1. Holography of Wilson-loop expectation values with local operator insertions Akitsugu Miwa ( Univ. of Tokyo, Komaba ) work in collaboration with Tamiaki Yoneya based on hep-th/0609007

2. introduction and motivation J.M.Maldacena(1997) AdS/CFT correspondence: present status and aim of this talk: generic loop: difficult circle, straight line: well studied small deformation of circle or straight line status this talk 4-dim, 4, SU(N), SYMIIB superstring on Wilson loop in AdS/CFT: S.J.Rey, J.Yee(1998), J.M. Maldacena(1998) Wilson loop string world sheet boundary AdS 5

3. deformation of loop and local operator insertion: Wilson loop: deformation of loop: large R-charge local operator insertions: N. Drukker S. Kawamoto (2006)

4. corresponding string world sheet: we consider:string world sheet with large angular momentum SO(6) symmetry: large R-charge gauge theory side R symmetry : SO(6): ( rotation of 6 scalars ) U(1) sub-group: rotation of a phase of string theory side isometry on S 5 : SO(6) U(1) sub-group: rotation in 5-6 plain 4 5 6

5. question: SYM side: string side:near boundaryseparated from boundary ? path of large angular momentum mode: classical path do not reach the boundary because of a potential barrier potential barrier path of large angular momentum mode

6. tunneling geodesic: S.Dobashi, H.Shimada, T.Yoneya (2002) we will study string world sheet around tunneling geodesic today’s talk: we want VEV of Wilson loop: path of localized angular momentum potential barrier tunneling solution correlation function of large R-charge local operators ex: 4 5 6

7. some comments string action: AdS/CFT correspondence: semi-classical approximation: large ‘t Hooft coupling in SYM semi-classical approximation of string theory requires large

8. plan of the talk introduction and motivation world sheet with large angular momentum evaluation of “area” of world sheets interpretation of results conclusion and future work

9. world sheet with large angular momentum S 5 part: solution: patch1 patch2 metric: path of localized angular momentum AdS 5 part: Hamiltonian constraint, EOM: boundary path of localized angular momentum

10. tunneling solution: S.Dobashi, H.Shimada, T.Yoneya (2002) tunneling solution (path of localized R-charge) world sheet we want: attached to a loop on the boundary propagating along the tunneling geodesic N.Drukker, S.Kawamoto (2006) comment on non-tunneling solution: If we use non-tunneling solution, it is difficult to check following relation:

11. full world-sheet solution: tunneling geodesic circle with radius

12. full world-sheet solution: corresponding Wilson loop:

13. evaluation of “area” of world sheets N.Drukker, D.Gross, H.Ooguri (1999) “area” of a world sheet two cutoff schemes target-space cutoff: world-sheet cutoff: w.s. cutoff can be rewritten as: divergences: singular behavior of the metric. infinite extension of the world sheet.

14. (const. depends on regularization schemes)results: consistent with ladder graph calculation

15. interpretation of results in our case consistent with our result ladder graph(planar graph without internal vertex): ladder graph (planar graph without internal vertex): D.Berenstein, R.Corrado, W. Fischler, J.M.Maldacena (1998), N.Drukker, D.J.Gross, H. Ooguri (1999), J.K.Erickson, G.W.Semenoff, K.Zarembo (2000), etc. not ladder circle: straight line: These results are known to be consistent with the calculation of the area of world sheet without angular momentum.

16. conclusion and future work future work: more complicated local operator insertions (more complicated deformation of string world sheet) conclusion: open string solution around the tunneling geodesic, the “area” of the world sheet in two cutoff schemes. We have studied Main results are ( consistent with ladder graph calculation ) and independent finite terms depend on cutoff schemes

17. Wilson loop in AdS/CFT S.J.Rey, J.Yee(1998); J.M.Maldacena(1998)conjecture: SYM side: Wilson loop string side: flat string world sheet

18. about Wilson loop localized R-charge

19. introduction and motivation J.M.Maldacena(1997) AdS/CFT correspondence: 4-dim, 4, SU(N), SYMIIB superstring on N D3-brane system flat boundary

20. present status and the aim of this talk: generic loop: difficult circle, straight line: well studied small deformation of circle or straight line status this talk Wilson loop in AdS/CFT: S.J.Rey, J.Yee(1998), J.M. Maldacena(1998) Wilson loop string world sheet boundary AdS 5 introduction and motivation