YITP, Kyoto University
Based on [arXiv:2007.06800]
collaboration withIbrahim Akal, Tadashi Takayanagi, Zixia Wei

Yuya Kusuki

Summary①[Akal, YK, Takayanagi, Wei]
Generalization

Original Holography

AdSd+1 = CFT𝑑Our Wedge Holography

AdS𝑑+1 = CFT𝑑−1

Motivation
 New laboratory New construction of CFT1
 Construction is more definite than doubly holography
 Joining gravity from AdS𝑑+1 and CFT𝑑−1⇒radiation setup
2/43

Summary②[Akal, YK, Takayanagi, Wei]

𝑥𝑟 = 𝑟0𝑟∞ =ETW brane𝑟1𝜖No asymptotic boundary⇒ No CFT dual?

𝑇 > 1/𝑅dS braneMotivation
 New laboratory
 dS braneworld
 Universe creation setup
3/43

Contents
 Introduction
 Basics of AdS/BCFT
 Quick lesson of BCFT
 Gravity dual

 Wedge holography Codimension two holography
 Another wedge holography

 Summary

4/43

Boundary CFT
boundaryBCFT

CFT

Boundary breaks conformal symmetry…
Conformal Symmetry of BCFT is part of conformal symmetry
preserving the bdy. position.

SO(2,d)SO(2,d-1)

5/43

Boundary CFT
Definite definition of BCFT2 is given by [Cardy].
Difference I: Symmetry

Symmetry of CFT is𝑽𝒊𝒓 × 𝑽𝒊𝒓

condition to preserve bdy.

On the other hand, bdy. condition of BCFT,

𝑻𝒙𝒚 = 𝟎⇔𝑳𝒏 − 𝑳ത −𝒏 = 𝟎

shows the dependency of 𝑉𝑖𝑟 and 𝑉𝑖𝑟 .
Instead, symmetry of BCFT is given by

𝑽𝒊𝒓
6/43

Boundary CFT
boundary𝜙

MirrorMirror method:𝜙(𝑧) 𝑏𝑑𝑦 = 𝜙 𝑧 𝜙 𝑧∗∝1𝑧 − 𝑧 ∗ 2ℎ𝜙with 𝑧 = 𝑥 + 𝑖𝑦.
The second eq. comes from standard conformal
Ward id.
7/43

Boundary CFT

boundary stateboundary
Conformal
map

Difference III: Basic contents
CFT is specified by
BCFT has extra contents,

With all of them, we can explicitly calculate any correlators
8/43

Boundary entropy[Affleck, Ludwig]

DoF of boundary is measured by boundary entropy.

0𝐵

Conformal map
Disk partition function

𝑆𝑏𝑑𝑦 = log 𝑔𝐵 , 𝑔𝐵 = 0 𝐵 ≃ 𝑍𝑑𝑖𝑠𝑘
Interesting point:• g-theorem [Friedan, Konechny]
𝑆𝑏𝑑𝑦 decreases under the RG flow like central charge⇒consistent with DoF measure interpretation
9/43

Gravity dual of BCFT

[Fujita, Tonni, Takayanagi]

boundaryETW brane

𝐀𝐝𝐒𝒅+𝟏𝐁𝐂𝐅𝐓𝒅

AdS with ETW brane

BCFT𝜕 ETW = bdy. of CFT
Induced metric: ℎ𝜇𝜈 = 𝑔𝜇𝜈 − 𝑛𝜇 𝑛𝜈 ,𝜌Extrinsic curvature: 𝐾𝜇𝜈 = ℎ𝜇 ℎ𝜈𝜆 ∇𝜌 𝑛𝜆Gravity action:
𝐼=−11
න 𝑔 𝑅 − 2Λ −
න ℎ 𝐾−𝑇16𝜋𝐺𝑁 𝑁
8𝜋𝐺𝑁 𝑄

Neumann b.c. is imposed on the brane (Einstein eq. of brane).

𝐾𝑎𝑏 − 𝐾ℎ𝑎𝑏 = −𝑇ℎ𝑎𝑏

10/43

Gravity dual of BCFT

[Fujita, Tonni, Takayanagi]

boundary
ETW brane

𝐀𝐝𝐒𝒅+𝟏𝐁𝐂𝐅𝐓𝒅BCFT

AdS with ETW brane𝜕 ETW = bdy. of CFTInteresting point of AdS/BCFT:
Boundary provides dynamical spacetime on ETW (see also
[Randull-Sandrum] )This is the reason why resent progresses about Page curve
utilize AdS/BCFT
11/43

Gravity dual of BCFT
Simple example:

𝒛𝒚Poincare patch:
2𝑑𝑠 = 𝑅2𝑑𝑧 2 − 𝑑𝑡 2 + 𝑑𝑥 2 + 𝑑𝑤 2𝑧2

𝝆BCFT𝒙Let us consider a BCFT on a UHP (x < 0).
It is useful to use the coordinate 𝑥 =𝑦𝜌,𝑧 =cosh𝑅𝑦
𝜌cosh𝑅,

𝜌 𝑑𝑦 2 − 𝑑𝑡 2 + 𝑑𝑤 2𝑑𝑠 = 𝑑𝜌 + 𝑅 cosh
𝑅𝑦2
2222

The bdy. position 𝜌 = 𝜌 𝑦 is fixed by the Neumann b.c.
𝐾𝑎𝑏 − 𝐾ℎ𝑎𝑏 = −𝑇ℎ𝑎𝑏

12/43

Gravity dual of BCFT
𝐾𝑎𝑏 − 𝐾ℎ𝑎𝑏 = −𝑇ℎ𝑎𝑏

A solution to the Neumann b.c. is𝜌 = 𝜌∗

𝑠. 𝑡.

𝑑−1
𝜌
𝑇=tanh ∗𝑅𝑅Dual AdS

𝝆∗BCFT𝝆

By an appropriate map from this UHP, the gravity
dual of a disk partition function (= boundary entropy)
is obtained.
𝜌∗
𝑆𝑏𝑑𝑦 = −𝐼𝑑𝑖𝑠𝑘 =4𝐺𝑁
𝑆𝑏𝑑𝑦 becomes large as the size of the space isincreased (i.e. 𝜌∗ is increased)⇒boundary becomes classical (explained later)

𝒙13/43

Ryu-Takayanagi formula[Ryu, Takayanagi]

ഥ can beEE can probe how much information about A
extracted from subregion A in CFT.

𝑆𝐴 = tr 𝜌𝐴 log 𝜌𝐴 ,𝜌𝐴 = tr𝐴ҧ 𝜌Gravity dual of EE is𝑆𝐴 = minΓ𝐴𝜕Γ𝐴 =𝜕𝐴Area Γ𝐴
4𝐺𝑁AdSd+1
Γ𝐴ACFTd⇒ Gravity calculation is easy
14/43

Ryu-Takayanagi formula
CFT calculation is also not so hard (replica trick).

1
𝑆𝐴 = tr 𝜌𝐴 log 𝜌𝐴 = limlog tr𝜌𝐴𝑛 ,𝑛→1 1 − 𝑛tr𝜌𝐴𝑛 =𝐴𝑛-sheets=𝜌𝐴 = tr𝐴ҧ 𝜌𝜎𝑛 𝜎ത𝑛𝐴
⟨𝜎𝑛 (𝑙)𝜎ത𝑛 (0)⟩15/43

Ryu-Takayanagi formula
CFT calculation is also not so hard (replica trick).

1𝑆𝐴 = tr 𝜌𝐴 log 𝜌𝐴 = limlog tr𝜌𝐴𝑛 ,𝑛→1 1 − 𝑛tr𝜌𝐴𝑛 =where𝑐ℎ𝜎𝑛 =24

As a result,𝜎𝑛 𝑙 𝜎ത𝑛 0

𝜌𝐴 = tr𝐴ҧ 𝜌

1= 2ℎ𝑙 𝜎𝑛1𝑛 − . The second eq. is just Ward id.
𝑛𝑐𝑙
𝑆𝐴 = log3
𝜖16/43

Ryu-Takayanagi formula
𝑆𝐴 = minΓ𝐴𝜕Γ𝐴 =𝜕𝐴Area Γ𝐴
4𝐺𝑁AdSd+1
Γ𝐴
AUsefulness Both gravity and CFT calculation are easy

CFTd

⇒ For this reason, we will use this to justify our new holography.
 Useful to probe global feature Which is part of gravity dual to subregion A in CFT?

Sub-region/sub-region duality (see [Suzuki, YK,Takayanagi, Umemoto] ) c-theorem [Myers-Sinha]⇒ How is this generalized to AdS/BCFT?

17/43

Ryu-Takayanagi formulaDifference RT surface can end on ETW brane (which satisfies

homologous cond. from string perspective.) DoF of boundary (boundary entropy) also contributes
to 𝑆𝐴 .
AdSd+1 Γ𝐴
𝑆𝐴 =minΓ𝐴𝜕Γ𝐴 =𝜕𝐴∪𝜕𝐵

Area Γ𝐴min
𝐵4𝐺𝑁BABCFTd
18/43

Ryu-Takayanagi formula
Simple example:Dual AdS

transitionAA𝑑Dual AdS

𝑙boundary entropy

𝑐𝑙 𝑐𝑙+𝑑
𝑆𝐴 = log + log+ 2 log 𝑔𝐵6𝜖 6𝜖𝑑𝑙𝑐𝑑
𝑆𝐴 = log
3𝜖This entanglement entropy can be completely reproduced
by CFT calculation (replica trick and mirror method)
19/43

Contents
 Introduction
 Basics of AdS/BCFT
 Quick lesson of BCFT
 Gravity dual

 Wedge holography Codimension two holography
 Another wedge holography

 Summary

20/43

Developments from AdS/BCFT[Akal, YK, Takayanagi, Wei]

Original Holography𝑑 + 1 dimensional AdS = 𝑑 dimensional CFTGeneralization

Our Wedge Holography𝑑 + 1 dimensional AdS = 𝒅 − 𝟏 dimensional CFT

21/43

Developments from AdS/BCFT[Akal, YK, Takayanagi, Wei]

Original Holography𝑑 + 1 dimensional AdS = 𝑑 dimensional CFT

𝐀𝐝𝐒𝒅+𝟏

22/43

Developments from AdS/BCFT[Akal, YK, Takayanagi, Wei]

Our Wedge Holography𝑑 + 1 dimensional AdS = 𝒅 − 𝟏 dimensional CFT
Cutting space by wedge

𝐀𝐝𝐒𝒅+𝟏

𝐂𝐅𝐓𝒅−𝟏
23/43

Wedge Holography

AdSd+1𝝆∗−𝝆∗𝝆
Definition of wedge:

CFTd−1222
𝜌𝑑𝑦−𝑑𝑡+𝑑𝑤
22 + 𝑅 2 cosh2
𝑑𝑠𝐴𝑑𝑆=𝑑𝜌𝑑+1
𝑅𝑦2𝜌 222= 𝑑𝜌 + cosh 𝑑𝑠𝐴𝑑𝑆𝑑
𝑅Restricted to wedge subspace −𝜌∗ ≤ 𝜌 ≤ 𝜌∗

⇒asymptotic boundary is 𝑑 − 1 dimensional
theoryOur proposal: This is CFT𝑑−1
24/43

Naive derivation?
𝐀𝐝𝐒𝒅+𝟏
𝐂𝐅𝐓𝒅−𝟏Braneworldholography

AdS/CFT
Asymptotic
boundary

𝐀𝐝𝐒𝒅

𝐀𝐝𝐒𝒅

25/43

Naive derivation?
𝐀𝐝𝐒𝒅+𝟏
𝐂𝐅𝐓𝒅−𝟏Braneworldholography

AdS/CFT

Unfortunately, not well-understoodso, here we do not use this.

𝐀𝐝𝐒𝒅

𝐀𝐝𝐒𝒅

26/43

Naive derivation?
𝐀𝐝𝐒𝒅+𝟏AdS/BCFT

𝐂𝐅𝐓𝒅−𝟏
Instead, we employ AdS/BCFT,
clear understanding,some justifications

𝐀𝐝𝐒𝒅

𝐀𝐝𝐒𝒅

27/43

Derivation from AdS/BCFT
𝐀𝐝𝐒𝒅+𝟏𝐁𝐂𝐅𝐓𝒅squeezing(dimensional reduction)

Definition:

Limit of BCFT (very clear!)
28/43

Derivation from AdS/BCFT
Definition:

Limit of BCFT (very clear!)⇒But is this really CFT?
Justifications:• Brane world holography + AdS/CFT (at least large 𝜌∗ )• Boundary of BCFT𝑑 can be thought of as CFT𝑑−1 , because OPE ofboundary state also satisfies axiom of CFT. In this sense, our CFT𝑑−1 canbe interpreted as two boundaries interacting with each other throughbulk.
• From now on, we see matching of gravity calculation and CFT
calculation for some physical quantities.
29/43

Free energy on SphereOne check of holography principle can be done by seeing freeenergy.
The on-shell action of AdSd+1 (without boundaries) has thefollowing form (see [Henningson, Skendteries] ),11
1
𝐼 = # 𝑑 + # 𝑑−2 + ⋯ + # 2 − # log 𝜖 + 𝑂 1𝜖𝜖𝜖111
𝐼 = # 𝑑 + # 𝑑−2 + ⋯ + # + 𝑂 1𝜖𝜖𝜖

(if 𝑑 = 𝑜𝑑𝑑)
(if 𝑑 = 𝑒𝑣𝑒𝑛)In fact, CFTd has the same form.

𝐼𝐴𝑑𝑆 𝑑+1 = 𝐼𝐶𝐹𝑇𝑑
This is one consistency check of the AdSd+1 /CFTd correspondence.
30/43

Free energy on SphereLet us consider AdSd+1 /CFTd−1 .
Of course, without the wedge, we can conclude

𝐼𝐴𝑑𝑆 𝑑+1 ≠ 𝐼𝐶𝐹𝑇𝑑−1
Therefore, the contribution from the wedge should have some nontrivial effects. We will see this.
For simplicity, let us focus on AdS4 /CFT2 .
We know
1 c𝐼𝐶𝐹𝑇2 = # 2 + 𝜒 Σ log 𝜖 + 𝑂(1)𝜖6Let’s calculate the gravity action of wedged AdS4 directly!
31/43

Free energy on SphereΣ
Metric of Euclidian AdSd+1 :−𝝆∗𝜌𝑑𝑠 2 = 𝑑𝜌2 + 𝑅2 cosh2 𝑑𝜂 2 + sinh2 𝜂 𝑑Ω2𝑑−1
𝑅𝑟
222= 𝑑𝑟 + 𝑅 sinh𝑑𝜃 2 + cos 2 𝜃 𝑑Ω2𝑑−1𝑄1

𝑅𝝆

𝝆∗
W𝑄2

Σ

𝒓

On-shell action gravity action on this wedge geometry,

11
1
𝐼=−
න 𝑔 𝑅 − 2Λ −නℎ 𝐾−𝑇 −
න ℎ𝐾
16𝜋𝐺𝑁 𝑊
8𝜋𝐺𝑁 𝑄1∪𝑄28𝜋𝐺𝑁 Σ
For the wedged AdS4 , we have
𝑅2
𝜌∗ 𝑅2
𝜌∗𝐼𝑊𝐴𝑑𝑆4 = −sinh +sinh log 𝜖 + 𝑂(1)2𝐺𝑁 𝜖 2𝑅 𝐺𝑁𝑅32/43Free energy on Sphere
𝑅2𝜌∗ 𝑅2𝜌∗𝐼𝑊𝐴𝑑𝑆4 = −sinh +sinh log 𝜖 + 𝑂(1)2𝐺𝑁 𝜖 2𝑅 𝐺𝑁𝑅The form for CFT2 is1 c
𝐼𝐶𝐹𝑇2 = # 2 + 𝜒 Σ log 𝜖 + 𝑂(1)𝜖6The 𝜖 dependence perfectly matches !!
The central charge (i.e. degrees of freedom) of the wedged
AdS4 is ( 𝜒 S 2 = 2)3𝑅2
𝜌∗𝑐=sinh𝐺𝑁𝑅
Next, we will check this result in an independent way.
33/43

Entanglement entropy
Entanglement entropy can be evaluated from both CFT andgravity side. Therefore, it is very useful to check our newholographic principle.
For simplicity, let us focus on AdS4 /CFT2 and consider a single
interval A with size 𝑙.
In this case, the entanglement entropy in CFT2 is known asc𝑙
𝑆𝐴 = log
3𝜖Let us compare this with gravity calculation.

A𝑙

34/43

Holographic EE
𝐀𝐝𝐒𝒅+𝟏𝐀𝐝𝐒𝒅+𝟏Γ𝐴(1)𝛾𝐴squeezing(2)𝛾𝐴

our HEE formulawell−known HEE formula

The holographic entanglement entropy for our holography isnaturally defined by recalling its definition,

𝑆𝐴 =min(2)1𝛾𝐴 ,𝛾𝐴(1,2)
𝜕𝛾𝐴 =𝜕𝐴min
Γ𝐴12Area Γ𝐴4𝐺𝑁𝜕Γ𝐴 =𝛾𝐴 ∪𝛾𝐴

35/43

Holographic EEThis formula in 4d case leads to𝑅2𝜌∗𝑙𝑆𝐴 =sinh log
𝐺𝑁𝑅𝜖This form matches with the CFT2 result.
We can extract the central charge from this result as

3𝑅2
𝜌∗𝑐=sinh𝐺𝑁
𝑅
This is completely same as the previous result!
Note: we can see this matching for more general cases (higherdimension and intervals).
36/43

Holographic EEAdStransition
AA CFTAdS

ACFTA

Double interval case:
We should find transition, like the standard HEE.
Implication:
CFT dual to Wedge geometry is similar toholographic CFT (i.e. sparse CFT)

Finding explicit constraints on the CFT data is interesting futurework!
37/43

Comments
 CFT𝑑−1 dual to AdS𝑑+1
3𝑅2
𝜌∗ 𝑐=sinh
𝐺𝑁𝑅 Sparseness (not the same as HKL sparseness)
 Large DoF of AdS𝑑+1 leads to infinite tower ofprimary operators, which are interpreted as KK
modes in brane world.
⇒NOT holographic CFT
 Zero size limit seems to be singular, but fromgravity perspective, quantity in CFT𝑑−1 are welldefined as wee saw.
38/43

Comments
 Braneworld holographyWedged AdSd+1
Joining gravity

Accessible

𝐂𝐅𝐓𝒅−𝟏

39/43

Comments
 Braneworld holography
From compactification,1𝑅𝜌∗= 𝑑+1 sinh𝑑𝑅𝐺𝐺𝑁𝑁

Joining gravity

3𝑅

3𝑅2𝑁𝑁

𝜌∗
𝑐𝐴𝑑𝑆3 × 2 =× 2 = (4) sinh
3𝑅
2𝐺𝐺

is consistent with the central charge
derived from EE.
40/43

double holography

Comments[Almheri, Mahajan,Maldacena, Zhao]

AdSd+1 Γ𝐴

radiation

BEE
Accessible

A
BCFTd

radiation

More realistic setup[Geng, Karch, Paradavila, Raju,Randall, Riojas, Shashi]

41/43

Summary①
Wedge Holography

𝐀𝐝𝐒𝒅+𝟏

AdS𝑑+1 = CFT𝑑−1
𝐂𝐅𝐓𝒅−𝟏
Justifications
 Brane world picture:
Brane world holography + AdS/CFT (at large 𝜌∗ )
 BCFT picture:
CFTs on two boundaries with bulk interaction
 Matching of physical quantities
explicit calculation of DoF and EE from both sides

42/43

Summary①
Wedge Holography

𝐀𝐝𝐒𝒅+𝟏

AdS𝑑+1 = CFT𝑑−1
𝐂𝐅𝐓𝒅−𝟏
Future direction
 Employing this new laboratory
 Identifying universal properies of CFT𝑑−1
 Joining gravity from AdS𝑑+1 and CFT𝑑−1⇒more general double holography setup
 More justification (e.g. realization via string theory)
43/43

Appendix(Another wedge holography)

44/43

W-holography at finite temp.
Let us consider more general case.
BTZ is simplest one other than Poincare AdS.
Metric of BTZ:
2𝑑𝑟22
𝑑𝑠 2 = − 𝑟 2 − 𝑟02 𝑑𝑡 2 + 𝑅2 22 + 𝑟 𝑑𝑥
𝑟 − 𝑟0

Coordinate transformation
𝑟0𝑡 ′ ± 𝑥 ′ = ±𝑒 𝑅 𝑥±𝑡

𝑟02
1 − 𝑟2,𝑟0 𝑟0 𝑥𝑧 = 𝑟 𝑒𝑅

Poincare:𝑑𝑠 2 = 𝑅2𝑑𝑧 2 − 𝑑𝑡′2 + 𝑑𝑥′2𝑧245/43

W-holography at finite temp.
General solution to Neumann bdy. in Poincare metric𝑧 − 𝛼 2 + 𝑥′ − 𝑝 2 − 𝑡′ − 𝑞 2 = 𝛽2𝛼Its tension is 𝑇 =𝛽𝑅Coordinate transformation

𝑅𝑇 < 1: A𝑑𝑆2 brane𝑟 𝑥 =𝑅𝑇 = 1: 𝑅2 brane
𝑅𝑇 > 1: 𝑑𝑆2 brane

𝑟0 𝑇𝑅𝑟𝑥
1 − 𝑇 2 𝑅2 sinh 0𝑅𝑟 𝑥𝑟 𝑥 =𝑟0 𝑥= 2𝑟0 𝑒 𝑅𝑟0 𝑇𝑅𝑟𝑥
𝑇 2 𝑅2 − 1 cosh 0𝑅46/43

W-holography at finite temp.
𝑥𝑥𝑟 = 𝑟0𝑟 = 𝑟01𝑟∞ =𝜖1𝑟∞ =𝜖ETW brane𝑟
𝑇 < 1/𝑅
AdS2 brane
AdS brane case:𝛽 sinh 𝜌∗𝜌∗𝐼𝑊 = −−4𝜋𝐺𝑁 𝜖2𝐺𝑁𝑟

𝑇 > 1/𝑅dS2 brane= 2 × 𝑆𝑏𝑑𝑦

Gravity action is consistent with anomaly of CFT1 .
DoF of CFT1 is equal to DoF of two AdS branes.

47/43

W-holography at finite temp.

ETW brane𝑟𝑇 < 1/𝑅
AdS2 brane𝑥𝑥𝑟 = 𝑟0𝑟 = 𝑟0
1𝑟∞ =𝜖1𝑟∞ =𝜖𝑟

𝑇 > 1/𝑅
dS2 branedS brane case?

No asymptotic boundary ⇒ No CFT dual ?
48/43

Contents
 Introduction
 Basics of AdS/BCFT
 Quick lesson of BCFT
 Gravity dual

 Wedge holography Codimension two holography
 Another wedge holography

 Summary

49/43Imaginary BCFTBTZ𝑥𝑟 = 𝑟0𝑟∞ =Bubble AdS

back to(Euclidean)
Poincare1𝜖𝑧𝑟
Profile of bubble brane is𝑧 − 𝛼 2 + 𝑥 2 + 𝑡 2 = 𝛽2with𝛼 > 𝛽Bubble AdS had not been considered in AdS/BCFT context.
Here, we give the CFT dual of this bubble AdS.
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Imaginary BCFT𝑧
𝑇 < 1/𝑅 ∶ AdS2 braneThis geometry has BCFT dual (known!).
Its bdy. is circle with radius𝑟 2 = 𝛽2 − 𝛼 2Note that 𝑇 < 1/𝑅 ⟺ 𝛽2 > 𝛼 2BCFT

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Imaginary BCFTanalyticcontinuation𝑧𝑧
𝑇 < 1/𝑅 ∶ AdS2 brane
𝑇 > 1/𝑅 implies

𝑇 > 1/𝑅 : dS2 brane𝑟 2 = 𝛽2 − 𝛼 2 < 0Imaginary radius is problematic, but in gravity picture,
this analytic continuation is not so problematic.
We formally define imaginary BCFT.
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ℑ𝑤

EE in BCFT
Let us recall thatmirror method cancompletely fix 1pt.correlator.

mirror image𝑥𝑟2𝑥𝑟ℜ𝑤

1-pt. correlator in BCFT is𝑟2
𝜙 𝑥 𝑑𝑖𝑠𝑘 ≃ 𝜙 𝑥 𝜙𝑥
up to conformal prefactor.

53/43

ℑ𝑤

EE in BCFT
Let us recall thatmirror method cancompletely fix 1pt.correlator.

mirror image𝑥𝑟2𝑥𝑟ℜ𝑤

1-pt. correlator in BCFT is𝜙 𝑥

𝑑𝑖𝑠𝑘 = 𝑔𝐵2ℎ𝜙𝑟𝑤 2 − 𝑟254/43

t

EE in BCFT
For example, EE in the right setup can beevaluated by𝜎𝑛 𝜏 + 𝑖𝑎

𝑑𝑖𝑠𝑘 = 𝑔𝐵2ℎ𝜎𝑛𝑟
𝜏 2 + 𝑎2 − 𝑟 2

where 𝜎𝑛 is twist op. with ℎ𝜎𝑛 =𝑐24𝑛−Ax

a1𝑛After analytic continuation 𝜏 → 𝑖𝑡,

1𝑆𝐴 𝑡 = lim
log 𝜎𝑛 𝜏 + 𝑖𝑎 𝑑𝑖𝑠𝑘𝑛→1 1 − 𝑛c𝑎2 − 𝑡 2 − 𝑟 2= log+ 𝑆𝑏𝑑𝑦
6𝜖𝑟where 𝑆𝑏𝑑𝑦 = log 𝑔𝐵 is boundary entropy

Lorentzian ver. ofEuclidean disk

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EE in Im-BCFT
Let us consider imaginary BCFT in physical setup

tAnalytic continuation
to imaginary radiusxtx

Aacreation of universe setupNaïve analytic continuation is imaginary.
To be more careful.

c𝑎2 − 𝑡 2 + 𝑟 2SA t = log+ 𝑆𝑏𝑑𝑦6𝑖𝜖𝑟56/43

𝜎𝑛 = hole with radius 𝜖

EE in Im-BCFT
𝑔𝐵 = ⟨𝜎𝑛 𝐵 =𝛽,𝐵𝛽,𝐵𝑍𝑛𝛽,𝐵𝑍1𝑅log
𝜖𝑍𝑛𝑛𝛽

𝐵

where 𝛽 is boundary state related to 𝜎𝑛 (which can be absorbed inregularization 𝜖)

Conformal map

𝛽𝛽𝛽𝛽

Cylinder with circumstance 2𝜋𝑛57/43

𝜎𝑛 = hole with radius 𝜖

EE in Im-BCFT𝛽,𝐵𝛽,𝐵𝑍𝑛𝑔𝐵 = ⟨𝜎𝑛 𝐵 =𝛽,𝐵𝑍1𝑅log
𝜖𝑍𝑛𝑛𝛽

𝐵

where 𝛽 is boundary state related to 𝜎𝑛 (which can be absorbed inregularization 𝜖)
1𝑅𝛽,𝐵−2𝜋𝑛 log 𝜖 𝐻
𝑍𝑛 = ⟨𝛽|𝑒𝐵𝜖→0𝑒1
𝑅−2𝜋𝑛 log 𝜖 𝐸0⟨𝛽|0⟩⟨0 𝐵

𝑅 is formal radius 𝑅 = 1 (→ 𝑖 later). As a result,
𝑅 −2ℎ𝜎𝑛⟨𝜎𝑛 𝐵 =( 𝛽 0 0 𝐵 1−𝑛𝜖Analytic continuation (𝑅 → 𝑖𝑅) leads toimportant point⟨𝜎𝑛 𝐼𝑚‐ 𝐵 = 𝑖 −2ℎ𝜎𝑛 ⟨𝜎𝑛 𝐵c𝑎2 − 𝑡 2 + 𝑟 2SA t = log+ 𝑆𝑏𝑑𝑦
6𝜖𝑟cancel imaginary part

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t

EE in Im-BCFT
CFT calculation is non-trivial but gravity
calculation is trivial (just a geodesic in a cutAdS).
⇒Consistency check from gravity
calculation is possible.
1𝑆𝐴 = lim
log 𝜎𝑛 𝑎, 𝑡 𝜎𝑛 𝑏, 𝑡𝑛→1 1 − 𝑛Abax

𝐼𝑚−𝑑𝑖𝑠𝑘𝑐𝑎2 +𝑟 2 −𝑡 2𝑐𝑏2 +𝑟 2 −𝑡 2log+ log+ 2𝑔𝐵6𝑟𝜖6𝑟𝜖=൞ 𝑐𝑏−𝑎log
3𝜖∶ disconnected∶ connected

CFT calculation based on our prescription completelymatches with the gravity calculation!
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BCFT Correlator
Instead of technicalconsideration, we can givea simple calculation rule ofa correlator in Im-BCFT.

ℑ𝑤

𝑟
reflected

𝑥−𝑟2

ℜ𝑤

𝑥1-pt. correlator in Im-BCFT is𝜙 𝑥

𝑑𝑖𝑠𝑘 =2ℎ𝜙𝑟
𝑤 2 + 𝑟2

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Lorentzian Im-BCFT

tt
dual
BCFT

x

AdS

x

Analytic continuation to Lorentzian signature provides
time-like boundary in CFT.
This setup is interesting for two reasons: universe creation setup
 dS braneworld (as mentioned before) by CFT
description as boundary statez

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Contents
 Introduction
 Basics of AdS/BCFT
 Quick lesson of BCFT
 Gravity dual

 Wedge holography Codimension two holography
 Another wedge holography

 Summary

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Summary②𝑥𝑟 = 𝑟0

dS brane/Imaginary BCFTduality

ETW brane𝑟1
𝑟∞ =𝜖𝑇 > 1/𝑅
dS brane

Future direction
 Employing this new laboratory
 More understanding of dS braneworld holography
 Universe creation setup
 More justification
63/43

Appendix(extra comments)

64/43

Developments from AdS/BCFTAsymptotic
boundary

𝐂𝐅𝐓𝒅

𝐀𝐝𝐒𝒅

Setup:
AdS𝑑 & CFT𝑑 are glued along the (asymptotic) boundary
This AdS𝑑 is dynamical.
Light can go through asymptotic boundary.
We can discuss the Page curve in this setup.
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Developments from AdS/BCFT
Asymptotic
boundary

𝐂𝐅𝐓𝒅

AdS/CFT𝐀𝐝𝐒𝒅

boundary

𝐂𝐅𝐓𝒅
𝐂𝐅𝐓𝒅−𝟏

Setup:
AdS𝑑 & CFT𝑑 are glued along the (asymptotic) boundary
AdS/CFT correspondence:
AdS𝑑 = CFT𝑑−1
This CFT𝑑−1 can be thought of as boundary object of CFT𝑑
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Developments from AdS/BCFT
Asymptotic
boundary

𝐂𝐅𝐓𝒅boundary

AdS/CFT𝐀𝐝𝐒𝒅𝐂𝐅𝐓𝒅
𝐂𝐅𝐓𝒅−𝟏AdS/BCFT

Braneworld holography[Randall, Sandrum]

ETW brane

𝐀𝐝𝐒𝒅+𝟏

These threepictures are same
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Developments from AdS/BCFT
By utilizing AdS/BCFT,
We can find the contribution from Island(effective quantum contribution),
which reproduce the Page curveradiation

EE

Island contributiontBraneworldholography

AdSd+1 Γ𝐴BA
BCFTddouble holography[Almheri, Mahajan,Maldacena, Zhao]

68/43