There and Back Again: Bulk-to-Defect via Ward Identities

In conformal field theory, the presence of a defect may break the global symmetry, giving rise to defect operators such as the tilts. In this work, we derive integral identities that relate correlation functions involving bulk and defect operators -- including tilts -- to lower-point bulk-defect correlators, based on a detailed analysis of the Lie algebra of the symmetry group before and after the defect-induced symmetry breaking. As explicit examples, we illustrate these identities for the 1/2 BPS Maldacena-Wilson loop in $\mathcal{N}=4$ SYM and for magnetic lines in the $O(N)$ model in $d=4-\varepsilon$ dimensions. We demonstrate that these identities provide a powerful tool both to check existing perturbative correlators and to impose nontrivial constraints on the CFT data.