Weak quasi-Hopf algebras, quantum groups, and conformal field theory

We have two approaches to chiral CFT, conformal nets and vertex operator algebras. Conformal nets is a variant of algebraic quantum field theory based on operator algebras initiated in the 1960’s by Haag-Kastler. Under certain conditions, the representation category of a conformal net is a unitary modular fusion category (Kawahigashi-Longo-Mu ̈ger). Vertex operator algebras approach was initiated in the 1980’s by Borcherds and Frenkel-Lepowsky-Meurman. They also give rise to modular fusion categories under certain conditions (Huang-Lepowsky, Huang). One has a notion of unitarity for a vertex operator algebra and a connection between vertex operator algebras and conformal nets under a certain assumption, which includes many known examples are known. Then one would like to study unitarizability of representation category of VOA and connections with corresponding representation category of conformal nets. Important results in this direction have recently been obtained by Gui. In the talk we mostly focus on unitarizability. Our approach is based on fusion categories, and more specifically certain structures called weak quasi-Hopf (wqh) algebras. In particular, we construct unitary wqh algebras from the fusion categories associated to quantum groups at roots of unity associated to a simple complex Lie algebra g and relate them to unitarizability of rep. categories of affine VOAs.