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

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Abstract

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

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Authors and Affiliations

  1. Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada

    Tian Lan & Xiao-Gang Wen

  2. Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Tian Lan

  3. Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, 03824, USA

    Liang Kong

  4. Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, 02138, USA

    Liang Kong

  5. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Xiao-Gang Wen

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  1. Tian Lan
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  2. Liang Kong
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Correspondence to Tian Lan.

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Communicated by Y. Kawahigashi

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Lan, T., Kong, L. & Wen, XG. Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries. Commun. Math. Phys. 351, 709–739 (2017). https://doi.org/10.1007/s00220-016-2748-y

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