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.