Construction of Two-Dimensional Topological Field Theories

In this thesis I thoroughly review the construction of topological quantum field theories (TQFT), with particular emphasis on the two dimensional examples. In the first part of the thesis, we consider the mathematical foundations of TQFT's by introducing notions such as categories, cobordisms and Frobenius algebras. In this setting, an n-dimensional topological quantum field theory is a functor between the categories of cobordisms and vector spaces. In the two dimensional case, we show that to give a TQFT is equivalent to giving a Frobenius algebra. In the last part of the thesis, we investigate the Dijkgraaf-Witten model, which is a TQFT over principal G-bundles with G being a finite group