文档介绍：4
Abstract
This thesis provides an introduction to the various category theory ideas employed in topological quan-
tum field theory (TQFT). These theories can be viewed as symmetric monoidal functors from topological
cobordism categories into the category of vector spaces. In two dimensions, they are classified by Frobenius
algebras. In three dimensions, and under certain conditions, they are classified by modular categories. These
are special kinds of categories in which topological notions such as braidings and twists play a prominent
role. There is a powerful graphical calculus available for working in such categories, which may be regarded
as a generalization of the Feynman diagrams method familiar in physics. This method is introduced and the
necessary algebraic structure is graphically motivated step by step.
A large subclass of two-dimensional TQFTs can be obtained from a lattice gauge theory construction
using triangulations. In these theories, the gauge group is finite. This construction is reviewed, from both
the original algebraic perspective as well as using the graphical calculus developed in the earlier chapters.
This finite gauge group toy model can be defined in all dimensions, and has a claim to being the simplest
non-trivial quantum field theory. We take the opportunity to show explicitly the calculation of the modular
category arising from this model in three dimensions, pare this algebraic data with the corresponding
data in two dimensions, computed both geometrically and from triangulations. We use this as an example
to introduce the idea of a quantum field theory as producing a tower of algebraic structures, each dimension
related to the previous by the process of categorification.
Uittreksel
Deze scriptie vormt een inleiding in het gebruik van de categorie theorie in topologische kwantumvelden
theorie (TQFT). TQFT’s kunnen worden beschouwd als symmetrische monoidale functors van topologische
cobordisme-categorie¨ennaar de categ