Weyl anomalies of four dimensional conformal boundaries and defects

Abstract

Motivated by questions about quantum information and classification of quantum field theories, we consider Conformal Field Theories (CFTs) in spacetime dimension d ≥ 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 ≥ 6; and Takayanagi’s AdS/BCFT with d = 5. We demonstrate that several of our examples obey the boundary/defect a-theorem, as expected.

Method of images in defect conformal field theories

“…A promising candidate is the averaged null energy condition (ANEC), which has been shown to hold in any relativistic unitary quantum field theory [41,42]. ANEC has played a prominent role in restricting central charges of CFTs [43], and it imposes new constraints for the types of defects if it holds in DCFTs (see e.g., [18,19,44]). In the presence of extended objects, the existing proof of ANEC is no longer applicable, but even in such a situation, our method would be beneficial to the proof as it boils down to a statement of a CFT correlation function.…”

“…From the DCFT point of view, the boundary of BCFT in d dimensions can be regarded as a (d − 1)-dimensional defect. DCFTs with lower-dimensional defects have attracted attention and been extensively investigated in recent works; boundary and defect conformal bootstrap [10][11][12], Lorentzian inversion formula for two-point functions in DCFT [13], classification of defect central charges [14][15][16][17][18][19], defect Ctheorems [20][21][22][23], integrable structures [24,25], to name a few. Further developments of DCFT in various fields may be found in [26] and the references therein.…”

A scaling limit for line and surface defects

“…It is interesting to note that in this computation the diagrams corresponding to the anomalous dimension of the scalars are suppressed. 6 Indeed, going to the next order in perturbation theory one would have 3 diagrams: one which is two copies of that in figure 7b); one which is a "ladder" (two copies like figure 7a) connected by an intermediate line) and the one-loop self-energy of the exchanged field in figure 7a). To understand those, it is perhaps easiest to use the Feynman rules from the original lagrangian.…”

“…From a more theoretical point of view, defects in a given bulk CFT allow to probe interesting Physics which has recently attracted much attention, including the discovery of new central charges and the study properties of RG flows (see e.g. [1][2][3][4][5][6][7][8] for a very partial list of some of the most recent developments). Moreover, in gauge theories, defects (or equivalently, extended operators) play a very important role in understanding central aspects including the symmetries and phases of a given theory (see e.g.…”

Interacting conformal scalar in a wedge

“…• Conformal anomalies and RG flows: Conformal anomalies and their associated coefficients, sometimes called central charges, play a major role in the classification of CFTs and in defining monotonic quantities along the RG flow, providing strong constraints on the possible outcomes of the latter. While in a CFT conformal anomalies are present only in even space dimensions, the situation is richer when boundaries and/or defects occur, and the possibility of having anomalies localised on them considerably extends the possible anomaly contributions [39][40][41][42][43]. Conformal anomalies of boundaries and defects have been widely studied in the literature, where monotonicity theorems have been proven [44][45][46][47][48] or conjectured [49], and various relations to boundary and bulk correlation functions have been found [50][51][52][53][54][55].…”