1. Gauge-Gravity Duality: A brief overview
Andrei Starinets
Rudolf Peierls Centre for Theoretical Physics
Oxford University
ICMS workshop
“Numerical relativity beyond astrophysics”
Edinburgh
12 July 2011

2. Some references:
O.Aharony, S.Gubser, J.Maldacena, H.Ooguri, Y.Oz, hep-th/
J.Casalderrey-Solana, H.Liu, D.Mateos, K.Rajagopal, U.Wiedemann, [hep-th]
D.T.Son and A.O.S., “Viscosity, Black Holes, and Quantum Field Theory”, [hep-th]
P.K.Kovtun and A.O.S., “Quasinormal modes and holography”, hep-th/

3. AdS/CFT and condensed matter physics
S. Hartnoll
“Lectures on holographic methods for condensed matter physics”,
[hep-th]
C. Herzog
“Lectures on holographic superfluidity and superconductivity”,
[hep-th]
M. Rangamani
“Gravity and hydrodynamics: Lectures on the fluid-gravity correspondence”,
[hep-th]
S.Sachdev
“Condensed matter and AdS/CFT”,
[hep-th]

4. What is string theory?

5. Equations such as
describe the low energy limit of string theory
As long as the dilaton is small, and thus the string interactions are suppressed,
this limit corresponds to classical 10-dim Einstein gravity coupled to certain
matter fields such as Maxwell field, p-forms, dilaton, fermions
Validity conditions for the classical (super)gravity approximation
- curvature invariants should be small:
- quantum loop effects (string interactions = dilaton) should be small:
In AdS/CFT duality, these two conditions translate into
and

6. 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)

7. (super)conformal field theory = coupling doesn’t run
supersymmetric YM theory
Gliozzi,Scherk,Olive’77
Brink,Schwarz,Scherk’77
Field content:
Action:
(super)conformal field theory = coupling doesn’t run

8. AdS/CFT correspondence
conjectured
exact equivalence
Generating functional for correlation
functions of gauge-invariant operators
String partition function
In particular
Classical gravity action serves as a generating functional for the gauge theory correlators

9. 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 on-shell action
as a functional of the b.c.
Warning: e.o.m. for different bulk fields may be coupled: need self-consistent solution
Then, taking functional derivatives of gives

10. 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

11. Example: charge diffusion
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:

12. 4-dim gauge theory – large N, strong coupling 10-dim gravity
M,J,Q
Holographically dual system
in thermal equilibrium
M, J, Q
T S
Gravitational background fluctuations
Deviations from equilibrium
????
and B.C.
Quasinormal spectrum

13. First-order transport (kinetic) coefficients
Shear viscosity
Bulk viscosity
Charge diffusion constant
Supercharge diffusion constant
Thermal conductivity
Electrical conductivity
* Expect Einstein relations such as to hold

14. Second-order hydrodynamics
Hydrodynamics is an effective theory, valid for sufficiently small momenta
First-order 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
Navier-Stokes 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 second-order terms
compatible with symmetries (e.g. conformal symmetry for conformal plasmas)

15. Second-order conformal hydrodynamics (in d dimensions)

16. Second-order transport (kinetic) coefficients
(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 non-conformal theories such as QCD, the total number of second-order transport
coefficients is quite large

17. Predictions of the second-order conformal hydrodynamics
Sound dispersion:
Kubo:

18. In quantum field theory, the dispersion relations such as
appear as poles of the retarded correlation functions, e.g.
- in the hydro approximation -

19. The role of quasinormal modes
G.T.Horowitz and V.E.Hubeny, hep-th/
D.Birmingham, I.Sachs, S.N.Solodukhin, hep-th/
D.T.Son and A.O.S., hep-th/ ; P.K.Kovtun and A.O.S., hep-th/
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

20. Sound and supersymmetric sound in
In 4d CFT
Sound mode:
Supersound mode:
Quasinormal modes in dual gravity
Graviton:
Gravitino:

21. Quasinormal spectra of black holes/branes
Schwarzschild black hole
(asymptotically flat)
AdS-Schwarzschild black brane

22. Sound dispersion in
analytic approximation
analytic approximation

23. First-order transport coefficients in N = 4 SYM
in the limit
Shear viscosity
Bulk viscosity
for non-conformal 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

24. Shear viscosity in SYM perturbative thermal gauge theory
S.Huot,S.Jeon,G.Moore, hep-ph/
Correction to : Buchel, Liu, A.S., hep-th/
Buchel, [hep-th]; Myers, Paulos, Sinha, [hep-th]

25. Universality of 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:
Extended to non-zero chemical potential:
Benincasa, Buchel, Naryshkin, hep-th/
Extended to models with fundamental fermions in the limit
Mateos, Myers, Thomson, hep-th/
String/Gravity dual to QCD is currently unknown

26. Three roads to universality of
The absorption argument
D. Son, P. Kovtun, A.S., hep-th/
Direct computation of the correlator in Kubo formula from AdS/CFT A.Buchel, hep-th/
“Membrane paradigm” general formula for diffusion coefficient + interpretation as lowest quasinormal frequency = pole of the shear mode correlator + Buchel-Liu theorem
P. Kovtun, D.Son, A.S., hep-th/ , A.S., [hep-th],
P.Kovtun, A.S., hep-th/ , A.Buchel, J.Liu, hep-th/

27. Computing transport coefficients from dual gravity
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., hep-th/ , P.Kovtun, A.S., hep-th/ )
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

28. Hydrodynamic properties of strongly interacting hot plasmas in 4 dimensions
can be related (for certain models!)
to fluctuations and dynamics of 5-dimensional black holes

29. Beyond near-equilibrium regime

30. Computing real-time correlation functions from gravity
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
real-time 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]