Consequences of symmetry-breaking on conformal defect data

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.

💡 Research Summary

The paper investigates how conformal defects, which break part of the bulk conformal and possibly global symmetry algebra, impose non‑trivial constraints on both defect and bulk data. Starting from the broken Ward identities associated with the non‑conservation of the stress tensor and global currents localized on the defect, the authors derive two universal integrated identities (equations (2.39) and (2.40)). These identities correspond to inserting one or two broken charges into an arbitrary correlation function containing bulk and/or defect operators.

By applying these identities to correlators that include the protected defect operators – the displacement operator D_a (breaking translations orthogonal to the defect), the tilt operator t_i (breaking internal global symmetries), and a supersymmetry‑breaking operator ψ_α dubbed “displacino” – the authors obtain a hierarchy of “defect soft theorems”. The single‑soft theorems give integral constraints on four‑point defect correlators involving a single protected operator, while the double‑soft theorems involve two such insertions and lead to new mixed sum rules (e.g. tilt‑displacement).

The paper then focuses on line defects (p = 1), where the defect conformal group reduces to SL(2,R) and correlators depend on a single cross‑ratio z. Using the “master functional” approach and Polyakov‑type bootstrap techniques, the integrated constraints are rewritten as dispersive sum rules. Positivity of the spectral density together with crossing symmetry yields rigorous bounds on the spectrum of defect operators. In particular, any line defect must contain an operator of dimension Δ belonging to either the singlet or traceless‑symmetric transverse spin sector with Δ ∈