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)
String picture
Fund. strings
( Susy, 10d, Q.G. )
mesons
π, ρ, ...
Quark model
QCD [ SU(3) ]
qq
q
q
Strong coupling
Large N-limit [SU(N)]
Effective strings
2
More precisely: N   ,   gYM
N fixed
(‘t Hooft coupl.)
Lowest order: sum of planar diagrams (infinite number)
AdS/CFT correspondence (Maldacena)
Gives a precise example of the relation between
strings and gauge theory.
Gauge theory
String theory
N = 4 SYM SU(N) on R4
IIB on AdS5xS5
radius R

String states w/ E 
Aμ , Φi, Ψa
Operators w/ conf. dim. 
gs  g ;
R / ls  ( g
2
YM
N  ,   g
R
2
YM
2
YM
N
fixed
N)
1/ 4
λ large → string th.
λ small → field th.
Semi-classical strings
Why?
S






d

d

G


X

X


X

X







1
 

We cannot quantize exactly S (+fermions etc.)
Semi-classical regime is strong coupling limit >>1
Two important cases of classical solutions
Surface ending on the boundary
(Maldacena; Rey, Yee)
Wilson loops
(= bdy. curve)
String inside AdS5xS5
Operator in the field theory
(Berenstein, Maldacena, Nastase;
Gubser, Klebanov, Polyakov)
Rotating Strings and twist 2 operators
Y  Y  Y  Y  Y  Y  R
2
1
2
2
2
3
2
4
sinh 2  ;  [ 3]
2
5
2
6
(Gubser,
Klebanov,
Polyakov)
2
cosh 2  ; t
ds   cosh  dt  d  sinh  d
2
2
2
2
2
2
[ 3]

E S
ln S , ( S   )
2
θ=ωt
O  Tr   S   ,
x  z  t
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:
L 11
3
L

ln W   cusp ln


  ln 2  ...  ln

4 



It agrees with the previously known result from the
rotating string. In fact one can see that both classical
solutions are related.
String version of the
Y  Y  Y  Y  Y  Y  R
2
1
2
2
Y1Y4  Y5Y6
cusp
2
3
2
4
2
5
2
6
2
Korchemsky-Marchesini
argument
Y1Y5  Y4 Y6
(Y2  Y3  0)
rot. string for infinite S limit.
Generalization to higher twist operators (MK)
O[ 2 ]  Tr   
S


O[ n ]  Tr  S / n   S / n   S / n    S / n  
In flat space such solutions are easily found in conf. gaug
x  A cos[(n  1)  ]  A (n  1) cos[  ]
y  A sin[(n  1)  ]  A (n  1) sin [  ]
Spiky strings in AdS:
 n 
E S 
ln S , ( S   )
 2  2
O  Tr  S / n   S / n   S / n    S / n  

 2   j 1   j   



S
dt  (cosh 2 1  1) j 
dt   4 1  ln sin 


2
8
2

  

j
j 

Spiky strings on a sphere: (Ryang )
Similar solutions exist for strings rotating on a sphere:
(top view)
The metric is:
ds 2   dt 2  d 2  sin 2  d 2
We use the ansatz: t   ,      ,    ( )
And solve for  ( ) . Field theory interpretation?
Giant magnon: (Hofman-Maldacena)
d  sin   2 sin 2   A 2

d
A
 2   2 sin 2 

d
sin 

 2 sin 2   A 2
d A cos
A
1
sin  
 sin 
(top view)
giant magnon
The energy and angular momentum of the giant magnon
solution diverge. However their difference is finite:
 A d


E J

sin
2

2  sin 

2
 A
cos
 ,    Angular distance between spikes
2

Interpolating expression:




2 
E  J  1  2 sin
 
2


1 



sin
,  >> 1

2

2 
,   1
2 sin
2
2
One more angular momentum:
(Dorey, Chen-Dorey-Okamura)
There are solutions with another angular momentum J2.
The energy is given by:
E  J1 

2 
J  2 sin
2

2
2

2 
 J2 
,   1
2 sin
2
2 J 2
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.
x  A cos[(n  1)  ]  A (n  1) cos[  ]
In flat space:
y  A sin[(n  1)  ]  A (n  1) sin [  ]
namely:
x  iy  X  x() e
Consider txS5: ds 2   dt 2 
i
,      
3
 dX
a 1
3
a
d Xa,
X
a 1
a
Xa  1
Use similar ansatz:
X a  xa () e
i a
 ra () e
i  a (  )  i a
The reduced e.o.m. follow from the lagrangian:
L  ( 2   2 ) x ' a x ' a  i a ( x ' a x a  x ' a xa )   a2 xa x a   ( xa x a  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 solutions
and 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 are
usually 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 known
one. In particular dressing the infinitely wrapped string
one gets the single spike. By dressing again one gets
two spikes etc.
This is a confirmation of the idea of these solutions as
excitations of the infinitely wrapped string.
Example of spike
scattering
Phase shift is
the same as the
giant magnon
(in terms of the
dressing 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 matching
closed 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