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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)
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.
Can we make the map between string and gaugetheory 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 conf. of spin chain mixing matrix op. on spin chain
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.
H S Sj jj
J
4
1
42 11
Rotation on AdS5 (Gubser, Klebanov, Polyakov)
Y Y Y Y Y Y R12
22
32
42
52
62 2
s in h ; [ ]2
3 cosh ;2 t
ds d t d d2 2 2 2 23
2 cosh sin h [ ]
E S S S
O Tr x z tS
ln , ( )
, θ = ω t
Generalization to higher twist operators
O TrnS n S n S n S n
[ ]/ / / / O Tr S
[ ]2
x A n A n
y A n A n
co s[( ) ] ( ) 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
ln , ( )
/ / / /
Beccaria, Forini, Tirziu, Tseytlin
Spiky strings in flat space Quantum case
x A n A n
y A n A n
co s[( ) ] ( ) co s [ ]
sin [( ) ] ( ) sin [ ]
1 1
1 1
Classical:
Quantum:
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 S=s1+…+sL
Conformal dimension 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 periodicchain with L sites. At one-loop:
We need to find the uk (real numbers)
For large spin, namely large number of roots, wecan 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 theL=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 anda>0. Each root has an associated wave number nw. We choose nw=-1 for u<0 and nw=n-1 for u>0.Solution?
d aDefine
and
We get on the cut:
Consider
i
-i
C
z
We get
Also:
Since we get
We also have:
Finally, we obtain:
Root density
We can extend the results to strong coupling usingthe 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 are taken to zero. - recovers scaling
●Particular limit: pp-wave type scaling
In string theory: this limit is seen when zooming near the boundary of AdS.
Spiky string solution in this background the same as spiky string solution in AdS in the limit:
solutions near the boundary of AdS – S is large
In pictures:
z
Spiky stringin global AdS
Periodic spike in AdS pp-wave
If we do not take number of spikes to infinity we get asingle spike:
How to get this pp-wave scaling at weak coupling ?
can get it from 1-loop BA 2-cut solution
This is leading order strong coupling in
while
●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 spikystrings 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-Jinfound using the sinh-Gordon model. Relation to elliptic curves description found byDorey and Losi and Dorey. Pp-wave limit for the all-loops two cuts-solution.Semiclassical methods?