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...
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| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2017
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| Online Access: | http://hdl.handle.net/1721.1/110209 https://orcid.org/0000-0002-5874-581X |
| _version_ | 1811079962841579520 |
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| author | Lan, Tian Kong, Liang Wen, Xiao-Gang |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Lan, Tian Kong, Liang Wen, Xiao-Gang |
| author_sort | Lan, Tian |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T11:23:17Z |
| format | Article |
| id | mit-1721.1/110209 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:23:17Z |
| publishDate | 2017 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1102092022-09-27T19:13:39Z Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries Lan, Tian Kong, Liang Wen, Xiao-Gang Massachusetts Institute of Technology. Department of Physics Lan, Tian Wen, Xiao-Gang 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 Canada Ontario. Ministry of Research Harvard University. Center of Mathematical Sciences and Applications 2017-06-23T15:23:59Z 2017-07-02T05:00:03Z 2016-09 2016-04 2017-02-25T04:47:01Z Article http://purl.org/eprint/type/JournalArticle 0010-3616 1432-0916 http://hdl.handle.net/1721.1/110209 Lan, Tian, Liang Kong, and Xiao-Gang Wen. “Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries.” Communications in Mathematical Physics 351, no. 2 (September 22, 2016): 709–739. https://orcid.org/0000-0002-5874-581X en http://dx.doi.org/10.1007/s00220-016-2748-y Communications in Mathematical Physics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ Springer-Verlag Berlin Heidelberg application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Lan, Tian Kong, Liang Wen, Xiao-Gang Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries |
| title | Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries |
| title_full | Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries |
| title_fullStr | Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries |
| title_full_unstemmed | Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries |
| title_short | Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries |
| title_sort | modular extensions of unitary braided fusion categories and 2 1d topological spt orders with symmetries |
| url | http://hdl.handle.net/1721.1/110209 https://orcid.org/0000-0002-5874-581X |
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