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 mathcal{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 mathcal{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.
Highlights
Which gives rise to a conformal field theory (CFT)
For conformal surface defects preserving N = (0, 2) supersymmetry, we prove a universal relation between the b-anomaly and the ’t Hooft anomaly for the U(1)r symmetry
To illustrate better the defect dilaton effective action which plays a crucial role in the proof, we find it educational to consider the simplest example of a nontrivial defect renormalization group (RG) flow, namely that between the Neumann and Dirichlet boundary conditions of a free scalar field in 3d
Summary
We start by reviewing the proof of the defect b-theorem [15] using the spurious dilaton [5, 6]. To illustrate better the defect dilaton effective action which plays a crucial role in the proof, we find it educational to consider the simplest example of a nontrivial defect RG flow, namely that between the Neumann and Dirichlet boundary conditions of a free scalar field in 3d. We will explicitly derive the dilaton effective action in this case, and find agreement with known results about the defect b-anomalies from heat kernel methods at the defect fixed points. Compared to (2.3), we conclude for the boundary RG flow of a free 3d scalar field4 This is consistent with the fixed point values of the b-anomalies computed from heat kernel methods [15].
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