Modular tensor categories from SL(2,Z) representations

The notion of modular tensor categories (MTC) emerges from the study of rational conformal field theory and topological quantum field theory (TQFT). They are closely related to various areas of research, including representation theory, low-dimensional topology and topological phases of matter. Of vital importance is the congruence representation of SL(2, Z) associated to an MTC, which not only reflects the structure of the MTC itself and the corresponding TQFT, but also has rich arithmetic properties such as Galois symmetry and integrality. Therefore, it is natural to study MTCs from the perspective of SL(2,Z) representations. 

In this talk, we will give a quick introduction to MTCs with an emphasis on their arithmetic properties. Then we will talk about the classification of MTCs from SL(2,Z) representations, including the recent result on the classification of transitive modular categories.