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Defect $a$-Theorem and $a$-Maximization

Conformal defects describe the universal behaviors of a conformal field theory (CFT) in the presence of a boundary or more general impurities. The coupled critical system is characterized by new conformal anomalies which are analogous to, and generalize those of standalone CFTs. Here we study the conformal $a$- and $c$-anomalies of four dimensional defects in CFTs of general spacetime dimensions greater than four. We prove that under unitary defect renormalization group (RG) flows, the defect $a$-anomaly must decrease, thus establishing the defect $a$-theorem. For conformal defects preserving minimal supersymmetry, the full defect symmetry contains a distinguished $U(1)_R$ subgroup. We derive the anomaly multiplet relations that express the defect $a$- and $c$-anomalies in terms of the defect (mixed) 't Hooft anomalies for this $U(1)_R$ symmetry. Once the $U(1)_R$ symmetry is identified using the defect $a$-maximization principle which we prove, this enables a non-perturbative pathway to the conformal anomalies of strongly coupled defects. We illustrate our methods by discussing a number of examples including boundaries in five dimensions and codimension-two defects in six dimensions. We also comment on chiral algebra sectors of defect operator algebras and potential conformal collider bounds on defect anomalies.