Anti-De Sitter space
Transcript of Anti-De Sitter space
Anti-De Sitter spaceA C o m m o n R o o m f o r S t r i n g s a n d A t o m s ?
Outline
Quark Gluon Plasma and Cold Atoms
The AdS/CFT Correspondence
The viscosity calculation
A holographic superfluid
Holography with Schrödinger Symmetry?
Elliptic flow in non-central collisions
Elliptic flow in non-central collisions
dN
d2pT dY=
dN
2πpT dpT dY[1 + 2v2(pT ) cos(2φ) + . . . ]
can be measured and calculated from hydrodynamic simulations
23
0 100 200 300 400N
Part
0
0.02
0.04
0.06
0.08
0.1
v2
PHOBOS
Glauber
!/s=10-4
!/s=0.08
!/s=0.16
0 1 2 3 4p
T [GeV]
0
5
10
15
20
25v
2 (p
erce
nt)
STAR non-flow corrected (est.)STAR event-plane
Glauber
!/s=10-4
!/s=0.08
!/s=0.16
0 100 200 300 400N
Part
0
0.02
0.04
0.06
0.08
0.1
v2
PHOBOS
CGC
!/s=10-4
!/s=0.08
!/s=0.16
!/s=0.24
0 1 2 3 4p
T [GeV]
0
5
10
15
20
25
v2 (p
erce
nt)
STAR non-flow corrected (est).STAR event-plane
CGC!/s=10
-4
!/s=0.08
!/s=0.16
!/s=0.24
FIG. 8: (Color online) Comparison of hydrodynamic models to experimental data on chargedhadron integrated (left) and minimum bias (right) elliptic flow by PHOBOS [85] and STAR [87],respectively. STAR event plane data has been reduced by 20 percent to estimate the removal
of non-flow contributions [87, 88]. The line thickness for the hydrodynamic model curves is anestimate of the accumulated numerical error (due to, e.g., finite grid spacing). The integrated v2
coefficient from the hydrodynamic models (full lines) is well reproduced by 12ep (dots); indeed, the
difference between the full lines and dots gives an estimate of the systematic uncertainty of thefreeze-out prescription.
experimental data from STAR with the hydrodynamic model is shown in Fig. 8.For Glauber-type initial conditions, the data on minimum-bias v2 for charged hadrons
is consistent with the hydrodynamic model for viscosities in the range η/s ∈ [0, 0.1], whilefor the CGC case the respective range is η/s ∈ [0.08, 0.2]. It is interesting to note thatfor Glauber-type initial conditions, experimental data for both the integrated as well as theminimum-bias elliptic flow coefficient (corrected for non-flow effects) seem to be reproducedbest7 by a hydrodynamic model with η/s = 0.08 " 1
4π . This number has first appeared in the
7 In Ref. [22] a lower value of η/s for the Glauber model was reported. The results for viscous hydrodynamics
shown in Fig. 8 are identical to Ref. [22], but the new STAR data with non-flow corrections became
[Romatschke, Luzum, Phys.Rev.C78:034915,2008]
η
s=
14π ?
From Quark Matter conference 2009, Session 12: AdS/CFT, Cold Atoms, Flow
John Thomas (Duke University)
1 10 100 1000T, K
0
50
100
150
200
Helium 0.1MPaNitrogen 10MPa
Water 100MPa
Viscosity bound
4! "
sh
Figure 2: The viscosity-entropy ratio for some common substances: helium, nitrogen and
water. The ratio is always substantially larger than its value in theories with gravity duals,
represented by the horizontal line marked “viscosity bound.”
experimentally whether the shear viscosity of these gases satisfies the conjectured bound.
This work was supported by DOE grant DE-FG02-00ER41132, the National Science
Foundation and the Alfred P. Sloan Foundation.
References
[1] S.W. Hawking, “Particle creation by black holes,” Commun. Math. Phys. 43, 199 (1975).
[2] J.D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7, 2333 (1973).
[3] G.T. Horowitz and A. Strominger, “Black strings and p-branes,” Nucl. Phys. B 360,
197 (1991).
[4] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, New York, 1987).
[5] G. ’t Hooft, “Dimensional reduction in quantum gravity,” gr-qc/9310026.
7
[Kovtun, Son, Starinets, Phys.Rev.Lett.94:111601,2005] (KSS bound)
AdS/CFT
String Theory: Quantum Gravity
Elementary Particles Strings
Closed Strings: Graviton
Open Strings: Gauge Bosons, Matter
Supersymmetry
Higher Dimensions (10, 11)
D-Branes
D-brane : place where open strings end
arbitrary dimension (up to 10)
Non-abelian Gauge theories
non-perturbative D-brane geometry
Decoupling
IIb String theory with N D3-branes
N=4 SUSY + Gravity + massive Stringstates
decouple by taking
leaves N=4 SUSY gauge theory
Geometry = Anti-deSitter space
α′ → 0
AdS/CFT
Maldacena 1998:
metric
IIb strings in AdS5 x S5 = SYM
strong/weak coupling duality
ds2 =r2
L2(−dt2 + d!x2) +
L2dr2
r2+ L2dΩ2
5
L4
α′2 = g2Y MN gs =
1N
L =1
g2Y M
tr
(−1
4FµνFµν + iψiσ
µDµψi +12DµΦijD
µΦij . . .
)
AdS CFT
Finite temperature
Black hole metric
Horizon
Hawking temperature = field theory temperature
dual to N=4 plasma
ds2 = r2(−f(r)dt2 + d!x2) +dr2
f(r)r2
f(r) = 1− π4T 4
r4
Finite Temperature
AdS CFT: Green Functions
field in AdS with fixed boundary value
acts as source for gauge invariant operatore.g. metric
allows calculation of n-point functions
in practice: large N and strong coupling: classical gravity (5D Einstein + cosmological constant)
ZAdS [Φ0] = 〈eR
Φ0O〉CFT
gµν → Tµν
retarded Gf’s
conformal weight
boundary
at horizon: infalling
retarded Green function
1√−g
∂r
(√−ggrr∂rφ
)− (k2 + m2)φ = 0
limr→∞
r∆−φ = φ0
∆(∆− d) = m2
φ(r) = [Ar−∆−(1 + . . . ) + Br−∆+(1 + . . . )]φ0
GR =B
A
retarded Gf’s
Linear response
Response to an external source j(t,x):
〈Φ(t,x)〉 = −∫
dτ d3ξ GR(t − τ,x − ξ) j(τ, ξ) , (1)
Im ν
Re ν
Hydro and beyond – p.12/28
GR has poles at A=0 : quasinormal frequencies
ωn = ±Ωn − iΓn Dissipation
Shear ViscosityThe theorists way of measuring viscosity:
gravitational waves!
Hydrodynamics predicts
D =η
sT
Compute from gravitational shear wave in AdS!
Either compute pole or set k=0 and use Kubo formula
〈TxyTxy〉 =iηω2
ω + iDk2
universal result: valid for all holographic field theories to leading order (even for anisotropic systems, Miesowicz coeff [K.L., J. Mas, JHEP 0707:088,2007] )
Conjectured lower bound (recent counter examples for sub-leading N contributions)
Consistent with low value at RHIC (and cold atoms?)
Shear Viscosityη
s=
14π
[G. Policastro, D.T. Son, A. Starinets, PRL 87:081601]
Quantum Criticality[C. Herzog, P. Kovtun, S. Sachdev, D.T. Son; Phys.Rev.D75:085020,2007]
• AdS4 (2+1) dim CFT
• study conductivity
• hydrodynamic-to-collisionless ω ∝ q2 ω ∝ q
0.2 0.4 0.6 0.8 1.0
!0.6
!0.5
!0.4
!0.3
!0.2
!0.1
σ(ω) = σ0
[J. Mas, M. Kaminski, K.L., J. Mas, J. Tarrio, to appear]
A Holographic Superfluid
abelian Higgs model in AdS4 Black Hole
charge scalar condenses for low T
charge manages to hoover over the horizon
chem. potential At -component
[S. Gubser, Phys.Rev.Lett.101:191601,2008][S. Hartnoll, C. Herzog, G. Horowitz, Phys.Rev.Lett.101:031601,2008]
L = −14F 2
µν −m2ΨΨ−DµΨDµΨ
At = µ − n
r+ O(1/r2)
Ψ =J
r+
〈O〉r2
+ O(1/r3)
A Holographic SuperfluidQuasinormal frequencies for Ψ unbroken phase 〈O〉 = 0
T < Tc
[I. Amado, M. Kaminski, K.L., JHEP 0905:021,2009]
〈O〉 ∝(
1− T
Tc
)1/2
A Holographic SuperfluidConductivity:
σn = limω→0
![σ(ω)] = exp(−∆/T ) ∆ =2 ωg ∆ ∼ 8Tc
A Holographic Superfluid2nd Sound: speed and attenuation
Further Results
p-wave superconductors (S.Gubser)
explicit string models (J. Erdmenger et al.)
models with pseudogap regions (J. Erdmenger et al.)
dynamical universality class: Model A with z=2 (K. Maeda et al.)
Meissner effect (1/2 of it: generation of currents)
(optimistic) hope: relevant for high Tc superconductors
Schrödinger Symmetry
Quantum critical point with z=2
Symmetry group of Schrödinger equation
Fermions at unitarity
Generators are
central extension
t→ λzt , "x→ λ"x
Translations Pi , Rotations Mij
Galilean boosts Ki , time translations H
Dilatations D , special conformal C
[Ki, Pj ] = −iδijN
Particle Number
Schrödinger Symmetry
Schrödinger algebra on the lightcone: x± = (x0 ± x3)
E2 − !p 2 = 0 −→ p− =p2⊥
2p+
Hamiltonian H = p−
Mass N = p+
Boosts Ki = M i+
Dilatations D = Dr + M+−
Special conformal C = K+r
Schrödinger Symmetry
ds2 = −β2r4dt2 + r2(−2dξdt + dx2⊥) +
dr2
r2
[D.T. Son, Phys. Rev. D 78 (2008) 046003][K. Balasubramanian,
J. McGreevy, Phys.Rev.Lett.
101:061601,2008]
• Schrödinger = isometries
• d+1+2 dimensions
(space , time, holographic and extra)
• particle number ~
compactify to get discrete spectrum
• non-relativistic causal structure
• singular free energy
• superfluid at low T?
• DLCQ of non-commutative theory?
∂ξ
F = −cN2V MT 2
(T
µ
)2
SummaryHolographic Quantum Field Theories
strongly coupled transport theory
useful for QGP
maybe useful for QCPs
relativistic superfluids, (Non-)Fermi liquids, ...
non-relativistic ?
holographic condensed matter physics ?
Additional Reading
Gauge-Gravity Duality: S. Gubser, A. Karch, arXiv:0901.0953 [hep-th]
Lectures on Holography and Condensed Matter Physics: S. Hartnoll, arXiv:0903.3246 [hep-th]
Lectures on holographic Superfluidity and Superconductivity, C. Herzog, arXiv:0904.1975 [hep-th]
Nearly Perfect Fluidity: QGP and Cold Atoms,T. Schaefer, D. Teaney, arXiv:0904.3107 [hep-ph]