The Classification of Two Dimensional topological Field Theories

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.