Weyl Anomalies of Four Dimensional Conformal Boundaries and Defects

Motivated by questions about quantum information and classification of quantum field theories, we consider Conformal Field Theories (CFTs) in spacetime dimension $d\geq 5$ with a conformally-invariant spatial boundary (BCFTs) or $4$-dimensional conformal defect (DCFTs). We determine the boundary or defect contribution to the Weyl anomaly using the standard algorithm, which includes imposing Wess-Zumino consistency and fixing finite counterterms. These boundary/defect contributions are built from the intrinsic and extrinsic curvatures, as well as the pullback of the ambient CFT's Weyl tensor. For a co-dimension one boundary or defect (i.e. $d=5$), we reproduce the $9$ parity-even terms found by Astaneh and Solodukhin, and we discover $3$ parity-odd terms. For larger co-dimension, we find $23$ parity-even terms and $6$ parity-odd terms. The coefficient of each term defines a "central charge" that characterizes the BCFT or DCFT. We show how several of the parity-even central charges enter physical observables, namely the displacement operator two-point function, the stress-tensor one-point function, and the universal part of the entanglement entropy. We compute several parity-even central charges in tractable examples: monodromy and conical defects of free, massless scalars and Dirac fermions in $d=6$; probe branes in Anti-de Sitter (AdS) space dual to defects in CFTs with $d \geq 6$; and Takayanagi's AdS/BCFT with $d=5$. We demonstrate that several of our examples obey the boundary/defect $a$-theorem, as expected.