On protected defect correlators in 3d $\mathcal{N}\ge 4$ theories

We study and compute supersymmetric observables for line defects in 3d $\mathcal{N}\ge 4$ theories. Our setup is a novel supersymmetric configuration involving line operators and local operators living on a linked circle. The algebra of the local operators is described by a topological quantum mechanics. For operators belonging to conserved current multiplets, we propose an exact formula for their correlation functions based on a Ward identity for integrated correlators. Our formula gives a general recipe to compute the bremsstrahlung function for any $\frac{1}{3}$-BPS lines in $\mathcal{N}=6$ SCFTs. We apply our relation to the $\frac{1}{2}$-BPS Wilson loop in the ABJM model, showing the validity of previous computations. Furthermore, our construction allows us to explore higher points correlators. As an example, we compute the two-point function of the stress tensor multiplet correlators in ABJM theory in the presence of the Wilson line. We also present some perturbative checks of our formulae.