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The Giant Magnon and Spike Solution Chanyong Park (CQUeST) Haengdang Workshop ’07, 11. 30 The Giant Magnon and Spike Solution Chanyong Park.

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Presentation on theme: "The Giant Magnon and Spike Solution Chanyong Park (CQUeST) Haengdang Workshop ’07, 11. 30 The Giant Magnon and Spike Solution Chanyong Park."— Presentation transcript:

1 The Giant Magnon and Spike Solution Chanyong Park (CQUeST) Haengdang Workshop ’07, 11. 30 The Giant Magnon and Spike Solution Chanyong Park

2 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 Park The Giant Magnon and Spike Solution

3 1. Introduction AdS/CFT correspondence string theory on dual to SYM Chanyong 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 Haengdang Workshop (Hanyang Univ.) 1. Review of the spin chain model

4 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) Chanyong Park 222 Haengdang Workshop (Hanyang Univ.) 1. Review of the spin chain model

5 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. Chanyong Park Haengdang Workshop (Hanyang Univ.) 1. Review of the spin chain model

6 To apply one should consider as a spin ½ chain identifying Z with a spin down and W with a spin 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. Chanyong Park Haengdang Workshop (Hanyang Univ.) 1. Review of the spin chain model

7 2. The giant magnon and the spike Warming up in flat space In the light cone gauge, the solution with where Chanyong Park 2. The giant magnon and the spike on S In world sheet ( ) In target space Haengdang Workshop (Hanyang Univ.) 2

8 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, Chanyong Park The world sheet momentum of the string excitation corresponds to the difference of the target space coordinate Haengdang Workshop (Hanyang Univ.) 2. The giant magnon and the spike on S 2

9 Rotating string on Nambu-Goto action with the target space-time metric Ansatz Chanyong Park Haengdang Workshop (Hanyang Univ.) 2. The giant magnon and the spike on S 2

10 Equation of motion From the first equation, c: integration const This solution satisfy all equations of motion. Chanyong Park oo Haengdang Workshop (Hanyang Univ.) 2. The giant magnon and the spike on S 2

11 Conserved quantities 1) the energy 2) the angular momentum 3) the angle difference( ~ the momentum of an excitation) Chanyong Park Haengdang Workshop (Hanyang Univ.) 2. The giant magnon and the spike on S 2

12 Depnding on the parameter region, we obtain two different configurations. Chanyong Park magnon spike Haengdang Workshop (Hanyang Univ.) 2. The giant magnon and the spike on S 2

13 1) magnon (case ii) the conserved quantities Chanyong Park Haengdang Workshop (Hanyang Univ.) 2. The giant magnon and the spike on S 2

14 1) spike (case iv) the dispersion relation for spike Chanyong Park Haengdang Workshop (Hanyang Univ.) 2. The giant magnon and the spike on S 2

15 3. 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 Chanyong Park Haengdang Workshop (Hanyang Univ.) 3. The string description for the magnon bound state 21

16 4. Conclusion 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 giant magnon with three spin. - Study the scattering of the magnon solution. - Find the dual integrable model corresponding to the spike solution. Chanyong Park Haengdang Workshop (Hanyang Univ.) 4. Conclusion and discussion


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