Exact methods for defects in conformal field theory

The defect operators admitted by a given quantum field theory (QFT) contain crucial information. E.g in 4d gauge theories some defects play the role of order parameters, classifying phases of the theory. Defects are also omnipresent in real-world laboratories. E.g. real systems typically have impurities and defects, which may change their properties. Since QFT can be used to describe such systems in the continuum limit, it is essential to systematically understand defects in QFT. This thesis explores defects in d-dimensional conformal field theories (CFT). CFTs arise naturally at fixed points of renormalisation group (RG) flows and describe real physical systems at criticality. We focus on p-dimensional defects, with p ≤ d-1, that preserve some of the system's conformal invariance. Conformal defects give rise to defect-localised contributions to the CFT's Weyl anomaly. Their coefficients are often called defect central charges. They control many physical observables, and obey interesting bounds, constraints, and relations, partially characterising the defect. We report novel and original results about conformal defects and their central charges across dimensions. Our results are exact and apply to large classes of defects. Firstly, we determine the form of the defect Weyl anomaly of a p=4 conformal defect in a CFT of arbitrary co-dimension q=d-4. We show how some of the new defect central charges appear in physical observables, and discuss bounds that they need to obey. We then illustrate these results with a set of simple, yet non-trivial, examples of defects in free CFTs. Using existing methods available in free field theories, we compute various correlation functions exactly for arbitrary p, and demonstrate how to extract defect central charges when p=2 and p=4. Moreover, we study novel defect RG flows which are found to obey monotonicity theorems. Finally, we develop novel techniques to compute central charges for superconformal defects in a large class of interacting superconformal field theories. Our methods rely on supersymmetric localisation, and thus are non-perturbative in the coupling constants. We illustrate our techniques in numerous examples