Boostrapping line defects with $O(2)$ symmetry

Line defects play an important role in our understanding of QFTs, explaining interesting phenomena in both condensed matter physics and high-energy theories, and giving access to new data and observables. I will discuss recent work in which we explore 1d conformal line defects with an additional O(2) symmetry using the numerical bootstrap. The starting point is an agnostic approach, where we perfom a systematic bootstrap study of correlation functions between two canonical defect operators: the displacement and the tilt. We then move on to study two specific defects: a monodromy line defect and a localized magnetic field line defect. I will highlight the results of the latter one, where we found a series of intriguing cusps which we investigate