Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries

A finite bosonic or fermionic symmetry can be described uniquely by a symmetric fusion category . In this work, we propose that 2+1D topological/SPT orders with a fixed finite symmetry are classified, up to quantum Hall states, by the unitary modular tensor categories over and the modular extensions of each . In the case , we prove that the set of all modular extensions of has a natural structure of a finite abelian group. We also prove that the set of all modular extensions of , if not empty, is equipped with a natural -action that is free and transitive. Namely, the set is an -torsor. As special cases, we explain in detail how the group recovers the well-known group-cohomology classification of the 2+1D bosonic SPT orders and Kitaev's 16 fold ways. We also discuss briefly the behavior of the group under the symmetry-breaking processes and its relation to Witt groups.