Abstract
We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular S-matrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 non-trivial prime UMTCs are given in Sect. 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Sect. 6.
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ZW thanks Nick Read for his insightful Comments on earlier versions.
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Communicated by Y. Kawahigashi
The first author is partially supported by NSA grant H98230-08-1-0020.
The second and third authors are partially supported by NSF FRG grant DMS-034772.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Rowell, E., Stong, R. & Wang, Z. On Classification of Modular Tensor Categories. Commun. Math. Phys. 292, 343–389 (2009). https://doi.org/10.1007/s00220-009-0908-z
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DOI: https://doi.org/10.1007/s00220-009-0908-z