as well as the correspondence between equivalence classes of excitations. Within the new framework, we describe the gapped boundaries via modular covariant partition functions straightforwardly. Yet, some subtleties have to be considered when we convert the solutions back to the original fermionic systems. An algorithm to find such partition functions is also discussed. Along the way, we raised a conjecture, "S+T-2 always have a rational basis."