AdS/CFT correspondence and hydrodynamics

1. 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 21-26, 2009

2. Plan I. Introduction and motivation • II. Hydrodynamics • hydrodynamics as an effective theory • linear response • transport properties and retarded correlation functions • III. AdS/CFT correspondence at finite temperature and density • holography beyond equilibrium • holographic recipes for non-equilibrium physics • the hydrodynamic regime • quasinormal spectra • some technical issues

3. Plan (continued) • IV. Some applications • transport at strong coupling • universality of the viscosity-entropy ratio • particle emission rates • relation to RHIC and other experiments Some references: D.T.Son and A.O.S., “Viscosity, Black Holes, and Quantum Field Theory”, 0704.0240 [hep-th] P.K.Kovtun and A.O.S., “Quasinormal modes and holography”, hep-th/0506184 G.Policastro, D.T.Son, A.O.S., “From AdS/CFT to hydrodynamics”, hep-th/0205052 G.Policastro, D.T.Son, A.O.S., “From AdS/CFT to hydrodynamics II: Sound waves”, hep-th/0210220

4. I. Introduction and motivation

5. 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 heavy-ion 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 non-conformal theories etc). We don’t know quantities such as for QCD

6. Heavy ion collision experiments at RHIC (2000-current) and LHC (2009-??) create hot and dense nuclear matter known as the “quark-gluon plasma” (note: qualitative difference between p-p and Au-Au collisions) Evolution of the plasma “fireball” is described by relativistic fluid dynamics (relativistic Navier-Stokes equations) Need to know thermodynamics(equation of state) kinetics(first- and second-order transport coefficients) in the regime of intermediate coupling strength: initial conditions(initial energy density profile) thermalization time(start of hydro evolution) freeze-out conditions(end of hydro evolution)

7. 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 [hep-th]

8. Quantum field theories at finite temperature/density Equilibrium Near-equilibrium transport coefficients emission rates ……… entropy equation of state ……. perturbative non-perturbative perturbative non-perturbative ???? Lattice kinetic theory pQCD

9. II. Hydrodynamics 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, long-time tails, and supersymmetry”, hep-th/0303010.

10. The hydrodynamic regime 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)

11. 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:

12. 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:

13. Example: momentum diffusion and sound Thermodynamic equilibrium: Near-equilibrium: Eigenmodes of the system of equations Shear mode (transverse fluctuations of ): Sound mode: For CFT we have and

14. What is viscosity? Friction in Newton’s equation: Friction in Euler’s equations

15. Viscosity of gases and liquids Gases (Maxwell, 1867): Viscosity of a gas is • independent of pressure • scales as square of temperature • inversely proportional to cross-section Liquids (Frenkel, 1926): • Wis the “activation energy” • In practice, A and W are chosen to fit data

16. “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.

17. 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

18. 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)

19. 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

20. Derivative expansion in hydrodynamics: first order Hydrodynamic d.o.f. = densities of conserved charges or (4 equations) (4 d.o.f.)

21. First-order conformal hydrodynamics (in d dimensions) Weyl transformations: In first-order hydro this implies: Thus, in the first-order (conformal) hydro:

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

23. Second-order Israel-Stewart conformal hydrodynamics Israel-Stewart

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

25. 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)

26. Linear response theory

27. Linear response theory (continued)

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

29. Computing transport coefficients from “first principles” Fluctuation-dissipation 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

30. Spectral function and quasiparticles A B A: scalar channel C B: scalar channel - thermal part C: sound channel

31. III. AdS/CFT correspondence at finite temperature and density

32. 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 Deviations from equilibrium Gravitational fluctuations ???? and B.C. Quasinormal spectrum

33. Dennis W. Sciama (1926-1999) P.Candelas & D.Sciama, “Irreversible thermodynamics of black holes”, PRL,38(1977) 1732

34. 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)

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

36. AdS/CFT correspondence conjectured exact equivalence Latest test: Janik’08 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

37. 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

38. 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

39. 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

40. 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]

41. Example: R-current correlator in in the limit Zero temperature: Finite temperature: Poles of = quasinormal spectrum of dual gravity background (D.Son, A.S., hep-th/0205051, P.Kovtun, A.S., hep-th/0506184)

42. The role of quasinormal modes G.T.Horowitz and V.E.Hubeny, hep-th/9909056 D.Birmingham, I.Sachs, S.N.Solodukhin, hep-th/0112055 D.T.Son and A.O.S., hep-th/0205052; P.K.Kovtun and A.O.S., hep-th/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

43. Classification of fluctuations and universality O(2) symmetry in x-y plane Shear channel: Sound channel: Scalar channel: Other fluctuations (e.g. ) may affect sound channel But not the shear channel universality of

44. Two-point correlation function of stress-energy tensor Field theory Zero temperature: Finite temperature: Dual gravity • Five gauge-invariant combinations • of and other fields determine • obey a system of coupled ODEs • Their (quasinormal) spectrum determines singularities of the correlator

45. Computing transport coefficients from dual gravity – various methods 1. Green-Kubo formulas (+ retarded correlator from gravity) 2. Poles of the retarded correlators 3. Lowest quasinormal frequency of the dual background 4. The membrane paradigm

46. Example: stress-energy 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)

47. Example 2 (continued): stress-energy 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:

48. IV. Some applications

49. 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

50. SYM New transport coefficients in Sound dispersion: Kubo: