Xingshan Cui 

Postdoctoral Fellow, Stanford Institute for Theoretical Physics, Stanford University

 

 

Abstract

The concept of group symmetries plays an important role in understanding topologically ordered states and phase transitions. In particular, 2d topologically ordered states with symmetries lead to symmetry fractionalization, defects, and gauging, which also have applications in topological quantum computing. Here we make an attempt to generalize group symmetries to what we call category symmetries utilizing the notion of linear Hopf monads, which is a far-reaching generalization of categorical Hopf algebras. We start with the mathematical side of the theory by defining category symmetries on modular categories which describe 2d topologically ordered states. A main goal is to develop a theory for extension and gauging category symmetries generalizing that of group symmetries. As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras.