TO WA R D S W E A K CO U P L I N G I N H O LO G R A P H Y WO R K - PowerPoint PPT Presentation

S A Š O G R OZ DA N OV I N ST I T U U T- LO R E N TZ F O R T H E O R E T I CA L P H Y S I C S L E I D E N U N I V E R S I TY TO WA R D S W E A K CO U P L I N G I N H O LO G R A P H Y WO R K I N CO L L A B O R AT I O N W I T H J . CA S A L D E R R E Y-S O L A N A , N . KA P L I S , N . P O OV U T T I K U L , A . STA R I N E T S A N D W. VA N D E R S C H E E OX F O R D, 6 . 3 . 2 0 1 7

2 O U T L I N E • from holography to experiment • coupling constant dependence and universality in hydrodynamics • thermalisation and higher-energy spectrum • heavy ion collisions • conclusion and future directions

3 F R O M H O LO G R A P H Y TO E X P E R I M E N T some strongly coupled field theories have a • dual gravitational description (AdS/CFT correspondence) Z [field theory] = Z [string theory] originally a duality in type IIB string theory • so far: universality at strong coupling • challenges • field content (no supersymmetry) • away from infinite N • away from infinite coupling • α 0 ∝ 1 / λ 1 / 2 λ ≡ g 2 Y M N

4 T WO C L A S S E S O F T H E O R I E S top-down dual of N=4 theory with ’t Hooft coupling corrections from • type IIB string theory ✓ ◆ 1 R − 1 1 Z d 10 x √− g 5 + γ e − 3 2 ( ∂φ ) 2 − 2 φ W + . . . 4 · 5! F 2 S IIB = 2 κ 2 10 α 0 /L 2 = λ � 1 / 2 γ = α 0 3 ζ (3) / 8 ρσδ + 1 W = C αβγδ C µ βγν C ρσ µ C ν 2 C αδβγ C µ νβγ C ρσ µ C ν α α ρσδ bottom-up curvature-squared theory with a special, non-perturbative • case of Gauss-Bonnet gravity Z 1 α 1 R 2 + α 2 R µ ν R µ ν + α 3 R µ νρσ R µ νρσ �⇤ d 5 x √− g R − 2 Λ + L 2 � ⇥ S R 2 = 2 κ 2 5  R 2 − 4 R µ ν R µ ν + R µ νρσ R µ νρσ �� Z R − 2 Λ + λ GB 1 d 5 x √− g L 2 � S GB = 2 κ 2 2 5

H Y D R O DY N A M I C S

6 H Y D R O DY N A M I C S QCD and quark-gluon plasma • low-energy limit of QFTs (effective field theory) • = ( ε + P ) u µ u ν + Pg µ ν � ησ µ ν � ζ r · u ∆ µ ν + . . . T µ ν � u λ , T, µ � r µ T µ ν = 0 tensor structures (phenomenological gradient expansions) • with transport coefficients (microscopic)

7 H Y D R O DY N A M I C S conformal (Weyl-covariant) hydrodynamics • T µ µ = 0 g µ ν → e − 2 ω ( x ) g µ ν T µ ν → e 6 ω ( x ) T µ ν infinite-order asymptotic expansion • ∞ O H X T µ ν = T µ ν X α n k n +1 ω = ( n ) n =0 n =0 classification of tensors beyond Navier-Stokes • first order: 2 (1 in CFT) - shear and bulk viscosities second order: 15 (5 in CFT) - relaxation time, … [Israel-Stewart and extensions] third order: 68 (20 in CFT) - [S. G., Kaplis, PRD 93 (2016) 6, 066012, arXiv:1507.02461] 


8 H Y D R O DY N A M I C S • diffusion and sound dispersion relations in CFT " # η 2 τ Π ( ε + P ) 2 � 1 η θ 1 k 4 + O ε + P k 2 � i k 5 � � shear: ω = � i 2 ε + P " # η 2 τ Π 8 ( ε + P ) 2 � 1 θ 1 + θ 2 ω = ± c s k � i Γ c k 2 ⌥ Γ c k 4 + O Γ c � 2 c 2 k 3 � i k 5 � � � � sound: s τ Π 2 c s 9 3 ε + P • loop corrections break analyticity of the gradient expansion (long-time tails), but are 1/N suppressed [Kovtun, Yaffe (2003)] • entropy current, constraints on transport and new transport coefficients (anomalies, broken parity) • non-relativistic hydrodynamics • hydrodynamics from effective Schwinger-Keldysh field theory with dissipation [Nicolis, et. al.; S. G., Polonyi; Haehl, Loganayagam, Rangamani; de Boer, Heller, Pinzani-Fokeeva; Crossley, Glorioso, Liu]

9 H Y D R O DY N A M I C S F R O M H O LO G R A P H Y holography can compute microscopic transport coefficients [Policastro, • Son, Starinets (2001)] low-energy limit of QFTs / low-energy gravitational perturbations in • backgrounds with black holes Green’s functions and Kubo formulae • R (0) − 1 RA (0 , x ) h cd ( x ) + 1 Z Z d 4 xG ab,cd d 4 xd 4 yG ab,cd,ef T ab = G ab ⌦ ↵ R (0) (0 , x, y ) h cd ( x ) h ef ( y ) + . . . RAA 2 8 quasi-normal modes [Kovtun, Starinets (2005)] • ∞ X α n k n +1 ω = n =0 fluid-gravity [Bhattacharyya, Hubeny, Minwalla, Rangamani (2007)] 
 •

10 N = 4 AT I N F I N I T E CO U P L I N G type IIB theory on S 5 dual to N=4 supersymmetric Yang-Mills at • infinite ’t Hooft coupling and infinite N c ✓ ◆ Z 1 R + 12 d 5 x √− g S = 2 κ 2 L 2 5 κ 5 = 2 π /N c black brane • ds 2 = r 2 du 2 − f ( u ) dt 2 + dx 2 + dy 2 + dz 2 � f ( u ) = 1 − u 2 0 � + 4 u 2 f ( u ) u use to find field theory stress-energy tensor to third order 
 •  � h D σ ab i + 1 T ab = ε u a u b + P ∆ ab � ησ ab + ητ Π 3 σ ab ( r · u ) h i R h ab i � 2 u c R c h ab i d u d + κ 20 c σ b i c + λ 2 σ h a c Ω b i c + λ 3 Ω h a c Ω b i c + X + λ 1 σ h a λ (3) n O n n =1

11 N = 4 AT I N F I N I T E CO U P L I N G transport coefficients [S. G., Kaplis, PRD 93 (2016) 6, 066012, arXiv:1507.02461] 
 • η = π s = 1 η 8 N 2 c T 3 4 π κ = N 2 c T 2 λ 1 = N 2 c T 2 λ 2 = − N 2 c T 2 τ Π = (2 − ln 2) ln 2 λ 3 = 0 2 π T 8 16 8 ≡ − θ 1 = − N 2 c T λ (3) + λ (3) + λ (3) 1 2 4 32 π ≡ − θ 2 = N 2 ✓ π 2 ◆ c T 12 + 18 ln 2 − ln 2 2 − 22 λ (3) + λ (3) + λ (3) 3 5 6 384 π 16 = N 2 ✓ π 2 ◆ c T 12 + 4 ln 2 − ln 2 2 λ (3) − λ (3) 1 16 π 17 = N 2 ✓ π 2 ◆ c T 12 + 2 ln 2 − ln 2 2 λ (3) 16 π λ (3) + 4 λ (3) + 4 λ (3) + 5 λ (3) + 5 λ (3) + 4 λ (3) − λ (3) 1 2 3 4 5 6 7 6 3 3 6 6 3 2 + 3 λ (3) + λ (3) − 2 λ (3) − 11 λ (3) − λ (3) + λ (3) 15 = N 2 c T 15 − 2 π 2 − 45 ln 2 + 24 ln 2 2 − λ (3) 8 9 10 11 12 13 � � 2 2 3 6 3 6 648 π

12 TO P- D O W N CO N ST R U C T I O N type IIB action with ’t Hooft coupling corrections • ✓ ◆ 1 R − 1 1 Z d 10 x √− g 5 + γ e − 3 2 ( ∂φ ) 2 − 2 φ W + . . . 4 · 5! F 2 S IIB = 2 κ 2 10 α 0 /L 2 = λ � 1 / 2 γ = α 0 3 ζ (3) / 8 ρσδ + 1 W = C αβγδ C µ βγν C ρσ µ C ν 2 C αδβγ C µ νβγ C ρσ µ C ν α α ρσδ dimensional reduction • ✓ ◆ Z 1 R + 12 d 5 x √− g S = L 2 + γ W 2 κ 2 5 black brane 
 • ds 2 = r 2 du 2 − f ( u ) Z t dt 2 + dx 2 + dy 2 + dz 2 � f ( u ) = 1 − u 2 0 � + Z u 4 u 2 f u 5 u 2 + 5 u 4 − 3 u 6 � 5 u 2 + 5 u 4 − 19 u 6 � � � Z t = 1 − 15 γ Z u = 1 + 15 γ

13 TO P- D O W N CO N ST R U C T I O N N=4 transport coefficients to second order [S. G., Starinets, JHEP 1503 (2015) • 007 arXiv:1412.5685] η = π s = 1 c T 3 (1 + 135 γ + . . . ) η ⇣ 1 + 15 ζ (3) λ − 3 / 2 + . . . ⌘ 8 N 2 4 π τ Π = (2 − ln 2) + 375 γ 4 π T + . . . 2 π T κ = N 2 c T 2 (1 − 10 γ + . . . ) 8 η λ 1 = N 2 c T 2 s (1 + 350 γ + . . . ) 16 λ 2 = − N 2 c T 2 (2 ln 2 + 5 (97 + 54 ln 2) γ + . . . ) 16 λ 3 = 25 N 2 c T 2 γ + . . . 2 ¯ h 4 π k B g 2 N c 0 [Kovtun, Son, Starinets (2005)]

14 B OT TO M - U P CO N ST R U C T I O N curvature-squared theory [S. G., Starinets, JHEP 1503 (2015) 007 arXiv:1412.5685] • Z 1 α 1 R 2 + α 2 R µ ν R µ ν + α 3 R µ νρσ R µ νρσ �⇤ d 5 x √− g R − 2 Λ + L 2 � ⇥ S R 2 = 2 κ 2 5 η = r 3 + (1 − 8 (5 α 1 + α 2 )) + O ( α 2 i ) 2 κ 2 5 ητ Π = r 2 − r 2 + (2 − ln 2) + (23 + 5 ln 2) ✓ 1 − 26 ◆ α 3 + O ( α 2 3 (5 α 1 + α 2 ) i ) 4 κ 2 12 κ 2 5 5 κ = r 2 − 25 r 2 ✓ ◆ 1 − 26 + + α 3 + O ( α 2 3 (5 α 1 + α 2 ) i ) 2 κ 2 6 κ 2 5 5 λ 1 = r 2 − r 2 ✓ 1 − 26 ◆ + + α 3 + O ( α 2 3 (5 α 1 + α 2 ) i ) 4 κ 2 12 κ 2 5 5 λ 2 = − r 2 − r 2 + ln 2 ✓ ◆ + (21 + 5 ln 2) 1 − 26 α 3 + O ( α 2 3 (5 α 1 + α 2 ) i ) 2 κ 2 6 κ 2 5 5 λ 3 = − 28 r 2 + α 3 + O ( α 2 i ) κ 2 5

15 B OT TO M - U P CO N ST R U C T I O N Gauss-Bonnet theory [S. G., Starinets, Theor. Math. Phys. 182 (2015) 1, 61-73] •  R 2 − 4 R µ ν R µ ν + R µ νρσ R µ νρσ �� Z R − 2 Λ + λ GB 1 d 5 x √− g L 2 � S GB = 2 κ 2 2 5 p γ = 1 − 4 λ GB η = s γ 2 / 4 π 1 ✓ 1 ✓ 5 + γ − 2 ◆ − 1  2 (1 + γ ) �◆ s = 1 η τ Π = 4 (1 + γ ) 2 log 4 π (1 − 4 λ GB ) 2 π T γ γ ! � 3 − 4 γ + 2 γ 3 � (1 + γ ) η λ 1 = 2 π T 2 γ 2 ✓ ✓ ◆  2 (1 + γ ) �◆ − 1 1 + γ − 2 + 1 λ 2 = − η 4 (1 + γ ) 2 log π T γ γ ! � 3 + γ − 4 γ 2 � (1 + γ ) λ 3 = − η π T γ 2 ! � 2 γ 2 − 1 � (1 + γ ) κ = η π T 2 γ 2 η 2 γ 2 + γ − 1 � � θ 1 = 8 π 2 T 2 γ

16 L I M I T S O F T H E G AU S S - B O N N E T T H E O R Y how can we interpret the extreme limits of the Gauss-Bonnet coupling? • exact spectrum in the extreme (anomalous) limit of • λ GB = 1 / 4 ⇣ ⌘ ⇣ ⌘ p p Scalar: w = − i 4 + 2 n 1 − 4 − 3 q 2 w = − i 4 + 2 n 2 + 4 − 3 q 2 , Shear: w = − 2 i (1 + n 1 ) , w = − 2 i (3 + n 2 ) ⇣ ⌘ ⇣ ⌘ p p Sound: w = − i 4 + 2 n 1 − 4 + q 2 w = − i 4 + 2 n 2 + 4 + q 2 , in the extreme ``weak” limit there is a curvature • λ GB → −∞ singularity, which needs a stringy resolution λ GB →−∞ S GB = λ GB L 2  � Z 4 Λ R 2 − 4 R µ ν R µ ν + R µ νρσ R µ νρσ − d 5 x √− g lim 4 κ 2 λ GB L 2 5 2 3 r r 4 r 2 L 2 r 2 1 − ˜ 4 − ˜ r 2 + ˜ ds 2 = r 4 dt 2 + dx 2 + dy 2 + dz 2 � p + � − λ GB d ˜ 5 L 2 q L 2 ˜ r 4 ˜ r 2 + ˜ 1 − r 4 ˜