Defects in Conformal Field Theory with O(N) Symmetry

Conformal field theories (CFTs) are essential tools in quantum field theory and statistical physics, providing a framework for understanding critical phenomena and renormalisation group flows, with further applications to other formal areas of theoretical physics. In recent years, extended excitations within these theories–such as conformal defects–have gained prominence as powerful probes of quantum field theories, offering insights into dualities and symmetries. Moreover, conformal defects are widely employed in statistical physics, where they model the critical behaviour of systems with impurities, finite-size effects, or interfaces. This thesis explores various aspects of the role that conformal defects play in CFTs. It begins by reviewing the fundamental concepts of CFTs, conformal defects, defect renormalisation group flows, and bootstrap methods. The focus then shifts to the study of specific conformal defects in CFTs with O(N) internal symmetry, though the techniques employed are readily applicable to more general contexts. The localised magnetic field and the magnetic impurity in the critical O(N) model are introduced and analysed extensively using the analytic bootstrap. Next, the concept of transdimensional defects is explored through concrete examples in free O(N) field theory and the critical O(N) model. Finally, several constructions of conformal defects in generalised free field theories and long-range O(N) models are examined