Semi-Classical strings as probes of AdS/CFT

M. Kruczenski

Purdue University

Based on:arXiv:0707.4254 R. Roiban, A. Tirziu, A. Tseytlin, M.K.arXiv:0705.2429 R. Ishizeki, M.K.arXiv:0710.2300 R. Ishizeki, M. Spradlin, A. Volovich, M.K.

Summary

● Introduction

String / gauge theory duality (AdS/CFT)

Semi-Classical strings and field theory operators

Semi-Classical strings and Wilson loops

● Rotating strings and twist 2 ops. (S. Gubser, I. Klebanov, A. Polyakov)● Light-like cusp W.L. and twist 2 ops. (M.K., Y. Makeenko)● Light like-cusp at one-loop. (R.Roiban, A.Tirziu, A.Tseytlin, M.K.)

● Spiky strings and higher twist operators (M.K.)

● Spiky strings on a sphere (Ryang)

● Giant magnons (Hofman-Maldacena)

● Single spike solutions on S2 and S3

( R.Ishizeki, M.K. )

● Multiple spikes solutions on S2 and S3 ( R. Ishizeki, M. Spradlin, A. Volovich, M.K. ) ● Conclusions

String/gauge theory duality: Large N limit (‘t Hooft)

mesons

String picture

π, ρ, ...

Quark model

Fund. strings

( Susy, 10d, Q.G. )

QCD [ SU(3) ]

Large N-limit [SU(N)]

Effective strings

q q

Strong coupling

q q

Lowest order: sum of planar diagrams (infinite number)

N g N fixedYM , 2More precisely: (‘t Hooft coupl.)

AdS/CFT correspondence (Maldacena)

Gives a precise example of the relation betweenstrings and gauge theory.

Gauge theory

N = 4 SYM SU(N) on R4

Aμ , Φi, Ψa

Operators w/ conf. dim.

String theory

IIB on AdS5xS5

radius RString states w/ E

R

g g R l g Ns YM s YM 2 2 1 4; / ( ) /

N g NYM , 2fixed

λ large → string th.λ small → field th.

Semi-classical stringsWhy?

We cannot quantize exactly S (+fermions etc.) Semi-classical regime is strong coupling limit

Two important cases of classical solutions

Surface ending on the boundary Wilson loops (Maldacena; Rey, Yee) (= bdy. curve)

String inside AdS5xS5 Operator in the field theory (Berenstein, Maldacena, Nastase; Gubser, Klebanov, Polyakov)

S d d G X X X X

1

Rotating Strings and twist 2 operators

Y Y Y Y Y Y R12

22

32

42

52

62 2

s inh ; [ ]2

3 cosh ;2 t

ds d t d d2 2 2 2 232 cosh sinh [ ]

E S S S

O Tr x z tS

2ln , ( )

, θ = ω t

(Gubser, Klebanov, Polyakov)

Light-like cusps and twist 2 ops. (MK, Makeenko)

The anomalous dimensions of twist two operators can also be computed by using the cusp anomaly of light-like Wilson loops (Korchemsky and Marchesini).

In AdS/CFT Wilson loops can be computed using surfaces of minimal area in AdS5 (Maldacena, Rey, Yee)

z

The result agrees with the rotating string calculation.

One loop correction to light-like cusp.(w/ R. Roiban, A. Tirziu, A. Tseytlin)

By doing a quadratic expansion around the classical solution we can compute the one-loop correction to the partition function. The result is finite and corrects the cusp anomaly:

It agrees with the previously known result from the rotating string. In fact one can see that both classical solutions are related.

cusp rot. string for infinite S limit.

Y Y Y Y Y Y R12

22

32

42

52

62 2

ln ln ln . . . lnWL L

cusp

1

4

1 32

Y Y Y Y Y Y Y Y Y Y1 4 5 6 1 5 4 6 2 3 0 ( )

String version of theKorchemsky-Marchesiniargument

Generalization to higher twist operators (MK)

O TrnS n S n S n S n

[ ]/ / / / O Tr S

[ ]2

x A n A n

y A n A n

cos[( ) ] ( ) co s [ ]

sin [( ) ] ( ) sin [ ]

1 1

1 1

In flat space such solutions are easily found in conf. gauge:

Spiky strings in AdS:

E Sn

S S

O Tr S n S n S n S n

2 2

ln , ( )

/ / / /

S d t d tjj

j j

j

22 1

84

21 12 1

(cosh ) ln s in

Spiky strings on a sphere: (Ryang )

Similar solutions exist for strings rotating on a sphere:

(top view)

The metric is:

We use the ansatz:And solve for . Field theory interpretation?

ds d t d d2 2 2 2 2 sin

t , , ( ) ( )

Giant magnon: (Hofman-Maldacena)

d

d A

A

sin sin

sin

2 2 2

2 2 2

d

d AA

sin

co ssin2 2 2

sinsin

A 1

(top view) giant magnon

The energy and angular momentum of the giant magnon solution diverge. However their difference is finite:

E JA d

A

2 2

2

2sinsin

cos ,

Angular distance between spikes

E J

12

21

12 2

1

22

22

sin

sin ,

sin ,

Interpolating expression:

One more angular momentum: (Dorey, Chen-Dorey-Okamura)

There are solutions with another angular momentum J2. The energy is given by:

E J J JJ

1 22

22

22

22

2 2 21

sin sin ,

Justifies interpolating formula for J2=1

More general solutions: (Russo, Tseytlin, MK)

Strategy: We generalize the spiky string solution and then take the giant magnon limit.

In flat space:x A n A n

y A n A n

co s[( ) ] ( ) co s [ ]

sin [( ) ] ( ) sin [ ]

1 1

1 1

Consider txS5: ds d t dX d X X Xa aa

a aa

2 2

1

3

1

3

1

,

namely: x iy X x e i ( ) ,

Use similar ansatz:

X x e r ea ai

ai ia a a ( ) ( ) ( )

The reduced e.o.m. follow from the lagrangian:

L x x i x x x x x x x xa a a a a a a a a a a a ( ) ' ' ( ' ' ) ( ) 2 2 2 1

If we interpret as time this is particle in a sphere subject to a quadratic potential and a magnetic field.The trajectory is the shape of the string

The particle is attracted to the axis but the magnetic field curves the trajectory

L describes an integrable system

Single spike solutions on S2: (w/ R. Ishizeki; Mosaffa, Safarzadeh)It turns out that looking at rigid strings rotating on a two-sphere one can find other class of solutionsand in particular another limiting solution:

Antiferromagnetic magnon?(see Roiban,Tirziu,Tseytlin)[Hayashi, Okamura, Suzuki,Vicedo]

(Goes around infinite times)

Now the energy and winding number diverge but the difference is finite. The angular momentum is finite

This describes excitations over an infinitely wound string. Adding one ang. momentum:

Multiple spike solutions (w/Ishizeki, Spradlin, Volovich)

Using methods from integrable systems which areusually used to find soliton solutions in two dimensional field theories we can construct more general solutions.We use dressing method (Zakharov, Mikhailov)This method gives new solutions by dressing a knownone. In particular dressing the infinitely wrapped stringone gets the single spike. By dressing again one getstwo spikes etc. This is a confirmation of the idea of these solutions asexcitations of the infinitely wrapped string.

Example of spike scattering

Phase shift is the same as the giant magnon(in terms of thedressing param.)

Conclusions:Classical string solutions are a powerful tool to study the duality between string and gauge theory. Both through Wilson loop computations and by matchingclosed strings with gauge invariant f.t. operators

We saw several examples:

● folded strings rotating on AdS5 and light-like cusps. ● light-like cusps at one-loop ● spiky strings rotating in AdS5 and S5

● giant magnons on S2 and S3 ● single spike solutions on S2

● multiple spike solutions on S2