1. Transport coefficients from string theory: an update Andrei Starinets Perimeter Institute Wien 2005 workshop

2. Collaboration: Dam Son Giuseppe Policastro Chris Herzog Alvaro Nunez Pavel Kovtun Alex Buchel Jim Liu Andrei Parnachev Paolo Benincasa References: hep-th/0205051 hep-th/0205052 hep-th/0302026 hep-th/0309213 hep-th/0405231 hep-th/0406124 hep-th/0506144 hep-th/0506184 hep-th/0507026

3. Prologue  Our goal is to understand thermal gauge theories, e.g. thermal QCD  Of particular interest is the regime described by fluid dynamics, e.g. quark-gluon plasma  This near-equilibrium regime is completely characterized by values of transport coefficients, e.g. shear and bulk viscosity  Transport coefficients are hard to compute from “first principles”, even in perturbation theory. For example, no perturbative calculation of bulk viscosity in gauge theory is available.

4. Prologue (continued)  Transport coefficients of some gauge theories can be computed in the regime described by string (gravity) duals – usually at large N and large ‘t Hooft coupling  Corrections can in principle be computed  Shear viscosity result is universal. Model- independent results may be of relevance for RHIC physics  Certain results are predicted by hydrodynamics. Finding them in gravity provides a check of the AdS/CFT conjecture

5. Timeline and status report  2001: shear viscosity for N=4 SYM computed  2002: prescription to compute thermal correlators from gravity formulated and applied to N=4 SYM; shear and sound poles in correlators are found  2002-03: other poles in N=4 SYM correlators identified with quasinormal spectrum in gravity  2003-04: universality of shear viscosity; general formula for diffusion coefficient from “membrane paradigm”; correction to shear viscosity  2004-05: general prescription for computing transport coefficients from gravity duals formulated; bulk viscosity and the speed of sound computed in two non-conformal theories; equivalence between AdS/CFT and the “membrane paradigm” formulas established; spectral density computed /preliminary/  2005-?? Nonzero chemical potential (with D.Son).

6. What is hydrodynamics? 0t |||| Hierarchy of times (example) Mechanical description Kinetic theory Hydrodynamic approximation Equilibrium thermodynamics Hierarchy of scales (L is a macroscopic size of a system)

7. Holography and hydrodynamics Gravitational fluctuationsDeviations from equilibrium Quasinormal spectrum Dispersion relations

8. Gauge-gravity duality in string theory Perturbative string theory: open and closed strings (at low energy, gauge fields and gravity, correspondingly) Nonperturbative theory: D-branes (“topological defects” in 10d) Complementary description of D-branes by open (closed) strings: perturbative gauge theory description OK perturbative gravity description OK

9. Hydrodynamics as an effective theory Thermodynamic equilibrium: Near-equilibrium: Eigenmodes of the system of equations Shear mode (transverse fluctuations of ): Sound mode: For CFT we haveand

10. Computing transport coefficients from “first principles” 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 AdS/CFT Fluctuation-dissipation theory (Callen, Welton, Green, Kubo)

11. Universality of Theorem: For any thermal gauge theory (with zero chemical potential), the ratio of shear viscosity to entropy density is equal to in the regime described by a corresponding dual gravity theory Remark: Gravity dual to QCD (if it exists at all) is currently unknown.

12. Universality of shear viscosity in the regime described by gravity duals Graviton’s component obeys equation for a minimally coupled massless scalar. But then. Since the entropy (density) is we get

13. Three roads to universality of  The absorption argument D. Son, P. Kovtun, A.S., hep-th/0405231  Direct computation of the correlator in Kubo formula from AdS/CFT A.Buchel, hep-th/0408095  “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/0309213, A.S., to appear, P.Kovtun, A.S., hep-th/0506184, A.Buchel, J.Liu, hep-th/0311175

14. Shear viscosity in SYM Correction to : A.Buchel, J.Liu, A.S., hep-th/0406264 P.Arnold, G.Moore, L.Yaffe, 2001

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

16. A viscosity bound conjecture P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231

17. 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

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

19. Bulk viscosity and the speed of sound in SYM is a “mass-deformed”(Pilch-Warner flow)  Finite-temperature version: A.Buchel, J.Liu, hep-th/0305064  The metric is known explicitly for  Speed of sound and bulk viscosity:

20. Relation to RHIC  IF quark-gluon plasma is indeed formed in heavy ion collisions  IF a hydrodynamic regime is unambiguously proven to exist  THEN hydrodynamic MODELS describe experimental results for e.g. elliptic flows well, provided  Bulk viscosity and speed of sound results are potentially interesting

21. Epilogue AdS/CFT gives insights into physics of thermal gauge theories in the nonperturbative regime Generic hydrodynamic predictions can be used to check validity of AdS/CFT General algorithm exists to compute transport coefficients and the speed of sound in any gravity dual Model-independent statements can presumably be checked experimentally

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