Abstract

Modular fusion categories (MFCs) arise naturally in many areas of mathematics and physicsAssociated with an MFC is a pair of complex matrices, called modular data, which are arguably themost important invariants of an MFC. The modular data of an MFC generate some uncanonicalcongruence representations of SL(2,Z). In this talk, we will discuss how modular data could bereconstructed or discovered from congruence representations of SL(2.Z). The talk is based on ajoint work with Eric Rowell, Zhenghan Wang and Xiao-Gang Wen.