Consequences of symmetry-breaking on conformal defect data

Abstract: Conformal defects spontaneously break part of the symmetry algebra of a bulk CFT. We show that the broken Ward identities imply very general sum rules on the defect CFT data as well as on the DOE data of bulk operators, which we call defect soft theorems. Our derivation is elementary, allowing us to easily reproduce and generalize constraints on displacement and tilt operators previously obtained in the literature as well as a plethora of new ones, including constraints on bulk-defect correlation functions. For line defects we rewrite constraints in dispersive sum rule form, showing they lead to exact, optimal bounds on the OPE data of the defect. We test these sum rules in concrete perturbative examples, finding perfect agreement with existing calculations and making new predictions for various dCFT data.