Holographic calculation of boundary terms in conformal anomaly

TL;DR

The paper analyzes how boundaries modify the integrated conformal anomaly via extrinsic-curvature terms, applying holographic BCFT to compute these boundary contributions in $d=3$ and $d=4$ for both Takayanagi's tension-based prescription and a minimal-surface prescription. It shows that the two approaches yield distinct structures in the holographic anomaly: Takayanagi's method introduces a tunable parameter $m$ that affects boundary charges, while the minimal-surface method gives a parameter-free result in $d=4$ that aligns with the free-field boundary data and preserves more supersymmetry. For ${ m N}=4$ SYM, the boundary anomaly depends on the split of scalars between Dirichlet and Robin conditions, with the minimal-surface prescription reproducing the expected boundary charges when supersymmetry is preserved. The work highlights interpretational issues for $m$ as boundary-condition data, discusses potential protection of boundary charges under renormalization theorems, and suggests that the minimal-surface prescription provides a robust framework for holographic BCFT and potential tests via boundary entanglement entropy.

Abstract

In the presence of boundaries the integrated conformal anomaly is modified by the boundary terms so that the anomaly is non-vanishing in any (even or odd) dimension. The boundary terms are due to extrinsic curvature whose exact structure in $d=3$ and $d=4$ has recently been identified. In this note we present a holographic calculation of those terms in two different prescriptions for the holographic description of the boundary CFT. We stress the role of supersymmetry when discussing the holographic description of ${\cal N}=4$ SYM on a $4$-manifold with boundaries.