Codimension-two holography for wedges

We propose a codimension-two holography between a gravitational theory on a $d+1$ dimensional wedge spacetime and a $d-1$ dimensional CFT which lives on the corner of the wedge. Formulating this as a generalization of $\mathrm{AdS}/\mathrm{CFT}$, we explain how to compute the free energy, entanglement entropy and correlation functions of the dual CFTs from gravity. In this wedge holography, the holographic entanglement entropy is computed by a double minimization procedure. Especially, for a four dimensional gravity ($d=3$), we obtain a two dimensional CFT and the holographic entanglement entropy perfectly reproduces the known result expected from the holographic conformal anomaly. We also discuss a lower dimensional example ($d=2$) and find that a universal quantity naturally arises from gravity, which is analogous to the boundary entropy. Moreover, we consider a gravity on a wedge region in Lorentzian AdS, which is expected to be dual to a CFT with a spacelike boundary. We formulate this new holography and compute the holographic entanglement entropy via a Wick rotation of the AdS/BCFT construction. Via a conformal map, this wedge spacetime is mapped into a geometry where a bubble-of-nothing expands under time evolution. We reproduce the holographic entanglement entropy for this gravity dual via CFT calculations.

Figure 1

A sketch of wedge holography.

Figure 2

A basic setup of wedge holography with UV regularization.

Figure 3

A sketch of derivation of wedge holography from AdS/BCFT. The left picture describes the gravity dual of a CFT on a semi-infinite plane. The right one shows the gravity dual of a CFT on an interval.

Figure 4

A sketch of compactified setup of wedge holography.

Figure 5

A sketch of calculation of holographic entanglement entropy in codimension-two holography (left) and in AdS/BCFT (right).

Figure 6

Other two versions of wedge holography setups. The left (and right) setup imposes the Dirichlet (and Neuman) boundary condition on the tip $\mathrm{\Sigma }$ of the wedge.

Figure 7

Gluing two AdS backgrounds.

Figure 8

Profiles of end-of-the-world branes in BTZ.

Figure 9

Small wedge regions of AdS/BCFT in the Euclidean (left) and Lorentzian (right) setup. The boundary surface $Q$ in the former and latter case has the tension $-1/L<T<0$ and $T<-1/L$, respectively.

Figure 10

Large wedge regions of AdS/BCFT in the Euclidean (left) and Lorentzian (right) setup. The boundary surface $Q$ in the former and latter case has the tension $0<T<1/L$ and $T>1/L$, respectively.

Figure 11

A bubble nucleation of universe (left) and a nucleation of bubble-of-nothing (right) in a Lorentzian AdS/BCFT setup, described by a boundary surface $Q$ with the tension $T\ge 1/L$ (left) and $T\le -1/L$ (right). The colored regions describe physical spacetimes.

Figure 12

The Euclidean CFT with the boundary $|w{|}^2=r^2$. Left: the doubling trick of the usual BCFT. The mirror image of the point $x$ is inserted onto the inversed point $r^2/x$. Right: the doubling trick of the analytic continued BCFT. The analytic continuation can be interpreted as the reflection of the mirror point.

Figure 13

The map from a disk (left) to a cylinder (right).

Figure 14

The vacuum block approximation of a two-point function in the holographic BCFT. There are two possibilities, like the s- and t-channel vacuum block approximation of a full plane four-point function with same operators.

Figure 15

The Euclidean setup corresponding to the right picture of Fig. 11 with $x_2=i{x}_0$. The end-of-the-world brane $Q$ is given by $x_1^2+x_2^2+(z-\alpha {)}^2={\beta }^2$.

Figure 16

The $x_2=0$ slice. It is shown how the disconnected geodesic (and its extension) intersects with the bubble and the AdS boundary $z=0$.

Figure 17

A sketch of $d=2$ wedge holography with another UV cutoff.