Surface defect, anomalies and b-extremization

Quantum field theories (QFT) in the presence of defects exhibit new types anomalies which play an important role constraining defect dynamics and renormalization group (RG) flows. Here we study surface their conformal (CFT) general spacetime dimensions. When is conformal, it characterized by a $b$-anomaly analogous to $c$-anomaly 2d CFTs. The $b$-theorem states that $b$ must monotonically decrease under RG flows was proven coupling spurious dilaton. We revisit proof deriving explicitly dilaton effective action for flow free scalar theory. For preserving ${\cal N}=(0,2)$ supersymmetry, prove universal relation between 't Hooft anomaly $U(1)_r$ symmetry. also establish $b$-extremization principle identifies superconformal symmetry from Together they provide powerful tool extract strongly coupled defects. To illustrate our method, determine $b$-anomalies number 3d, 4d 6d SCFTs. comment on manifestations these correlation functions.