Exact Results in Quantum Theory

Many different tools are available when studying quantum field theories: one can start from a classical action, one can constrain the correlation functions via symmetries, one can identify indices or topological subsectors which remain invariant under deformations, etc. The relations between these methods are often indirect and conjectural. In the example of two-dimensional field theories, I would like to explain how one-dimensional defect lines appear from various point of view, and how they can be valuable in linking different descriptions of quantum field theory.