the giant magnon and spike solution

The Giant Magnon and Spike Solution

Chanyong Park (CQUeST)

Haengdang Workshop

’07, 11. 30

The Giant Magnon and Spike Solution Chanyong ParkChanyong Park

1. Review of the spin chain model

2. The giant magnon and the spike on

3. The string description for the magnon bound state

4. Conclusion and discussion

Haengdang Workshop (Hanyang Univ.)

Chanyong ParkChanyong ParkThe Giant Magnon and Spike Solution

1. Introduction

AdS/CFT correspondence

string theory on dual to SYM

Chanyong ParkChanyong Park

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



According to the AdS/CFT correspondence,

isometry of R-symmetry group of N=4 SYM

- J : the angular momentum describing the rotation on the equator of in the string theory side.

- 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 (E=1 and J=1 for each Z)


2 2 2



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 o

p.




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.




2. The giant magnon and the spike

Warming up in flat space

In the light cone gauge , the solution with where

Chanyong ParkChanyong Park2. The giant magnon and the spike on S

In world sheet ( )In target space


2

Now, consider two localized excitation carrying world sheet

momentum p and –p respectively.

Note that 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 giant magnon and the spike on S2

Rotating string on

Nambu-Goto action

with the target space-time metric

Ansatz




Equation of motion

From the first equation,

c: integration const

This solution satisfy all equations of motion.


o o



Conserved quantities

1) the energy

2) the angular momentum

3) the angle difference( ~ the momentum of an excitation)




Depnding on the parameter region,

we obtain two different configurations.


magnon

spike



1) magnon (case ii)

the conserved quantities




1) spike (case iv)

the dispersion relation for spike





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


- 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 giant magnon with three spin. - Study the scattering of the magnon solution. - Find the dual integrable model corresponding to the spike solution.




the giant magnon and spike solution

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