't Hooft Defects and Wall Crossing in SQM

In this paper we study the contribution of monopole bubbling to the expectation value of supersymmetric 't Hooft defects in Lagrangian theories of class $\mathcal{S}$ on $\mathbb{R}^3\times S^1$. This can be understood as the Witten index of an SQM living on the world volume of the 't Hooft defect that couples to the bulk 4D theory. The computation of this Witten index has many subtleties originating from a continuous spectrum of scattering states along the non-compact vacuum branches. We find that even after properly dealing with the spectral asymmetry, the standard localization result for the 't Hooft defect does not agree with the result obtained from the AGT correspondence. In this paper we will explicitly show that one must correct the localization result by adding an extra term to the standard Jeffrey-Kirwan residue formula. This extra term accounts for the contribution of ground states localized along the non-compact branches. This extra term restores both the expected symmetry properties of the line defect expectation value and reproduces the results derived using the AGT correspondence.