Communications in Mathematical Physics

, Volume 292, Issue 2, pp 343–389

On Classification of Modular Tensor Categories

Open Access
Article

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.

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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