Communications in Mathematical Physics

, Volume 292, Issue 2, pp 343–389 | Cite as

On Classification of Modular Tensor Categories

  • Eric Rowell
  • Richard Stong
  • Zhenghan WangEmail author
Open Access


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.


Galois Group Fusion Rule Galois Extension Tensor Category Fusion Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



ZW thanks Nick Read for his insightful Comments on earlier versions.

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationU.S.A.
  2. 2.Center for Communications ResearchSan DiegoU.S.A.
  3. 3.Microsoft Station QUniversity of CaliforniaSanta BarbaraU.S.A.

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