Holography of Wilson-loop expectation valuesHolography of Wilson-loop expectation values

with local operator insertionswith local operator insertions Akitsugu Miwa ( Univ. of Tokyo, Komaba )

work in collaboration with Tamiaki Yoneya

based on hep-th/0609007

introduction and motivationJ.M.Maldacena(1997)AdS/CFT correspondence:AdS/CFT correspondence:

present status and aim of this talk: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), SYM IIB superstring on

Wilson loop in AdS/CFT:Wilson loop in AdS/CFT: S.J.Rey, J.Yee(1998), J.M. Maldacena(1998)

Wilson loop string world sheet

boundary

AdS5

deformation of loop and local operator insertion:deformation of loop and local operator insertion:

Wilson loop:

deformation of loop:

large R-charge local operator insertions:large R-charge local operator insertions: N. Drukker S. Kawamoto (2006)

corresponding string world sheet:corresponding string world sheet:

we consider: string world sheet with large angular momentum

SO(6) symmetry: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 S5 : SO(6)

U(1) sub-group: rotation in 5-6 plain

4

5 6

question:question:

SYM side:

string side: near boundary separated from boundary?

path of large angular momentum mode:path of large angular momentum mode:

classical path do not reach the boundarybecause of a potential barrier

potential barrier

path of large angular momentum mode

tunneling geodesic:tunneling geodesic: S.Dobashi, H.Shimada, T.Yoneya (2002)

we will study string world sheet around tunneling geodesic

today’s talk: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

some comments

stringstring action:action:

AdS/CFT correspondence:AdS/CFT correspondence:

semi-classical approximation:semi-classical approximation:

large ‘t Hooft coupling in SYM

semi-classical approximation of string theory requires large

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

SS55 part: part:solution: patch1

patch2

metricmetric::

path of localized angular momentum

AdSAdS55 part: part:

Hamiltonian constraint, EOM:

boundary

path of localized angular momentum

tunneling solution:tunneling solution: S.Dobashi, H.Shimada, T.Yoneya (2002)

tunneling solution (path of localized R-charge)

world sheet

we want:we want:

attached to a loop on the boundary

propagating along the tunneling geodesic

N.Drukker, S.Kawamoto (2006)comment on non-tunneling solution:comment on non-tunneling solution:

If we use non-tunneling solution, it is difficult to check following relation:

full world-sheet solution:full world-sheet solution:

tunneling geodesic

circle with radius

full world-sheet solution:full world-sheet solution:

corresponding Wilson loop:corresponding Wilson loop:

evaluation of “area” of world sheets

N.Drukker, D.Gross, H.Ooguri (1999)““area” of a world sheetarea” of a world sheet

two cutoff schemestwo cutoff schemes

target-space cutoff:

world-sheet cutoff:

w.s. cutoff can be rewritten as:

divergences:divergences:

singular behavior of the metric .

infinite extension of the world sheet .

(const. depends on regularization schemes)

results:results:

consistent with ladder graph calculation

interpretation of results

in our casein our case

consistent with our result

ladder graphladder graph (planar graph without internal vertex):(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 laddercircle:

straight line:

These results are known to be consistent with the calculation of the area of world sheet without angular momentum.

conclusion and future work

future work:future work:

more complicated local operator insertions (more complicated deformation of string world sheet)

conclusion: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

Wilson loop in AdS/CFTS.J.Rey, J.Yee(1998); J.M.Maldacena(1998)

conjecture:conjecture:

SYM side:SYM side:Wilson loop

string side:string side:

flat

string world sheet

about Wilson loop

localized R-charge

introduction and motivation

J.M.Maldacena(1997)AdS/CFT correspondence:AdS/CFT correspondence:

4-dim, 4, SU(N), SYM IIB superstring on

N D3-brane system

flat

boundary

present status and the aim of this talk: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:Wilson loop in AdS/CFT: S.J.Rey, J.Yee(1998), J.M. Maldacena(1998)

Wilson loop string world sheet

boundary

AdS5

introduction and motivation