The Classification of Two Dimensional topological Field Theories

TL;DR

This paper classifies two-dimensional extended topological quantum field theories using the Cobordism Hypothesis, constructs an example with Dijkgraaf-Witten theory, and proves its uniqueness among framed theories.

Contribution

It provides explicit classifications and constructions of 2D extended TQFTs, including a detailed example with Dijkgraaf-Witten theory and a uniqueness proof.

Findings

Classifies framed and oriented 2D extended TQFTs using the Cobordism Hypothesis.

Constructs 2D Dijkgraaf-Witten theory as an explicit example.

Shows Dijkgraaf-Witten theory is the unique extended framed 2D TQFT.

Abstract

The goal of this paper is to introduce some of the major ideas behind extended topological quantum field theories with an emphasis on explicit examples and calculations. The statement of the Cobordism Hypothesis is explained and immediately used to classify framed and oriented extended two dimensional topological quantum field theories. The passage from framed theories to oriented theories is equivalent to giving homotopy fixed points of an $SO(n)$ action on the space of field theories. This paper then constructs extended two dimensional Dijkgraaf-Witten theory (also called finite gauge theory) as an example of a two dimensional extended field theory by assigning invariants at the level of points and extending up. Finally, it is concluded that Dijkgraaf-Witten theory is the only example of an extended framed two dimensional topological quantum field theory by showing that any field theory is equivalent to Dijkgraaf-Witen theory for some cyclic group.