Holography of Wilsonloop expectation values with local operator

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

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

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

SO(6) symmetry: 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) 4 U(1) sub-group: rotation in 5 -6 plain 5 6 corresponding string world sheet: large R-charge we consider: string world sheet with large angular momentum

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

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

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

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

world sheet with large angular momentum metric: S 5 part: solution: patch 1 patch 2 path of localized angular momentum Ad. S 5 part: Hamiltonian constraint, EOM: path of localized angular momentum boundary

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

full world-sheet solution: tunneling geodesic circle with radius

full world-sheet solution: corresponding Wilson loop:

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

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

interpretation of results ladder graph (planar graph without internal vertex): not ladder circle: straight line: These results are known to be consistent with the calculation of the area of world sheet without angular momentum. 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. in our case consistent with our result

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

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

about Wilson loop localized R-charge

introduction and motivation Ad. S/CFT correspondence: J. M. Maldacena(1997) N D 3 -brane system boundary flat 4 -dim, 4, SU(N), SYM IIB superstring on

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