Abstract: Rational conformal field theory and (2+1)-d topological field theory naturally give rise representations of SL(2,Z), and these reprentations have the remarkable property that their kernels are congruence subgroups of SL(2,Z). The congruence kernel property has long been conjectured by physicists, and was finally proved via the study of algebraic properties of modular tensor categories. In this talk, we will review the notion of modular tensor categories and the classification of Nobs and Wolfart on congruence representations of SL(2,Z). Then we will illustrate how these congruence representations help us in the study of modular tensor categories and their topological field theories.
