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

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

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

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.

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

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

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.

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

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

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

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

Maldacena conjecture : Maldacena, Witten Euclidean signature, generalizable to Minkowski signature

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

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

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

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

Free energy = temperature X (the gravity action without metric fluctuations) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998), hep-th/ 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:

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)

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

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

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,

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

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:

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

III. Remarks: N=4 SYM is not QCD, since 1). It is supersymmetric 2). It is conformal ( no confinement ) 3). No fundamental quarks ) 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 ) 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

Deconfinement phase transition: Herzog, PRL98, (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.

Thank You!