On Monopole Bubbling Contributions to 't Hooft Loops

Monopole bubbling contributions to supersymmetric 't Hooft loops in 4d $\mathcal{N}=2$ theories are computed by SQM indices. As recently argued, those indices are hard to compute due to the presence of Coulomb vacua that are not captured by standard localization techniques. We propose an algorithmic method to compute the full bubbling contributions that circumvent this issue, by considering SQM with more matter fields and isolating the bubbling terms as residues in flavor fugacities. The enlarged SQMs are read from brane configurations realizing the bubbling sector of a given 't Hooft loop. We apply our technique to loop operators in $\mathcal{N}=2$ conformal SQCD theories. In addition we embed this discussion in the larger setup of a 5d-4d system interacting along a line, associated to the brane systems previously discussed. The bubbling terms arise from residues of specific instanton sectors of 5d line operators in this context.