1. Fluctuation Partition Function of a Wilson Loop in a Strongly Coupled N=4 SYM Plasma Defu Hou (CCNU), James T.Liu (U. Michigan) and Hai-cang Ren (Rockefeller & CCNU) USTC ， 7 、 2008

2. Contents: I.AdS/CFT correspondence and Wilson loops II.Semi-classical expansion III. Some examples IV. Remarks

3. I. AdS/CFT correspondence and Wilson loops ( Maldacena; Witten ) AdS/CFT corerspondence N_c 3-branes on AdS boundary AdS_5 XS^5 bulk

4. AdS/CFT corerspondence Symmetry matching in AdS/CFT Field theory symmetryString theory symmetry SU(N_c)N_c 3-branes 4D conformal groupAdS_5 isometry, SO(2,4) R-symmetry, SU(4)S^5 isometry, SO(6)

5. AdS/CFT corerspondence Leading order results: Equation of state ( Witten ): Viscosity ratio (Policastro,Son & Starinets ): Jet quenching (Liu,Rajagopal & Wiederman): And many others.

6. AdS/CFT correspondence But N=4 SYM is not QCD! It is supersymmetric; It is conformal ( no confinement ); No fundamental quarks; It is large N_c. ---- 1, 2 may not be serious issues for QGP; ---- Attempts to add quark flavors; ---- Attempts to introduce IR cutoff.

7. Gravity dual of a Wilson loop: = the gauge potential of N=4 SYM ; C = a loop on the AdS boundary z=0; The metric of -Schwarzschild

8. Gravity dual of a Wilson loop: C AdS boundary z=0 AdS bulk z x t

9. Gravity dual of a Wilson loop: Heavy quark self energy Quark-antiquark potential J. Madacena; S. J. Rey et. al. Jet-quenching parameter H. Liu et. al. x x t t C on the boundaryImplied physical quantity

10. Comparison with RHIC physics: Finite coupling correction:. ---- b[C] comes from the fluctuation of the string world sheet around the one of minimum area. ---- Has been considered by Forste, Ghoshal, Theisen and by Drukker, Gross, Tesytlin at T=0. ---- Generalization to nonzero T. Finite N_c correction: String interaction, very difficult.

11. II. Semi-classical expansion: Classical solution: Target space metric World sheet metric

12. II. Semi-classical expansion: Quadratic fluctuations: where theta=fermionic coordinate. Need to explore the full super multiplet of the world sheet. extracted from Metsaev-Tseytlin action

13. Bosonic fluctuations: Decompose the into its eight tangent components and two longitudinal ones: We find that

14. Fermionic fluctuations: -symmetry: whereisdependent and Gauge fixing:

15. Fermionic fluctuations: Choose the 10-beins’s such that two of them, aretangent to the embedding world sheet, whereare the world sheet zweibeins and spin connection. For the world sheets considered below K does not contribute (not in general!) ---- Write ----Packinto eight 2-component Majorana spinors. where Finally

16. III. Examples: O x z t Embedding ： World sheet metric:with Zweibeins: Spin connections: Curvature: A straight string Z_h

17. Transverse fluctuations: s=5, 6, 7, 8, 9 Substituting into the Nambu-Goto action we find where We have Partition function:

18. A pair of parallel lines: Oz Embedding: Z_0

19. World sheet metric: with Zweibeins: Spin connections: Curvature: The world sheet tangent vectors: The transverse bosonic fluctuation:

20. Substituting into the Nambu-Goto action we find where the same as the case without the black hole We have Partition function:

21. IV. Remarks UV divergence: ------ Quadratic divergence is cancelled between bosons and fermions. Z=1 for zero world sheet curvature and zero target space curvature. ------ Logarithmic divergence: ------ The black hole does not introduce new UV divergences Analog of an ordinary field theory: A nonzero temperature does not introduce new UV divergences.

22. Method for computing the determinant ratio (Kruczenski & Tirziu) Given under Dirichlet boundary condition Generalization to more complicated loops, such as A pair of oblique parallel lines (boosted quark-antiquark potential); A pair of light-like parallel lines (jet-quenching).

23. Thank You!