Fluctuation Partition Function of a Wilson Loop in a Strongly Coupled N=4 SYM Plasma

1. USTC，7、2008 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)

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

3. AdS_5XS^5 bulk N_c 3-branes on AdS boundary • AdS/CFT correspondence and Wilson loops • (Maldacena; Witten) AdS/CFT corerspondence

4. AdS/CFT corerspondence Symmetry matching in AdS/CFT

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 The metric of -Schwarzschild

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

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

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: extracted from Metsaev-Tseytlin action where theta=fermionic coordinate. Need to explore the full super multiplet of the world sheet.

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

14. Fermionic fluctuations: -symmetry: where is dependent and Gauge fixing:

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

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

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

18. O z A pair of parallel lines: Z_0 Embedding:

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!