  In CFTs, the partition function of a line defect with a cusp depends
logarithmically on the size of the line with an angle-dependent coefficient:
the cusp anomalous dimension. In the first part of this work, we study the
general properties of the cusp anomalous dimension. We relate the small cusp
angle limit to the effective field theory of defect fusion, making predictions
for the first couple of terms in the expansion. Using a concavity property of
the cusp anomalous dimension we argue that the Casimir energy between a line
defect and its orientation reversal is always negative ("opposites attract").
We use these results to determine the fusion algebra of Wilson lines in
$\mathcal{N}=4$ SYM as well as pinning field defects in the Wilson-Fisher fixed
points. In the second part of the paper we obtain nonperturbative numerical
results for the cusp anomalous dimension of pinning field defects in the Ising
model in $d=3$, using the recently developed fuzzy-sphere regularization. We
also compute the pinning field cusp anomalous dimension in the $O(N)$ model at
one-loop in the $\varepsilon$-expansion. Our results are in agreement with the
general theory developed in the first part of the work, and we make several
predictions for impurities in magnets.
