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
On the Giant Magnon and Spike Solutions in String Theories
Bum-Hoon Lee (Sogang University, Seoul, Korea)
December 16-21
D-branes and Gauge D-branes and Gauge TheoriesTheories
http://cquest.sogang.ac.kr CQUeST
#16 Supersymmetric # Nc Dp Branes inYM theories in p+1 dim. String theory
Ex. d=3+1, N=4 SU(Nc) SYM
fermionsaI ),(A: Fields * a
#Nc parallel D3-branes
1, … , 6
#Nc
D1
F1
(string frame)
(harmonic function)
( for D-brane )
radius S5 = radius AdS5 = R
For , , can trust the supergravity solution
AdS5 x S5 GeometryIn near horizon limit
with
For D3 branes,
Dp-brane solution in Supergravity
Contents
1. Motivation : AdS-CFT (Holography)
2. giant magnon and spikes (AdS5 x S5)
3. giant magnon and spikes (AdS4 x CP3)
4. Summary and discussion
1. AdS/CFT correspondence(Closed/Open string dulaty)
-The gravity theory on
- Symmetry SO(2,4) * SO(6) Isometry group
-N=4 SYM on theboundary 4d space - Symmetry
(same) SO(2,4) * SO(6) conf. * R-sym
full string theory closed string theory sugra approx.perturbative Yang-Mills theory nonperturbative
AdS/CFT Dictionary• 4D CFT (QCD) 5D AdS
• Spectrum : - 4D Operator 5D string states - Dim. of [Operator] 5D mass
• Current conservation 5D gauge symmetry
• Large Q small z• Confinement (IR) cutoff zm
• Resonances Kaluza-Klein states
According to the AdS/CFT correspondence,
isometry of R-symmetry group of N=4 SYM
Z, W, X : three complex scalar fields of SYM describing coordinates of the internal space with |Z| + |W| + |X| =1. (Z and Z: the plane on which the equator of lies)
J in SYM : # of Z fields J : the angular momentum describing the rotation on the equator of in the string theory side.
Consider the limit
2 2 2
As an example, consider the SU(2) part only (with Z and W )
-energy and R-charge E=1 and J=1 for Z and E=1 and J=O for W
for case ii) E - J = 1 + correction anomalous dim.
the spectrum of string states
string with infinite E and J
1) state (E-J=0)
2) the giant magnon (E-J=0)
the spectrum of operators in SYM
long chain operator
1)
2)
Impurity or magnon
On the gauge theory side (related to spin chain model)
By Minahan and Zarembo the one-loop anomalous dimension of operators ( : # of Z and W) composed of scalars in N=4 SYM theory follows from solving the spin chain model
The one loop anomalous dim. eigenvalue of the 1-loop dilatation operator acting on
these op.
To apply one should consider asa spin ½ chain identifying Z with a spin down and W with
aspin up
the dispersion relation for the magnon
in the large ‘t Hooft coupling limit,
Now, we study which spectrum of the string side corresponds to this magnon solution in SYM.
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)
2. The giant magnon and the spike
magnon in flat space
In the light cone gauge , the solution with where
2. The giant magnon and the spike on S
In world sheet ( )In target space
2
Hofman & Maldacena (2006)
- (closed) string excitation : two excitations carrying world sheetmomentum p and –p respectively.
two trajectories (blue and green) lie in the different values of ,
The world sheet momentum of the string excitation corresponds to thedifference of the target space coordinate
2
- the open string case :
a single excitation with momentum p along an infinite
string.
~ p
- Strings on the AdS5 x S5
Metric on S5 Parametrization
Action :
Solution
Dispersion Relation
2
Spike in flat spacetime
In conformal gauge
in flat Minkowski
solution
(Eq. of motion )
(constraints )
Dispersion relation
n = 3 n = 10
Gauge Theory Operator
Spiky strings in AdS
Ansatz
Metric
solution
Dispersion relation
Action
Rotating string on
Nambu-Goto action
with the target space-time metric
Ansatz
2Magnons and Spikes on AdS5 x S5
Equation of motion
From the first equation,
c: integration const
This solution satisfy all equations of motion.
o o
2
Conserved quantities
1) the energy
2) the angular momentum
3) the angle difference( ~ the momentum of an excitation)
3. The giant magnon and the spike on S2
Depnding on the parameter region,
we obtain two different configurations.
magnon
spike
2
1) magnon (case ii)
the conserved quantities
2
1) spike (case iv)
the dispersion relation for spike
2
(*). The string description for the magnon bound state
The dispersion relation for the magnon bound state
- Q-magnon bound state the elementary magnon in this subsector :
In string theory side, this dispersion relation corresponds to that of the giant magnon carrying two independent angular momentum, J and J describing the string moving on 21
Spike on R x S2 with NS-NS B field
• metric
• action
• ansatz
Solution (Dispersion Relation)• giant graviton
• spike solution
Rotating String on Melvin Deform AdS3 x S3
• metric
• action
• ansatz
Solution (Dispersion Relation)• small B
Three-spin Spiky string onAdS3 x S3
• metric
• action
• ansatz
Solution (Dispersion Relation)• circular string on AdS
• Helical string on AdS
3.AdS –CFT for M2 Branes in M theory
2+1 dim. CFT (ABJM Theory)
Gravity on
Rotating String Solution on RxS2xS2
Metric for AdS4 x CP3
Metric for R x S2 x S2
Ansatz
Solution
Giant Magnon & Spike (finite size)
Dispersion Relation
Spike Solution
Dispersion Relation
finite size effect
Giant magnon
Spike
Parameter
Action
Ansatz
Solution
Dispersion relation with the finite size effect
Multi Magnons on R x S2
Solution
Dispersion relation with the finite size effect
Multi Spikes on R x S2
4. Summary and discussion
- It was shown that the magnon in the spin chain can be described by the giant magnon solution in string theory.
- Furthermore, the magnon bound state is also described by a giant magnon with two angular momentum
- Investigate the 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
Summary - continued
- Magnon like solutions for strings in I-brane background
- Spiky Strings on AdS4 x CP3
- much of the AdS / CFT still need to be confirmed such as finding the dual integrable model corresponding
to the spike solution.