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 $${\mathcal{E}}$$E. In this work, we propose that 2+1D topological/SPT orders with a fixed finite symmetry $${\mathcal{E}}$$E are classified, up to $${E_8}$$E8 quantum Hall states, by the unitary modular tensor categories $${\mathcal{C}}$$C over $${\mathcal{E}}$$E and the modular extensions of each $${\mathcal{C}}$$C. In the case $${\mathcal{C}=\mathcal{E}}$$C=E, we prove that the set $${\mathcal{M}_{ext}(\mathcal{E})}$$Mext(E) of all modular extensions of $${\mathcal{E}}$$E has a natural structure of a finite abelian group. We also prove that the set $${\mathcal{M}_{ext}(\mathcal{C})}$$Mext(C) of all modular extensions of $${\mathcal{E}}$$E, if not empty, is equipped with a natural $${\mathcal{M}_{ext}(\mathcal{C})}$$Mext(C)-action that is free and transitive. Namely, the set $${\mathcal{M}_{ext}(\mathcal{C})}$$Mext(C) is an $${\mathcal{M}_{ext}(\mathcal{E})}$$Mext(E)-torsor. As special cases, we explain in detail how the group $${\mathcal{M}_{ext}(\mathcal{E})}$$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 $${\mathcal{M}_{ext}(\mathcal{E})}$$Mext(E) under the symmetry-breaking processes and its relation to Witt groups.

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