Dijkgraaf-Witten Theory for Surfaces

In this paper I describe Dijkgraaf-Witten theory as it applies to oriented surfaces. The first section provides physical motivation from the background of quantum field theory along with necessary definitions and ingredients of a topological quantum field theory. These include category theory, fiber bundles, covering spaces and bordisms. In the second section we make clear the connection between bordisms and principal G-bundles. We then describe the machinery of Dijkgraaf-Witten theory and explicitly construct a topological quantum field theory for the circle. This leads to a brief interlude on representation theory which we use to give constructions on elementary bordisms and this section concludes with an extension to constructions on oriented surfaces of any genus. The final section contains cloncluding remarks, and we see how two equivalent interpretations of a TQFT on a closed oriented manifold gives a nice reproduction of Mednykh's formula.