Codimension-two holography for wedges

Ibrahim Akal,1 Yuya Kusuki,1 Tadashi Takayanagi ,1,2,3 and Zixia Wei 1

1Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University,Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

2Inamori Research Institute for Science, 620 Suiginya-cho, Shimogyo-ku, Kyoto 600-8411 Japan3Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo,

Kashiwa, Chiba 277-8582, Japan

(Received 4 November 2020; accepted 9 November 2020; published 2 December 2020)

We propose a codimension-two holography between a gravitational theory on a dþ 1 dimensional wedgespacetime and a d − 1 dimensional CFT which lives on the corner of the wedge. Formulating this as ageneralization of AdS=CFT, we explain how to compute the free energy, entanglement entropy andcorrelation functions of the dual CFTs from gravity. In this wedge holography, the holographic entanglemententropy is computed by a doubleminimization procedure. Especially, for a four dimensional gravity (d ¼ 3),we obtain a two dimensional CFTand the holographic entanglement entropy perfectly reproduces the knownresult expected from the holographic conformal anomaly. We also discuss a lower dimensional example(d ¼ 2) and find that a universal quantity naturally arises from gravity, which is analogous to the boundaryentropy.Moreover, we consider a gravity on a wedge region in Lorentzian AdS, which is expected to be dualto a CFT with a spacelike boundary. We formulate this new holography and compute the holographicentanglement entropy via a Wick rotation of the AdS/BCFT construction. Via a conformal map, thiswedge 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.

DOI: 10.1103/PhysRevD.102.126007

I. INTRODUCTION

The AdS=CFT correspondence [1–3] provides a non-perturbative definition of quantum gravity on an anti–de Sitter space (AdS) in terms of a lower dimensionalconformal field theory (CFT). Since one of the final goals instring theory is to understand how our universe is created, itis important to extend the idea of holography [4,5] to morerealistic spacetimes.Till now, there have been several progresses in this

direction by generalizing or deforming the AdS=CFTcorrespondence. For example, this includes the brane worldholography [6–8] which proposes a holographic dualitybetween AdS with a finite geometric cutoff in the radialdirection and a CFT coupled to dynamical gravity. Byfurther developing the brane world approach, a holographicdescription of de Sitter gravity was proposed in [9–11]. Forholographic duals of de Sitter gravity, there have also beenworked out more direct approaches called dS=CFT byexplicitly changing the sign of the cosmological constantand by regarding the future/past infinity as the holographic

dual boundary where the dual CFT looks nonunitary[12–14]. There also exists a general approach called sur-face/state correspondence, which is applicable to hologra-phy in any spacetime [15,16], regarding codimension-twoand codimension one surface as a quantum state and aquantum circuit, respectively.Another approach to extend the standard AdS=CFT is to

introduce boundaries on the manifold where the CFT lives.When a part of conformal symmetry is preserved, such atheory is called a boundary conformal field theory (BCFT).Holographic duals of BCFTs, called AdS/BCFT, can beconstructed by cutting the AdS along an end-of-the-worldbrane with backreactions taken into account [17–20]. Onecharacteristic feature of AdS/BCFT is that the holographicentanglement entropy [21–25] can have a contribution fromminimal surfaces ending on the end-of-the-world brane,which has recently been called an island. Indeed, bycombining the AdS/BCFT with the brane world hologra-phy, the island phenomena [26,27] in effective gravitytheories have been derived holographically in [28] (also see[29–36] for more progresses on the applications of braneworld holography and AdS/BCFT to this subject).The main aim of the present paper is to combine these

ideas to explore a novel holographic duality—we call thiswedge holography—between a dþ 1 dimensional wedgespace W and a d − 1 dimensional CFT on Σ as sketched inFig. 1. Similar ideas of higher codimension holography

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.

PHYSICAL REVIEW D 102, 126007 (2020)Editors' Suggestion

2470-0010=2020=102(12)=126007(28) 126007-1 Published by the American Physical Society

were already discussed in a brane world setup [37], thedS/dS correspondence [9,10], a dimensional reduction ofde-Sitter space [38] and also in the context of gravity edgemodes [39].After we show that our wedge holography can be

understood as a certain limit of AdS/BCFT, we will studythis wedge holography by calculating the free energy, theconformal anomaly, the entanglement entropy and corre-lation functions. The results of these computations confirmthe proposed holographic relation. In particular, in thed ¼ 3 case, we show that the conformal anomaly computedfrom the gravity on-shell action agrees with that computedfrom the holographic entanglement entropy. We alsoanalyze Lorentzian wedge spacetimes which is dual to aCFT on a manifold with a spacelike boundary via a Wickrotation of an Euclidean AdS/BCFT setup. We see that thisopens up an interesting possibility of nucleations ofbubbles-of-nothing in AdS and its CFT dual.This paper is organized as follows: In Sec. II, we present

a general formulation of wedge holography, includingcomputations of the free energy, entanglement entropyand correlation functions. In Sec. III, we study the detailsby focusing on d ¼ 3. In Sec. IV, we present details in thecase of d ¼ 2. In Sec. V, we give a holographic connectionbetween wedge holography and the gravity dual of joiningtwo CFTs. In Sec. VI, we analyze wedge holography for theBTZ spacetime. In Sec. VII, we discuss Lorentzian wedgespacetimes whose CFT duals have spacelike boundaries. InSec. VIII, we summarize our conclusions and discuss futureproblems. In Appendix A, we present a short review of theSchwarzian action. In Appendix B, we present gravitycalculations with another choice of UV cutoff.

II. WEDGE HOLOGRAPHYIN GENERAL DIMENSIONS

Here, we would like to formulate our codimension-twoholography, called wedge holography, in any dimensionsby keeping only a wedge region in the standard AdS=CFTcorrespondence. Consider a Poincare AdSdþ1 with the AdSradius L, whose metric can be written in the following twodifferent choices of the coordinates:

ds2 ¼ L2

�dz2 − dt2 þ dx2 þP

d−2i¼1 dξ

2i

z2

�ð2:1Þ

¼ dρ2þL2 cosh2ρ

L

�dy2−dt2þP

d−2i¼1 dξ

2i

y2

�; ð2:2Þ

where the coordinates ðz; xÞ are related to ðρ; yÞ via

z ¼ ycosh ρ

L

; x ¼ y tanhρ

L: ð2:3Þ

The Euclidean solution is simply obtained by settingt ¼ −itE.Now, we consider a dþ 1 dimensional wedge geometry,

called Wdþ1, in this background defined by the followinglimited region in the ρ coordinate introduced in (2.2):

−ρ� ≤ ρ ≤ ρ�; ð2:4Þwhere we call the two boundaries of the wedge i.e.,ρ ¼ −ρ� and ρ ¼ ρ�, the d dimensional suface Q1 andQ2, respectively. As usual in AdS=CFT, we regard the UVcutoff of CFT as the geometric cutoff z ≥ ϵ. This generatesan extra d dimensional boundary surface Σ given by z ¼ ϵand jxj ≤ ϵ sinh ρ�

L . This setup of the wedge geometry isdepicted in Fig. 2.We define the classical gravity on this wedge Wdþ1 by

imposing the Neumann boundary condition in AdS/BCFT[18,19]:

Kab − Khab ¼ −Thab; ð2:5Þon Q1 and Q2 where hab and Kab are the induced metricand the extrinsic curvature on the surfaces Q1;2, while T isthe tension parameter. Since we have

Kab ¼1

Ltanh

ρ�L· hab;

K ≡ habKab ¼dLtanh

ρ�L; ð2:6Þ

we choose the tension to be

T ¼ d − 1

Ltanh

ρ�L: ð2:7Þ

FIG. 1. A sketch of wedge holography.

FIG. 2. A basic setup of wedge holography with UVregularization.

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-2

On the asymptotically AdS surface Σ, we impose thestandard Dirichlet boundary condition which fixes themetric on Σ, denoted by γαβ, to be given by (2.1).

A. Wedge holography

Thinking of the above setup, we now would liketo argue for the following codimension-two holographiccorrespondence.

A classical gravity on the dþ 1 dimensional wedgeWdþ1, defined with the above boundary conditions, is dualto a (effectively) d − 1 dimensional CFT located on Σ, asdepicted in Fig. 2. Notice that since the width in x directionof the surface Σ is OðϵÞ, i.e., infinitesimally small, we caneffectively regard Σ as being d − 1 dimensional, namelyR1;d−2, spanned by ðt; ξ1; · · ·; ξd−2Þ, which we write asΣd−1. Our proposal is summarized as follows.

Wedgeholography proposal

Classical gravity onwedgeWdþ1 ≃ ðQuantumÞ gravity on twoAdSd ð¼Q1 ∪ Q2Þ≃ CFTonΣd−1

ð2:8Þ

This holography can be understood by two steps. Forsimplicity, let us take the strict UV limit ϵ ¼ 0 as in Fig. 1.First, the classical gravity on the dþ 1 dimensional wedgeWdþ1 is dual to a (quantum) gravity on its d dimensionalboundary ∂Wdþ1 ¼ Q1 ∪ Q2 via the brane world holog-raphy [6–8]. Since Q1 and Q2 are d dimensional AdSspacetimes, the quantum gravity theories on them areexpected to be dual to d − 1 dimensional CFTs which liveon ∂Q1 ¼ ∂Q2 ¼ Σ. We combine these two CFTs on Σ andtreat them as a single d − 1 dimensional CFT. This argu-ment explains the previous proposal of the codimension-two holography.In the middle expression of (2.8), i.e., d dimensional

gravity, one may think there is also a contribution from theCFTd on a small interval. However, we can neglect such acontribution. To see this, we can look, for example, at theanomaly for d ¼ 3 which we will discuss later. Theanomaly is proportional to ρ� and this cannot be explainedby the fields on the small interval, which should not dependon ρ�.One may also understand our codimension-two holog-

raphy by decomposing the wedge geometry into an interval½−ρ�; ρ�� in the ρ direction and the transverse AdSd. Aftera compactification on an interval we can apply theAdSd=CFTd−1 duality to find the codimension-two holog-raphy. However, our dþ 1 dimensional description ofwedge holography has several advantages. First of all, aswe will see below the nature of wedge holography issensitive to the choice of UV cut off surface as a function ofρ, which is implicit after the compactification. Indeedbelow we choose the UV cut off surface with a non-trivialρ dependence. Another advantage is that we can employ thefull symmetry of the AdSdþ1 to have analytical controlsover computations of physical quantities, even when thebulk fields have nontrivial profiles in the ρ direction.Moreover, the structure of the wedge geometry leavesthe ρ direction explicit in the bulk and we expect this helpsus to understand nonlocal physics such as entanglement onthe compact manifold.

B. Derivation from AdS/BCFT

We can derive the previous codimension-two holographyby taking a suitable limit of the AdS/BCFT. Consider agravity dual of a d dimensional CFT on a manifold Σ withboundaries ∂Σ. The AdS/BCFT construction [18–20]argues that the gravity dual of the CFT on Σ is given bya gravity on a dþ 1 dimensional manifold M with extra ddimensional boundaries Q (so called end-of-the-worldbrane), in addition to the standard AdS boundary Σ suchthat ∂M ¼ Q ∪ Σ. This manifold M is determined bysolving the Einstein equation and by imposing theNeumann-type boundary condition (2.5), parameterizedby the tension T, on the surface Q. When the boundaryΣ is a semi-infinite plane x > 0 in (2.1), the correspondingconstruction is sketched in the left panel of Fig. 3.When we consider a gravity dual of a d dimensional CFT

on a strip 0 ≤ x ≤ w, there are two possibilities in general,which may be called as the confined solution and decon-fined solution, as in the Hawking-Page transition. In theconfined phase, a mass gap is generated from the stronginteractions in the holographic CFT and the d dimensionalsurfaceQ is connected in the bulk as explicitly examined in[18–20]. However, since we are interested only in gravityduals with scale invariance in this paper, we will notconsider this confined case below. We would like to focuson the deconfined phase. For example, only this phase isallowed when we impose an appropriate supersymmetricboundary condition (analogous to the R-sector of openstrings) on ∂Σ which helps the system to be conformallyinvariant even in the presence of the boundary. In thisdeconfined case, the surface Q consists of two discon-nected planes Q1 and Q2, both of which satisfy theboundary condition (2.5). Note that Q1 and Q2 are parallelwith those in Fig. 2.If we take the limit w → 0 of the vanishing width of strip

Σ, we can regard Σ as a d − 1 dimensional space Rd−1. Thegravity dual of AdS/BCFT is now reduced to the wedgegeometry in Fig. 1 or its regularized version shown in

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-3

Fig. 2. This explains our proposal of the codimension-twoholography. Note that we expect the original CFTd on Σ isnow reduced to a d − 1 dimensional CFT in the zero widthlimit w → 0. This phenomenon is quite usual. For example,a massless free scalar on a d dimensional strip ½0; w� × Rd−1

is reduced to a d − 1 dimensional massless free scalar onRd−1 in the limit w → 0.

C. Free energy in noncompact space

As a fundamental quantity which characterizes theholographic correspondence, we would like to evaluatethe free energy first. This is evaluated from the on-shellgravity action on the Euclidean wedge geometry ofFig. 2:

IW ¼−1

16πGN

ZW

ffiffiffig

p ðR− 2ΛÞ− 1

8πGN

ZQ1∪Q2

ffiffiffih

pðK−TÞ

−1

8πGN

ZΣ

ffiffiffiγ

pK: ð2:9Þ

We calculate the explicit values using the metric (2.2)and (2.1):

R¼−ðdþ1Þd

L2; Λ¼−

dðd−1Þ2L2

;

KjQ ¼ dLtanh

ρ�L; T ¼ d−1

Ltanh

ρ�L; KjΣ ¼

dL: ð2:10Þ

The gravity action is evaluated as follows

IW ¼ d8πGNL2

ZW

ffiffiffig

p−

tanh ρ�L

8πGNL

ZQ1∪Q2

ffiffiffih

p−

d8πGNL

ZΣ

ffiffiffiγ

p

¼ d · Vd−1Ld−2

4πGN

Zρ�

0

dρ

�cosh

ρ

L

�dZ

∞

ϵ coshρL

dyyd

−Ld−1Vd−1 tanh

ρ�L

4πGN

�cosh

ρ�L

�dZ

∞

ϵ coshρ�L

dyyd

−d · Ld−1Vd−1 sinh

ρ�L

4πGNϵd−1

¼ Vd−1Ld−1

4πGNϵd−1

�d

ðd − 1ÞLZ

ρ�

0

dρ coshρ

L−

1

d − 1sinh

ρ�L− d sinh

ρ�L

�

¼ ð1 − dÞ · Vd−1Ld−1

4πGNϵd−1 sinh

ρ�L; ð2:11Þ

where we defined Vd−1 ¼RdtEdξ1 · · · dξd−2. Indeed, the

scaling IG ∝ Vd−1ϵ1−d is consistent with the vacuum

energy of a d − 1 dimensional CFT (CFTd−1). Also wefind that the degrees of freedom of the CFTd−1 areestimated as

∼Ld−1

GNsinh

ρ�L: ð2:12Þ

However, when there are cusp like codimension-twosingularities on the boundaries, we need to add theHayward term [40,41] to the standard gravity action IW

in (5.2). For each cusp at Σ, the corresponding Haywardterm IH is written as

IH ¼ 1

8πGN

ZΣðΘ − πÞ ffiffiffi

γp

; ð2:13Þ

where γ is the induced metric on Σ and Θ is the cusp anglebetween two surfaces. In our setup shown in the right panelof Fig. 3, Σ and Θ are the d − 1 dimensional intersectionand the angle between Q1;2 and Σ. This angle is given byθ þ π

2, where θ is

FIG. 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.

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-4

tan θ ¼ sinhρ�L: ð2:14Þ

The reason why we add the Hayward term is because weneed to have a sensible variation on Σ

δðIW þ IHÞ ¼1

8πGN

ZΣðΘ − πÞδ ffiffiffi

γp

; ð2:15Þ

when we impose the on-shell conditions, i.e., the Einsteinequation and the boundary condition (2.5) on Q1;2. TheDirichlet boundary condition on Σ (and thus on Σ) leads toδ

ffiffiffiγ

p ¼ 0 and we therefore have δðIW þ IHÞ ¼ 0. In oursetup, this is evaluated as follows

IH ¼ ð2θ − πÞ Vd−1Ld−1

8πGNϵd−1 : ð2:16Þ

Finally, the total gravity action is given by

IW þ IH ¼ ð1 − dÞ Vd−1Ld−1

4πGNϵd−1 sinh

ρ�L

þ ð2θ − πÞ Vd−1Ld−1

8πGNϵd−1 : ð2:17Þ

Though one may think that it is strange that the total actiondoes not vanish even when the wedge region gets squeezedto zero size, ρ� ¼ 0, the Hayward term contribution IH isimportant to appropriately take into account the gravityedge mode degrees of freedom [39].It is useful to compare this (2.17) with the standard

gravity partition function on the Poincare AdSd with thecutoff z ¼ ϵ, which is dual to a d − 1 dimensional CFTon Rd−1 (we impose the Dirichlet boundary condition onz ¼ ϵ):

IP ¼ −1

16πGðdÞN

ZAdSd

ffiffiffig

p ðR − 2ΛÞ − 1

8πGðdÞN

ZRd−1

ffiffiffih

pK

¼ d − 1

8πGðdÞN L2

ZAdSd

ffiffiffig

p−

d − 1

8πGðdÞN L

ZRd−1

ffiffiffih

p

¼ ð2 − dÞ Vd−1Ld−2

8πGðdÞN ϵd−1

: ð2:18Þ

This takes the same form as in (2.17).

D. Free energy in compact space

It is also useful to calculate the free energy in a setupdual to a CFTd−1 on Sd−1. In particular, this is useful todetermine the conformal anomaly in the next section.For this, we employ the following metric expression ofEuclidean AdSdþ1:

ds2 ¼ dr2 þ L2 sinh2rLðdθ2 þ cos2 θdΩ2

d−1Þ; ð2:19Þ

where dΩd−1 describes the area element of Sd−1. Let usintroduce another coordinate system ðρ; ηÞ instead of ðr; θÞvia

coshrL¼ cosh η cosh

ρ

L;

sinhrLsin θ ¼ sinh

ρ

L: ð2:20Þ

This leads to the equivalent expression of the metric

ds2 ¼ dρ2 þ L2 cosh2ρ

Lðdη2 þ sinh2 ηdΩ2

d−1Þ: ð2:21Þ

We define the wedge geometry Wdþ1 as the dþ 1dimensional subspace of the Euclidean AdSdþ1 (2.21)given by jρj ≤ ρ�, as depicted in Fig. 4. Therefore, Q1

andQ2 are defined by ρ ¼ ρ� and ρ ¼ −ρ�, respectively. Inthe UV limit, the surface Σ, where the dual CFTd−1 lives, isthe d − 1 dimensional surface given by ðr; θÞ ¼ ð∞; 0Þ. Toregulate the UV divergences, we introduce the cutoff suchthat r < r∞ð→ ∞Þ and the surface Σ is identified withr ¼ r∞ and jθj ≤ θ�, where

sinhρ�L

¼ sin θ� sinhr∞L

: ð2:22Þ

In the limit r∞ → ∞, we obtain

θ�≃2sinhρ�Le−

r∞L þ2

�sinh

ρ�Lþ2

3sinh3

ρ�L

�e−

3r∞L : ð2:23Þ

Note also that the extrinsic curvature of the surface r ¼ r∞is found as

KjΣ ¼ dL tanh r∞

L

: ð2:24Þ

FIG. 4. A sketch of compactified setup of wedge holography.

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-5

Under this UV regularization, the η coordinate in(2.21) follows the regularization near the AdS boundaryas η < η∞ such that

cosh η∞ coshρ

L¼ cosh

r∞L

: ð2:25Þ

This is solved as

eη∞ ≃1

cosh ρL

er∞L þ

�1

cosh ρL

− coshρ

L

�e−

r∞L : ð2:26Þ

The gravity action is calculated as follows

IW ¼ d8πGNL2

ZW

ffiffiffig

p−

tanh ρ�L

8πGNL

ZQ1∪Q2

ffiffiffih

p−

d8πGNL tanh ρ�

L

ZΣ

ffiffiffiγ

p

¼ dLd−2 · VolðSd−1Þ4πGN

Zρ�

0

dρcoshdρ

L

Zη∞ðrÞ

0

sinhd−1η −Ld−1 · VolðSd−1Þ · tanh ρ�

L

4πGNcoshd

ρ�L·Z

η�∞

0

dηsinhd−1η

−d · Ld−1 · VolðSd−1Þ

4πGN tanh r∞L

sinhdr∞L

·Z

θ�

0

dθcosd−1θ: ð2:27Þ

Here VolðSd−1Þ is the volume of a unit radius of d − 1dimensional sphere given by

VolðSd−1Þ ¼ 2πd2

Γðd2Þ : ð2:28Þ

In the limit r∞ → ∞, we can expand the action as

IW ¼ Ld−1

4πGN·�

2πd=2

Γðd=2Þ�· 21−d · JW; ð2:29Þ

where JW is given by

JW ¼ ð1 − dÞs�eðd−1Þr∞=L

þ��

d2 − 4d − 1 −4

d − 3

�s� þ

2

3ðd − 1Þðd − 2Þs3�

�× eðd−3Þr∞=L þOðeðd−5Þr∞=LÞ; ð2:30Þ

where we set s� ¼ sinh ρ�L for simplicity. Note that the

leading contribution Oðeðd−1Þr∞=LÞ agrees with the non-compact result (2.11) by identifying er∞=L ∼ 1

ϵ.In addition, there is the Hayward term contribution

(2.13), as we have evaluated in the previous example ofthe non-compact CFT dual. Since this calculation isstraightforward and does not contribute to what we areinterested in for the upcoming arguments, we simply omitthis here.

E. d dimensional gravity viewpoint

If we regard the dþ 1 dimensional gravity onWdþ1 as ad dimensional gravity on AdSd via the compactificationalong ρ direction as in the brane world holography [6–8],we find the effective d dimensional Newton constant byreducing the original dþ 1 dimensional Einstein-Hilbertaction in terms of the d dimensional one:

1

16πGN

ZW

ffiffiffig

pR

¼ 1

16πGN

Zρ�

−ρ�dρ

�cosh

ρ

L

�d−3 Z

AdSd

ffiffiffiffiffiffiffigðdÞ

qRðdÞ

≡ 2

16πGðdÞN

ZAdSd

ffiffiffiffiffiffiffigðdÞ

qRðdÞ; ð2:31Þ

where gðdÞ is the metric of AdSd given by ds2 ¼L2y−2ðdy2 − dt2 þP

d−2i¼1 dξ

2i Þ and RðdÞ is its scalar curva-

ture. We inserted the factor 2 in the final expression becausethere are two surfaces Q1 and Q2 where both are identicalto AdSd.This leads to the relation

1

GðdÞN

¼ 1

GN

Zρ�

0

dρ

�cosh

ρ

L

�d−2

: ð2:32Þ

In the small wedge limit ρ� ≪ L, we find

GN

LGðdÞN

≃ρ�Lþ d − 2

6

�ρ�L

�3

þ · · ·: ð2:33Þ

In the opposite limit ρ� ≫ L, where the wedge getslarger and gets closer to the full AdSdþ1, the d dimensionalNewton constant (2.32) in terms of the CFTmetric, behavesas follows1

GN

LGðdÞN

¼ 22−d

d − 2ed−2L ρ� ≫ 1: ð2:34Þ

1Notice that here we normalized d dimensional metric andthe corresponding Newton constant GðdÞ

N in terms of the CFTmetric which is obtained from the bulk metric via ds2bulk ¼L2

y2 ðdy2 þ ds2CFTÞ.

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-6

F. Holographic entanglement entropy

When a gravity dual is given, a useful and universalprobe of the geometry is the holographic entanglemententropy [21–23]. Let us consider how we can calculate theholographic entanglement entropy in our wedge hologra-phy. We choose a d − 2 dimensional subsystem A in thed − 1 dimensional space Σd−1 at a time slice, chosen to bet ¼ 0 (remember the setup of Fig. 2). We would like tocalculate the holographic entanglement entropy which isequal to the entanglement entropy SA in the dual CFTd−1.We argue for the following holographic formula whichinvolves the two step minimization:

SA ¼ MinγAs:t:∂γð1ÞA ¼∂γð2ÞA ¼∂A

�MinΓAs:t:∂ΓA¼γð1ÞA ∪γð2ÞA

�AðΓAÞ4GN

��:

ð2:35Þ

As in the left picture of Fig. 5, we choose d − 2 dimensional

surfaces γð1ÞA onQ1 and γð2ÞA onQ2 so that ∂γð1;2ÞA ¼∂A. Then

we extend the d − 2 dimensional compact surface γð1ÞA ∪ γð2ÞAto the bulk wedge Wdþ1 as a d − 1 dimensional surface ΓA

such that ∂ΓA ¼ γð1ÞA ∪ γð2ÞA . First we fix the shape of γð1;2ÞAand minimize the area of ΓA, denoted by AðΓAÞ. Next weminimize this area by changing the shape of γð1;2ÞA . Thisprocedure selects a singleminimal surface and this area givesthe holographic entanglement entropy dual to SA in the d − 1dimensional CFT. This is what the formula (2.35) means.Notice that this formula guarantees the property of strongsubadditivity as in the usual holographic entanglemententropy [42]. In the Lorentzian time dependent backgroundswe just need to replace the minimizations with the extrem-alizations as in [23].We can derive the formula (2.35) by taking the zero

width limit w → 0 of the AdS/BCFT with the two boun-daries (refer to the right picture of Fig. 3). This is depictedin the right picture of Fig. 5. Remember that in the AdS/BCFT [18,19], we calculate the holographic entanglemententropy by minimizing the area among surfaces ΓA whichend on the boundary surface Q as well as the boundary ofthe subsystem A.Consider the holographic entanglement entropy when

the subsystem A is given by d − 2 dimensional round diskPd−2i¼1 ξ

2i ≤ l2 with the radius l. To find the correct minimal

surface ΓA, we need to first solve the partial differential

equation for arbitrary choices of γð1;2ÞA , which requires a lotof numerical computations.

Let us first focus on the case where the wedge is verysmall i.e., ρ� ≪ L. In this case, the wedge geometry isapproximated by the direct product of AdSd, whosecoordinate is given by ðy; t; ξ1; · · ·; ξd−2Þ, and an intervaljρj ≤ ρ� as the warp factor stays almost constant cosh ρ

L ≃ 1.

Therefore, it is clear that both γð1ÞA and γð2ÞA should also beminimal surfaces on Q1 and Q2, i.e., the AdSd. Thus wecan identify ΓA with the wedge part of the sphere

x2 þ z2 þXd−2i¼1

ðξiÞ2 ¼ l2; ð2:36Þ

which is equivalent to

y2 þXd−2i¼1

ðξiÞ2 ¼ l2; ð2:37Þ

where the ρ dependence drops out.Furthermore, we would like to argue that the surface

(2.37) is the correct minimal surface defined by (2.35) forany ρ�. Though it is well known that this surface solves theminimal surface equation in the Poincare AdS, we need toconfirm that the area of the minimal surface with variouschoices of boundaries γð1;2Þ is minimized when γð1;2Þ areboth the same semi circle. To see this, let us set d ¼ 3 justfor notational simplicity and consider a perturbation aroundthis semisphere solution

y ¼ffiffiffiffiffiffiffiffiffiffiffiffiffil2 − ξ2

pþ fðξ; ρÞ; ð2:38Þ

where we impose fð�l; ρÞ ¼ 0 as the surface should end onthe boundary of subsystem A in CFTd−1.We find that under this perturbation the area of the

surface looks like

A ¼Z

dξdρffiffiffiffiG

p¼ −

�L cosh ρ

L

lðl2 − ξ2Þ12 ξfðξÞ�ξ¼l

ξ¼−lþZ

dρdξ

�Ll cosh ρ

L

l2 − ξ2

þ L cosh ρL

2l3ðl2 − ξ2Þ2�2l4f2 þ L2l2

�cosh2

ρ

L

�ðl2 − ξ2Þð∂ρfÞ2 þ 2l2ξðl2 − ξ2Þf∂ξf þ ðl2 − ξ2Þ3ð∂ξfÞ2Þ

��: ð2:39Þ

FIG. 5. A sketch of calculation of holographic entanglemententropy in codimension-two holography (left) and in AdS/BCFT(right).

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-7

The first term represents the linear perturbation and thistakes the form of total derivative as the profile f ¼ 0satisfies the minimal surface equation. This surface con-tribution vanishes due to the boundary condition. From theform of quadratic terms of f, the local minimum is clearlyidentical to the solution without ρ dependence, i.e., theminimal surface (or geodesic) in AdS3. Thus we canconfirm that the surface y ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffil2 − ξ2

pshould be the exact

minimal surface we want even when ρ� is finite.Now, using this minimal surface solution, we can

evaluate the minimal area AðΓAÞ and the holographicentanglement entropy reads

SA ¼ AðΓAÞ4GN

¼ Ld−2

2GN· VolðSd−3Þ

Zρ�

0

dρ

�cosh

ρ

L

�d−2

·Z

l

ϵ coshρL

dylðl2 − y2Þd2−2

yd−2; ð2:40Þ

where VolðSd−3Þ is the volume of a unit radius d − 3dimensional sphere given by

VolðSd−3Þ ¼ 2πd−22

Γðd−22Þ : ð2:41Þ

In the limit ρ�=L ≪ 1, the holographic entanglemententropy looks like

SA ¼ AðΓAÞ4GN

≃ρ�4GN

· ½Aðγð1ÞA Þ þ Aðγð2ÞA Þ�; ð2:42Þ

where Aðγð1;2ÞA Þ is the minimal surface area inQ1;2, given by

Aðγð1ÞA Þ ¼ Aðγð2ÞA Þ

¼ Ld−2 · VolðSd−1Þ ·Z

l

ϵdy

lðl2 − y2Þd2−2yd−2

: ð2:43Þ

This agrees with the (standard) holographic entanglemententropy in CFTd−1 calculated from the minimal surface inPoincare AdSd, by relating the d dimensional Newtonconstant to GN in dþ 1 dimension as

ρ�GN

≃1

GðdÞN

; ð2:44Þ

when ρ� ≪ L. This agrees with (2.33).Now, let us go further to perform the full computation in

(2.40) for finite ρ� using a power expansion with respect tol=ϵ. As a first step, we get

Zl

ϵ coshρL

dylðl2 − y2Þd2−2

yd−2¼ p1

�cosh

ρ

L

�−dþ3

�lϵ

�d−3

þ p3

�cosh

ρ

L

�−dþ5

�lϵ

�d−5

þ � � �

…þ�pd−3ðcosh ρ

LÞ−1ðlϵÞ þ pd−2 þOðϵlÞ d∶ even

pd−4ðcosh ρLÞ−2ðlϵÞ2 þ q logðlϵÞ þOð1Þ d∶ odd

ð2:45Þ

¼(Pðd−2Þ=2

k¼1 p2k−1ðcosh ρLÞ−dþ2kþ1ðϵlÞd−2k−1 þ pd−2 þOðϵlÞ d∶ evenPðd−3Þ=2

k¼1 p2k−1ðcosh ρLÞ−dþ2kþ1ðϵlÞd−2k−1 þ q logðlϵÞ þOð1Þ; d∶ odd

ð2:46Þ

where the coefficients are as follows:

p1 ¼ ðd − 3Þ−1; p3 ¼ −ðd − 4Þ=½2ðd − 5Þ�; � � �pd−2 ¼ ð2 ffiffiffi

πp Þ−1Γððd − 2Þ=2ÞΓðð3 − dÞ=2Þ ðif d is evenÞ;

q ¼ffiffiffiπ

pΓðð4 − dÞ=2ÞΓððd − 1Þ=2Þ ðif d is oddÞ: ð2:47Þ

Accordingly,

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-8

SA ¼ AðΓAÞ4GN

¼ Ld−2

2GN· VolðSd−3Þ

Zρ�

0

dρ

�cosh

ρ

L

�d−2

·Z

l

ϵ coshρL

dylðl2 − y2Þd2−2

yd−2;

¼ πd−22 Ld−2

GNΓðd−22 Þ�Z

ρ�

0

dρ

�cosh

ρ

L

�p1

�lϵ

�d−3

þZ

ρ�

0

dρ

�cosh

ρ

L

�3

p3

�lϵ

�d−5

þ � � ��

¼ πd−22 Ld−1

GNΓðd−22 Þ�sinh ρ�

L

d − 3

�lϵ

�d−3

−ðd − 4Þ2ðd − 5Þ

�sinh 3ρ�

L

12þ 3 sinh ρ�

L

4

��lϵ

�d−5

þ � � �

� � � þ�pd−2

R ρ�=L0 dηðcosh ηÞd−2 þOðϵlÞ; d∶ even

qR ρ�=L0 dηðcosh ηÞd−2 · logðlϵÞ þOð1Þ; d∶ odd

�: ð2:48Þ

First, notice that the above scaling profile of holographicentanglement entropy agrees with general results [22,43] ind − 1 dimensional CFTs. Moreover, by comparing theconstant terms in even d cases (or the log terms in oddd cases) with those in the conventional holographicentanglement entropy from AdSd=CFTd−1, we can obtainthe same relationship as (2.32) between the two Newtonconstants in AdSdþ1 and AdSd, respectively. This obser-vation gives a cross check to (2.32).Some low dimensional results are shown as follows. For

d ¼ 3, i.e., the AdS4=CFT2 case,

SA ¼ L2

Gð4ÞN

sinhρ�Llog

lϵþOð1Þ; ð2:49Þ

whose details will be discussed later in Sec. III B. Ford ¼ 4, i.e., the AdS5=CFT3 case, we have

SA ¼�πL3

Gð5ÞN

sinhρ�L

�lϵ−πL3

Gð5ÞN

�1

2

ρ�Lþ 1

4sinh

2ρ�L

�

þO�ϵ

l

�: ð2:50Þ

For d ¼ 5, i.e., the AdS6=CFT4 case, we obtain

SA ¼�πL4

Gð6ÞN

sinhρ�L

��lϵ

�2

−πL4

Gð6ÞN

�sinh3ρ�

L

12þ3sinhρ�

L

4

�log

lϵ

þOð1Þ: ð2:51Þ

G. Bulk scalar field and spectrum

1. Dimensional reduction

Consider the following metric in AdSdþ1 (we set L ¼ 1)

ds2 ¼ dρ2 þ cosh2 ρdy2 þ dxidxi

y2; ð2:52Þ

describing the wedge bulk spacetime characterized by−ρ� ≤ ρ ≤ ρ�. We consider a massive Klein-Gordon field

ϕ≡ ϕðρ; y; xiÞ with mass m. For the latter, we aim atsolving the corresponding wave equation,

1ffiffiffig

p ∂μðffiffiffig

pgμν∂νϕÞ −m2ϕ ¼ 0 ð2:53Þ

near the end-of-the-world branes Q1 and Q2 enclosing theAdSdþ1 wedge. Remember that in the AdS=CFT corre-spondence, the mass m of the scalar field is dual to a scalaroperator with the conformal dimension Δ ¼ d=2þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðd=2Þ2 þm2

pin the CFTd.

Employing the coordinates above, the equation ofmotion takes the explicit form

1

coshdðρÞ ∂ρðcoshdðρÞ∂ρϕÞ þy2

cosh2ðρÞ�∂2yϕþ

Xd−1i¼1

∂2iϕ

�

−d − 2

cosh2ðρÞ y∂yϕ −m2ϕ ¼ 0: ð2:54Þ

Next, we impose a dimensional reduction in the ρ direction,so that we have the wave equation on AdSd described by

ds2 ¼ dy2þdxidxiy2 ,

y2�∂2yϕþ

Xd−1i¼1

∂2iϕ

�− ðd − 2Þy∂yϕ −M2ϕ ¼ 0; ð2:55Þ

where M corresponds to the Kaluza-Klein mass. We cannow use (2.55) in order to rewrite (2.54) as

1

coshdðρÞ∂ρðcoshdðρÞ∂ρϕÞþM2ϕ

cosh2ðρÞ−m2ϕ¼ 0: ð2:56Þ

We want to solve this equation. For doing so, we considerthe following standard ansatz

ϕ ¼ ϕðρÞφðy; xiÞ: ð2:57Þ

For accuracy, now we restrict our discussion to d ¼ 2.Results in higher dimensional setups can be obtained in a

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-9

similar way. Note that the ds appearing below in the sectionare set to be d ¼ 2.For the dimensionally reduced system on AdSd, by

following the described procedure, we would end up witha modified conformal dimension in CFTd−1 of the form

Δ ¼ d − 1

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðd − 1Þ2

4þM2

r: ð2:58Þ

The latter would then determine the correlation functions.Having said this, let us consider the Dirichlet boundary

condition on the surfaces Q1 and Q2:

ϕðρ�Þ ¼ ϕð−ρ�Þ ¼ 0: ð2:59Þ

Imposing the first equality in (2.59) on the wave function ϕ,we find the following ρ dependent part

ϕþðρÞ¼½PθηðρÞQθ

ηðρ�Þ−Pθηðρ�ÞQθ

ηðρÞ�×ffiffiffiffiffiffiffiffiffiffiffi1− ρ2

pQθ

ηðρ�Þ; ð2:60Þ

where, for simplicity, we have defined

ρ¼ tanhðρÞ; ρ� ¼ tanhðρ�Þ;

θ¼ffiffiffiffiffiffiffiffiffiffiffiffiffim2þ1

p; η¼Π−1

2; Π¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4M2þ1

p: ð2:61Þ

The function Pmn denotes the associated Legendre function

of the first kind, whereas Qmn corresponds to the associated

Legendre function of the second kind. It is useful to notethat when

θ ¼ ηþ 1; ð2:62Þ

the expression in (2.60) does vanish.In order to satisfy the second boundary condition in

(2.59), we should solve the following equation

ϕþðρ ¼ −ρ�Þ ¼ 0 ð2:63Þ

for the variable η. The final result, rewritten as M using(2.61), will depend on the AdSdþ1 mass m and branelocation ρ�.We may alternatively impose the Neumann boundary

condition, i.e.,

ϕ0ðρ�Þ ¼ ϕ0ð−ρ�Þ ¼ 0: ð2:64Þ

For this case, the exact solution for the ρ dependent partsatisfying the first condition in (2.64) takes the following,more complicated form

ϕþðρÞ ¼ ½ð2θ − Π − 1Þ½PθηðρÞQθ

1þηðρ�Þ − Pθ1þηðρ�ÞQθ

ηðρÞ�þ ðΠρ� − ρ�Þ½Pθ

ηðρÞQθηðρ�Þ − Pθ

ηðρ�ÞQθηðρÞ��

×

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ρ2

pð2θ − Π − 1ÞQθ

1þηðρ�Þ þ ðΠρ� − ρ�ÞQθηðρ�Þ

:

ð2:65Þ

It can be seen that when

2θ − Π ¼ 1 ⇔ θ ¼ ηþ 1; ð2:66Þ

the solution (2.65) does vanish as before. In the following,we discuss the mass spectrum in more detail when ρ� istaken to be very small.Before doing so, let us note that for both boundary

conditions, we can find the mass spectrum in the dimen-sionally reduced systemMn as a function of m and ρ�. Thetwo point function of the dual scalar operator OðxÞ in theCFTd−1 then behaves as

hOðx1ÞOðx2Þi ¼1

jx1 − x2j2Δ; ð2:67Þ

where the conformal dimension is given by (2.58). Insteadof the full numerical analysis, we below give the massspectrum in the small ρ� limit by bringing the problem intoa Schrödinger-like form. As we will see, for the Neumannboundary condition, the lowest mass in AdSd is given byM0 ¼ m, while in the Dirichlet case, the lowest mass isheavy, i.e., M1 ¼ Oð1=ρ�Þ.

2. Schrödinger analysis

By introducing a rescaled function ψðρÞ, such that

ψðρÞ ¼ coshðρÞϕðρÞ; ð2:68Þ

we can transform the equation of motion (2.56) into thefollowing Schrödinger-like form

−∂2ρψ þ VðρÞψ ¼ 0; ð2:69Þ

where

VðρÞ ¼ 1þm2 −M2

cosh2 ρþ tanh2 ρ: ð2:70Þ

Dirichlet boundary condition.—When we impose theDirichlet boundary condition ψðρ�Þ ¼ ψð−ρ�Þ ¼ 0 or(2.59), the problem is similar to the quantum mechanicsin a box. In particular, when ρ� is very small, we canapproximate the potential (2.70) as

VðρÞ ≃ 1þm2 −M2: ð2:71Þ

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-10

Since in this small ρ� approximation, we can solve theboundary condition as

ψðρÞ ∝ sin

�nπ2

�ρ

ρ�þ 1

��; ðn ¼ 1; 2; 3;…Þ; ð2:72Þ

the spectrum of M can be found as follows

Mn ≃

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þm2 þ n2π2

4ρ2�

s; ðn ¼ 1; 2; 3;…Þ: ð2:73Þ

Therefore, the lowest excitation givesM1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þm2þ π2

4ρ2�

q≃

π2ρ�. In this case, we should be careful because of an artificial

zero point of ϕðρÞ that arises when 12ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4M2

p− 1Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þm2p

. Note that the latter observation precisely agreeswith the vanishing condition in (2.62) and (2.66).

Neumann boundary condition.—If we impose theNeumannboundary condition (2.64), then when ρ� is very small, wefind the obvious lowest mode

ϕðρÞ ¼ const: ð2:74Þ

This leads to the lowest spectrum M0 ¼ m. For excitedmodes, we find

ψðρÞ ∝ cos

�nπ2

�ρ

ρ�þ 1

��; ðn ¼ 1; 2; 3;…Þ; ð2:75Þ

which leads to the same spectrum as in (2.73).

III. WEDGE HOLOGRAPHY FOR d = 3

Here we focus on d ¼ 3 case of the wedge holography,where the dual CFT is two dimensional and study how thisholographic duality works in detail.

A. Conformal anomaly

First we would like to study the behavior of free energyof the dual theory on S2 and extract the value of centralcharge from the conformal anomaly as an extension ofholographic weyl anomaly [44]. By setting d ¼ 3 in thefree energy (2.29) with (2.30), we obtain the followingbehavior in the limit r∞ ¼ − log ϵ → ∞:

IW ¼ −L2

2GNϵ2sinh

ρ�Lþ L2

GNsinh

ρ�L· log ϵþOð1Þ: ð3:1Þ

Note thatweneglect theHayward contribution IH as this doesnot contribute to the conformal anomaly. Since the Euclideanaction of a two dimensional CFT include the conformalanomaly term as I2dCFT ¼ #ϵ−2 þ c

6χðΣÞ log ϵþOð1Þ, where

c is the central charge of the two dimensional CFTand χðΣÞ is

the Euler character of the two dimensional manifold Σ. SinceχðS2Þ ¼ 2 we obtain

c ¼ 3L2

GNsinh

ρ�L: ð3:2Þ

B. Holographic entanglement entropy

Next we would like to analyze the holographic entan-glement entropy in the d ¼ 3 case. We consider thenoncompact setup in Sec. II F. The dual two dimensionalCFT is defined on R2 and we define the entanglemententropy SA by choosing the subsystem A on Σ as theinterval −l ≤ w ≤ l at a time t ¼ 0. By setting d ¼ 3 in thegeneral result (2.40) we obtain the holographic entangle-ment entropy SA when A is a length 2l interval:.

SA¼L

2GN

Zρ�

−ρ�dρ

�cosh

ρ

L

�Zl

ϵcoshρL

dyl

yffiffiffiffiffiffiffiffiffiffiffiffiffil2−y2

p¼ L2

GNsinh

ρ�Llog

2lϵ−4L2

Zρ�=L

0

dw coshðwÞ log coshðwÞ;

ð3:3Þ

which agrees with the well-known form [45] in a twodimensional CFT SA ¼ c

3log 2l

ϵ þ ðconst:Þ. This leads to theprecisely same identification of the central charge as that of(3.2). We can view this as a quantitative test of the wedgeholography proposal.It is also intriguing to consider more general choices of

the subsystem A. When A consists of two disjoint intervals,we can again calculate the holographic entanglemententropy (2.35) using two spherical minimal surfaces asin (2.37). Since the resulting entanglement entropy com-puted from such spheres takes the form of the logarithmicfunction of l as in (3.3), the result of holographic entan-glement entropy for the two intervals also simply repro-duces the known results of holographic CFTs [46,47]. Inthis case, we know that there is a phase transitionphenomenon between the connected and disconnectedminimal surfaces. The same is true when A consists ofmultiple disjoint intervals. In this way, the holographicentanglement entropy in wedge holography perfectlyreproduces know results in holographic CFTs.

C. Three dimensional gravity viewpoint

If we apply the Brown-Henneaux formula [48] to thed ¼ 3 dimensional gravity on Q1 and that on Q2 with

the Newton constantGð3ÞN , we find the central charges of the

dual two dimensional CFTs

c1 ¼ c2 ¼3L

2Gð3ÞN

: ð3:4Þ

Since the relation (2.32) at d ¼ 3 leads to

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-11

1

Gð3ÞN

¼ LGN

sinhρ�L; ð3:5Þ

we can confirm that the central charge(3.2) is the sum of theabove two central charges c ¼ c1 þ c2 as we expect.

IV. WEDGE HOLOGRAPHY FOR d = 2

Here we would like to focus on d ¼ 2 setup of the wedgeholography depicted in Fig. 2. In this case, the dual CFTlives in d − 1 ¼ 1 dimension and therefore one maywonder whether this theory turns out to be a Schwarziantheory as in the JT gravity [49] or some other theory. Noticethat since our setup is based on three dimensional gravityrather than two dimensional one, we can assume a standardEinstein gravity, which looks different from the situation intwo dimension.Since the wedge W3 is a part of the AdS3 geometry, we

have the advantage that all solutions to the Einsteinequation with the Neumann boundary condition on Q1

and Q2 is locally expressed as the metric (2.1) or (2.2). Inthe Lorentzian signature the metric is written as

ds2 ¼ L2

�dz2 − dt2 þ dx2

z2

�

¼ dρ2 þ L2 cosh2ρ

L

�−dt2 þ dy2

y2

�: ð4:1Þ

The surfaces Q1 and Q2 are again identified with ρ ¼ −ρ�and ρ ¼ ρ�. We can change the shape of Σ by deforming thechoice of the UV cutoff. We specify the form of Σ by

z ¼ gðtÞ: ð4:2Þ

A. Bulk on-shell action

The total gravity action in the Lorentzian signature lookslike

IG¼ 1

16πGN

ZW

ffiffiffiffiffiffi−g

p ðR−2ΛÞþ 1

8πGN

ZQ1∪Q2

ffiffiffiffiffiffi−γ

p ðK−TÞ

þ 1

8πGN

ZΣ

ffiffiffiffiffiffi−γ

pK: ð4:3Þ

Note that in our setup we have

R ¼ −6

L2; Λ ¼ −

1

L2; T ¼ 1

Ltanh

ρ�L: ð4:4Þ

The induced metric on Σ is

ds2 ¼ L2−ð1 − _g2Þdt2 þ dx2

g2; ð4:5Þ

and its extrinsic curvature Kab ¼ ð∇aNbÞΣ for the normalvector Na

ðNz;Nt; NxÞ ¼ g

Lffiffiffiffiffiffiffiffiffiffiffiffi1 − _g2

p ð−1;−_g; 0Þ; ð4:6Þ

has the trace value

KjΣ ¼ 2 − 2_g2 − gg

Lð1 − _g2Þ3=2 : ð4:7Þ

Thus we find

IW ¼ −1

4πGNL2

ZW

ffiffiffiffiffiffi−g

p þ tanh ρ�L

8πGNL

ZQ1∪Q2

ffiffiffiffiffiffi−h

pþ 1

8πGN

ZΣ

ffiffiffiffiffiffi−γ

pKΣ

¼ −1

2πGN

Zρ�

0

dρcosh2ρ

L

Zdt

Z∞

gðtÞ coshρL

dyy2

þ L sinh ρ�L cosh ρ�

L

4πGN

Zdt

Z∞

gðtÞ coshρ�L

dyy2

þ L4πGN

Zdt

ZgðtÞ sinhρ�L

0

dx

�2 − 2_g2 − ggg2ð1 − _g2Þ

�

¼ L sinh ρ�L

4πGN

Zdtg

�1 −

gg1 − _g2

�: ð4:8Þ

We introduce the Weyl scaling factor as follows:

e2φ ¼ 1 − _g2

g2: ð4:9Þ

Assuming the usual UV cutoff property g ≪ 1, we canexpand

g ≃ e−φ −1

2e−3φ _φ2 þ · · ·: ð4:10Þ

This leads to

IW≃Lsinhρ�

L

4πGN

Zdtg¼Lsinhρ�

L

4πGN

Zdt

�eφþ1

2_φ2e−φ

�; ð4:11Þ

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-12

where we performed a partial integration. The kinetic termof φ in (4.11) is proportional to the Schwarzian action (referto the Appendix A for a brief review of Schwarzian action).

B. Hayward term contribution

However, it is still too early to conclude as there isanother contribution from the Hayward term (2.13) as wewill see below. We write the angle between Q1;2 and Σ byηþ π

2, where η is given by

sin η ¼ Nð1Þ · Nð2Þ ¼ tanh ρ�Lffiffiffiffiffiffiffiffiffiffiffiffi

1 − _g2p : ð4:12Þ

Nð1Þ and Nð2Þ are the normal vector on Σ and Q2, explicitlygiven by

ðNð1Þz; Nð1Þt; Nð1ÞxÞ ¼ g

Lffiffiffiffiffiffiffiffiffiffiffiffi1 − _g2

p ð−1;−_g; 0Þ;

ðNð2Þz; Nð2Þt; Nð2ÞxÞ ¼ gL cosh ρ�

L

�− sinh

ρ�L; 0; 1

�: ð4:13Þ

We can evaluate the (Lorentzian) Hayward term asfollows

IH ¼ L4πGN

Zdt

ffiffiffiffiffiffiffiffiffiffiffiffi1 − _g2

pg

�π

2− η

�

¼ L8GN

Zdteφ −

L4πGN

Zdteφη: ð4:14Þ

By expanding (4.12) assuming _g ≪ 1, we find

η ¼ η0 þ1

2sinh

ρ�L· _g2 þ · · ·; ð4:15Þ

where η0 is defined by sin η0 ¼ tanh ρ�L. Thus we can

approximate (4.14) as follows

IH≃L

4πGN

Zdt

��π

2−η0

�eφ−

1

2sinh

ρ�L· _φ2e−φ

�: ð4:16Þ

C. Total gravity action

By adding (4.11) and (4.16), we obtain the total con-tribution to the free energy:

IW þ IH ≃L

4πGN

Zdt

�sinh

ρ�Lþ π

2− η0

�eφ: ð4:17Þ

Thus we can conclude that there is no contribution whichlook like the Schwarzian action. One may worry that thiscancellation can depend on the details of UV regulariza-tion. However, as we show in the Appendix B, we againfind no Schwarzian action term in another choice ofthe regularization where the Hayward contribution is

vanishing. Also this is consistent with the fact that asopposed to the JT gravity, our three dimensional Einsteingravity manifestly preserves the conformal symmetry. Insummary, we expect the dual CFT1 is not a Schwarziantheory but a conformal quantum mechanics or a certaintopological theory.If we were to start with the Dirichlet boundary condition

also on both Q1 and Q2, then we would not need to add theterm − T

8πGN

R ffiffiffiffiffiffi−γpin (4.3). In this case, the bulk contri-

bution vanishes and we find the extra contribution inaddition to (4.11):

ΔIW ≃L sinh ρ�

L

4πGN

Zdt

�eφ þ 1

2_φ2

�

¼ L sinh ρ�L

4πGN

Zdteφð1 − Schðt; uÞÞ; ð4:18Þ

where we defined the coordinate u by eφdt ¼ du andSchðt; uÞ is the Schwarzian action.This suggests the following interpretation. The latter

Dirichlet setup of our wedge holography is dual to twoCFTs on AdS2 coupled to a quantum mechanics on thedefect. This defect quantum mechanics is effectivelydescribed by the Schwarzian action (4.18). In the originalNeumann setup, the CFTs on AdS2 are coupled to quantumgravity. Therefore, it absorbs some degrees of freedom andthis cancels the previous degrees of freedom of defectquantum mechanics given by the Schwarzian action.

V. WEDGE HOLOGRAPHY AS INTERACTINGTWO CFTs

Here we would like to consider the limit where the size(i.e., width) of Σ is strictly vanishing and its holographicinterpretation. In the Poincare AdSdþ1, it is straightforwardto realize this limit by choosing the boundary surfaces Q1

and Q2 to be

Q1∶ x¼−sinhρ�Lðz− ϵÞ; Q2∶ x¼ sinh

ρ�Lðz−ϵÞ: ð5:1Þ

In this limit, we obtain the setup of the left picture inFig. 6, where the surface Σ is now d − 1 dimensional. Herewe impose the Neumann boundary condition on the ddimensional surfacesQ1 andQ2 and the Dirichlet boundarycondition on Σ. This corresponds to a limit of the wedgeholography with the vanishing size of Σ.On the other hand, as a different setup, we can also

impose the Neumann boundary condition also on Σ i.e., theright picture in Fig. 6, which is holographic dual to agravity on a manifold defined by two AdSd glued alongtheir boundaries. If we apply the brane-world holography[6–8], the gravity on the wedge Wdþ1 is dual to the(quantum) gravity on two copies of AdSd glued alongtheir boundaries. Therefore, by applying the AdS=CFT

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-13

holography once more, we expect that this theory isequivalent to two d − 1 dimensional CFTs on Σd−1,interacting as the common metric is dynamical and fluc-tuating owing to the Neumann boundary condition on Σd−1.Wewill analyze these two setups with different boundary

conditions below.

A. Dirichlet boundary condition on ΣThe gravity action for the region surrounded by (5.1) can

be computed as in our previous analysis. Actually we findthat the gravity action IW defined by

IW ¼ −1

16πGN

ZW

ffiffiffig

p ðR − 2ΛÞ

−1

8πGN

ZQ1∪Q2

ffiffiffih

pðK − TÞ ð5:2Þ

is vanishing in this wedge background:

IW ¼ 1

16πGN

ZW

2dL2

ffiffiffig

p−

tanh ρ�L

8πGNL

ZQ1∪Q2

ffiffiffiγ

p

¼ dVd−1Ld−1

4πGN

Z∞

ϵ sinhρ�L

dxZ

∞

xsinh

ρ�L

dzzdþ1

−tanh ρ�

L Vd−1Ld

4πGNL

�cosh

ρ�L

�dZ

∞

ϵ coshρ�L

dyyd

¼ Vd−1Ld−1 sinh ρ�L

4ðd − 1ÞπGNϵd−1 −

Vd−1Ld−1 sinh ρ�L

4ðd − 1ÞπGNϵd−1 ¼ 0: ð5:3Þ

This result differs from (2.11) as the latter includes thecontribution from the small but nonzero size of Σ.Also the Hayward contribution (2.13), where in the

present setup we have

IH ¼ 1

4πGN

ZΣ

�θ −

π

2

� ffiffiffiγ

p ð5:4Þ

is evaluated to be the same as (2.16). Since IW ¼ 0, the fullgravity action in this Dirichlet case is given by (2.16).If we apply the brane world holography, we will find a

copy of AdSd attached along their boundaries Σd−1.Performing another AdS=CFT, the setup is dual to twoCFTd−1 s on Σd−1. Since we impose the Dirichlet boundaryon Σ, there is no dynamical gravity coupled to them. Onemay worry that they look decoupled, which may contradictwith the fact that in the wedge geometry Wdþ1, the twoasymptotic regions are connected. However, the two CFTsare interacting through the large N CFTd degrees offreedom, though they are not via a dynamical gravityon Σ.

B. Neumann boundary condition on ΣTo impose a Neumann boundary condition which fixes

the value of θ, we need to add to the Hayward term (5.4) acosmological constant term localized on Σ:

IC ¼ κ

8πGN

ZΣ

ffiffiffiγ

p: ð5:5Þ

The vanishing variation condition δðIW þ IH þ ICÞ ¼ 0leads to the following condition (Neumann boundarycondition on Σ):

2θ − π þ κ ¼ 0: ð5:6Þ

Thus for any θ we want to realize, we can choose κ so thatthe Neumann boundary condition is satisfied. In this casewith the Neumann boundary condition (5.6), we find thatthe total action vanishes

IW þ IH þ IC ¼ 0: ð5:7Þ

This vanishing action is not surprising. We can easily see,for example, that the pure AdS with the cutoff z ≥ ϵ, wherewe impose the Neumann boundary condition (2.5) on thecutoff surface, the gravitational action vanishes, by addingthe contribution T

8πGðdÞN

RRd−1

ffiffiffih

pto the Dirichlet computa-

tion (2.18).In this Neumann case, the classical gravity on the dþ 1

dimensional wedge region is dual to a gravity on the ddimensional space Q1 ∪ Q2. Since Q1 and Q2 are bothAdSd, this gravity lives on a space obtained by gluing twoAdSd along their boundary Rd−1 as depicted in Fig. 7.In the limit θ ≃ π

2(i.e., ρ� ≫ L), we expect the gravity on

AdSd gets weakly coupled in terms of the CFT metric.By approximating this d dimensional gravity by theEinstein gravity, the bulk gravity action can be expressedas follows:

FIG. 6. Other two versions of wedge holography setups. Theleft (and right) setup imposes the Dirichlet (and Neuman)boundary condition on the tip Σ of the wedge.

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-14

Igravity ¼ −1

16πGðdÞN

ZQ1

ffiffiffig

p ðR − 2ΛðdÞÞ

−1

8πGN

ZΣ

ffiffiffih

pðKð1Þ − TÞ

−1

16πGðdÞN

ZQ2

ffiffiffig

p ðR − 2ΛðdÞÞ

−1

8πGN

ZΣ

ffiffiffih

pð−Kð2Þ − TÞ; ð5:8Þ

where ΛðdÞ ¼ − ðd−1Þðd−2Þ2L2 . On their common boundary Σ,

we impose the Neumann boundary condition which con-nect the two AdSd regions:

hð1Þαβ ¼ hð2Þαβ ð≡γabÞ;ðKð1Þ

αβ − Kð1ÞγαβÞ − ðKð2Þαβ − Kð2ÞγαβÞ ¼ −2T γαβ; ð5:9Þ

where hð1Þαβ and Kð1Þαβ are the metric and extrinsic curvature

on Σ in the left AdSd (i.e., Q1), while hð2Þαβ andKð2Þ

αβ are those

in the right one Q2. T is the tension of each boundary,which gives totally 2T

8πGðdÞN

RΣ

ffiffiffiγ

pcontribution to the total d

dimensional gravity action.As in Sec. II E, we relate the dþ 1 wedge geometry to

AdSd via ds2wedge ¼ dρ2 þ cosh2 ρL ds

2AdSd

. Then the extrin-sic curvature on Σ for the AdSd gravity is estimated as

Kð1Þαβ ¼ −Kð2Þ

αβ ¼ d − 1

L; ð5:10Þ

which leads to the value of the tension

T ¼ d − 2

L; ð5:11Þ

by solving the boundary condition (5.9). In this limitρ� ≫ L (or equally θ ≃ π

2), we can indeed confirm the

boundary cosmological constant term in the d dimensionalgravity agrees with that in the dþ 1 dimensional one IC:

2T

8πGðdÞN

ZΣ

ffiffiffiγ

p≃

κ

8πGN

ZΣ

ffiffiffiγ

p; ð5:12Þ

where we substituted γαβ ¼ cosh ρ�L · γαβ, κ ¼ π − 2θ,

cos θ ¼ 1coshρ�L

and (2.34).

We can apply the AdS=CFT once more to the ddimensional gravity on two AdSd regions glued each other.If we impose the Neumann boundary condition on Σ, whichleads to dynamical gravity on Σ, the holographic dual isgiven by the CFT1 and CFT2, which are now interacting viathe common metric field γαβ. This interacting two CFTs canbe explicitly described by the total CFT action

Stot ¼ SCFTðΦð1Þ; γÞ þ SCFTðΦð2Þ; γÞ − TZN

ffiffiffiγ

p; ð5:13Þ

where we path-integrate over the metric γαβ. HereSCFTðΦð1;2Þ; γÞ are the CFTd−1 actions in a curved back-ground with the metric γαβ. In the linear level analysis, we

expect a double trace interactionR ffiffiffi

γp

Tð1Þαβ T

ð2Þαβ, where

Tð1;2Þαβ are energy stress tensor in the two CFTs. It will be an

intriguing future problem to study directly the d dimen-sional gravity dual of this two interacting d − 1 dimen-sional CFTs.

VI. BTZ AND WEDGE HOLOGRAPHY

As an example of wedge holography at finite temper-ature, we would like to consider that in the BTZ black holespacetime. For this we first investigate the profiles of end ofthe world-branes in the BTZ background where we imposethe Neumann boundary condition for the metric, namelysolutions to (2.5). Since the BTZ geometry can be locallyequivalent to the Poincare AdS3 via a coordinate trans-formation, let us start with the Poincare AdS3.

A. End-of-the-world branes in Poincare AdS3

General solutions to the Neumann boundary condition(2.5) in the Poincare AdS3 (4.1) are given in the form

ðz − αÞ2 þ ðx − pÞ2 − ðt − qÞ2 ¼ β2; ð6:1Þ

where α, p, q and β are arbitrary real valued constants (wetake β ≥ 0). In this solution, the tension T is found to be

T ¼ α

βL; ð6:2Þ

assuming that we choose the inside of the region (6.1) as aphysical space for which we consider the AdS/BCFTholography. If instead we choose the outside region, thetension is given by (6.2) with −1 multiplied.In particular, by taking the limits where ðα; p; qÞ goes to

infinity, we can find the plane solutions of the form

FIG. 7. Gluing two AdS backgrounds.

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-15

αzþ px − qt ¼ const: ð6:3Þ

This plane boundary surface Q was already employedmany times in this paper as our basic setup of wedgeholography as in Fig. 2.

B. End-of-the-world branes in BTZ

The BTZ metric

ds2 ¼ −ðr2 − r20ÞdT2 þ L2dr2

r2 − r20þ r2dX2; ð6:4Þ

can be mapped into the Poincare AdS3 via the coordinatetransformation

t − x ¼ −er0L ðX−TÞ

ffiffiffiffiffiffiffiffiffiffiffiffi1 −

r20r2

r; tþ x ¼ e

r0L ðXþTÞ

ffiffiffiffiffiffiffiffiffiffiffiffi1 −

r20r2

r;

z ¼ r0rer0LX: ð6:5Þ

By mapping the solution (6.1) at p ¼ q ¼ 0 and thenperforming appropriate translation along X direction ineach case, we obtain the static profile of the end-of-the-world brane in BTZ. When LT < 1, we obtain (refer toFig. 8):

rðXÞ ¼ r0TLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − T2L2

psinh r0X

L

: ð6:6Þ

When LT ¼ 1 we have

rðXÞ ¼ 2r0e−r0XL : ð6:7Þ

When LT > 1 we find

rðXÞ ¼ r0TLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2L2 − 1

pcosh r0X

L

: ð6:8Þ

Note that equally, the translated profile rðX − X�Þ in theabove solutions is also solution to (2.5). In each case ofLT < 1, LT ¼ 1 and LT > 1, we find the world volume is

identical to AdS2, R2 and dS2, respectively. If we set p,q ≠ 0 in (6.1), then we get more profiles but they are time-dependent, which we will study elsewhere as a futureproblem. Below, in this section, we focus on LT < 1,which corresponds to a timelike boundary in a two dimen-sional CFT, to construct a basic example of wedgeholography in BTZ. However, in Sec. VII, we will studythe case LT > 1 in Poincare AdS3, which corresponds to aspacelike boundary in a two dimensional CFT.

C. Gravity action

We would like to evaluate the gravity action for a wedgegeometry in the Euclidean BTZ space (note the periodicitytE ∼ tE þ β with β ¼ 2πL

r0):

ds2 ¼ ðr2 − r20Þdt2E þ L2dr2

r2 − r20þ r2dx2; ð6:9Þ

defined by the region surrounded by the surfaces of theprofile (6.6)

−r0TLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−T2L2p

sinhr0xL; <rðxÞ< r0TLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−T2L2p

sinhr0xL

; ð6:10Þ

whose boundaries define Q1 and Q2. We assumed0 < TL < 1. We put a UV cutoff as

r < r∞ ¼ 1

ϵ; ð6:11Þ

which introduces the surface Σ, whose (infinitesimallysmall) width Δx is estimated as

Δx ≃2L2Tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − L2T2

p ϵþOðϵ3Þ: ð6:12Þ

We can also evaluate the trace of extrinsic curvatureon Σ as

KjΣ ¼ 2r2∞ − r20Lr∞

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2∞ − r20

p ≃2

LþO

�r40r4∞

�: ð6:13Þ

Also the trace of the extrinsic curvature onQ1 andQ2 readsKjQ ¼ 2T. The induced metric on Q looks like

ds2 ¼ r20½1 − ð1 − L2T2Þcosh2 r0xL �

ð1 − L2T2Þsinh2 r0xL

dt2E

þ L2T2r20ð1 − L2T2Þsinh2 r0x

L ½1 − ð1 − L2T2Þcosh2 r0xL �

dx2:

ð6:14Þ

We also introduce x� as the coordinate value at theintersection of Q2 and the horizon r ¼ r0 which is thesolution toFIG. 8. Profiles of end-of-the-world branes in BTZ.

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-16

sinhr0x�L

¼ LTffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − L2T2

p : ð6:15Þ

Finally, the gravity action is estimated as follows:

IW ¼ 1

4πGNL2

ZW

ffiffiffig

p−

T8πGN

ZQ1∪Q2

ffiffiffih

p−

1

4πGNL

ZΣ

ffiffiffiγ

p

¼ β

4πGNL2

�Δx

Z1=ϵ

r0

Lrdrþ 2

Zx�

Δx=2dx

ZrðxÞ

r0

Lrdr�

−Tβ

4πGN

Zx�

Δx=2

LTr20ð1 − L2T2Þsinh2 r0x

L

−βΔx

4πGNLr∞

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2∞ − r20

q

¼ −βLT

4πGNϵffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − L2T2

p −βr20x�4πGNL

: ð6:16Þ

If we write T ¼ 1L sinh

ρ�L then we can rewrite the above

result as

IW ¼ −β sinh ρ�

L

4πGNϵ−

ρ�2GN

: ð6:17Þ

Notice that the finite term is equal to minus of the twice ofthe boundary entropy [50] (for a surface with the tensionT ¼ 1

L sinhρ�L ) computed in [18–20]:

Sbdy ¼ρ�4GN

: ð6:18Þ

This makes sense as Σ can be regarded as a merger of twoboundaries, each has the tension T ¼ 1

L sinhρ�L . This result

tells us that the holographic dual theory of the gravity onthe wedgeW is one dimensional theory (quantum mechan-ics) which has the degree of freedom given by the aboveboundary entropy.

VII. SPACELIKE BOUNDARIES IN BCFT ANDGRAVITY DUALS

We would like to explore gravity duals of spacelikeboundaries in two dimensional Lorentzian CFTs. Thiscorresponds to the values of tension jTjL > 1 in thesolutions (6.1) with the tensions(6.2). In the bulk gravity,they correspond to timelike end of world branes, whoseinduced metrics coincide with de Sitter spaces (refer to [51]for a recent application of such branes to brane worldscenario). As we will see below, we can regard this as a newclass of wedge holography in a Lorentzian AdS.

A. Euclidean AdS/BCFT and entanglement entropy

Let us first start with the familiar Euclidean AdS/BCFTsetup where the gravity geometry is given by the followingregion inEuclideanPoincareAdS3 ds2¼L2

z2 ðdτ2þdx2þdz2Þ:

z > ϵ; τ > λz: ð7:1Þ

as depicted in the left picture of Fig. 9 (λ > 0) and Fig. 10(λ < 0). The boundary τ ¼ λz defines the surfaceQ (i.e., theend-of-the-world brane) and the parameter λ is related to thetension as follows

λ ¼ sinhρ�L; TL ¼ − tanh

ρ�R: ð7:2Þ

This is solved as a function of the tension T

TL ¼ −λffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ λ2p : ð7:3Þ

Therefore the tension takes the values in the range jTj ≤ 1=L.In this case, let us consider the geodesic which starts fromðτ; zÞ ¼ ðτ0; ϵÞ on the AdS boundary and ends at a point onQ. The minimal geodesic length is realized when the latterpoint is at ðτ; zÞ ¼ ð τ0ffiffiffiffiffiffiffiffi

1þλ2p ; λτ0ffiffiffiffiffiffiffiffi

λ2þ1p Þ and its length leads to the

contribution to HEE:

FIG. 9. Small wedge regions of AdS/BCFT in the Euclidean(left) and Lorentzian (right) setup. The boundary surface Q inthe former and latter case has the tension −1=L < T < 0 andT < −1=L, respectively.

FIG. 10. Large wedge regions of AdS/BCFT in the Euclidean(left) and Lorentzian (right) setup. The boundary surface Q in theformer and latter case has the tension 0 < T < 1=L and T > 1=L,respectively.

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-17

SA ¼ c6log

�2τ0ϵ

ðffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 þ 1

p− λÞ

�: ð7:4Þ

At the initial time τ0 ¼ OðϵÞ → 0, the entanglement entropyvanishesSA as the boundary state does not have any real spaceentanglement up to the cutoff scale ϵ [52]. Notice also that thetwo dimensional geometry of the boundary surfaceQ is givenby the hyperbolic space H2 (Euclidean AdS2) with the AdSradius

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ λ2

pL.

B. Wick rotation to Lorentzian AdS/BCFT

Now we perform the Wick rotation τ ¼ it. Thisleads to the Lorentzian Poincare AdS3 metic ds2 ¼L2

z2 ð−dt2 þ dx2 þ dz2Þ. By introducing λ ¼ iλ, we find

TL ¼ −λffiffiffiffiffiffiffiffiffiffiffiffi

λ2 − 1p ; ð7:5Þ

where we assume that the surface Q given by t ¼ λz istimelike. Indeed, in gravity we usually do not admitspacelike boundaries. Also notice that now the tensiontakes the values of

jTj > 1=L: ð7:6Þ

For λ > 0 (TL < −1), we can find a spacelike geodesic byanalytical continuation from the previous Euclidean one(7.2), as in the right of Fig. 9. This leads to

SA ¼ c6log

�2t0ϵðλ −

ffiffiffiffiffiffiffiffiffiffiffiffiλ2 − 1

pÞ�: ð7:7Þ

For λ < 0 (TL > 1) there is no spacelike geodesicwhich ends on Q from a point ðt; zÞ ¼ ðt0; ϵÞ on theAdS boundary. This is depicted in the right of Fig. 10.Below we study further these two setups, mainly focus-

ing on the first one. In these cases, the two dimensionalgeometry of the boundary surface Q is given by the de

Sitter space dS2 with the dS radiusffiffiffiffiffiffiffiffiffiffiffiffiλ2 − 1

pL.

C. Spacelike boundaries with λ > 0

Now let us focus on the spacelike CFT boundary withλ > 0 (i.e., the right of Fig. 9) and discuss the interpretationof the entropy (7.7). We argue that the wedge Lorentziangeometry (7.1) is dual to a BCFT defined by a LorentzianCFT with a spacelike boundary at t ¼ 0.However, we should distinguish our setup from that

given by a time evolution of a UV regulated boundary state:

jΨðtÞiHH ¼ N e−iHte−βH=4jBi; ð7:8Þ

whose gravity dual is known to be the BTZ geometry with atimelike boundary surface given in [53]. As opposed to our

setup, the boundary surface Q in this geometry does notend on the Lorentzian AdS boundary and therefore does notcorrespond to a Lorentzian BCFT with a spacelike boun-dary, though we might regard this as a regularized spacelikeboundary. Indeed the time evolution of entanglemententropy for (7.8) is given by the expression [53]:

SA ¼ c6log

�β

ϵcosh

�πt0β

��; ð7:9Þ

which is totally different from (7.7).Instead our setup is expected to be dual to the time

evolution of a nonregularized boundary state

jΨðtÞiour ¼ N e−iHtjB0i; ð7:10Þ

where jB0i denotes a nonregulated boundary state. It is alsouseful to consider the behavior of boundary entropy [50](or g-function). In the Euclidean setup (7.2) the boundaryentropy is found as

Sbdy¼ logg¼c6log

� ffiffiffiffiffiffiffiffiffiffiffiffiλ2þ1

p−λ

¼c6log

ffiffiffiffiffiffiffiffiffiffiffi1−T1þT

r: ð7:11Þ

In our Lorentzian continuation, this boundary entropy getscomplex valued

Sbdy ¼ log g ¼ c6log

ffiffiffiffiffiffiffiffiffiffiffiffi1 − T1þ T

r

¼ c6log

h−i�λ −

ffiffiffiffiffiffiffiffiffiffiffiffiλ2 − 1

p i¼ c

6log

hλ −

ffiffiffiffiffiffiffiffiffiffiffiffiλ2 − 1

p i−πc12

i: ð7:12Þ

However, owing to this, the total entropy becomes positivevalued as in (7.7). This suggests the dual Lorentzian BCFTassumes an exotic choice of boundary state. In Sec. VII E,we will show how we obtain such a boundary state from aordinary one via a Wick rotation of a given CFT.Finally we would like to ask the precise meaning of the

entropy (7.7). One possibility is that we regard (7.7) as theholographic pseudo entropy [54], where the initial statejψ1i is chosen to (7.10), while the final state is the CFTvacuum jψ2i ¼ j0i. In the replica calculation the pseudoRenyi entropy is evaluates as

SðnÞA ¼ 1

1 − nlogh0jσnσnjB0i: ð7:13Þ

For example, in a massless free Dirac fermion CFTwe canevaluate this explicitly (see the calculation in [55])

SðnÞA ¼ 1

6

�1þ 1

n

�log

�tx

ϵffiffiffiffiffiffiffiffiffiffiffiffiffiffit2 − x2

p�; ð7:14Þ

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-18

where the subsystem A is chosen to be the interval ½0; x� attime t. Note that at late time t ≫ x, we find the standardresult SA ¼ c

3log x

ϵ. On the other hand, when t < x we havecomplex valued entropy and in the limit t → 0 we find thebehavior SA ∼ c

6log t

ϵ þ πi12which agrees with (7.7) up to the

imaginary constant. This appearance of a complex valueitself is not surprising because the pseudoentropy takescomplex values in general. However, we expect theimaginary part disappears in the large c limit of holographicCFTs because the boundary entropy gets complex valuedsuch that this cancels the imaginary contribution to thedisconnected geodesic.In addition to the above interpretation in terms of

pseudoentropy, we also expect that (7.7) can also beinterpreted as a standard entanglement entropy. Thisfollows from the covariant holographic entanglemententropy [23], whose derivation was given in [56]. Sincethe entanglement entropy gives a basic probe of timeevolution of quantum states, it is useful to ask how thequantum state at time t dual to our wedge geometry lookslike. As the boundary surface Q extends from the UVboundary toward the IR horizon under the time evolution,we expect that this state looks like adding layers from theUV layers in a scale invariant way which ends up with thefull MERA tensor network [57,58]. The time evolution ofentanglement entropy for this simple model can qualita-tively explain the logarithmic growth in (7.7).

D. Spacelike boundaries with λ < 0

Now let us focus on the spacelike boundary λ < 0 (i.e.,the right of Fig. 10). In this wedge geometry, which is muchlarger than the λ > 0 case, there is no spacelike geodesicwhich connects a boundary point at z ¼ ϵ and a point on thesurface Q. Instead there is a timelike geodesic between thepoints.2 Notice that the case λ < 0 and λ > 0 are separatedby the case where the surfaceQ is spacelike and therefore itis natural that the interpretations of them are quite different.The brane world holography implies that the gravity on thislarge wedge region for λ < 0 is dual to a CFT on an upperhalf plane plus a (quantum) gravity on dS2. This is incontrast with the λ > 0 case where we argued that thegravity is dual to just a CFTon an upper half plane. In otherwords, the wedge holography for λ < 0 has an additionaldegrees of freedom corresponding to the two dimensionaldS gravity. In this sense, understandings of the physics inthe λ < 0 case has a potential to answer the long standingproblem of holography of dS gravity.Consider the time evolution of quantum state in CFT in

this large wedge spacetime. For t < 0 we can regard thequantum state describes a time evolution as a MERA-liketensor network. Starting from the IR disentangled state, we

gradually add each layer with quantum entanglement, andfinally we realize the full MERA network which describesthe CFT ground state. Therefore we have a genuineCFT vacuum j0i at t ¼ 0. For later time, t > 0 the timeevolution is trivial because e−iHtj0i ¼ j0i under the CFTHamiltonian H. This interpretation is consistent with theabsence of spacelike geodesic which connect the AdSboundary andQ. Indeed, for t > 0 the correct entanglemententropy should be given by the familiar connected geodesic(i.e., a half circle shape) which leads to SA ¼ c

3log l

ϵ andthere is no need for a presence of disconnected geodesics. Itwill be an intriguing future problem to explore more on thisand pursuit a possible connection to a de Sitter holography.

E. Bubble nucleations in AdS

Let us get back to the general profile of the boundarysurface Q (6.1) in AdS/BCFT. The spacetime boundary inBCFT corresponds to a timelike surface in AdS with theunusual values of the tension (7.6). This corresponds to theparameter region jαj > jβj. In the Euclidean AdS, thisboundary surface is floating in the bulk AdS withoutintersecting with the AdS boundary. Its Lorentzian con-tinuation x0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix21 − β2 þ α2

pdescribes the bubble nucle-

ation. The inside and outside of the Euclidean sphere Q aretime evolved as in the left and right picture of Fig. 11,which describes a bubble nucleation of universe and anucleation of bubble-of-nothing, respectively. In thisLorentzian signature, the intersection between Q andAdS boundary is eventually created as in Fig. 11.In the Euclidean CFT, we can describe this Euclidean

BCFT by the following disk with an imaginary radius

jwj2 ¼ β2 − α2 < 0: ð7:15Þ

This imaginary radius corresponds to the fact that there isno actual intersection between Q and the AdS boundary inthe Euclidean setup. By taking Lorentzian analyticalcontinuation we can see that the choice (7.15) in theAdS/BCFT leads to gravity side results which agree withthe CFT calculations, as we will see below.

1. BCFT description via analytical continuation

In the following, we will formulate this nucleation of thebubble-of-nothing in AdS from the CFT viewpoint andevaluate the entanglement entropy in the bubble nucleationprocess. We focus on the setup of right picture of Fig. 11,which is related to the setup of subsection VII C (i.e.,λ > 0) via a global conformal transformation. On the otherhand, the left one in Fig. 11 is related to subsection VII D(i.e., λ < 0) and we can analyze this case in a similar way,though there is no disconnected geodesic which contributesthe holographic entanglement entropy.Let us first consider the Euclidean BCFT on jwj2 ≥ r2

i.e., a region outside a radius r disk. To evaluate a correlator2For another interpretation of presence of a timelike geodesic

as a complex valued entanglement entropy, refer to [59–61].

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-19

in this setup, we usually make use of the doubling trick[62], as shown in the left of Fig. 12. The kinematics of aBCFT n-point function is fixed by this 2n-point functionwith mirror points on the full plane. Thus, we can evaluatea 1-point function with the circle boundary,

hϕðwÞidisk ∝�

rðjwj2 − r2Þ

�2h; ð7:16Þ

where h is the conformal weight of ϕ. In general, the BCFTn-point function depends on the details of the boundary andthus has a nonuniversal and complicated form, which canbe expressed as the sum of chiral conformal blocks with thecoefficients related to the bulk-boundary 2-point func-tions [63].To proceed our analysis, we focus on a particular

boundary. Since we finally would like to justify ourCFT calculation by comparing with a simple gravitycalculation, we assume our CFT boundary to be dual toa purely gravitational end-of-the-world brane. In that case,we can approximate the BCFT correlator by the chiralvacuum block in the semiclassical limit, like the usualprescription in the large c CFT [47]. In other words, we canthink of the BCFT correlator as the chiral correlator(equivalently, the chiral block) on the full plane. This is

the usual procedure to evaluate the BCFT correlator withthe circle boundary in a holographic CFT.Let us consider our analytic continuation r2 → −r2. This

unusual BCFTon a disk with an imaginary radius can also betreated by the doubling trick with a slight change.We put themirror operatornoton the inversedpoint buton the inversedþreflected point (see the right of Fig. 12). Following the aboveprescription, this is also just the semiclassical correlator onthe full plane, therefore, we expect this BCFT is reasonable.Note thatwecannot find theboundary in thisBCFT, that is forexample, if one operator approaches the origin from outsidethe circle, it needs an infinite energy to cross the circle barrierin the usual BCFT, on the other hand, in our BCFT, one findsno singularity even when it crosses the circle.One may ask whether this CFT really leads a reasonable

result. To answer this, we will evaluate the entanglemententropy in the setup (the right of Fig. 11) following theabove prescription.3 It is well-known that the entanglement

FIG. 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 ≥ 1=L (left) and T ≤ −1=L (right). The colored regions describe physicalspacetimes.

FIG. 12. The Euclidean CFTwith the boundary jwj2 ¼ r2. Left: the doubling trick of the usual BCFT. The mirror image of the point xis inserted onto the inversed point r2=x. Right: the doubling trick of the analytic continued BCFT. The analytic continuation can beinterpreted as the reflection of the mirror point.

3Even though there is a single boundary in this BCFT, we canregard (7.17) as a regular entanglement entropy (or equally thepseudo entropy with the initial and final state identical) because att ¼ 0 there is a (Euclidean) time reversal symmetry such that theupper path-integral is identical to the lower one.

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-20

entropy can be calculated by a correlator with twistoperators. If we set a subsystem as A ¼ ½a; b�, then theentanglement entropy is given by

SAðtÞ ¼ limn→1

1

1 − nloghσnða; tÞσnðb; tÞiIm-disk; ð7:17Þ

where we mean Im -disk as the imaginary boundary (theright of Fig. 12) with the radius ir with r2 ¼ α2 − β2 > 0.For simplicity, we first calculate the entanglement entropyfor a subsystem A ¼ ½a;∞Þ. This is given by the 1-pointfunction with the twist operator,

hσnðwÞidisk ¼�

rðjwj2 − r2Þ

�2hn

; ð7:18Þ

where w ¼ τ þ ia (τ ¼ it) and hn ¼ c24ðn − 1

nÞ is theconformal weight of the twist operator. A naive analyticcontinuation to the imaginary radius leads to

hσnðwÞiIm-disk ¼�

irða2 þ r2 − t2Þ

�2hn

; ð7:19Þ

and then we obtain the entanglement entropy as

SAðtÞ ¼c6log

a2 þ r2 − t2

riϵ: ð7:20Þ

This imaginary entanglement entropy is problematic. Toresolve this problem, we have to take into account theproperties of the imaginary BCFT correlator more carefully.For the purpose of the resolution, we follow the more

rigorous evaluation of the entanglement entropy [64]. Fromthe viewpoint of the replica partition function, the twistoperator is thought of as a specific boundary of radius ϵ. Wedescribe the 1-point function on a unit dist by this partitionfunction Zβ;B

n with two boundaries, the first β is theboundary associated to the twist operator and the secondB is the original boundary,

hσnð0Þiunit-disk ¼ hσnjBi ¼Zβ;Bn

ðZβ;B1 Þn : ð7:21Þ

From now on, we introduce a formal radius R for the unitdisk in order to take the effect of the analytic continuationr2 → −r2 into account. This replica partition function canbe mapped into a cylinder (with coordinate u, see Fig. 13)by the map,

u ¼ i logðwÞ: ð7:22Þ

The length of this cylinder is log Rϵ and its circumference is

2πn. This replica partition function can be written by

Zβ;Bn ¼ hβje− 1

2πn logRϵHjBi; ð7:23Þ

where H is the Hamiltonian of our CFT. Since this cylinderis very long (because of the infinitesimal regulator ϵ), wecan approximate this amplitude as

Zβ;Bn ¼ e−

12πn log

RϵE0hβj0ih0jBi; ð7:24Þ

where E0 ¼ − cπ6is the vacuum energy. Thus, we obtain

hσnjBi ¼�Rϵ

�−2hnðhβj0ih0jBiÞ1−n: ð7:25Þ

From this result, we find that the analytic continuationleads to

hσnjIm-Bi ¼�iϵ

�−2hnðhβj0ih0jBiÞ1−n; ð7:26Þ

where Im-B is the imaginary boundary state. This term isknown as the boundary entropy [50]. Taking this term intoaccount, our calculation is improved as

hσnðwÞiIm-disk ¼ hσnjIm-Bi�

irða2 þ r2 − t2Þ

�2hn

¼ ðhβj0ih0jBiÞ1−n�

ϵrða2 þ r2 − t2Þ

�2hn

;

ð7:27Þ

and the entanglement entropy is improved by

SAðtÞ ¼c6log

a2 þ r2 − t2

rϵþ gB; ð7:28Þ

where gB is the boundary entropy gB ¼ logh0jBi and thecontribution hβj0i is absorbed by the redefinition of theregulator ϵ as usual. Consequently, we find the “cancella-tion” between the imaginary factor of the leading part of theentanglement entropy and the imaginary factor of theboundary entropy, and then the imaginary problem isperfectly resolved, therefore, we argue that our formulationof this imaginary BCFT is reasonable in this sense.

FIG. 13. The map from a disk (left) to a cylinder (right).

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-21

Let us move on to the more general case where thesubsystem is given by an interval A ¼ ½a; b� witha < −r < r < b. If we assume that in the gravity dual,the end-of-the-world brane is only coupled to the metric,we can approximate the two-point function (7.17) as [34]

hσnða;tÞσnðb;tÞiIm-disk

⟶n→1

(h0jBi2ðn−1Þð rϵða2þr2−t2ÞÞ2hnð rϵ

ðb2þr2−t2ÞÞ2hn ; disconnected;ð ϵðb−aÞÞ4hn ; connected:

ð7:29Þ

Here there are two possibilities of the vacuum blockapproximation. One is the disconnected channel approxi-mation (the left of Fig. 14) and the other is the connectedchannel approximation (the right of Fig. 14). As a result, weobtain

SAðtÞ ¼� c

6loga2þr2−t2

rϵ þ c6logb2þr2−t2

rϵ þ 2gB; disconnected;c3logb−a

ϵ ; connected;

ð7:30Þ

where we remind r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiα2 − β2

p> 0. If we neglect the

contribution from the boundary degrees of freedom i.e.,gB ¼ 0, for simplicity, the transition time t� from theconnected geodesic to disconnected one is

t� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða − rÞ2 þ ðb − rÞ2

2

r: ð7:31Þ

In particular, if we set a ¼ −b, then we have t� ¼ b − r.Note that if the time t approaches t ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ r2

p(more

formally, jt −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ r2

pj → ϵ) with a ¼ −b, then the entan-

glement entropy vanishes as one can see in (7.30). This isnatural because, in that time, the subsystem A coincideswith the spacelike boundary where the two dimensionalspacetime of the CFT ends.

2. Gravity dual description

In the former subsection, we performed CFT computa-tions via a doubling trick with a Wick rotated boundary tocompute the entanglement entropy in the setup given by theright picture of Fig. 11. Here in this subsection, we willperform a holographic calculation on the gravity side andconfirm that the two results indeed match with each other.Let us first consider the subsystem A ¼ ½a;∞Þ. The

holographic entanglement entropy is given by

SA ¼ SdisA ¼ La

4GN; ð7:32Þ

where La is the length of the so-called disconnectedgeodesic which is the spacelike geodesic extending fromðt; x; zÞ ¼ ðt; a; ϵÞ and ending on the end-of-the-worldbrane Q. This calculation of entanglement entropy inAdS/BCFT [18,19] is similar to those for the joining/splitting local quenches [65–67] (see also [68–70]). Notethat the Newton constant GN in AdS3 is related to thecentral charge c in CFT2 by 1=ð4GNÞ ¼ c=6.Consider the corresponding Euclidean setup of the right

picture in Fig. 11 with x2 ¼ ix0. The bulk is given by thePoincare AdS outside of the bubble:

x21 þ x22 þ ðz − αÞ2 ≥ β2: ð7:33Þ

This is shown in Fig. 15.Now, let us find out the disconnected geodesic between

ðx2; x1; zÞ ¼ ð0; a; ϵÞ and the end-of-the-world brane. Thedisconnected geodesic should lie on the x2 ¼ 0 slice due tothe x2 ↔ −x2 symmetry in our setup. Therefore, thegeodesic’s ending point on the brane can be denoted asðx2; x1; zÞ ¼ ð0; ξ; ζÞ. The distance between ðx2; x1; zÞ ¼ð0; a; ϵÞ; ð0; ξ; ζÞ is given by

cosh−1�ða − ξÞ2 þ ζ2 þ ϵ2

2ϵζ

�≃ log

�ða − ξÞ2 þ ζ2

ϵζ

�:

ð7:34Þ

Varying ðξ; ζÞ under the restriction ξ2 þ ðζ − αÞ2 ¼ β2, wecan see that (7.34) takes extremal value at

FIG. 14. The vacuum block approximation of a two-pointfunction in the holographic BCFT. There are two possibilities,like the s- and t-channel vacuum block approximation of a fullplane four-point function with same operators.

FIG. 15. The Euclidean setup corresponding to the right pictureof Fig. 11 with x2 ¼ ix0. The end-of-the-world brane Q is givenby x21 þ x22 þ ðz − αÞ2 ¼ β2.

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-22

ðξ; ζÞ ¼�−

2aβðα − βÞa2 þ ðα − βÞ2 ;

ðα − βÞða2 þ α2 − β2Þa2 þ ðα − βÞ2

�;

ð7:35Þ

�2aβðαþβÞa2þðαþβÞ2 ;

ðαþβÞða2þα2−β2Þa2þðαþβÞ2

�: ð7:36Þ

If we carefully look at the x2 ¼ 0 slice, we can see fromFig. 16 that (7.36) is the point where the disconnectedgeodesic ends. Plugging this back into (7.34), we can getthe length of the disconnected geodesic

La ¼ log

�a2 þ α2 − β2

ðαþ βÞϵ�: ð7:37Þ

The geodesic itself is given by the following equation

�z2 þ ðx1 − aÞðx1 þ α2−β2

a Þ ¼ 0;

x2 ¼ 0:ð7:38Þ

Here note that ðx2; x1Þ ¼ ð0; ðα2 − β2Þ=aÞ is nothing butthe inversed þ reflected point of ðx2; x1Þ ¼ ð0; aÞ shown inthe right of Fig. 12.Thanks to the rotation symmetry of the Euclidean setup,

these results can be easily extended to the disconnectedgeodesic starting from ðx2; x1; zÞ ¼ ðτ; a; ϵÞ. In this case,the length is

La ¼ log

�a2 þ τ2 þ α2 − β2

ðαþ βÞϵ�; ð7:39Þ

and the geodesic follows

(z2 þ

�affiffiffiffiffiffiffiffiffia2þτ2

p x1 þ τffiffiffiffiffiffiffiffiffia2þτ2

p x2 −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ τ2

p ��affiffiffiffiffiffiffiffiffia2þτ2

p x1 þ τffiffiffiffiffiffiffiffiffia2þτ2

p x2 þ α2−β2ffiffiffiffiffiffiffiffiffia2þτ2

p�

¼ 0;

ax2 − τx1 ¼ 0:ð7:40Þ

Performing analytic continuation x2 ¼ ix0, τ ¼ it,

La ¼ log

�a2 − t2 þ α2 − β2

ðαþ βÞϵ�; ð7:41Þ

and the geodesic follows

�z2 ¼ ðt−x0Þðða2−t2Þx0þðα2−β2ÞtÞ

t2 ;

x1 ¼ at x0:

ð7:42Þ

Here we can explicitly write down the tangent vector of thisgeodesic and check it is spacelike. Accordingly, the holo-graphic entanglement entropy of subsystem A ¼ ½a;∞Þ isgiven by

SA ¼ La

4GN¼ c

6log

�a2 − t2 þ α2 − β2

ðαþ βÞϵ�: ð7:43Þ

This perfectly matches with (7.28) with the boundaryentropy identified as

FIG. 16. The x2 ¼ 0 slice. It is shown how the disconnected geodesic (and its extension) intersects with the bubble and the AdSboundary z ¼ 0.

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-23

gB ¼ c6log

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiα2 − β2

pαþ β

�¼ c

6log

� ffiffiffiffiffiffiffiffiffiffiffiα − β

αþ β

s �: ð7:44Þ

The assumption 0 ≤ β < α in our setup implies that theboundary entropy gB should always be nonpositive. Thisvalue of gB agrees with the real part of our earlier result(7.12) for the plane shape boundary surface.We can also consider a more general case where

the subsystem is given by A ¼ ½a; b�. In this case, theholographic entanglement entropy is given by

SA ¼ minfSconA ; SdisA g; ð7:45Þ

SconA ¼ Lab

4GN; SdisA ¼ La þ Lb

4GN: ð7:46Þ

where Lab is the length of the so-called connected geodesicwhich is spacelike and connects ðx2; x1; zÞ ¼ ðt; a; ϵÞ andðx2; x1; zÞ ¼ ðt; b; ϵÞ. The connected geodesic and thedisconnected geodesic correspond to the connected channeland the disconnected channel on the CFT side shown inFig. 14 respectively. This holographic formula gives us thesame results with (7.30) and hence we would like to omitthe discussion here.

VIII. CONCLUSION AND DISCUSSION

In this paper, we have proposed a codimension-twoholography, called wedge holography, between gravity on adþ 1 dimensional wedge regionWdþ1 in AdS, surroundedby two end-of-the-world branes and a d − 1 dimensionalCFT on its corner Σd−1. We can derive our wedgeholography via two different routes. One is to employthe brane world holography and apply holography twice.The other one is to take a limit of the AdS/BCFTformulation. After we have formulated this holography,we have calculated the total free energy, holographicentanglement entropy and spectrum of conformal dimen-sions. In particular, we have studied the d ¼ 2 and d ¼ 3cases in detail. We have shown that the expected correlationfunctions in d − 1 dimensional CFT can be recovered froma bulk scalar field on the dþ 1 dimensional wedge space.The forms of free energy and entanglement entropy agreewith general expectations for d − 1 dimensional CFTs.In d ¼ 3, we have computed the central charge from the

conformal anomaly. We have independently evaluatedthe holographic entanglement entropy and shown thatthe result perfectly matches with the known result ofentanglement entropy in two dimensional CFTs. Thecentral charge computed from the holographic entangle-ment entropy agrees with that from the conformal anomaly.Moreover, we have confirmed that our wedge holographyleads to phase transition behaviors of entanglement entropyfor disconnected subsystems which are expected from

holographic CFTs. These provide nontrivial tests of ourwedge holography.In d ¼ 2, the dual theory is supposed to be a quantum

mechanics. However, our gravity calculation shows that noSchwarzian action appears. Moreover, we find that the freeenergy at finite temperature is given by a universal quantityanalogous to the boundary entropy (or g function), which isexpected to describe the degrees of freedom of ourconformal quantum mechanics. Since our wedge geometryis a part of AdS3, we expect that the ground state preservesthe SLð2; RÞ conformal symmetry as opposed to theSchwarzian quantum mechanics and its dual JT gravity.Though the dual theory can be a genuine conformalquantum mechanics, we need to remember that there isa Kaluza-Klein tower of primary operators as the gravity isactually three dimensional. It will be an interesting futureproblem to explore this more and identify the dual theory.If we impose the Neumann boundary condition on Σ

instead of the Dirichlet condition, we find a third inter-pretation of our wedge setup in terms of two d − 1dimensional CFTs on Σ which are interacting via d − 1dimensional gravity. It is an interesting future problem toexplore this gravity dual of interacting two CFTs obtainedby gluing two AdSd geometries in more depth.Finally, we have studied another class of wedge geom-

etries, which is expected to be dual to a spacelike boundaryin a Lorentzian CFT. Even though we have analyzed theAdS3 case, our studies can be generalized to higherdimensions in a straightforward way. In this class ofexamples, the end-of-the-world brane takes the form ofde Sitter spacetime and the tension takes unusual valuesjTjL > 1. We have argued that the wedge geometry withT < −1=L is dual to a BCFT with a spacelike boundary,while that with T > 1=L is dual to a combined systemof a BCFT with a spacelike boundary and a gravity onde Sitter space. Our result of holographic entanglemententropy is consistent with this interpretation. We also findthat more general profiles of end-of-the-world branes,obtained from global conformal maps, can describe eithera bubble nucleation of universe or a nucleation of a bubble-of-nothing. These provide new setups of AdS/BCFTgoing beyond the standard ones with the values of tensionjTjL < 1. Notice that in them, boundaries of the Lorentzianspacetimes where CFTs are defined, are spacelike, whilethe end-of-the-world branes in the bulk are timelike. Wehave also given a CFT description of these new setupswhich we originally found from the gravity viewpoint.Interestingly, we encounter unusual analytical continuationof the standard Euclidean CFT such that the radius ofdisk is imaginary. Though this imaginary continuationgives a complex value for the boundary entropy, the finalvalues of entanglement entropy turn out to be real,matching with the holographic result. It will be anintriguing future direction to further explore this class ofboundaries in CFTs.

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-24

ACKNOWLEDGMENTS

We are grateful to Pawel Caputa, Kotaro Tamaoka andTomonori Ugajin for useful discussions. We would like tothank Raphael Bousso, Hao Geng and Andreas Karch verymuch for valuable comments on this paper. I. A. is supportedby the Japan Society for the Promotion of Science (JSPS)and the Alexander von Humboldt (AvH) foundation. I. A.and T. T. are supported by Grant-in-Aid for JSPS FellowsNo. 19F19813. Y. K. and Z.W. are supported by the JSPSfellowship. Y. K. is supported by Grant-in-Aid for JSPSFellows No. 18J22495. T. T. is supported by the SimonsFoundation through the “It from Qubit” collaboration. T. T.is supported by Inamori Research Institute for Science andWorld Premier International Research Center Initiative(WPI Initiative) from the Japan Ministry of Education,Culture, Sports, Science and Technology (MEXT). T. T.is supported by JSPS Grant-in-Aid for Scientific Research(A) No. 16H02182 and by JSPS Grant-in-Aid forChallenging Research (Exploratory) 18K18766. Z. W. issupported by the ANRI Fellowship and Grant-in-Aid forJSPS Fellows No. 20J23116.

Note added.—When we finished computations and werewriting up this paper, we got aware of the recent paper [71],where a similar codimension-two holography and a calcu-lation of holographic entanglement entropy via doubleminimizations were independently considered.

APPENDIX A: A BRIEF REVIEW OF THESCHWARZIAN ACTION

The Schwarzian derivative is defined by (_t ¼ dtdu)

Schðt; uÞ ¼ _t−2�t…_t−

3

2ðtÞ2

�: ðA1Þ

It satisfies the following relations under coordinatetransformations

Schðt; uÞ ¼ Schðt; uÞ þ�dtdt

�2

· Schðt; tÞ;

Schðt; uÞ ¼ Schðu; uÞ þ�dudu

�2

· Schðt; uÞ;

Schðt; uÞ ¼ −�dtdu

�4

Schðu; tÞ: ðA2Þ

We have Schðt; uÞ ¼ 0 iff

tðuÞ ¼ auþ bcuþ d

: ðA3Þ

Also the equation of motion for the Schwarzian action,i.e., ISch¼

RduSchðt;uÞ, is proportional to _t−1 d

du Schðt; uÞ,

which means that Schðt; uÞ is a constant. Thus, thesolutions to this equation of motion are either (A3) or

tðuÞ ¼ pþ q tanðruþ sÞ: ðA4ÞIf we set

dtdu

¼ e−φðuÞ; ðA5Þ

then we find

Schðt; uÞ ¼ −1

2_φ2 − φ ¼ e−2φ

�−φ00 þ 1

2ðφ0Þ2

�; ðA6Þ

where note that φ0 ¼ ∂tφ ¼ eφ _φ. In this case, theSchwarzian action can be expressed as

ZduSchðt; uÞ ¼

Zdte−φ

�−φ00 þ 1

2ðφ0Þ2

�: ðA7Þ

APPENDIX B: GRAVITY ACTION FOR d = 2WITH ANOTHER UV CUTOFF

Here, we evaluate the gravity action (in Lorentziansignature) for the choice of the UV cutoff given by (referto Fig. 17)

y ≥ fðtÞ: ðB1ÞNotice that this is different from the regularization weemployed in Sec. IV. The total gravity action looks like

IG ¼ 1

16πGN

ZWedge

ffiffiffiffiffiffi−g

p ðR − 2ΛÞ

þ 1

8πGN

ZQ1∪Q2

ffiffiffiffiffiffi−γ

p ðK − TÞ

þ 1

8πGN

ZΣ

ffiffiffiffiffiffi−γ

pK: ðB2Þ

Note that in our setup we have

R ¼ −6

L2; Λ ¼ −

1

L2; T ¼ 1

Ltanh

ρ�L: ðB3Þ

FIG. 17. A sketch of d ¼ 2 wedge holography with another UVcutoff.

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-25

We find

1

16πGN

ZWedge

ffiffiffiffiffiffi−g

p ðR − 2ΛÞ

¼ −1

4πGN

�ρ� þ L sinh

ρ�Lcosh

ρ�L

�ZdtfðtÞ ;

1

8πGN

ZQ1∪Q2

ffiffiffiffiffiffi−γ

p ðK − TÞ

¼ L4πGN

sinhρ�Lcosh

ρ�L

ZdtfðtÞ : ðB4Þ

Thus, the sum of these two contributions simplifies as

1

16πGN

ZWedge

ffiffiffiffiffiffi−g

p ðR − 2ΛÞ þ 1

8πGN

ZQ1∪Q2

ffiffiffiffiffiffi−γ

p ðK − TÞ

¼ −ρ�4πGN

ZdtfðtÞ : ðB5Þ

To calculate the contribution from the Σ surface, we needto calculate the extrinsic curvature Kab ¼ ð∇aNbÞΣ. Theoutgoing (unit normalized) normal vector is given by

ðNt;Ny;NρÞ¼ fðtÞLcosh ρ

R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−f0ðtÞ2

p ð−f0ðtÞ;−1;0Þ: ðB6Þ

The trace of extrinsic curvature is computed as

K ¼ 1

L cosh ρL

·1 − f02 − ff00

ð1 − f02Þ3=2 : ðB7Þ

Therefore, we find

1

8πGN

ZΣ

ffiffiffiffiffiffi−γ

pK ¼ ρ�

4πGN

Zdt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − f02

pf

·1 − f02 − ff00

ð1 − f02Þ3=2 :

ðB8Þ

Since the induced metric on Σ looks like

ds2 ¼ −L21 − f02

f2dt2; ðB9Þ

we introduce the Weyl scaling factor as e2φ ¼ 1−f02f2 .

Assuming the usual UV cutoff property f ≪ 1, we canmake the expansion

fðtÞ ≃ e−φðtÞ þOðe−3φÞ: ðB10Þ

This leads to

1

8πGN

ZΣ

ffiffiffiffiffiffi−γ

pK ≃

ρ�4πGN

Zdte−φ

�e2φ −

1

2ðφ0Þ2 þ φ00

�:

ðB11Þ

By using the expression for the Schwarzian action (A7), thelatter action (B11) can be rewritten as4

1

8πGN

ZΣ

ffiffiffiffiffiffi−γ

pK ≃

ρ�4πGN

Zdteφð1 − Schðt; uÞÞ: ðB12Þ

However, if we add the bulk contribution (B5) andevaluate the total gravity action on this wedge, then theSchwarzian action above is canceled, i.e.,

IG ≃ρ�

4πGN

Zdte−φð−ðφ0Þ2 þ φ00Þ ¼ 0; ðB13Þ

where we have performed partial integration.So far we ignored the Hayward term. Indeed, this gives

the trivial contribution as the angle between Q1;2 and Σ isθ ¼ π

2:

IH ¼ 1

4πGN

ZΣðπ − θÞ ffiffiffi

γp ¼ 1

8GN

Zdt

ffiffiffiffiffiffiffiffiffiffiffiffi1 − _g2

pg

¼ 1

8GN

Zdteφ: ðB14Þ

In this way, we again find that there is no Schwarzian termin the total gravity action as long as we impose theNeumann boundary condition on Q1 and Q2.On the other hand, if we assume the Dirichlet boundary

condition on bothQ1 andQ2, then we do not need to add theterm − 1

8πGN

R ffiffiffiffiffiffi−γpT in (B2). Then, the bulk contribution

vanishes and we find that the Schwarzian term appears as

IG ≃ρ�

4πGN

Zdteφð1 − Schðt; uÞÞ; ðB15Þ

where the coordinate u is defined such that ds2 ¼−L2ð1−f02f2 Þdt2 ¼ −L2du2.

4Actually, here we have rescaled ϵeφ → eφ.

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-26

[1] J. M. Maldacena, The large N limit of superconformal fieldtheories and supergravity, Int. J. Theor. Phys. 38, 1113(1999).

[2] S. Gubser, I. R. Klebanov, and A.M. Polyakov, Gaugetheory correlators from noncritical string theory, Phys. Lett.B 428, 105 (1998).

[3] E. Witten, Anti-de Sitter space and holography, Adv. Theor.Math. Phys. 2, 253 (1998).

[4] G. ’t Hooft, Dimensional reduction in quantum gravity,Conf. Proc. C 930308, 284 (1993).

[5] L. Susskind, The world as a hologram, J. Math. Phys. (N.Y.)36, 6377 (1995).

[6] L. Randall and R. Sundrum, A Large Mass Hierarchyfrom a Small Extra Dimension, Phys. Rev. Lett. 83, 3370(1999).

[7] L. Randall and R. Sundrum, An Alternative to Compacti-fication, Phys. Rev. Lett. 83, 4690 (1999).

[8] A. Karch and L. Randall, Locally localized gravity, J. HighEnergy Phys. 05 (2001) 008.

[9] M. Alishahiha, A. Karch, E. Silverstein, and D. Tong, ThedS/dS correspondence, AIP Conf. Proc. 743, 393 (2004).

[10] M. Alishahiha, A. Karch, and E. Silverstein, Hologravity,J. High Energy Phys. 06 (2005) 028.

[11] X. Dong, E. Silverstein, and G. Torroba, De Sitter holog-raphy and entanglement entropy, J. High Energy Phys. 07(2018) 050.

[12] A. Strominger, The dS/CFT correspondence, J. High EnergyPhys. 10 (2001) 034.

[13] J. M. Maldacena, Non-Gaussian features of primordialfluctuations in single field inflationary models, J. HighEnergy Phys. 05 (2003) 013.

[14] D. Anninos, T. Hartman, and A. Strominger, Higher spinrealization of the dS=CFT correspondence, Classical Quan-tum Gravity 34, 015009 (2017).

[15] M. Miyaji and T. Takayanagi, Surface/state correspondenceas a generalized holography, Prog. Theor. Exp. Phys. 7(2015) 073B03.

[16] T. Takayanagi, Holographic spacetimes as quantum circuitsof path-integrations, J. High Energy Phys. 12 (2018) 048.

[17] A. Karch and L. Randall, Open and closed string inter-pretation of SUSY CFT’s on branes with boundaries, J. HighEnergy Phys. 06 (2001) 063.

[18] T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett.107, 101602 (2011).

[19] M. Fujita, T. Takayanagi, and E. Tonni, Aspects ofAdS/BCFT, J. High Energy Phys. 11 (2011) 043.

[20] M. Nozaki, T. Takayanagi, and T. Ugajin, Central chargesfor BCFTs and holography, J. High Energy Phys. 06 (2012)066.

[21] S. Ryu and T. Takayanagi, Holographic Derivation ofEntanglement Entropy from AdS=CFT, Phys. Rev. Lett.96, 181602 (2006).

[22] S. Ryu and T. Takayanagi, Aspects of holographic entan-glement entropy, J. High Energy Phys. 08 (2006) 045.

[23] V. E. Hubeny, M. Rangamani, and T. Takayanagi, Acovariant holographic entanglement entropy proposal,J. High Energy Phys. 07 (2007) 062.

[24] T. Faulkner, A. Lewkowycz, and J. Maldacena, Quantumcorrections to holographic entanglement entropy, J. HighEnergy Phys. 11 (2013) 074.

[25] N. Engelhardt and A. C. Wall, Quantum extremal surfaces:Holographic entanglement entropy beyond the classicalregime, J. High Energy Phys. 01 (2015) 073.

[26] G. Penington, Entanglement wedge reconstruction and theinformation paradox, J. High Energy Phys. 09 (2020) 002.

[27] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield,The entropy of bulk quantum fields and the entanglementwedge of an evaporating black hole, J. High Energy Phys.12 (2019) 063.

[28] A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, ThePage curve of Hawking radiation from semiclassical geom-etry, J. High Energy Phys. 03 (2020) 149.

[29] M. Rozali, J. Sully, M. Van Raamsdonk, C. Waddell, and D.Wakeham, Information radiation in BCFT models of blackholes, J. High Energy Phys. 05 (2020) 004.

[30] H. Z. Chen, Z. Fisher, J. Hernandez, R. C. Myers, and S.-M.Ruan, Information flow in black hole evaporation, J. HighEnergy Phys. 03 (2020) 152.

[31] A. Almheiri, R. Mahajan, and J. E. Santos, Entanglementislands in higher dimensions, SciPost Phys. 9, 001 (2020).

[32] Y. Kusuki, Y. Suzuki, T. Takayanagi, and K. Umemoto,Looking at shadows of entanglement wedges, arXiv:1912.08423.

[33] V. Balasubramanian, A. Kar, O. Parrikar, G. Srosi, and T.Ugajin, Geometric secret sharing in a model of Hawkingradiation, arXiv:2003.05448.

[34] J. Sully, M. Van Raamsdonk, and D. Wakeham, BCFTentanglement entropy at large central charge and the blackhole interior, arXiv:2004.13088.

[35] H. Geng and A. Karch, Massive Islands, J. High EnergyPhys. 09 (2020) 121.

[36] H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes, andJ. Sandor, Quantum extremal Islands made easy, part I:Entanglement on the brane, J. High Energy Phys. 10 (2020)166.

[37] R. Bousso and L. Randall, Holographic domains of anti-de Sitter space, J. High Energy Phys. 04 (2002) 057.

[38] C. Arias, F. Diaz, R. Olea, and P. Sundell, Liouvilledescription of conical defects in dS4, Gibbons-Hawkingentropy as modular entropy, and dS3 holography, J. HighEnergy Phys. 04 (2020) 124.

[39] T. Takayanagi and K. Tamaoka, Gravity edges modes andhayward term, J. High Energy Phys. 02 (2020) 167.

[40] G. Hayward, Gravitational action for space-times withnonsmooth boundaries, Phys. Rev. D 47, 3275 (1993).

[41] D. Brill and G. Hayward, Is the gravitational actionadditive?, Phys. Rev. D 50, 4914 (1994).

[42] M. Headrick and T. Takayanagi, A holographic proof of thestrong subadditivity of entanglement entropy, Phys. Rev. D76, 106013 (2007).

[43] H. Casini, M. Huerta, and R. C. Myers, Towards a derivationof holographic entanglement entropy, J. High Energy Phys.05 (2011) 036.

[44] M. Henningson and K. Skenderis, The holographic Weylanomaly, J. High Energy Phys. 07 (1998) 023.

[45] C. Holzhey, F. Larsen, and F. Wilczek, Geometric andrenormalized entropy in conformal field theory, Nucl. Phys.B424, 443 (1994).

[46] M. Headrick, Entanglement Renyi entropies in holographictheories, Phys. Rev. D 82, 126010 (2010).

CODIMENSION-TWO HOLOGRAPHY FOR WEDGES PHYS. REV. D 102, 126007 (2020)

126007-27

[47] T. Hartman, Entanglement entropy at large central charge,arXiv:1303.6955.

[48] J. Brown andM. Henneaux, Central charges in the canonicalrealization of asymptotic symmetries: An example fromthree-dimensional gravity, Commun. Math. Phys. 104, 207(1986).

[49] J. Maldacena, D. Stanford, and Z. Yang, Conformal sym-metry and its breaking in two dimensional Nearly Anti-de-Sitter space, Prog. Theor. Exp. Phys. 12 (2016) 12C104.

[50] I. Affleck and A.W. Ludwig, Universal Noninteger ‘GroundState Degeneracy’ in Critical Quantum Systems, Phys. Rev.Lett. 67, 161 (1991).

[51] A. Karch and L. Randall, Geometries with mismatchedbranes, J. High Energy Phys. 09 (2020) 166.

[52] M. Miyaji, S. Ryu, T. Takayanagi, and X. Wen, Boundarystates as holographic duals of trivial spacetimes, J. HighEnergy Phys. 05 (2015) 152.

[53] T. Hartman and J. Maldacena, Time evolution of entangle-ment entropy from black hole interiors, J. High EnergyPhys. 05 (2013) 014.

[54] Y. Nakata, T. Takayanagi, Y. Taki, K. Tamaoka, and Z. Wei,Holographic pseudo entropy, arXiv:2005.13801.

[55] T. Takayanagi and T. Ugajin, Measuring black hole for-mations by entanglement entropy via coarse-graining,J. High Energy Phys. 11 (2010) 054.

[56] X. Dong, A. Lewkowycz, and M. Rangamani, Derivingcovariant holographic entanglement, J. High Energy Phys.11 (2016) 028.

[57] G. Vidal, Entanglement Renormalization, Phys. Rev. Lett.99, 220405 (2007).

[58] B. Swingle, Entanglement renormalization and holography,Phys. Rev. D 86, 065007 (2012).

[59] Y. Sato, Comments on entanglement entropy in the dS=CFTcorrespondence, Phys. Rev. D 91, 086009 (2015).

[60] K. Narayan, de Sitter space and extremal surfaces forspheres, Phys. Lett. B 753, 308 (2016).

[61] K. Narayan, On dS4 extremal surfaces and entanglemententropy in some ghost CFTs, Phys. Rev. D 94, 046001(2016).

[62] J. Cardy, Boundary conformal field theory, arXiv:hep-th/0411189v2.

[63] J. L. Cardy andD. C. Lewellen, Bulk and boundary operatorsin conformal field theory, Phys. Lett. B 259, 274 (1991).

[64] J. Cardy and E. Tonni, Entanglement Hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. (2016)123103.

[65] P. Calabrese and J. Cardy, Entanglement and correlationfunctions following a local quench: A conformal fieldtheory approach, J. Stat. Mech. (2007) P10004.

[66] T. Ugajin, Two dimensional quantum quenches and holog-raphy, arXiv:1311.2562.

[67] T. Shimaji, T. Takayanagi, and Z. Wei, Holographic quan-tum circuits from splitting/joining local quenches, J. HighEnergy Phys. 03 (2019) 165.

[68] P. Caputa, T. Numasawa, T. Shimaji, T. Takayanagi, and Z.Wei, Double local quenches in 2D CFTs and gravitationalforce, J. High Energy Phys. 09 (2019) 018.

[69] M. Mezei and J. Virrueta, Exploring the membrane theory ofentanglement dynamics, J. High Energy Phys. 02 (2020)013.

[70] A. G. Cavalcanti and D. Melnikov, Cut and glue quenches inthe AdS/BCFT model, arXiv:2006.04623.

[71] R. Bousso and E. Wildenhain, Gravity/ensemble duality,Phys. Rev. D 102, 066005 (2020).

AKAL, KUSUKI, TAKAYANAGI, and WEI PHYS. REV. D 102, 126007 (2020)

126007-28