Topological Quantum Field Theories from Spherical Fusion Categories

Abstract

The main topic of this work is the Turaev-Viro construction of a 3-dimensopnal topological quantum field theory (TQFT). We begin by giving a description of spherical fusion categories an it's associated graphical calculus. Then, with a given spherical fusion category, we construct the Turaev-Viro TQFT on triangulated manifolds by labeling the triangulation and using the graphical calculus to assign either a morphism vector space or a given morphism in said space. Lastly, we construct two classes of examples of spherical fusion categories which appear from finite depth subfactors and planar algebras.