Central Charges for BCFTs and Holography

TL;DR

This work investigates logarithmic terms in BCFT partition functions, linking even-dimensional Weyl anomalies to central charges and odd-dimensional boundary central charges, and proposes a c-theorem-like monotonicity for c_{bdy} under boundary RG flows. Using AdS/BCFT, the authors derive a codimension-two boundary term, compute a topological log term in AdS$_3$/BCFT$_2$ independent of boundary deformations, and construct perturbative gravity solutions for AdS$_4$/BCFT$_3$ with arbitrary boundary shapes, showing breakdown of Fefferman–Graham near general boundaries. They provide field-theory perturbative checks of c_{bdy} monotonicity in BCFT_d with odd d and explicit scalar-field examples, supporting the higher-dimensional g-theorem conjecture. These results illuminate how boundaries influence holographic duals and RG flows in BCFTs.

Abstract

In this paper, we study the logarithmic terms in the partition functions of CFTs with boundaries (BCFTs). In three dimensions, their coefficients give the boundary central charges, which are conjectured to be monotonically decreasing functions under the RG flows. We present a few supporting evidences from field theory calculations. In two dimensions, we give a holographic construction (AdS/BCFT) for an arbitrary shape of boundary and calculate its logarithmic term as well as boundary energy momentum tensors, confirming its consistency with the Weyl anomaly. Moreover, we give perturbative solutions of gravity duals for the three dimensional BCFTs with any shapes of boundaries. We find that the standard Fefferman-Graham expansion breaks down for generic choices of BCFT boundaries.

Figures (3)

• Figure 1: A schematic setup of AdS/BCFT. The CFT lives on $M$, which has the boundary $P\equiv \partial M$. Its gravity dual is denoted by $N$ and its asymptotically AdS boundary is $M$. The boundary $P$ is extended into the bulk AdS, which constitutes the boundary $Q$.
• Figure 2: A summary of notations on the manifolds and their metrics in this paper. Notice that $N$ is the original spacetime where the gravity dual lives. $M$ is its AdS boundary and $Q$ is the other part of the boundary of $N$. $P$ is defined by $P=\partial M=\partial Q$.
• Figure 3: The setup of BCFT$_3$ to calculate the boundary central charge $c_{bdy}$.