报告题目：Levin-Wen Models and tensor categories

报 告 人：Lan Tian (兰天）金沙总站6165地址物理系

报告时间：Lecture 1: Jan 3, 14:00-15:30，

Lecture 2: Jan 5, 14:00-15:30

报告地点：高等研究院 科学馆322报告厅

报告摘要：I will give an introduction to the mathematical theory of tensor category and its applications to Levin-Wen models, which describe a large family of non-chiral topological orders. As we know, the representation theory of groups can be used to classify phases, excitations or particles when the system adopts some kind of symmetry. We found symmetry is, however, not enough in some cases, such as topological order in which the relevant mathematical structures are tensor categories.

I will first start with some basic things of general category theory. Then I will introduce the notion of a tensor category and a module category over it. There are a lot of structures that one can add to a tensor category. I will introduce a few of them, which are relevant to us, such as the unitarity, semi-simpleness, rigidity, pivotal and spherical structures. These structures enable us to express the tensor category in terms of directed labeled graphs, which also appeared in Levin-Wen Models.

The representation theory gives us an approach to describe the excitations in Levin-Wen Model. We don't deal with the excitation directly; instead, we study the "boundary" of the excitation. With the tensor category structures, these "boundaries" form algebras and modules over algebras. We can classify the excitations by studying these algebras. This also corresponds to structures of the tensor category and its module category.