AdS/CFT correspondence and hydrodynamics. Andrei Starinets. Oxford University. From Gravity to Thermal Gauge Theories: the AdS/CFT correspondence. Fifth Aegean Summer School Island of Milos Greece. September 2126, 2009. Plan. I. Introduction and motivation. II. Hydrodynamics
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Andrei Starinets
Oxford University
From Gravity to Thermal Gauge Theories: the AdS/CFT correspondence
Fifth Aegean Summer School
Island of Milos
Greece
September 2126, 2009
I. Introduction and motivation
Some references:
D.T.Son and A.O.S., “Viscosity, Black Holes, and Quantum Field Theory”, 0704.0240 [hepth]
P.K.Kovtun and A.O.S., “Quasinormal modes and holography”, hepth/0506184
G.Policastro, D.T.Son, A.O.S., “From AdS/CFT to hydrodynamics”, hepth/0205052
G.Policastro, D.T.Son, A.O.S., “From AdS/CFT to hydrodynamics II: Sound waves”, hepth/0210220
Over the last several years, holographic (gauge/gravity duality) methods
were used to study strongly coupled gauge theories at finite temperature
and density
These studies were motivated by the heavyion collision programs at RHIC
and LHC (ALICE, ATLAS) and the necessity to understand hot and dense
nuclear matter in the regime of intermediate coupling
As a result, we now have a better understanding of thermodynamics
and especially kinetics (transport) of strongly coupled gauge theories
Of course, these calculations are done for theoretical models such as
N=4 SYM and its cousins (including nonconformal theories etc).
We don’t know quantities such as for QCD
Heavy ion collision experiments at RHIC (2000current) and LHC (2009??)
create hot and dense nuclear matter known as the “quarkgluon plasma”
(note: qualitative difference between pp and AuAu collisions)
Evolution of the plasma “fireball” is described by relativistic fluid dynamics
(relativistic NavierStokes equations)
Need to know
thermodynamics(equation of state)
kinetics(first and secondorder transport coefficients)
in the regime of intermediate coupling strength:
initial conditions(initial energy density profile)
thermalization time(start of hydro evolution)
freezeout conditions(end of hydro evolution)
Energy density vs temperature for various gauge theories
Ideal gas
of quarks
and gluons
Ideal gas
of hadrons
Figure: an artistic impression from Myers and Vazquez, 0804.2423 [hepth]
Quantum field theories at finite temperature/density
Equilibrium
Nearequilibrium
transport coefficients
emission rates
………
entropy
equation of state
…….
perturbative
nonperturbative
perturbative
nonperturbative
????
Lattice
kinetic theory
pQCD
L.D.Landau and E.M.Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1987
D.Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions,
Benjamin/Cummings, New York, 1975
P.K. Kovtun and L.G.Yaffe, “Hydrodynamic fluctuations, longtime tails, and supersymmetry”,
hepth/0303010.
Hierarchy of times (e.g. in Bogolyubov’s kinetic theory)
0
t




Mechanical
description
Hydrodynamic
approximation
Equilibrium
thermodynamics
Kinetic
theory
Hierarchy of scales
(L is a macroscopic size of a system)
The hydrodynamic regime (continued)
Degrees of freedom




0
t
Hydrodynamic
approximation
Equilibrium
thermodynamics
Mechanical
description
Kinetic
theory
Coordinates,
momenta of
individual
particles
Coordinate
and time
dependent
distribution functions
Local densities
of conserved charges
Globally conserved
charges
Hydro regime:
Hydrodynamics: fundamental d.o.f. = densities of conserved charges
Need to add constitutive relations!
Example: charge diffusion
Conservation law
Constitutive relation
[Fick’s law (1855)]
Diffusion equation
Dispersion relation
Expansion parameters:
Example: momentum diffusion and sound
Thermodynamic equilibrium:
Nearequilibrium:
Eigenmodes of the system of equations
Shear mode (transverse fluctuations of ):
Sound mode:
For CFT we have
and
Gases (Maxwell, 1867):
Viscosity of a gas is
Liquids (Frenkel, 1926):
“For the viscosity…expansion was developed by Bogolyubov in 1946 and this remained the standard reference
for many years. Evidently the many people who quoted Bogolyubov expansion had never looked in detail at more
than the first two terms of this expansion. It was then one of the major surprises in theoretical physics when
Dorfman and Cohen showed in 1965 that this expansion did not exist. The point is not that it diverges,
the usual hazard of series expansion, but that its individual terms, beyond a certain order, are infinite.
Shear viscosity
Bulk viscosity
Charge diffusion constant
Supercharge diffusion constant
Thermal conductivity
Electrical conductivity
* Expect Einstein relations such as to hold
Hydrodynamics is an effective theory, valid for sufficiently small momenta
Firstorder hydro eqs are parabolic. They imply instant propagation of signals.
This is not a conceptual problem since hydrodynamics becomes “acausal” only
outside of its validity range but it is very inconvenient for numerical work on
NavierStokes equations where it leads to instabilities [Hiscock & Lindblom, 1985]
These problems are resolved by considering next order in derivative expansion,
i.e. by adding to the hydro constitutive relations all possible secondorder terms
compatible with symmetries (e.g. conformal symmetry for conformal plasmas)
(for theories conformal at T=0)
Relaxation time
Second order trasport coefficient
Second order trasport coefficient
Second order trasport coefficient
Second order trasport coefficient
In nonconformal theories such as QCD, the total number of secondorder transport
coefficients is quite large
Derivative expansion in hydrodynamics: first order
Hydrodynamic d.o.f. = densities of conserved charges
or
(4 equations)
(4 d.o.f.)
Firstorder conformal hydrodynamics (in d dimensions)
Weyl transformations:
In firstorder hydro this implies:
Thus, in the firstorder (conformal) hydro:
Secondorder IsraelStewart conformal hydrodynamics
IsraelStewart
Supersymmetric sound mode (“phonino”) in
Hydrodynamic mode
(infinitely slowly relaxing
fluctuation of the charge
density)
Hydro pole
in the retarded
correlator of the
charge density
Conserved
charge
Sound wave pole:
Supersound wave pole:
Lebedev & Smilga, 1988 (see also Kovtun & Yaffe, 2003)
In quantum field theory, the dispersion relations such as
appear as poles of the retarded correlation functions, e.g.
 in the hydro approximation 
Fluctuationdissipation theory
(Callen, Welton, Green, Kubo)
Kubo formulae allows one to calculate transport
coefficients from microscopic models
In the regime described by a gravity dual
the correlator can be computed using
the gauge theory/gravity duality
A
B
A: scalar channel
C
B: scalar channel  thermal part
C: sound channel
at finite temperature and density
strong coupling
10dim gravity
M,J,Q
Holographically dual system
in thermal equilibrium
M, J, Q
T S
Deviations from equilibrium
Gravitational fluctuations
????
and B.C.
Quasinormal spectrum
P.Candelas & D.Sciama, “Irreversible thermodynamics of black holes”, PRL,38(1977) 1732
From brane dynamics to AdS/CFT correspondence
Open strings picture:
dynamics of coincident D3 branes
at low energy is described by
Closed strings picture:
dynamics of coincident D3 branes
at low energy is described by
conjectured
exact equivalence
Maldacena (1997); Gubser, Klebanov, Polyakov (1998); Witten (1998)
supersymmetric YM theory
Gliozzi,Scherk,Olive’77
Brink,Schwarz,Scherk’77
(super)conformal field theory = coupling doesn’t run
conjectured
exact equivalence
Latest test: Janik’08
Generating functional for correlation
functions of gaugeinvariant operators
String partition function
In particular
Classical gravity action serves as a generating functional for the gauge theory correlators
AdS/CFT correspondence: the role of J
For a given operator , identify the source field , e.g.
satisfies linearized supergravity e.o.m. with b.c.
The recipe:
To compute correlators of , one needs to solve the bulk
supergravity e.o.m. for and compute the onshell action
as a functional of the b.c.
Warning: e.o.m. for different bulk fields may be coupled: need selfconsistent solution
Then, taking functional derivatives of gives
Holography at finite temperature and density
Nonzero expectation values of energy and charge density translate into
nontrivial background values of the metric (above extremality)=horizon
and electric potential = CHARGED BLACK HOLE (with flat horizon)
temperature of the dual gauge theory
chemical potential of the dual theory
The bulk and the boundary in AdS/CFT correspondence
UV/IR: the AdS metric is invariant under
z plays a role of inverse energy scale in 4D theory
z
5D bulk
(+5 internal dimensions)
strings
or supergravity fields
0
gauge fields
4D boundary
To extract transport coefficients and spectral functions from dual gravity,
we need a recipe for computing Minkowski space correlators in AdS/CFT
The recipe of [D.T.Son & A.S., 2001]and [C.Herzog & D.T.Son, 2002] relates
realtime correlators in field theory to Penrose diagram of black hole in dual gravity
Quasinormal spectrum of dual gravity = poles of the retarded correlators in 4d theory
[D.T.Son & A.S., 2001]
Example: Rcurrent correlator in
in the limit
Zero temperature:
Finite temperature:
Poles of = quasinormal spectrum of dual gravity background
(D.Son, A.S., hepth/0205051, P.Kovtun, A.S., hepth/0506184)
G.T.Horowitz and V.E.Hubeny, hepth/9909056
D.Birmingham, I.Sachs, S.N.Solodukhin, hepth/0112055
D.T.Son and A.O.S., hepth/0205052; P.K.Kovtun and A.O.S., hepth/0506184
I. Computing the retarded correlator: inc.wave b.c. at the horizon, normalized to 1 at the boundary
II. Computing quasinormal spectrum: inc.wave b.c. at the horizon, Dirichlet at the boundary
O(2) symmetry in xy plane
Shear channel:
Sound channel:
Scalar channel:
Other fluctuations (e.g. ) may affect sound channel
But not the shear channel
universality of
Field theory
Zero temperature:
Finite temperature:
Dual gravity
of the correlator
1. GreenKubo formulas (+ retarded correlator from gravity)
2. Poles of the retarded correlators
3. Lowest quasinormal frequency of the dual background
4. The membrane paradigm
Example: stressenergy tensor correlator in
in the limit
Zero temperature, Euclid:
Finite temperature, Mink:
(in the limit )
The pole
(or the lowest quasinormal freq.)
Compare with hydro:
In CFT:
Also,
(Gubser, Klebanov, Peet, 1996)
Example 2 (continued): stressenergy tensor correlator in
in the limit
Zero temperature, Euclid:
Finite temperature, Mink:
(in the limit )
The pole
(or the lowest quasinormal freq.)
Compare with hydro:
in the limit
Shear viscosity
Bulk viscosity
for nonconformal theories see
Buchel et al; G.D.Moore et al
Gubser et al.
Charge diffusion constant
Supercharge diffusion constant
(G.Policastro, 2008)
Thermal conductivity
Electrical conductivity
Sound and supersymmetric sound in
In 4d CFT
Sound mode:
Supersound mode:
Quasinormal modes in dual gravity
Graviton:
Gravitino:
Strong coupling: A.S., hepth/0207133
Weak coupling: S. Hartnoll and P. Kumar, hepth/0508092
Assuming validity of the gauge/gravity duality,
all transport coefficients are completely determined
by the lowest frequencies
in quasinormal spectra of the dual gravitational background
(D.Son, A.S., hepth/0205051, P.Kovtun, A.S., hepth/0506184)
This determines kinetics in the regime of a thermal theory
where the dual gravity description is applicable
Transport coefficients and quasiparticle spectra can also be
obtained from thermal spectral functions
perturbative thermal gauge theory
S.Huot,S.Jeon,G.Moore, hepph/0608062
Correction to : Buchel, Liu, A.S., hepth/0406264
Buchel, 0805.2683 [hepth]; Myers, Paulos, Sinha, 0806.2156 [hepth]
Weak coupling:
Strong coupling:
* Charge susceptibility can be computed independently:
D.T.Son, A.S., hepth/0601157
Einstein relation holds:
Theorem:
For a thermal gauge theory, the ratio of shear viscosity
to entropy density is equal to
in the regime described by a dual gravity theory
Remarks:
Benincasa, Buchel, Naryshkin, hepth/0610145
Mateos, Myers, Thomson, hepth/0610184
Graviton’s component obeys equation for a minimally
coupled massless scalar. But then .
Since theentropy (density) is
we get
D. Son, P. Kovtun, A.S., hepth/0405231
P. Kovtun, D.Son, A.S., hepth/0309213, A.S., 0806.3797 [hepth],
P.Kovtun, A.S., hepth/0506184, A.Buchel, J.Liu, hepth/0311175
Minimum of in units of
P.Kovtun, D.Son, A.S., hepth/0309213, hepth/0405231
T.Schafer, condmat/0701251
(based on experimental results by Duke U. group, J.E.Thomas et al., 200506)
Chernai, Kapusta, McLerran, nuclth/0604032
Viscosity is ONE of the parameters used in the hydro models
describing the azimuthal anisotropy of particle distribution
Elliptic flow reproduced for
e.g. Baier, Romatschke, nuclth/0610108
Perturbative QCD:
Chernai, Kapusta, McLerran, nuclth/0604032
SYM:
Elliptic flow with color glass condensate initial conditions
Luzum and Romatschke, 0804.4015 [nucth]
Elliptic flow with Glauber initial conditions
Luzum and Romatschke, 0804.4015 [nucth]
Viscosity/entropy ratio in QCD: current status
Theories with gravity duals in the regime
where the dual gravity description is valid
(universal limit)
Kovtun, Son & A.S; Buchel; Buchel & Liu, A.S
QCD: RHIC elliptic flow analysis suggests
QCD: (Indirect) LQCD simulations
H.Meyer, 0805.4567 [hepth]
Trapped strongly correlated
cold alkali atoms
T.Schafer, 0808.0734 [nuclth]
Liquid Helium3
ReissnerNordstromAdS black hole
with three R charges
(Behrnd, Cvetic, Sabra, 1998)
(see e.g. Yaffe, Yamada, hepth/0602074)
J.Mas
D.Son, A.S.
O.Saremi
K.Maeda, M.Natsuume, T.Okamura
We still have
from supersymmetric YangMills plasma
S. CaronHuot, P. Kovtun, G. Moore, A.S., L.G. Yaffe, hepth/0607237
Photon emission from SYM plasma
Photons interacting with matter:
To leading order in
Mimic
by gauging global Rsymmetry
Need only to compute correlators of the Rcurrents
(Normalized) photon production rate in SYM for various values of ‘t Hooft coupling
Now consider strongly interacting systems at finite density
and LOW temperature
Probing quantum liquids with holography
Lowenergy elementary
excitations
Specific heat
at low T
Quantum liquid
in p+1 dim
Quantum Bose liquid
phonons
Quantum Fermi liquid
(Landau FLT)
fermionic quasiparticles +
bosonic branch (zero sound)
Departures from normal Fermi liquid occur in
 3+1 and 2+1 –dimensional systems with strongly correlated electrons
 In 1+1 –dimensional systems for any strength of interaction (Luttinger liquid)
One can apply holography to study strongly coupled Fermi systems at low T
Fermi Liquid Theory: 195658
The simplest candidate with a known holographic description is
at finite temperature T and nonzero chemical potential associated with the
“baryon number” density of the charge
There are two dimensionless parameters:
is the baryon number density
is the hypermultiplet mass
The holographic dual description in the limit
is given by the D3/D7 system, with D3 branes replaced by the AdS
Schwarzschild geometry and D7 branes embedded in it as probes.
Karch & Katz, hepth/0205236
AdSSchwarzschild black hole (brane) background
D7 probe branes
The worldvolume U(1) field couples to the flavor current at the boundary
Nontrivial background value of corresponds to nontrivial expectation value of
We would like to compute
 the specific heat at low temperature
 the charge density correlator
The specific heat (in p+1 dimensions):
(note the difference with Fermi and Bose systems)
The (retarded) charge density correlator
has a pole corresponding to a propagating mode (zero sound)
 even at zero temperature
(note that this is NOT a superfluid phonon whose attenuation scales as )
New type of quantum liquid?
Other avenues of (related) research
Bulk viscosity for nonconformal theories (Buchel, Gubser,…)
Nonrelativistic gravity duals (Son, McGreevy,… )
Gravity duals of theories with SSB (Kovtun, Herzog,…)
Bulk from the boundary (Janik,…)
NavierStokes equations and their generalization from gravity (Minwalla,…)
Quarks moving through plasma (Chesler, Yaffe, Gubser,…)