Boundary conformal field theory, holography and bulk locality

We study bulk locality in a scalar effective field theory (EFT) in AdS background in presence of an end-of-the-world (EOW) brane. The holographic dual description is given in terms of a boundary conformal field theory (BCFT). We compute the two point correlation function of scalar operators in the BCFT using the one-loop Witten diagrams and compare its analytic structure with the constraints imposed by boundary conformal symmetry. We find that the loop-corrected correlator derived from a local bulk description is not fully compatible with BCFT expectations. This result places nontrivial constraint on bulk locality in holographic BCFT constructions and identifies BCFT correlators as sensitive probes of quantum bulk dynamics in presence of boundaries.

💡 Research Summary

The paper investigates the compatibility of bulk locality with holographic boundary conformal field theories (BCFTs) by explicitly computing one‑loop Witten diagrams in an AdS background that contains an end‑of‑the‑world (EOW) brane. The authors begin by recalling the Heemskerk‑Penedones‑Polchinski‑Sully (HPPS) program, which shows that in ordinary AdS/CFT the analytic structure of CFT four‑point functions forces the dual bulk effective field theory (EFT) to be local up to a finite number of derivatives at each order in the 1/N expansion. They then argue that the two‑point function of scalar operators in a BCFT plays an analogous role: because the presence of a planar boundary reduces the symmetry group, the correlator depends on a single cross‑ratio ξ, and the region ξ∈(−1,0) must exhibit a logarithmic singularity of the form ξⁿ log ξ. The coefficient of this log term encodes the anomalous dimensions of double‑trace operators and therefore provides a sharp test of bulk locality.

To perform the holographic calculation, the authors consider a scalar field Φ of mass m² propagating in Euclidean AdS_{d+1} with a flat EOW brane extending along the radial direction. The bulk action contains cubic (λ₃Φ³) and quartic (λ₄Φ⁴) interactions, as well as higher‑derivative terms localized on the brane (e.g. λ₅(∇Φ)⁴). Dirichlet or Neumann boundary conditions are imposed on the brane, encoded by a sign ν=±1. Using the method of images, they construct modified bulk‑to‑bulk and bulk‑to‑boundary propagators G^{BB}_ν and G^{B∂}_ν that are linear combinations of the standard AdS propagators and their reflected images. The image prescription automatically implements the chosen boundary condition.

The one‑loop contributions to the BCFT two‑point function arise from two classes of Witten diagrams: (i) a “bubble” diagram generated by the quartic vertex λ₄Φ⁴, and (ii) a “triangle” diagram generated by the cubic vertex λ₃Φ³. Both involve integrating over the bulk point X the product of two bulk‑to‑boundary propagators and a bulk‑to‑bulk propagator evaluated at coincident points. The authors evaluate these integrals using Schwinger parameterisation for the bulk‑to‑boundary propagators and a Mellin‑Barnes representation for the bulk‑to‑bulk propagator. Divergences associated with coincident bulk points are regularised by discarding the self‑energy piece and keeping only the image contribution, which is justified by the presence of the brane.

After performing the integrals, the resulting ξ‑dependence of the correlator contains hypergeometric functions ₂F₁(−ξ) multiplied by powers of ξ, together with logarithmic terms that arise from the image contributions. Crucially, the logarithms appear as ξⁿ log(1+ξ) or as combinations of log ξ with additional rational functions, rather than the pure ξⁿ log ξ structure required by BCFT crossing symmetry. Moreover, the branch cut of the hypergeometric function lies at ξ=−1, whereas the BCFT logarithmic singularity must start at ξ=0. Consequently, the coefficients of the log terms do not match the expected combination a_n^{(0)} γ_n that encodes anomalous dimensions of double‑trace operators.

The mismatch persists for both Neumann and Dirichlet boundary conditions and for all examined interaction vertices, including higher‑derivative brane‑localized terms. The authors therefore conclude that a strictly local bulk EFT, defined by a finite set of local vertices and standard boundary conditions, cannot reproduce the full analytic structure of BCFT two‑point functions beyond tree level. This constitutes a BCFT analogue of the HPPS locality test, but with a qualitatively different outcome: the presence of an EOW brane forces a breakdown of sharp bulk locality already at one‑loop order.

In the discussion, the authors emphasize that this result does not invalidate the AdS/BCFT correspondence itself; rather, it signals that additional ingredients are needed on the bulk side. Possible resolutions include (a) introducing non‑local interactions that are localized on the brane, (b) adding explicit defect degrees of freedom that couple to the bulk field, or (c) abandoning a purely local EFT description in favor of a more general bulk theory (e.g., stringy corrections or higher‑spin fields). They suggest that BCFT two‑point functions, being technically simpler than four‑point functions yet still sensitive to loop effects, provide a powerful diagnostic for the emergence of spacetime locality in holographic settings with boundaries.

Overall, the paper delivers a clear and detailed computation showing that bulk locality, as defined by a local EFT, is more fragile in the presence of boundaries than previously appreciated. It opens several avenues for future work, such as exploring alternative boundary conditions, incorporating brane‑localized fields, or extending the analysis to higher‑point functions and different bulk matter content.