Communications in Mathematical Physics

, Volume 292, Issue 2, pp 343–389

On Classification of Modular Tensor Categories

Open AccessArticle

DOI: 10.1007/s00220-009-0908-z

Cite this article as:
Rowell, E., Stong, R. & Wang, Z. Commun. Math. Phys. (2009) 292: 343. doi:10.1007/s00220-009-0908-z


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.

Download to read the full article text

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.