Codimension-n Holography for Cones

TL;DR

Cone holography proposes a codimension-$n$ holographic duality between gravity in a $(d+1)$-dimensional conical bulk and a $(d+1-n)$-dimensional defect CFT, obtained as a zero-volume limit of AdS/dCFT and interpreted as a dual to edge modes on defects. For a key class of solutions, the framework is shown to be equivalent to AdS/CFT with Einstein gravity, supporting the validity of the construction in the classical limit, while generically the presence of infinite KK towers on the branes yields deviations from standard AdS/CFT. The authors validate the proposal through holographic Weyl anomaly, Rényi entropy, and correlation-function calculations, finding results in line with defect CFT expectations and establishing a holographic c-theorem. They also analyze Neumann boundary conditions and the spectrum of massive brane modes, showing that low-energy physics on the brane can reduce to Einstein gravity when tensions are small. Overall, cone holography extends wedge holography and connects to defect CFTs, entanglement measures, and AdS/BCFT/dCFT frameworks, with broad potential applications and rich avenues for future work.

Abstract

We propose a novel codimension-n holography, called cone holography, between a gravitational theory in $(d+1)$-dimensional conical spacetime and a CFT on the $(d+1-n)$-dimensional defects. Similar to wedge holography, the cone holography can be obtained by taking the zero-volume limit of holographic defect CFT. Remarkably, it can be regarded as a holographic dual of the edge modes on the defects. For one class of solutions, we prove that the cone holography is equivalent to AdS/CFT, by showing that the classical gravitational action and thus the CFT partition function in large N limit are the same for the two theories. In general, cone holography and AdS/CFT are different due to the infinite towers of massive Kaluza-Klein modes on the branes. We test cone holography by studying Weyl anomaly, Entanglement/Rényi entropy and correlation functions, and find good agreements between the holographic and the CFT results. In particular, the c-theorem is obeyed by cone holography. These are strong supports for our proposal. We discuss two kinds of boundary conditions, the mixed boundary condition and Neumann boundary condition, and find that they both define a consistent theory of cone holography. We also analyze the mass spectrum on the brane and find that the larger the tension is, the more continuous the mass spectrum is. The cone holography can be regarded as a generalization of the wedge holography, and it is closely related to the defect CFT, entanglement/Rényi entropy and AdS/BCFT(dCFT). Thus it is expected to have a wide range of applications.

Figures (10)

• Figure 1: (left) Wedge holography from AdS/BCFT; (right) Geometry of wedge holography.
• Figure 2: (left) $\text{BCFT}_d$ on $M$ and the edge mode as effective $\overline{\text{CFT}}_{d-1}$ on $\Sigma=\partial M$; (right) In the zero-volume limit $M\to 0$, $\text{BCFT}_d$ on $M$ disappears and only the edge modes $\text{CFT}_{d-1}=\overline{\text{CFT}}_{d-1}\oplus\overline{\text{CFT}}_{d-1}$ on $\Sigma$ survive.
• Figure 3: Zero-volume limit of dCFT. $\hat{M}$ is a d-dimensional manifold where dCFT is defined, $P$ is the boundary of $\hat{M}$ and $D$ is a codim-m defect at the center of $\hat{M}$. The metric is given by (\ref{['metricdCFT']}), $ds^2=dz^2+\frac{z^2}{q^2} d\Omega_{m-1}^2+\sum_{\hat{i}=1}^{d-m} dy_{\hat{i}}^2$ with $0\le z\le z_0$. The defect $D$ and the boundary $P$ are located at $z=0$ and $z=z_0$, respectively. Note that the geometry of $P$ is chosen to be $S_{m-1}\times R_{d-m}$ so that it coincides with the defect $D=R_{d-m}$ in the zero-volume limit $\hat{M}\to 0$ with $z_0\to0$ and $S_{m-1}\to 0$. Here $z_0$ is the radius of the sphere $S_{m-1}$. In such limit, only the edge modes of dCFT survive.
• Figure 4: Cone holography from AdS/BCFT and AdS/dCFT. dCFT lives in the manifold $\hat{M}$ with a boundary $P$ and a codim-m defect D at the center. The boundary P and codim-m defect D are extended to an end-of-world brane $Q$ and a codim-m brane $E$ in the bulk, respectively. $C$ (orange) is the bulk spacetime bounded by $Q$ and $\hat{M}$, $M$ (gray) is the AdS boundary. In the limit $\hat{M}\to 0$, the bulk spacetime $C$ becomes a cone and we obtain the cone holography from AdS/BCFT and AdS/dCFT.
• Figure 5: Geometry of cone holography: $M$ is a d-dimensional manifold (gray plane), $D$ is a codim-m defect (blue point) in $M$, where $m=n-1$. $M$ is extended to a (d+1)-dimensional asymptotically AdS space $N$, and $D$ is extended to a (d+1)-dimensional cone $C$ (orange) in the bulk. The cone $C$ is bounded by a codim-1 brane $Q$ (boundary of orange cone), i.e., $\partial C=Q$. The geometries of $Q$ and $E$ are set to be $\text{AdS}_{d+2-n}\times \text{S}_{n-2}$ and $\text{AdS}_{d+2-n}$ so that they shrink to the same defect $D=\partial Q=\partial E$ on the AdS boundary $M$. The cone holography proposes that a gravity theory in the (d+1)-dimensional cone $C$ is dual to a CFT on (d+1-n)-dimensional defect $D$.