Topological quantum field theories

Abstract

The aim of this work is to explain what a topological quantum field theory (TQFT) is and the relation between TQFTs in dimension 2 and Frobenius algebras. More precisely, it is shown that there exists an equivalence of categories between the category of 2-dimensional TQFTs on the one hand, and the category of commutative Frobenius algebras on the other. The work begins with a review of the basic concepts of category theory and, in particular, of the theory of symmetric monoidal categories, a categorical analogue of the abelian monoids. In the next two chapters the categories of cobordisms and of commutative Frobenius algebras are introduced. Both are examples of monoidal categories. Finally, in the last chapter TQFTs are defined and the above mentioned theorem proved. At the end of the work, some explicit examples of TQFTs in dimension 2 are described and the corresponding topological invariants of surfaces computed.