Transcript The Giant Magnon and Spike Solution

Yerevan State University / Leibniz Universität Hannover
Supersymmetry in Integrable Systems - SIS'10
International Workshop, 24-28 August 2010, Yerevan, Armenia
Integrability in AdS/CFT and finite
size effects in dyonic Magnons
Bum-Hoon Lee
Center for Quantum Spacetime
Sogang University
Seoul, Korea
Based on
B.-H.L, R. Nayak, K. Panigrahi, C. Park
On the giant magnon and spike solutions for strings on AdS(3) x S**3.
JHEP 0806:065,2008. arXiv:0804.2923
J. Kluson, B.-H.L, K. Panigrahi, C. Park,
Magnon like solutions for strings in I-brane background.
JHEP 0808;032, 2008, arXiv:0806.3879
B.-H.L, K. Panigrahi, C. Park , Spiky Strings on AdS4 x CP3,
JHEP 0811:066,2008 , arXiv:0807.2559
B.-H.L, C. Park , Unbounded Multi Magnon and Spike, arXiv:0812.2727
C.Ahn, M. Kim, B-H.L., Quantum finite-size effects for dyonic magnons in the
AdS_4 x CP^3. arXiv:1007.1598 [hep-th], to appear in JHEP,
Contents
1. Duality of AdS-CFT
Anomalous Dimensions of the Operators in CFT
= Energy of the string states
2. Guage Theory Operators and Integrability
3. Classical String solutions (giant magnon and spikes)
and Integrability
4. Algebraic curves, Exact S-matrix,
and finite size corrections
5. Summary and discussion
I. Duality of AdS-CFT
Maldacena 97
* Dp branes carry tension (energy) and charge (source for p+2 form)
 Gravity in AdS space (dim = ((p+1)+1) )
* Dp brane’s low energy dynamics by fluctuating open strings
 Yang-Mills in (p+1) dim. (CFT)
Ex) #Nc D3 branes :
3+1 dim N=4 SYM
Conformal Field Theory
in the 3+1 dim “boundary”
open string
L
-
=
=후ㅎ
1
1
1
Tr ( Fu F u  D   D   I  [ I ,  J ]2 )  Ferm ions
2
4 gYM
4
2
I ,J
A ,  I ,  :
(i)
1, …, 6
0, 1, …, 3
: Nc×Nc
(j)
N
4+1 dim Gravity
in AdS
4+1 dim “bulk”,
close string
AdS5
x
S5
(Nc)
Aij
X ijI
mtx, adj. repn. of U(Nc)
Conformal x R-symm : SO(4, 2) x SO (6) Isometry of (AdS5) x (S5)
Perturbative if
gN << 1
Reliable if
gN >> 1
AdS/CFT Dictionary (for AdS5 x S5)
Witten 98
Gubser-Klebanov-Polyakov 98
• 4D CFT (QCD)
 5D AdS
• Spectrum :
- 4D Operator
 5D string states
-
Dim. of [Operator]
 5D mass
• Not easy to confirm AdS/CFT in practice
 string theory side : reliable for large tHooft coupling
quantization in AdS b.g. not known, etc.
YM theory side : reliable perturbation only for small coupling
operator mixing, etc.
2). AdS –CFT for M2 Branes in M theory
Gravity on
2+1 dim. N=6 C.-S. QFT
(ABJM Theory)
Gauge Field (Chern-Simons)
Scalars :
( =1,2)
Fermions
Aharony, Bergman, Jafferis &Maldacena,
arXiv:0806.1218
• Not easy to confirm AdS/CFT
 string theory side : reliable for large tHooft coupling
YM theory side
quantization in AdS b.g. not known, etc.
: reliable perturbation only for small coupling
operator mixing, etc.
• Operators with large charges J -> additional parameter
Ex)
Both
BMN, hep-th/0202021
and
may be evaluated in powers of
• Two sets of operators :
1) Chiral primary op. & descendents
-> nonrenormalization theorem
2) op. w/ large charges  classical string
Ex) chiral primary operator w/ large R-charge Tr phi^L
<-> pointlike string rotating on a big circle of S5 with v=c
some “impurities” (BMN operators) <-> almost pointlike
etc.
Integrability plays crucial role
• Semiclassical strings of string theory with world-sheet sigma model
corresponds to large operators with high excitations in gauge theory
String Theory Side :
Bena, Polchinski, Roiban
- Integrability in string theory hep-th/0305116
- string sigma model on AdS5xS5 admit Lax representation
- Exists various methods for string solutions.
- Algebraic curve methods,
- solution through Pohlmeyer reduction, etc.
- Computation of
is straightforward
Gauge Theory Side :
- Operator mixing matrix (that grows exponetially with the size)
is dentified with the hamiltonian of
an integrable spin chain Minahan, Zarembo 0212208
- the anomalous dimension
from the integrability (by algebraic Bethe ansatz,
exact scattering matrix, etc.)
2. Large Operators in Gauge Theory and Integrability
Ex) N=4 SYM :
Z, W, X : three complex scalar fields of SYM describing
coordinates of the internal space
2
with |Z|2 + |W|
+ 2|X| =1.
(Z and Z-bar : the plane on which the equator of
lies)
J = # of Z fields in SYM
= ang. Mom. of string rotating along the equator of
Consider the limit
.
SU(2) sector in 1-loop)(with Z and W)Minahan and Zarembo (2002)
-energy and R-charge
E=1 and J=1 for Z and
E=1 and J=O for W
• Identifying Z with a spin down and W with a spin up
Ground state
Excited state
(
: # of Z and W, J1 + J2 = J )
• Dilatation operator is related to the Hamiltonian of the
integrable XXX Heisenberg spin chain model
• Hence, the one-loop anomalous dimension of operators
eigenvalue of the spin chain Hamiltonian
• which can be solved by the Algebraic Bethe Ansatz, etc.
Ex) single magnon
Operator
Spin chain configuration
the dispersion relation for the magnon
Note : the all loop dispersion relation conjectured for the magnon
In the large ‘t Hooft coupling limit, the dispersion relation becomes
This is the same as that of the giant magnon
in the string sigma model
Algebraic Bethe Ansatz for spin Hamiltonian
(SU(2) sector)
• Operators with R-charges (J1= L-J, J2=J)
• Bethe equations
or
• Cyclicity
• Anomalous dimension
• Scattering matrix
(
: rapidities )
Scaling limit
• Bethe equatioin for the Large operator (scaling)
• Distribution of the Bethe roots - density or resolvent
• Scaling limit of Bethe equations
• Momentum condition
• Anomalous dimension
Comments
• Integrability also for N=6 ABJM model ( AdS4 x CP3)
- excitations Ai,Bi  Two decoupled Heisenberg XXX Hamiltonian
Ex)
Comments -continued
• There exist many other types of operators
Ex) (Single Trace operators, with higher twists)
: The anomalous dimension is dominated by the contribution
of the derivatives
 Dual description in terms of rotating strings with n cusps (Conjecture)
• Dilatation operators and Bethe Ansatz in higher orders
- in 2-loops – Beisert, Kristjansen & Staudacher, hep-th/0303060
- 3-loops - Beisert, Kristjansen & Staudacher, hep-th/0303060
Beisert hep-th/0308074, 0310252
Klose & Plefka, hep-th/0310232
- Higher loops – Serban & Staudacher, hep-th/0401057
• finite size effects
wrapping interactions at loop order higher than length,
2
3. Classical String Solutions - Giant magnons & spikes
• The dual description in the string theory side
Hofman & Maldacena (2006)
The giant magnon
Ex) magnon in flat space
In the light cone gauge
the solution with
where
In world sheet (
)
,
In target space
2
- (closed) string excitation :
two excitations carrying world sheet
momentum p and –p respectively.
two trajectories (blue and green) lie in the
values of
,
different
The world sheet momentum of the string excitation corresponds to the
difference of the target space coordinate
- the open string case :
a single excitation
with momentum p
along an infinite string.
~ p
2
Magnon on the AdS5 x S5 - string rotating on S2 ⊂ S5
Metric on S5
Parametrization
Action :
Solution
Dispersion Relation
Note : Match with the all loop dispersion relation in the gauge theory if
take the large tHooft coupling limit
Spike in flat spacetime
in flat Minkowski
In conformal gauge
(Eq. of motion )
(constraints )
solution
Dispersion relation
n=3
Gauge Theory Operator
n = 10
Magnon bound states – dyonic giant magnons
- the giant magnon with two angular momenta, J1 and J2
- the string moving on an RxS3 subspace of AdS5 x S5
Hofman-Maldacena limit (Hofman-Maldacena hep-th/0604135)
J1, E  infinity, E-J1, J2, lambda = finite
String equations with Virasoro constraints
is equvalent to the complex sine Gordon equation
The dispersion relation
Chen-Dorey-Okamura ‘06
Note : Operator of the Gauge theory
Pohlmeyer ‘76
comments
• Magnons and Spikes
- in S5, AdS5, and AdS5 x S5
- in different background
e.g., Melvin background, NS-NS B field, etc.
- with 1, 2, and 3 angular momenta
- multi magnons and spikes
• Solutions in AdS4 x CP3 – three kinds of giant magnons
- small magnon
: CP1 & CP2 magnon
- Pair of small magnon : RP2 and RP3 magnon
- Big magnon
: dressed solution
Comments -continued
• Dispersion relations for various solutions obtained
• Finite size corrections
Giant magnon
Spike
Classical Integrability of string sigma model
• Focus on an SU(2) reduction of the full sigma-model to the
subsector of string moving on
• The string action in the conformal gauge
• Equation of motion with Virasoro constraints
• Eq. of motion in the weak coupling limit
or
where
This is the equation of the classical Heisenberg model, which is
completely integrable.
• Equivalentl to the consistency condition [L, M] = 0
for the following auxiliary linear problem
• The monodromy matrix : parallel transport of the flat connection (L, M)
with
• The trace of the monodromy matrix : independent of tau_0
 infinite set of integrals of motion
• unimodular and unitary when the spectral parameter is real
• Eigenvalues
determine the quasi-momentum p(x)
• The string action can be written as
where
• The equation of motion
• The equation of motion as the zero curvature condition
where
• Or as the consistency condition for the following linear problem
• Monodromy matrix
• resolvent
is an analytic function on the physical sheet, and can be written as
• Define
• Integral equation for the density
etc.
4. Algebraic curves, Exact S-matrix & finite size corrections
• Integrability in the spectral problem of AdS/CFT
- Gauge theory  Integrable spin chain  small g  all loop
- String sigma model  Lax representation
• All loop Bethe ansatz and exact S-matrix (for L  infinity)
• At finite L, there are corrections
• We consider the finite size effects at strong coupling regime
• Two independent approaches using integrability in both sides
- Algebraic curve  semiclassical effects in string theory
- Exact S-matrix  Luesher F-term correction
• All three kinds of giant magnons (small (on CP1 & CP2), Pair of
small (on RP2 and RP3) and Big (dressed solution) can be
reproduced in algebraic curve
5. summary & Discussion
• AdS/CFT  Dimension of an operator in Gauge Theory
= Energy of the corresponding string state
• Ex) the magnon in the spin chain corresponds to the giant
magnon solution in string theory, etc.
• Furthermore, the magnon bound state is also described
by a giant magnon with two angular momentum
• The integrability plays an important role and is shown to exist in
both sides of the gauge and string sides.
• Also mentioned solutions of
Spikes on R x S2 with B field
Rotating String on Melvin deformed AdS3 x S3
Three spin spiky solutions on AdS3 x S3
-> circular/helical strings on AdS
- Multi magnon and spike solutions
• Classical strings can be reprented by algebraic curve.
Ex) Various magnons (small, pair of small, big) in AdS4 x CP3
Summary & Discussion -continued
 Much of the AdS / CFT still need to be confirmed such as finding
the dual integrable model corresponding to the spike solution, etc.