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 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
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
OX F O R D, 6 . 3 . 2 0 1 7
S A Š O G R OZ DA N OV
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
hydrodynamics
2
dual gravitational description (AdS/CFT correspondence)
Z[field theory] = Z[string theory]
3
α0 ∝ 1/λ1/2 λ ≡ g2
Y MN
type IIB string theory
SIIB = 1 2κ2
10
Z d10x√−g ✓ R − 1 2 (∂φ)2 − 1 4 · 5!F 2
5 + γe− 3
2 φW + . . .
◆ γ = α03ζ(3)/8 α0/L2 = λ1/2
W = CαβγδCµβγνC ρσµ
α
Cν
ρσδ + 1
2CαδβγCµνβγC ρσµ
α
Cν
ρσδ
4
case of Gauss-Bonnet gravity
SR2 = 1 2κ2
5
Z d5x√−g ⇥ R − 2Λ + L2 α1R2 + α2RµνRµν + α3RµνρσRµνρσ⇤
SGB = 1 2κ2
5
Z d5x√−g R − 2Λ + λGB 2 L2 R2 − 4RµνRµν + RµνρσRµνρσ
with transport coefficients (microscopic)
6
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]
µ = 0
ω =
OH
X
n=0
αnkn+1
T µν =
∞
X
n=0
T µν
(n)
7
suppressed [Kovtun, Yaffe (2003)]
broken parity)
[Nicolis, et. al.; S. G., Polonyi; Haehl, Loganayagam, Rangamani; de Boer, Heller, Pinzani-Fokeeva; Crossley, Glorioso, Liu]
shear: ω = i η ε + P k2 i " η2τΠ (ε + P)2 1 2 θ1 ε + P # k4 + O
sound: ω = ±csk iΓck2 ⌥ Γc 2cs
sτΠ
" 8 9 η2τΠ (ε + P)2 1 3 θ1 + θ2 ε + P # k4 + O
8
Son, Starinets (2001)]
backgrounds with black holes
⌦ T ab
R (0)
↵ =Gab
R (0) − 1
2 Z d4xGab,cd
RA (0, x)hcd(x) + 1
8 Z d4xd4yGab,cd,ef
RAA
(0, x, y)hcd(x)hef(y) + . . .
ω =
∞
X
n=0
αnkn+1
9
infinite ’t Hooft coupling and infinite Nc
S = 1 2κ2
5
Z d5x√−g ✓ R + 12 L2 ◆
κ5 = 2π/Nc
ds2 = r2 u
+ du2 4u2f(u)
f(u) = 1 − u2
T ab = εuaub + P∆ab ησab + ητΠ
hDσabi + 1
3σab (r · u)
h Rhabi 2ucRchabidud i + λ1σha
cσbic + λ2σha cΩbic + λ3Ωha cΩbic + 20
X
n=1
λ(3)
n On 10
η s = 1 4π
λ(3)
1
+ λ(3)
2
+ λ(3)
4
≡ −θ1 = −N 2
c T
32π λ(3)
3
+ λ(3)
5
+ λ(3)
6
≡ −θ2 = N 2
c T
384π ✓π2 12 + 18 ln 2 − ln2 2 − 22 ◆ λ(3)
1
− λ(3)
16 = N 2 c T
16π ✓π2 12 + 4 ln 2 − ln2 2 ◆ λ(3)
17 = N 2 c T
16π ✓π2 12 + 2 ln 2 − ln2 2 ◆
λ(3)
1
6 + 4λ(3)
2
3 + 4λ(3)
3
3 + 5λ(3)
4
6 + 5λ(3)
5
6 + 4λ(3)
6
3 − λ(3)
7
2 + 3λ(3)
8
2 + λ(3)
9
2 − 2λ(3)
10
3 − 11λ(3)
11
6 − λ(3)
12
3 + λ(3)
13
6 − λ(3)
15 = N 2 c T
648π
8 N 2
c T 3
τΠ = (2 − ln 2) 2πT κ = N 2
c T 2
8 λ1 = N 2
c T 2
16 λ2 = −N 2
c T 2
8 ln 2 λ3 = 0
11
SIIB = 1 2κ2
10
Z d10x√−g ✓ R − 1 2 (∂φ)2 − 1 4 · 5!F 2
5 + γe− 3
2 φW + . . .
◆ γ = α03ζ(3)/8 α0/L2 = λ1/2
W = CαβγδCµβγνC ρσµ
α
Cν
ρσδ + 1
2CαδβγCµνβγC ρσµ
α
Cν
ρσδ
S = 1 2κ2
5
Z d5x√−g ✓ R + 12 L2 + γW ◆
ds2 = r2 u
+ Zu du2 4u2f
f(u) = 1 − u2
Zt = 1 − 15γ
Zu = 1 + 15γ
12
007 arXiv:1412.5685]
η = π 8 N 2
c T 3 (1 + 135γ + . . . )
τΠ = (2 − ln 2) 2πT + 375γ 4πT + . . . κ = N 2
c T 2
8 (1 − 10γ + . . . ) λ1 = N 2
c T 2
16 (1 + 350γ + . . . ) λ2 = −N 2
c T 2
16 (2 ln 2 + 5 (97 + 54 ln 2) γ + . . . ) λ3 = 25N 2
c T 2
2 γ + . . .
η s = 1 4π ⇣ 1 + 15ζ(3)λ−3/2 + . . . ⌘
¯ h 4πkB η s g2Nc
[Kovtun, Son, Starinets (2005)]
13
SR2 = 1 2κ2
5
Z d5x√−g ⇥ R − 2Λ + L2 α1R2 + α2RµνRµν + α3RµνρσRµνρσ⇤
η = r3
+
2κ2
5
(1 − 8 (5α1 + α2)) + O(α2
i )
ητΠ = r2
+ (2 − ln 2)
4κ2
5
✓ 1 − 26 3 (5α1 + α2) ◆ − r2
+ (23 + 5 ln 2)
12κ2
5
α3 + O(α2
i )
κ = r2
+
2κ2
5
✓ 1 − 26 3 (5α1 + α2) ◆ − 25r2
+
6κ2
5
α3 + O(α2
i )
λ1 = r2
+
4κ2
5
✓ 1 − 26 3 (5α1 + α2) ◆ − r2
+
12κ2
5
α3 + O(α2
i )
λ2 = −r2
+ ln 2
2κ2
5
✓ 1 − 26 3 (5α1 + α2) ◆ − r2
+ (21 + 5 ln 2)
6κ2
5
α3 + O(α2
i )
λ3 = −28r2
+
κ2
5
α3 + O(α2
i )
14
SGB = 1 2κ2
5
Z d5x√−g R − 2Λ + λGB 2 L2 R2 − 4RµνRµν + RµνρσRµνρσ
η = sγ2/4π τΠ = 1 2πT ✓1 4 (1 + γ) ✓ 5 + γ − 2 γ ◆ − 1 2 log 2 (1 + γ) γ ◆ λ1 = η 2πT (1 + γ)
2γ2 ! λ2 = − η πT ✓ −1 4 (1 + γ) ✓ 1 + γ − 2 γ ◆ + 1 2 log 2 (1 + γ) γ ◆ λ3 = − η πT (1 + γ)
γ2 ! κ = η πT (1 + γ)
! θ1 = η 8π2T 2 γ
p 1 − 4λGB
η s = 1 4π (1 − 4λGB)
15
16
λGB = 1/4
Scalar: w = −i ⇣ 4 + 2n1 − p 4 − 3q2 ⌘ , w = −i ⇣ 4 + 2n2 + p 4 − 3q2 ⌘ Shear: w = −2i (1 + n1) , w = −2i (3 + n2) Sound: w = −i ⇣ 4 + 2n1 − p 4 + q2 ⌘ , w = −i ⇣ 4 + 2n2 + p 4 + q2 ⌘
singularity, which needs a stringy resolution
lim
λGB→−∞ SGB = λGBL2
4κ2
5
Z d5x√−g R2 − 4RµνRµν + RµνρσRµνρσ − 4Λ λGBL2
p −λGB 2 4− ˜ r2 L2 r 1 − ˜ r4
+
˜ r4 dt2 + L2 ˜ r2 q 1 −
˜ r4
+
˜ r4
d˜ r2 + ˜ r2 L2
3 5
λGB → −∞
arXiv:1611.07053]
17
S = 1 2κ2
5
Z d5x√−g [R − 2Λ + LGB] + Z d5x√−gLA
LA = 1 4FµνF µν + α4RFµνF µν + α5RµνFµρF ρ
ν + α6RµνρσFµνFρσ + α7 (FµνF µν)2
+ α8rµFρσrµF ρσ + α9rµFρσrρF µσ + α10rµF µνrρFρν + α11F µνFνρF ρσFσµ
LA = − 1 4FµνF µν + β1L2 (RFµνF µν − 4RµνFµρF ρ
ν + RµνρσFµνFρσ)
+ β2L2 (FµνF µν)2 + β3L2F µνFνρF ρσFσµ
D = (1 + γGB)(1 + 2β) ⇣ β + p β2 − γ2
GB
⌘ 6(β − 1) h β ⇣ β + p β2 − γ2
GB
⌘ − γ2
GB
i ( p (1 − γ2
GB) (β2 − γ2 GB) ln
" γGB 1 + p 1 − γ2
GB
# −
GB
" γGB β + p β2 − γ2
GB
#)
γGB = p 1 − 4λGB β = 1 + 48β1
D 6= D = 0 D 6= D = 0
(2015) 007 arXiv:1412.5685]
i
πT (1 − γGB)
GB
γ2
GB
= −40λ2
GBη
πT + O
GB
[Gursoy, Tarrio (2015); S. G., Poovuttikul, JHEP 1609 (2016) 046 arXiv: 1603.08770]
✓ hδJµi hδJµ
5 i
◆ = ✓ σJB σJω σJ5B σJ5ω ◆ ✓ Bµ ωµ ◆
σJ5B = −2γµ σJB = −2γµ5, σJ5ω = κµ2
5 + γµ2 + 2λ(2πT)2
σJω = 2γµ5µ
S = Z d5x√−g {L [Aa, Va, gab, φi] + LCS [Aa, Va, gab]}
LCS [Aa, Va, gab] = ✏abcdeAa ⇣ 3 FA,bcFA,de + FV,bcFV,de + Rp
qbcRq pde
⌘
19
[S. G., Kaplis, Starinets, JHEP 1607 (2016) 151 arXiv:1605.02173]
21
∂F ∂t + pi m ∂F ∂ri − ∂U(r) ∂ri ∂F ∂pi = C[F]
F(t, r, p) = F0(r, p) [1 + ϕ(t, r, p)]
∂ϕ ∂t = −pi m ∂ϕ ∂ri + ∂U(r) ∂ri ∂ϕ ∂pi + L0[ϕ] ϕ(t, r, p) = etL ϕ0(r, p) = 1 2πi
γ+i∞
Z
γ−i∞
est Rsds ϕ0(r, p)
Rs = (sI − L)−1
22
ϕ(t, p) = e−νth(p)
−νh = L0[h]
ϕ(t, p) = X
n
Cne−νnthn(p)
dominant:
τR = 1/νmin
τR ≥ C ~ kBT
GR(ω, q) = C∆ π Γ2(∆ − 1) sin π∆
✓∆ 2 + i(ω − q) 4πT ◆ Γ ✓∆ 2 + i(ω + q) 4πT ◆
× " cosh q 2T − cos π∆ cosh ω 2T + i sin π∆ sinh ω 2T #
ω = ±q − i4πT ✓ n + ∆ 2 ◆
τR = 1 2π∆ ~ kBT
23
∂F ∂t + pi m ∂F ∂ri − ∂U(r) ∂ri ∂F ∂pi = −F − F0 τR
cute:
∆ = 2 ⇒ “η/s” = 1/4π
mode calculations) [Hartnoll, Kumar (2005)]
q q
Re ω Im ω Re ω Im ω
24
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x ◆ ◆ ◆ Re ω Im ω x x x x x x x x x x x x x x x x x x x x x x x x ◆ ◆ ◆ Re ω Im ω
``weak limit”
γ → ∞ λGB → −∞
``anomalous limit” λGB → 1/4
25
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x ◆ ◆ ◆ Re ω Im ω x x x x x x x x x x x x x x x x x x x x x x x x ◆ ◆ ◆ Re ω Im ω ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ ○ □ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ ○ □ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ ○ □ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ○ □
26
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x ◆ ◆ ◆ Re ω Im ω x x x x x x x x x x x x x x x x x x x x x x x x ◆ ◆ ◆ Re ω Im ω ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ○ □ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ○ □ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ○ □ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ○ □
27
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x ◆ ◆ ◆ Re ω Im ω x x x x x x x x x x x x x x x x x x x x x x x x ◆ ◆ ◆ Re ω Im ω
η/s ∼ const τRT
○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○○○○○○○○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □□□□□□□□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ○ □
28
29
30
31
1611.07053]
32
Gtt,tt(ω, q) = 3 √ 2π4T 4 (1 + γGB)3/2κ2
5
(1 − ω/ωg) − iγ2
GBωq2/πT
(3ω2 − q2) (1 − ω/ωg) + iγ2
GBωq2/πT
!
Gxz,xz(ω, q) = √ 2π3T 3γ2
GB
(1 + γGB)3/2κ2
5
✓ ω2 iω − iω2/ωg − γ2
GBq2/4πT
◆
ω1 = −i γ2
GB
4πT q2 ω2 = ωg + i γ2
GB
4πT q2
ω1,2 = ± 1 √ 3 q − i γ2
GB
6πT q2 ω3 = ωg + i γ2
GB
3πT q2 ωg = − 8πTi γGB (γGB + 2) − 3 + 2 ln ⇣
2 γGB+1
⌘ ≈ −8πTi γ2
GB
33
34
channel)
35
36
37
38
coupled field theory like QCD?
[S. G., van der Schee (2016) arXiv:1610.08976]
stopping of narrow shocks (88% higher energy density on lightcone) and decreased energy density of wide shocks
40
41
thydThyd = {0.41 − 0.52λGB, 0.43 − 6.3λGB}
at λGB = −0.2 :
{25%, 290%}
42
43
computed from weakly coupled, perturbative physics?
dependent holography to understand intermediate coupling
experiments?
44