Operator algebras, bi-unitary connections and tensor networks

Introduction

Tensor categories have emerged as a new type of "quantum symmetry" generalizing classical group symmetry in various fields of mathematics and physics such as quantum groups, quantum topological invariants, quantum information, vertex operator algebras, conformal field theory and topological order in condensed matter physics. The Jones theory of subfactors in operator algebras give a powerful method to study such symmetries. A bi-unitary connection is a tool to describe such tensor categories using finite dimensional unitary matrices and particularly suited to study tensor networks in 2-dimensional topological order. I will present this theory without assuming knowledge on operator algebras.

Lecturer Intro

Yasuyuki Kawahigashi is a Professor at the University of Tokyo. He was an invited speaker at ICM 2018. His specialty is operator algebra theory, especially subfactor theory in the theory of von Neumann algebras, and algebraic quantum field theory, as well as these and other fields (quantum groups, three-dimensional topology, conformal field theory, solvable lattice models, vertex operator algebras). He is in editorial boards for many journals such as Comm. Math. Phys.