1. Transport coefficients in strongly coupled gauge theories: insights from string theory Andrei Starinets Perimeter Institute for Theoretical Physics

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.  The goal is to compute transport coefficients (e.g. shear and bulk viscosity) and the speed of sound in thermal QCD  This is of interest for hydrodynamic models applied to RHIC data (elliptic flow)  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.) Lattice approach cannot be used directly.

4.  Transport coefficients and the speed of sound can be computed from string theory for SOME thermal gauge theories in nonperturbative regime  This calculation is based on approach known as “gauge/gravity duality” or “AdS/CFT correspondence” AdS = Anti de Sitter space CFT = Conformal Field Theory

5. 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

6. Computing transport coefficients from “first principles” Kubo formulae allow 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)

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

8. 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.

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

10. 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

11. 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:

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

13. 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

14. 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

15. Example: supersymmetric Yang-Mills theory at finite temperature Fields: The theory is conformal (coupling doesn’t run) In the limit the theory is described by a gravity dual (AdS-Schwarzschild black hole)