Topological Field Theories with Defects

Abstract

Topological field theories are important due to their numerous applications in theoretical physics and modern mathematics. A useful way to study topological field theories is by using category theory. In this dissertation we considered different types of topological filed theories in dimension two. R.Dijkraaf found that the category 2Cobc of surfaces is free on a commutative Frobenius algebra. G.Segal and G.Moore showed that the category 2Cobo-c of open surfaces is described by a Cardy pair. We extended these results to the following: the category 2Cob2o of two coloured ribbon graphs is free on a pair of Frobenius algebras and a Frobenius bimodule. In a separable case, a Frobenius bimodule and its dual form a Morita context. We also found that the category 2Cob2o-c of two coloured open surfaces is the free monoidal category on a bimodule btween two Cardy pairs. We moved then to the category 2Cobd of surfaces with defect of codimension 1. We gave a dicribtion of Cardy correspondences in this category and ended up with a conjecture stating that the category 2Cobd is free on a Cardy correspondence in a separable case, i.e. all Frobenius algebras are separable. We gave an example of a 2 dimensional topological field theory which we call a set-theoretic 2 dimensional topological field theory. The target of this functor is the category of correspondences Corr. We showed that any Frobenius monoid in Corr is a multi-fusion ring.