Representations of vertex operator algebras and braided finite tensor categories

TL;DR

This paper reviews the progress over twenty years in constructing braided finite tensor categories from vertex operator algebras, highlighting challenges, methods, applications, and open problems.

Contribution

It summarizes key developments, identifies main difficulties, and discusses solutions and applications in the theory of vertex operator algebras and tensor categories.

Findings

Progress in constructing braided finite tensor categories from vertex operator algebras

Identification of main difficulties and methods to overcome them

Discussion of applications and open problems

Abstract

We discuss what has been achieved in the past twenty years on the construction and study of a braided finite tensor category structure on a suitable module category for a suitable vertex operator algebra. We identify the main difficult parts in the construction, discuss the methods developed to overcome these difficulties and present some further problems that still need to be solved. We also choose to discuss three among the numerous applications of the construction.