Boundary monopole bubbling and Macdonald kernels for non-minuscule 't Hooft lines in $\mathcal{N}=4$ SYM

We study half-BPS boundary 't Hooft lines of non-minuscule magnetic charge in four-dimensional $\mathcal{N}=4$ $U(N)$ super Yang–Mills theory with the regular Nahm-pole boundary condition. In contrast to minuscule charges, non-minuscule boundary 't Hooft lines receive monopole bubbling contributions. For all one-row charges $\lambda =(r,0,\dots ,0)$ we compute the bubbling-corrected defect half-index and identify the boundary 't Hooft operator with the spherical DAHA element $\mathbf{e}{h}_r(Y)\mathbf{e}$. Its difference-operator expansion gives the screened magnetic sectors, while the Macdonald kernel proves equality with the S-dual Neumann Wilson-line half-index. As a consequence we obtain the identity for all dominant magnetic charges of $U(2)$. The boundary fixed-point formula realizes the same coefficients and gives explicit non-minuscule examples in ranks two and three.