Surface defects in quantum field theories usually reveal their Weyl and gravitational anomalies as contact terms exactly on the defect. The new study shows that these anomalies can also be read out from a distance using integrated bulk stress‑tensor correlators. This result follows from defect Ward identities and a minimal‑subtraction regulator that removes short‑distance singularities. The derived sum rules express the anomaly coefficients – symbols B and κ – as integrals over bulk points that are far from the defect. For three‑dimensional conformal field theories with boundaries the authors prove a universal lower bound on the boundary Weyl anomaly: b is larger than −(3π²/4) times the displacement coefficient c_D. The analysis, however, applies strictly to unitary defects in spacetimes of dimension less than six; higher‑dimension operators would cause non‑integrable divergences, and the minimal‑subtraction scheme is essential for the formulas to hold.

This advances our theoretical understanding of how anomalies are encoded in quantum field theories, but the methods rely on abstract mathematical identities and do not yet translate into experimental measurements. The bound on the boundary anomaly also depends on the assumption that no gapless degrees of freedom are decoupled from the boundary.