Modular categories and the Witt group

          @misc{ pirsa_PIRSA:17070064,
            doi = {10.48660/17070064},
            url = {https://pirsa.org/17070064},
            author = {Mueger, Michael},
            keywords = {Quantum Foundations, Quantum Information},
            language = {en},
            title = {Modular categories and the Witt group},
            publisher = {Perimeter Institute},
            year = {2017},
            month = {jul},
            note = {PIRSA:17070064 see, \url{https://pirsa.org}}
          }
          

Abstract

The aim of this talk is to give an introduction to modular categories, touching both basics and recent developments. I will begin with a quick reminder concerning tensor categories, in particular braided and symmetric ones, and notions like duality, fusion and spherical categories. We'll meet algebras in tensor categories, categories of modules, module categories and their connection. I will then focus on modular categories and some basic structure theory. We will consider two ways of obtaining modular categories: modularization and the Drinfeld center. (The important third one, quantum groups at root of unity, is too complicated to be discussed in any depth.) The Drinfeld center will be used to define Witt equivalence of modular categories and the Witt group. Several equivalent characterization of Witt equivalence, using module categories, will be discussed. The Witt group will be crucial for any (future) classification of modular categories, as well as for application to physics in condensed matter physics and conformal field theory.