Conformal defects break part of the symmetry of a bulk CFT. The broken Ward identities lead to very general sum rules on the defect CFT data as well as on the data of bulk operators in the presence of a defect. We call these sum rules "defect soft theorems", and they hold generally for defects which break conformal symmetry, flavor symmetry, or supersymmetry. In this talk I will focus on line defects for which we can rewrite the constraints in dispersive sum rule form, and show how the defect soft theorems impose constraints on the defect spectrum and OPE coefficients.
This talk is based on https://arxiv.org/pdf/2509.26561 with B. Girault and M. Paulos, and on work in progress with G. Bliard, J. Julius, M. Paulos and N. Suchel.
