Quantum Gravity and finite cut-off AdS Pawel Caputa YITP Kyoto - - PowerPoint PPT Presentation

Pawel Caputa

Quantum Gravity and finite cut-off AdS

QIST2019, 24.06.2019

YITP Kyoto

Based on:

“Sphere partition functions and cut-off AdS” JHEP 1905 (2019) 112 with Shouvik Datta (UCLA) and Vasudev Shyam (PI)

also earlier work:

“Airy function and 4d quantum gravity” JHEP 1806 (2018) 106 with Shinji Hirano (WITS)

Part of the TT story at this workshop: Shyam, Datta, Rolf, Soni, Silverstein, Verlinde, Song, Nomura, Apolo,… (I will try not to overlap too much)

Plan:

• Summary
• TT flow and sphere partition functions
• QGR: Wheeler-DeWitt equation
• It from Qubit physics and TT
• Open Questions

Take-home (reminder):

AdS/CFT: Tool to understand hol. CFT (QFTs) using Quantum Gravity

QGR in AdS

(Quantum) Gravity is the most efficient way of doing (hol.) QFTs!

Holographic CFT

Summary:

Classically: It can be interpreted as a (TT) flow equation in holographic CFTs that reproduces non-trivial features of finite cut-off AdS/CFT. Quantum: Wheeler-DeWitt equation in minisuperspace becomes a differential equation that captures information about finite-N partition functions. “Radial Hamiltonian” constraint is a powerful tool in AdS/CFT (any dim.)! This talk: Sphere partition functions and RT formula at finite cut-off. This talk: ABJM sphere partition function (Airy) that contains all orders contributions from perturbative quantum gravity.

“It from GR” (Gauss-Codazzi or quantum WDW)

K2 − KijKij = ˜ R + d(d − 1) l2 .

πij = pγ κ2 (Kγij Kij) , πi

i =

pγ κ2 (d 1)K,

With canonical momenta κ2 pγ  πijπij 1 d 1(πi

i)2

• +

pγ κ2  ˜ R + d(d 1) l2

• = 0.

In the (radial) ADM decomposition Einstein equations can be written as ([Skenderis,Papadimitriou’04])

K2 KijKij = ˜ R + 2κ2Td+1d+1, riKi

j rjK

= κ2Tjd+1, ˙ Ki

j + K Ki j

= ˜ Ri

j κ2

✓ T i

j

1 d 1T σ

σ δi j

◆

Just pure gravity (cc) the “radial Hamiltonian” constraint (QGR: WDW eq. see later) ds2 = dr2 + γij(r, x)dxidxj

K2 − KijKij = ˜ R + d(d − 1) l2 .

ˆ T i

i = lκ2

2  ˆ Tij ˆ T ij 1 d 1 ⇣ ˆ T i

i

⌘2

• l

2κ2 ˜ R.

GC can be rewritten as a “flow equation” And translated/interpreted as a large N QFT flow ( etc.)

ˆ Tij = Tij + adCij,

γij ! r2

cγb ij,

T

Given a general holographic energy-momentum tensor hTiji = 1 κ2  Kij Kγij d 1 l γij

• adCij

and ad =

l (d−2)κ2

[Taylor’18]

AdS/CFT flow equation from Gauss-Codazzi

Pure gravity

[Balasubramanian,Kraus’99] [Taylor’18] [Hartman et al.’18] [McGough’18]

hT i

i i = dλhXdi

[Kraus,Liu,Marolf’18]

Large N Flow equation

T i

i

= −d  TijT ij − 1 d − 1(T i

i )2 + 2↵d

• d−2

d

✓ TijCij − 1 d − 1T i

i Ci i

◆ + ✓ ↵d

• d−2

d

◆2 ✓ CijCij − 1 d − 1(Ci

i)2

◆ + 1 d ↵d

• d−2

d

✓(d − 2) 2 ˜ R + Ci

i

◆#

In terms of the “renormalized” EM tensor

λ = lκ2 2drd

c

, αd λ

d−2 d

⌘ lrd−2

c

(d 2)κ2,

with “dictionary” for λ, αd Matching with anomalies and known holographic setups we have e.g. α3 = NABJM 6 21/3π2/3, α4 = NSYM 27/2π , α6 = N(2,0) 24π ,

[Hartman,Kruthoff,Shaghoulian,Tajdini’18]

ˆ Tij = Tij + adCij,

[PC,Datta,Shyam’19]

IDEA: We can define (effective) “TT-deformed” hol. CFTs dual to AdS with finite cut-off by the above flow equation.

Holographic sphere partition functions in cut-off AdS

One of the oldest and very interesting holographic probes Even-d: sensitive to a-type holographic anomalies Closely related to entanglement entropy “Good” holographic measure of degrees of freedom See also applications in dS/dS [Gorbenko,Silverstein,Torroba’18] Precision tests of AdS/CFT (localization)

Holographic sphere partition functions and EM tensor

I(d+1)

• n−shell = − 1

2κ2 Z

M

dd+1x√g (R − 2Λ) + 1 κ2 Z

∂M

ddx√γK + Sct,

Sct = 1 κ2 Z

∂M

ddx√γ " d − 1 l c(1)

d

+ c(2)

d l

2(d − 2) ˜ R + c(3)

d l3

2(d − 4)(d − 2)2 ✓ ˜ Rij ˜ Rij − d 4(d − 1) ˜ R2 ◆# (2.2)

Gravity Action Holographic counter-terms (up to d=6) T d

ij[r] ≡ − 2

√γ δI(d+1)

• n−shell[r]

δγij ,

log ZSd[r] = −I(d+1)

• n−shell[r]

ds2 = l2 dr2 l2 + r2 + r2dΩ2

d 2 d ≡ l2 dr2

l2 + r2 + γij(r, x)dxidxj,

On-shell solution Holographic Partition functions E-M tensor (Brown-York)

[Emparan,Johnson,Myers’99]

[PC,Datta,Shyam’19]

The large N sphere partition function in finite cut-off holography

log ZSd[rc] = dSdrd

c

κ2l  l rc

2F1

✓ 1 2, d 1 2 , d + 1 2 , r2

c

l2 ◆ + c(1)

d

(d 1) d + c(2)

d (d 1)

2(d 2) l2 r2

c

c(3)

d (d 1)

8(d 4) l4 r4

c

#

T d

ij[rc] = (d − 1)

κ2l " c(1)

d

+ c(2)

d l2

2r2

c

− c(3)

d l4

8r4

c

− s 1 + l2 r2

c

# γij. ≡ ω[rc]γij

Holographic energy-momentum tensor They satisfy

rc∂rc log ZSd[rc] = Z ddxpγhT i

i i = rd cSd d ω[rc],

Sphere partition function in cut-off AdS

Holographic Field Theory

4 hTiji = ωdγij

For the spheres the symmetry fixes This way the flow equation becomes quadratic equation for ωd

d ωd = dλ  d d 1ω2

d + 2αd

λ

d−2 d

1 d 1 Ci

iωd 1

dλ αd λ

d−2 d fd(R)

• ω(±)

d

= d 1 2dλ B @1 2αdλ

2 d

d 1 Ci

i ±

v u u t 1 2αdλ

2 d

d 1 Ci

i

!2 + 4αdλ

2 d

d 1 fd(R) 1 C A

The two solutions are (- to match anomalies) with explicit form of and holographic dictionary we reproduce the gravity bulk computation from the TT flow equation!

Cij

[PC,Datta,Shyam’19]

Wheeler-DeWitt equation in mini-superspace

ds2 = N2(r)dr2 + a2(r)dΩ2

d,

Quantum Gravity Path-Integral in the “minisuperspace” approximation

SEH + SGH = −d(d − 1)Sd 2κ2 Z dr  q02 4N + N

• qd3 + l2qd2

H = N ˆ H = − 2κ2 Sdd(d − 1)N " p2 − ✓d(d − 1)Sd 2κ2l ◆2 l2qd−3 + qd−2 #

ˆ HΨ[q] = " ~2 d2 dq2 − ✓d(d − 1)Sd 2κ2l ◆2 l2qd−3 + qd−2 # Ψ[q] = 0.

Quantum Wheeler-DeWitt equation Quantum Gravity action (q=a^2 for a canonical kinetic term) Hamiltonian

[PC,S.Hirano’18]

Wheeler-DeWitt equation: (HJ) WKB solution

ˆ HΨ[q] = " ~2 d2 dq2 − ✓d(d − 1)Sd 2κ2l ◆2 l2qd−3 + qd−2 # Ψ[q] = 0.

ΨWKB(q) ≈ exp  ± ✓d(d − 1)Sd 2κ2l~ ◆ Z q p l2qd−3 + qd−2 dq

• Using the WKB expansion we have the leading order solution (HJ)

q = r2

c

ΨWKB[rc]

◆ = e−(Ion−shell

GR

[rc]−Sct[rc])

Extra canonical transformation to include the counter-terms needed

[Freidel’08] [PC,S.Datta,V.Shyam’19]

Quantum Gravity “radial Hamiltonian” yields the WDW equation (q=a^2)

[deBoer,Verlinde,Verlinde’99]

Which is precisely the “bare” gravity on-shell action with finite cut-off

ABJM and the Airy function

ZABJM(S3) ∼ Ai "✓πN 2 √ 2λ ◆2/3 ✓ 1 − 1 24λ − λ 3N 2 ◆#

Non-perturbative + ~instant. All orders pert. QGR.!

Ai [x] ∼ 1 x1/4 e− 2

3 x3/2 ✓

1 − 5 48x3/2 + 385 4608x3 + ... ◆

[Bhattacharyya,Grassi,Marino,Sen’12]

x−3/2 ∼ N −3/2 ∼ GN

[Marino,Putrov’12] [Bergman,Hirano’09] [Hatsuda,Moriyama,Okuyama’12] [Klebanov,Tseytlin’96] [Fuji,Hirano,Moriyama’11]

Classical GR 1 loop QGR 2 loop QGR 3 loop QGR

[Dabholkar,Drukker,Gomes’14]

Localization in SUGRA

WDW and Airy

WdW equation in AdS in 4 dim. => Airy equation! General (quantum) solution

Ψ(q) = C1Ai ( 3πℓ2 4GN )

2 3

(ℓ−2q + 1) + C2Bi ( 3πℓ2 4GN )

2 3

(ℓ−2q + 1)

[ d2 dq2 − 9π2 16G2

Nℓ2 (q + ℓ2)] Ψ(q) = 0

Holographic dictionary (ABJM) and the decaying part for larger N (Ai, C2=0)

3S3l2 κ2 = πN 2 √ 2λ

We find the relation (Pure Gravity!)

ZABJM(S3) ∼ Ai(x) = ΨW DW (q → 0)

[PC,S.Hirano’18]

! ? Works for 1/2 BPS Wilson Loops!

Is bulk really a Tensor Network?

Surface/state-correspondence?

[Miyaji&Takayanagi’15]

?

Is TT-bar one realization of this correspondence?

Finite cut-of RT from Sphere Partition Functions

dΩ2

d = n2 sin2 θdτ 2 + dθ2 + cos2 θdΩ2 d−2

Given the partition function for a CFT on a replicated geometry Trick: S1 = (1 n∂n) log Zn[r]|n→1 = ⇣ 1 r d∂r ⌘ log ZSd[r] ( S1 = ( Sd

8πGN rd−1

2F1

✓1 2, d 1 2 , d + 1 2 , r2 l2 ◆

So using “bare” partition functions at scale r

[Donnelly,Shyam’18] [PC,Datta,Shyam’19]

This matches RT for (finite cut-off) surface with induced metric RT no c.t.!

ds2 = l2 dρ2 + sinh2 ρdΩ2

d−2

• SA =

A 4GN = ld−1Sd−2 4GN Z ρ dρ sinhd−2 ρ

[Banerjee,Bhattacharyya,Chakraborty’19] [Murdia,Nomura,Rath,Salzeta’19]

Is it related to EE in TT deformed theory? Meaning on EE in TT?

Some open questions:

Meaning of entanglement in TT deformed CFTs? Better understanding of the flow equation from QFT side? Deforming ABJM or N=4 SYM ? 1/N and quantum gravity? Lessons from WDW? What are the “bulk” (universal GR) constraints on Holographic TN?

Thank You! Stay Tuned!