Spin Impurities, Wilson Lines and Semiclassics

We consider line defects with large quantum numbers in conformal field theories. First, we consider spin impurities, both for a free scalar triplet and in the Wilson-Fisher $O(3)$ model. For the free scalar triplet, we find a rich phase diagram that includes a perturbative fixed point, a new nonperturbative fixed point, and runaway regimes. To obtain these results, we develop a new semiclassical approach. For the Wilson-Fisher model, we propose an alternative description, which becomes weakly coupled in the large spin limit. This allows us to chart the phase diagram and obtain numerous rigorous predictions for large spin impurities in $2+1$ dimensional magnets. Finally, we also study $1/2$-BPS Wilson lines in large representations of the gauge group in rank-1 $\mathcal{N}=2$ superconformal field theories. We contrast the results with the qualitative behavior of large spin impurities in magnets.