Categorical Aspects of Topological Quantum Field Theories

TL;DR

This thesis explores the categorical structures underlying topological quantum field theories, emphasizing their classification, graphical calculus, and a finite gauge group lattice model that illustrates their algebraic and topological properties across dimensions.

Contribution

It introduces a comprehensive categorical framework for TQFTs, including graphical calculus, classification results, and a finite gauge group lattice model applicable in all dimensions.

Findings

Classification of 2D TQFTs via Frobenius algebras

Explicit calculation of modular categories from the lattice model in 3D

Demonstration of the tower of algebraic structures across dimensions

Abstract

This thesis provides an introduction to the various category theory ideas employed in topological quantum field theory. These theories are 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 topological field theories 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, and compare 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.