On Classification of Modular Tensor Categories

We classify all unitary modular tensor categories (UMTCs) of rank a parts per thousand currency sign 4. There are a total of 35 UMTCs of rank a parts per thousand currency sign 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 a parts per thousand currency sign 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.