Transcript ppt - Purdue University

Large spin operators in
string/gauge theory duality
M. Kruczenski
Purdue University
Based on: arXiv:0905.3536
(L. Freyhult, A. Tirziu, M.K.)
Miami 2009
Summary
● Introduction
String / gauge theory duality (AdS/CFT)
Classical strings and their dual field theory operators:
Folded strings and twist two operators.
Spiky strings and higher twist operators.
Quantum description of spiky strings in flat space.
● Spiky strings in Bethe Ansatz
Mode numbers and BA equations at 1-loop
● Solving the BA equations
1 cut at all loops and 2 cuts at 1-loop.
AdS-pp-wave limit.
● Conclusions and future work
Extending to all loops we find a precise matching
with the results from the classical string solutions.
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.
Can we make the map between string and gauge
theory precise? Nice idea (Minahan-Zarembo, BMN).
Relate to a phys. system, e.g. for strings rotating on S3
›
Tr( X X…Y X X Y )
operator
mixing matrix

H
2
4
| ↑ ↑…↓ ↑ ↑ ↓
conf. of spin chain
op. on spin chain
J

j 1
1   
  S j  S j 1 
4

Ferromagnetic Heisenberg model !
For large number of operators becomes classical and
can be mapped to the classical string.
It is integrable, we can use BA to find all states.
Rotation on AdS5 (Gubser, Klebanov, Polyakov)
Y  Y  Y  Y  Y  Y  R
2
1
2
2
2
3
2
4
sinh 2  ;  [ 3]
2
5
2
6
2
cosh 2  ; t
ds   cosh  dt  d  sinh  d
2
2
2
2
2
2
[ 3]

E S
ln S , ( S   )

θ=ωt
O  Tr   S   , x   z  t
Generalization to higher twist operators
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 
O  Tr  S / n   S / n   S / n    S / n  
Beccaria, Forini,
Tirziu, Tseytlin
Spiky strings in flat space Quantum case
Classical:
Quantum:
x  A cos[(n  1)  ]  A (n  1) cos[  ]
y  A sin[(n  1)  ]  A (n  1) sin [  ]
Strings rotating on AdS5, in the field theory side are
described by operators with large spin.
Operators with large spin in the SL(2) sector
Spin chain representation
si non-negative integers.
Spin
Conformal dimension
S=s1+…+sL
E=L+S+anomalous dim.
Again, the matrix of anomalous dimensions can be
thought as a Hamiltonian acting on the spin chain.
At 1-loop we have
It is a 1-dimensional integrable spin chain.
Bethe Ansatz
S particles with various momenta moving in a periodic
chain with L sites. At one-loop:
We need to find the uk (real numbers)
For large spin, namely large number of roots, we
can define a continuous distribution of roots with a
certain density.
It can be generalized to all loops
(Beisert, Eden, Staudacher
E = S + (n/2) f() ln S
Belitsky, Korchemsky, Pasechnik described in detail the
L=3 case using Bethe Ansatz.
Large spin means large quantum numbers so one can
use a semiclassical approach (coherent states).
Spiky strings in Bethe Ansatz
BA equations
Roots are distributed on the real axis between d<0 and
a>0. Each root has an associated wave number nw.
We choose nw=-1 for u<0 and nw=n-1 for u>0.
Solution?
i
Define
d
and
We get on the cut:
Consider
z
C
-i
a
We get
Also:
Since
we get
We also have:
Finally, we obtain:
Root density
We can extend the results to strong coupling using
the all-loop BA (BES).
We obtain
In perfect agreement with the classical string result.
We also get a prediction for the one-loop correction.
Two cuts-solutions and a pp-wave limit
When S is finite (and we consider also R-charge J) the
simplest solution has two cuts where the roots are
distributed with a density satisfying:
where, as before:
The result for the density is (example):
Here n=3, d=-510, c=-9.8, b=50, a=100, S=607, J=430
It is written in terms of elliptic integrals.
● Particular limit: 1 – cut solution
- can be obtained when parameters
zero.
- recovers
scaling
are taken to
● Particular limit: pp-wave type scaling
In string theory: this limit is seen when zooming near
the boundary of AdS.
solutions near the boundary of AdS – S is large
Spiky string solution in this background the same as
spiky string solution in AdS in the limit:
In pictures:
z
Spiky string
in global AdS
Periodic spike
in AdS pp-wave
If we do not take number of spikes to infinity we get a
single spike:
while
This is leading order strong coupling in
How to get this pp-wave scaling at weak coupling ?
can get it from 1-loop BA 2-cut solution
● Take
while keeping
fixed.
pp-wave scaling:
1-loop anomalous dimension complicated function of
only 3 parameters
If
are also large:
1-loop anomalous dimension simplifies:
Conclusions
We found the field theory description of the spiky
strings in terms of solutions of the BA equations.
At strong coupling the result agrees with the
classical string result providing a check of our proposal
and of the all-loop BA.
Future work
Relation to more generic solutions by Jevicki-Jin
found using the sinh-Gordon model.
Relation to elliptic curves description found by
Dorey and Losi and Dorey.
Pp-wave limit for the all-loops two cuts-solution.
Semiclassical methods?