Topological defects in conformal field theory: from vertex operator algebras to conformal nets

The concept of global symmetry in quantum field theory has been generalised in many different directions in the last ten years. Many of these generalisations are based on the concept of topological defects supported on submanifolds of the space-time. The formal structure underlying topological defects is best understood in the context of conformal field theories in two dimensions, where the only relevant submanifolds are lines. The goal of the thesis is to study the concept of topological defect lines within two alternative axiomatizations of two dimensional conformal field theory. The first setup applies to theories on a two-dimensional space-time with Euclidean signature, and is based on the theory of vertex operator algebras and its representations. The second approach is relevant for theories in Lorentzian 1+1 dimensional space-time, and is based on conformal nets, i.e. von Neumann algebras of local observables, subject to constraints related to conformal symmetry. The equivalence of the two approaches, through Wick rotation, is non-trivial, and has been proved only for certain classes of theories. The thesis aims at comparing the description of topological defects in these two different approaches in some simple examples of conformal field theories.