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

Quantum Gravity and finite cut-off AdS Pawel Caputa YITP Kyoto QIST2019, 24.06.2019

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

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

“It from GR” (Gauss-Codazzi or quantum WDW) ds 2 = dr 2 + γ ij ( r, x ) dx i dx j In the (radial) ADM decomposition Einstein equations can be written as ( [Skenderis,Papadimitriou’04] ) K 2 � K ij K ij ˜ R + 2 κ 2 T d +1 d +1 , = r i K i κ 2 T jd +1 , j � r j K = ✓ ◆ 1 ˙ ˜ K i j + K K i R i T i σ δ i j � κ 2 = j � d � 1 T σ j j Just pure gravity (cc) the “radial Hamiltonian” constraint R + d ( d − 1) K 2 − K ij K ij = ˜ . l 2 p γ p γ π i With canonical momenta π ij = κ 2 ( K γ ij � K ij ) , i = κ 2 ( d � 1) K, p γ κ 2  �  � 1 R + d ( d � 1) ˜ i ) 2 π ij π ij � d � 1( π i + = 0 . p γ κ 2 l 2 (QGR: WDW eq. see later)

AdS/CFT flow equation from Gauss-Codazzi [McGough’18] [Kraus,Liu,Marolf’18] [Taylor’18] [Hartman et al.’18] Pure gravity R + d ( d − 1) K 2 − K ij K ij = ˜ . l 2 [Balasubramanian,Kraus’99] Given a general holographic energy-momentum tensor h T ij i = � 1  K ij � K γ ij � d � 1 � l and a d = � a d C ij γ ij ( d − 2) κ 2 κ 2 l ˆ GC can be rewritten as a “flow equation” T ij = T ij + a d C ij , i = � l κ 2  1 ⌘ 2 � l T ij � ⇣ ˆ T ij ˆ ˆ ˆ 2 κ 2 ˜ T i T i � R. [Taylor’18] i 2 d � 1 c γ b γ ij ! r 2 ij , T And translated/interpreted as a large N QFT flow ( etc.) h T i i i = � d λ h X d i

[Hartman,Krutho ff ,Shaghoulian,Tajdini’18] Large N Flow equation [PC,Datta,Shyam’19] ˆ In terms of the “renormalized” EM tensor T ij = T ij + a d C ij ,  1 i ) 2 + 2 ↵ d ✓ 1 ◆ T ij T ij − T ij C ij − T i d − 1( T i d − 1 T i i C i = − d � i i d − 2 � d ✓ ↵ d ◆# ◆ 2 ✓ 1 ◆ + 1 ✓ ( d − 2) ↵ d ˜ C ij C ij − i ) 2 d − 1( C i R + C i + i d − 2 d − 2 2 d � � � d d with “dictionary” for λ , α d λ = l κ 2 lr d − 2 α d c ⌘ , ( d � 2) κ 2 , d − 2 2 dr d λ d c Matching with anomalies and known holographic setups we have e.g. α 6 = N (2 , 0) N ABJM α 4 = N SYM α 3 = 6 2 1 / 3 π 2 / 3 , 2 7 / 2 π , 24 π , 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 Precision tests of AdS/CFT (localization) Even-d: sensitive to a-type holographic anomalies “Good” holographic measure of degrees of freedom Closely related to entanglement entropy See also applications in dS/dS [Gorbenko,Silverstein,Torroba’18]

Holographic sphere partition functions and EM tensor Gravity Action on − shell = − 1 Z d d +1 x √ g ( R − 2 Λ ) + 1 Z d d x √ γ K + S ct , I ( d +1) 2 κ 2 κ 2 M ∂ M [Emparan,Johnson,Myers’99] Holographic counter-terms (up to d=6) " ◆# c (2) c (3) d l 3 S ct = 1 d − 1 ✓ Z d l d d d x √ γ R ij ˜ c (1) ˜ ˜ ˜ R 2 + R + R ij − d κ 2 2( d − 4)( d − 2) 2 2( d − 2) 4( d − 1) l ∂ M (2.2) On-shell solution d ≡ l 2 dr 2 ds 2 = l 2 dr 2 2 l 2 + r 2 + r 2 d Ω 2 l 2 + r 2 + γ ij ( r, x ) dx i dx j , d Holographic Partition functions E-M tensor (Brown-York) δ I ( d +1) on − shell [ r ] ij [ r ] ≡ − 2 log Z S d [ r ] = − I ( d +1) on − shell [ r ] T d , √ γ δγ ij

Sphere partition function in cut-off AdS [PC,Datta,Shyam’19] The large N sphere partition function in finite cut-off holography , � r 2 log Z S d [ r c ] = � dS d r d  ✓ ◆ � l � 1 2 , d � 1 , d + 1 c c 2 F 1 κ 2 l l 2 r c 2 2 # + c (2) � c (3) l 2 l 4 ( d � 1) d ( d � 1) d ( d � 1) + c (1) d r 2 r 4 d 2( d � 2) 8( d � 4) c c Holographic energy-momentum tensor s " # + c (2) − c (3) d l 2 d l 4 1 + l 2 ij [ r c ] = ( d − 1) c (1) T d γ ij . ≡ ω [ r c ] γ ij − d κ 2 l 2 r 2 8 r 4 r 2 c c c They satisfy d d x p γ h T i Z i i = � r d r c ∂ r c log Z S d [ r c ] = � c S d d ω [ r c ] ,

Holographic Field Theory [PC,Datta,Shyam’19] 4 h T ij i = ω d γ ij For the spheres the symmetry fixes This way the flow equation becomes quadratic equation for ω d  � d d + 2 α d 1 i ω d � 1 α d d � 1 ω 2 d � 1 C i d ω d = d λ d f d ( R ) d − 2 d − 2 d λ λ λ d The two solutions are (- to match anomalies) 0 1 v ! 2 u 2 2 2 = d � 1 @ 1 � 2 α d λ 1 � 2 α d λ + 4 α d λ d d d u ω ( ± ) d � 1 C i d � 1 C i d � 1 f d ( R ) i ± B C t d i 2 d λ A C ij with explicit form of and holographic dictionary we reproduce the gravity bulk computation from the TT flow equation!

Wheeler-DeWitt equation in mini-superspace [PC,S.Hirano’18] Quantum Gravity Path-Integral in the “minisuperspace” approximation ds 2 = N 2 ( r ) dr 2 + a 2 ( r ) d Ω 2 d , Quantum Gravity action (q=a^2 for a canonical kinetic term)  q 0 2 q d � 3 + l � 2 q d � 2 �� S EH + S GH = − d ( d − 1) S d Z � 4 N + N dr 2 κ 2 Hamiltonian " # ◆ 2 � 2 κ 2 ✓ d ( d − 1) S d l 2 q d − 3 + q d − 2 � H = N ˆ p 2 − H = − S d d ( d − 1) N 2 κ 2 l Quantum Wheeler-DeWitt equation " # ◆ 2 � ~ 2 d 2 ✓ d ( d − 1) S d l 2 q d − 3 + q d − 2 � ˆ H Ψ [ q ] = Ψ [ q ] = 0 . dq 2 − 2 κ 2 l

[deBoer,Verlinde,Verlinde’99] Wheeler-DeWitt equation: (HJ) WKB solution [PC,S.Datta,V.Shyam’19] Quantum Gravity “radial Hamiltonian” yields the WDW equation (q=a^2) " # ◆ 2 � ~ 2 d 2 ✓ d ( d − 1) S d l 2 q d − 3 + q d − 2 � ˆ H Ψ [ q ] = Ψ [ q ] = 0 . dq 2 − 2 κ 2 l Using the WKB expansion we have the leading order solution (HJ) ◆ Z q  ✓ d ( d − 1) S d � l 2 q d − 3 + q d − 2 dq p q = r 2 Ψ WKB ( q ) ≈ exp ± 2 κ 2 l ~ c 0 Which is precisely the “bare” gravity on-shell action with finite cut-off ◆� = e − ( I on − shell [ r c ] − S ct [ r c ] ) Ψ WKB [ r c ] GR [Freidel’08] Extra canonical transformation to include the counter-terms needed

ABJM and the Airy function [Fuji,Hirano,Moriyama’11] [Marino,Putrov’12] [Hatsuda,Moriyama,Okuyama’12] "✓ π N 2 ◆ 2 / 3 ✓ ◆# 1 λ Z ABJM ( S 3 ) ∼ Ai 1 − + ~instant. √ 24 λ − 3 N 2 2 λ [Bergman,Hirano’09] Non-perturbative All orders pert. QGR.! [Klebanov,Tseytlin’96] Classical GR 1 3 x 3 / 2 ✓ 5 385 ◆ x 1 / 4 e − 2 Ai [ x ] ∼ 1 − 48 x 3 / 2 + 4608 x 3 + ... x − 3 / 2 ∼ N − 3 / 2 ∼ G N 2 loop QGR 3 loop QGR 1 loop QGR [Bhattacharyya,Grassi,Marino,Sen’12] [Dabholkar,Drukker,Gomes’14] Localization in SUGRA

WDW and Airy [PC,S.Hirano’18] WdW equation in AdS in 4 dim. => Airy equation! [ N ℓ 2 ( q + ℓ 2 ) ] Ψ ( q ) = 0 d 2 9 π 2 dq 2 − 16 G 2 General (quantum) solution 2 2 ( 4 G N ) ( 4 G N ) 3 3 3 πℓ 2 3 πℓ 2 ( ℓ − 2 q + 1 ) ( ℓ − 2 q + 1 ) Ψ ( q ) = C 1 Ai + C 2 Bi Holographic dictionary (ABJM) and the decaying part for larger N (Ai, C2=0) 3 S 3 l 2 = π N 2 κ 2 √ 2 λ We find the relation (Pure Gravity!) Z ABJM ( S 3 ) ∼ Ai( x ) = Ψ W DW ( q → 0) ! ? Works for 1/2 BPS Wilson Loops!

Is bulk really a Tensor Network? ? [Miyaji&Takayanagi’15] Surface/state-correspondence? Is TT-bar one realization of this correspondence?

Finite cut-of RT from Sphere Partition Functions [Donnelly,Shyam’18] [PC,Datta,Shyam’19] [Banerjee,Bhattacharyya,Chakraborty’19] Given the partition function for a CFT on a replicated geometry d = n 2 sin 2 θ d τ 2 + d θ 2 + cos 2 θ d Ω 2 d Ω 2 d − 2 Trick: 1 � r ⇣ ⌘ S 1 = (1 � n ∂ n ) log Z n [ r ] | n → 1 = log Z S d [ r ] ( d ∂ r So using “bare” partition functions at scale r RT no c.t.! [Murdia,Nomura,Rath,Salzeta’19] , � r 2 S 1 = ( S d ✓ 1 2 , d � 1 , d + 1 ◆ r d − 1 2 F 1 l 2 8 π G N 2 2 This matches RT for (finite cut-off) surface with induced metric ds 2 = l 2 � d ρ 2 + sinh 2 ρ d Ω 2 � d − 2 Z ρ = l d − 1 S d − 2 A d ρ sinh d − 2 ρ S A = 4 G N 4 G N 0 Is it related to EE in TT deformed theory? Meaning on EE in TT?