Global Symmetry and Integral Constraint on Superconformal Lines in Four Dimensions

We study properties of point-like impurities preserving flavor symmetry and supersymmetry in four-dimensional ${\cal N} = 2$ field theories. At large distances, such impurities are described by half-BPS superconformal line defects. By working in the $\text{AdS}_2\times \text{S}^2$ conformal frame, we develop a novel and simpler way of deriving the superconformal Ward identities relating the various two-point functions of flavor current multiplet operators in the presence of the defect. We use these relations to simplify a certain integrated two-point function of flavor current multiplet operators that, in Lagrangian theories, can be computed using supersymmetric localization. The simplification gives an integral constraint on the two-point function of the flavor current multiplet superconformal primary with trivial integration measure in the $\text{AdS}_2\times \text{S}^2$ conformal frame. We provide several consistency checks on our Ward identities.