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 E. In this work, we propose that 2+1D topological/SPT orders with a fixed finite symmetry E are classified, up to E8 quantum Hall states, by the unitary modular tensor categories C over E and the modular extensions of each C. In the case C=E, we prove that the set Mext(E) of all modular extensions of E has a natural structure of a finite abelian group. We also prove that the set Mext(C) of all modular extensions of E, if not empty, is equipped with a natural Mext(C)-action that is free and transitive. Namely, the set Mext(C) is an Mext(E)-torsor. As special cases, we explain in detail how the group Mext(E) 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 Mext(E) under the symmetry-breaking processes and its relation to Witt groups.National Science Foundation (U.S.) (Grant No. DMR-1506475)National Science Foundation (U.S.) (Grant No.NSFC 11274192)Templeton Foundation (No. 39901)Canada. Industry CanadaOntario. Ministry of ResearchHarvard University. Center of Mathematical Sciences and Application