# iTHEMS Colloquium given by Professor Yasuyuki Kawahigashi

iTHEMS Colloquium was held on 4 Oct. 2019, inviting Prof. Yasuyuki Kawahigashi from the University of Tokyo (he is also a senior visiting scientist of iTHEMS). The title was "Topological phases of matter and operator algebras", and he explained his researches on operator algebra and their relations with theoretical physics. The central subject of the talk was modular tensor category (MTC). A subfactor (an inclusion of simple von Neumann algebra) has the symmetry of a tensor category, in an analogous way to the Galois theory of fields. MTC is an important class of tensor category having an interesting commutativity of the tensor product, and is also a useful tool to describe the anyonic statistics of quasi-particles. A powerful source of MTC is conformal field theory, and a comparison of two mathematical approaches (conformal net and vertex operator algebra) is a recent hot topic. In this talk Prof. Kawahigashi displayed three interesting examples in which a mathematical research, originated from its own motivation, helps the study of mathemtaical physics later. The first is a conjecture by Lan-Wang-Wen about the mathematical formulation of gapped domain wall, which is rejected by a knowledge in subfactor theory. The second is alpha-induction in the theory of MTC, which is now understood as a mathematical formulation of anyon condensation. The third is a work by Bultinck et. al. on tensor network, which turned out to be parallel with the subfactor theory (such as tube algebra and flat connection) studied by Ocneanu, Haagerup and Prof. Kawahigashi in 80-90's. Because the talk was organized coherently, we can enjoy a lot of topics in 90 minutes.