1. AdS/CFT Correspondence and Some Applications An amateur’s point of view Hai-cang Ren ( Rockefeller & CCNU )

2. Contents I. AdS/CFT correspondence II. Some applications III. Remarks

3. I. AdS/CFT correspondence The inversion invariance of a massless field theory in 4D : scalar field: vector field: spinor field:

4. The conformal group in 4D: Poincare transformations 10 generators: Dilatation =Inversion x inversion 1 generators: D Special conformal transformation = inversion x translation x inversion: 4 generators : Lie algebra of the 15 generators: other commutators vanish.

5. Under the conformal group: The conformal symmetry at quantum level requires since ClassicalQuantum masslessYesNo massless QCDYesNo N=4 Supersymmetric Yang-Mills Yes

6. The conformal group and O(2,4) O(2,4) = rotation group of M(2,4) M(2,4) = 6D Minkowski space of signature (-, -, +, +, +, +): Introduce 4D Lorentz transformation: O(1,3) subgroup among ’s’s 4D Dilatation: O(2) rotation 4D Translation (infinitesimal):O(2,4) transformation 4D Special conformal transformation (infinitesimal): O(2,4) transformation

7. AdS 5 : Isometry group: O(2,4) A hyperboloid in M(2,4) Metric: or where Space of a constant curvature Throughout this lecture, we set the AdS radius L=1.

8. AdS 5 -Schwarzschild A black hole at Hawking temperature(Plasma temperature) Curvature ： The same Ricci tensor and curvature scalar as AdS 5

9. The metric : The isometry group: O(2,4) X O(6) ------- O(6) is isomorphic to SU(4), the symmetry group of the R-charge of N=4 SUSY YM ------- A superstring theory can be established in

10. Large Nc field theory : t’Hooft Power counting of a Feynman diagram at large N c: Dominated by the diagram with lowest g, -------- the planar diagram ~ a string world sheet

11. Large Nc field theory : A planar diagram A non-planar diagram A handle free world sheet A world sheet with a handle

12. Maldacena conjecture : Maldacena, Witten ------ Euclidean signature, generalizable to Minkowski signature

13. N=4 SYMType IIB string theory N_c colorsN_c 3-branes 4d conformal groupAdS_5 isometry group R-charge SU(4)S^5 isometry group N_c 3branes AdS_5 X S^5 bulk AdS boundary z=0 z Matching the symmetries

14. For most applications : Minkowski signature: The role of the Gibbons-Hawking term

15. Example: Recipe for calculating stress tensor correlators: --------- Write --------- Solve the 5d Einstein equation subject the boundary condition --------- Expandin power series of --------- Extract the coefficients. Near the black hole horizon: Euclidean signatureMinkowski signature Decaying mode onlyIncoming mode for retarded correlators Outgoing mode for advanced correlators

16. II Some applications to N=4 SUSY YM Plasma: Equation of state in strong coupling: Plasma temperature = Hawking temperature Near Schwarzschild horizon Continuating to Euclidean time, To avoid a conic singularity at, the period of Recalling the Matsubara formulation

17. Free energy = temperature X (the gravity action without metric fluctuations) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998), hep-th/9803131. Consider a 4D Euclidean space of spatial volume V_3 at The EH action of AdS-Schwarzschild: The EH action of plain AdS ----- To eliminate the conic singularity, ----- To match the proper length in Euclidean time Plasma free energy: Plasma entropy:

18. Bekenstein-Hawking entropy: ------ The metric on the horizon ： ------ The gravitational constant of the dual: agree with the entropy extraced from the gravity action. Gubser, Klebanov & Pest, PRD54, 3915 (1996)

19. The ratio 3/4: The plasma entropy density at The free field limit: the contents of N=4 SUSY YM numberentropy density gauge potential1 real scalars6 Weyl spinors4 The lattice QCD yields

20. Shear viscosity in strong coupling: Kubo formula Gravity dual: the coefficient of the y x The friction force per unit area Policastro, Son and Starinets, JHEP09, 043 (2002) where term of the gravity action where

21. The metric fluctuation Classification according to O(2) symmetry between x and y No mixing betweenand others! Substituting into Einstein equationand linearize The Laplace equation of a scalar field in the axial gauge,

22. Calculation details: ------ Nonzero components of the Christofel (up to symmetris): ------ Nonzero components of the Ricci tensor: Linear expansion:

23. The solution: Heun equation (Fucks equation of 4 canonical singularities) ------trivial when energy and momentum equatl to zero; ------low energy-momentum solution can be obtained perturbatively. The boundary condition at horizon: The incoming solution at low energy and zero momentum:

24. Viscosity ratio: Elliptic flow of RHIC: Lattice QCD: noisy V_4 = 4d spacetime volume

25. III. Remarks: N=4 SYM is not QCD, since 1). It is supersymmetric 2). It is conformal ( no confinement ) 3). No fundamental quarks ---- 1) and 2) may not be serious issues since sQGP is in the deconfined phase at a nonzero temperature. The supersymmetry of N=4 SYM is broken at a nonzero T. ---- 3) may be improved, since heavy fundamental quarks may be introduced by adding D7 branes. ( Krach & Katz ) Introducing an infrared cutoff ---- AdS/QCD: ----- Regge behavior of meson spectrum ---- confinement; ----- Rho messon mass gives ----- Lack of string theory support. Karch, Katz, Son & Stephenov

26. Deconfinement phase transition: Herzog, PRL98, 091601 (2007) Hadronic phase: Plasma phase: Hawking-Page transition: ---- First order transition with entropy jump ---- Consistent with large N_c QCD because of the liberation of quark-gluon degrees of freedom.

27. Thank You!