On the Giant Magnon and Spike Solutions in String Theories

1. December 16-21 On the Giant Magnon and Spike Solutions in String Theories 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, JHEP0811:066,2008, arXiv:0807.2559 B.-H.L, C. Park , Unbounded Multi Magnon and Spike, arXiv:0812.2727 Bum-Hoon Lee (Sogang University, Seoul, Korea)

2. Ex. d=3+1, N=4 SU(Nc) SYM #Nc parallel D3-branes 1, … , 6 #Nc D-branes and Gauge Theories #16 Supersymmetric # Nc Dp Branes in YM theories in p+1 dim. String theory D1 F1 http://cquest.sogang.ac.kr CQUeST

3. Dp-brane solution in Supergravity ( for D-brane ) (string frame) (harmonic function) For D3 branes, with In near horizon limit AdS5 x S5 Geometry radius S5 = radius AdS5 = R For , , can trust the supergravity solution

4. 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

5. -The gravity theory on - Symmetry SO(2,4) * SO(6) Isometry group -N=4 SYM on the boundary 4d space Symmetry (same) SO(2,4) * SO(6) conf. * R-sym 1. AdS/CFT correspondence(Closed/Open string dulaty) full string theory closed string theory sugra approx. perturbative Yang-Mills theory nonperturbative

6. 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

7. 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

8. 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 • 2) Impurity or magnon

9. 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.

10. 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.

11. 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)

12. 2. The giant magnon and the spike on S 2 2. The giant magnon and the spike magnon in flat space In the light cone gauge , the solution with where In target space In world sheet ( ) Hofman & Maldacena (2006)

13. 2 - (closed) string excitation : two excitations carrying world sheet momentum 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 the difference of the target space coordinate ~ p - the open string case : a single excitation with momentum p along an infinite string.

14. 2 - Strings on the AdS5 x S5 Metric on S5 Parametrization Action : Solution Dispersion Relation

15. Spike inflat spacetime in flat Minkowski In conformal gauge (Eq. of motion ) (constraints ) solution Dispersion relation

16. n = 3 n = 10 Gauge Theory Operator

17. Spiky strings in AdS Metric Action Ansatz solution Dispersion relation

18. Magnons and Spikes on AdS5 x S5 2 Rotating string on Nambu-Goto action with the target space-time metric Ansatz

19. 2 o o Equation of motion From the first equation, c: integration const This solution satisfy all equations of motion.

20. 3. The giant magnon and the spike on S 2 Conserved quantities 1) the energy 2) the angular momentum 3) the angle difference( ~ the momentum of an excitation)

21. 2 Depnding on the parameter region, we obtain two different configurations. magnon spike

22. 2 1) magnon (case ii) the conserved quantities

23. 2 1) spike (case iv) the dispersion relation for spike

24. (*). 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 1 2

25. Spike on R x S2 with NS-NS B field • metric • action • ansatz

26. Solution (Dispersion Relation) • giant graviton • spike solution

27. Rotating String on Melvin Deform AdS3 x S3 • metric • action • ansatz

28. Solution (Dispersion Relation) • small B

29. Three-spin Spiky string onAdS3 x S3 • metric • action • ansatz

30. Solution (Dispersion Relation) • circular string on AdS • Helical string on AdS

31. 3.AdS –CFT for M2 Branes in M theory Gravity on 2+1 dim. CFT (ABJM Theory)

32. Rotating String Solution on RxS2xS2 Metric for AdS4 x CP3 Metric for R x S2 x S2

33. Ansatz Solution

34. Giant Magnon & Spike (finite size) Dispersion Relation

35. Spike Solution Dispersion Relation

36. finite size effect Giant magnon Spike

37. Multi Magnons on R x S2 Parameter Action Ansatz Solution Dispersion relation with the finite size effect

38. Multi Spikes on R x S2 Solution Dispersion relation with the finite size effect

39. 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

40. 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.

41. Thank You !