Surface Defect, Anomalies and $b$-Extremization

TL;DR

This paper investigates surface defect anomalies in conformal field theories across various dimensions, establishing the $b$-extremization principle and deriving the defect dilaton effective action, with applications to strongly coupled theories.

Contribution

It introduces the $b$-extremization principle for superconformal $U(1)_r$ symmetry and derives the defect dilaton effective action, advancing the understanding of defect anomalies in CFTs.

Findings

Proves the $b$-theorem for defect RG flows.

Establishes a universal relation between $b$-anomaly and 't Hooft anomaly.

Determines $b$-anomalies for surface defects in 3d, 4d, and 6d SCFTs.

Abstract

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