# Topological Quantum Field Theories – University of Copenhagen

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# Topological Quantum Field Theories

Specialeforsvar ved Klaus Albert Caning

Titel: Topological Quantum Field Theories

Abstract: This paper presents 2 constructions of topological quantum field theories. We construct the 2-dimensional TQFT that was first constructed in cite{fukuma1994lattice} by Fukuma, Hosono, Kawai (FKH); and we construct the Turaev-Viro 3-dimensional TQFT following the paper \cite{barrett1996invariants} by Barrett and Westbury. The first chapter is a brief introduction to some constructions and definitions, which are central in the following chapters. In the second chapter we construct a 2-dimensional TQFT from a semi-simple algebra. This chapter follows the interpretation of (FKH) by Baez and Wise \cite{BaezWise2004fuku} but provides more details. Chapter 3 introduce spherical cateogies, which are a special type of monoidal categories with duals. These will be impor-tant in the 3-dimensional construction. In chapter 4 we construct the Turaev-Viro TQFT from a strict finite semi-simple spherical category.\\ In each construction we define a map that send triangulated cobordisms to linear maps. A main part of this paper is spend proving that this map are independent of triangulation of the interior of cobordisms, and thus (a restriction of) this map is a functor from the category of cobordisms with triangulated boundary to the category of finite-dimensional vector spaces. This functor will depend on the triangulation on the boundary, so we have to compose it with a right inverse to the forgetful functor from the category of cobordisms with triangulation on the boundary to the category of cobordisms to obtian a TQFT. Each right inverse to the forgetful functor will be fully faithful and essentially surjective, hence the TQFT is determined up to isomorphism.

Vejleder:  Rune Hauseng
Censor:    Iver M. Ottosen, AAU