A weak quasi-Hopf algebra approach to unitary structures in conformal field theory

Algebraic quantum field theory is an approach to QFT based on operator algebras initiated in the 1960’s by Haag-Kastler. I In this approach, chiral conformal field theory is described by a family of operator algebras A(I) on a given Hilbert space parameterized by open intervals on the circle and subject to certain axioms. By Doplicher-Haag-Roberts theory, a representation of these operator algebras is described by an endomorphism, and composition of endomorphisms defines the structure of a strict tensor category. Moreover this category has a braiding and a natural unitary structure. Kawahigashi-Longo-M¨uger give a condition on finiteness of the Jones index of a certain inclusion of factors which imply that the category is a modular fusion category. Vertex operator algebras give another approach to chiral conformal field theory, based on the notion of vertex operators in an algebraic framework. It was initiated in the 1980’s by Borcherds and Frenkel-Lepowsky-Meurman. Let V be a vertex operator algebra. Under certain conditions, Rep(V) is a modular fusion category (Huang-Lepowsky, Huang). Carpi-Kawahigashi-Longo-Weiner introduce a notion of unitarity for a vertex operator algebra and give a connection between vertex operator algebras and conformal nets under a certain assumption, called strong locality. They develop a sufficient condition which applies to many known examples, including the affine vertex operator algebras. Then one would like to study unitarity 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 via a direct approach. The aim of my talk is to discuss these questions, and we focus on unitarizability, in the framework of abstract fusion categories. Our approach is based on certain structures called weak quasi-Hopf algebras (wqh) that may be regarded as general group like structures among fusion categories. Moreover, we shall see that wqh arise in a natural way for the fusion categories associated to quantum groups at roots of unity associated to a general simple complex Lie algebra g.