1. String solitons in the M5-brane worldvolume with a Nambu-Poisson structure and Seiberg-Witten map Tomohisa Takimi (NTU) Ref ) Kazuyuki Furuuchi, T.T JHEP08(2009)050 [arXiv: 0906.3172 ]

2. M-theory M-theory: candidate to unify the 5 superstring theory and 11 SUGRA Even the fundamental degree of freedom has not been understood well

3. Recent progress in the ‘M-theory’ study BLG model ABJM model Studies of these models would lead us to the deeper understanding about the M-theory Dealing with multiple membranes It is important to study these models. (Aharony-Bergman-Jafferis and Maldacena, JHEP 0810:091,2008) (Bagger-Lambert, Phys. Rev. D 77, 065008 (2008)) (A. Gustavsson,Nucl.Phys.B811:66-76,2009)

4. BLG model. N=8 3-d SUSY ‘gauge’ model described by the Lie 3-algebra Originally Formulation of Multiple M2-branes

5. Action Multiple M2-branes in (1+10) dimensions.

6. BLG model. Originally Formulation of Multiple M2-branes But also single M5-brane might be included! JHEP 0806:105,2008. Ho-Matsuo Ho-Imamura-Matsuo and Shiba JHEP 0808:014,2008

7. Study of ‘the M5-brane’ (Our main purpose)

8. M5 If we take the of M2 dimensional Lie 3-algebra Expansion around the background (M5-brane BPS configuration) 1. 2. Represent the 3-algebra by the Nambu-Poisson bracket 3. BLG can be a ‘M5-brane world-volume’ theory

9. Known property of this M5-brane ‘The M5-brane’ possesses some ‘Non-commutativity’ with NC-parameter in the worldvolume coordinate Coincides with the weak limit ( : : a non-commutative parameter) of the D4-brane world-volume theory with non-commutative geometry (written by the Poisson bracket) After double dimensional reduction to IIA string theory

10. First order of the Seiberg-Witten Map is already proposed in Ho-Imamura-Matsuo and Shiba JHEP 0808:014,2008 In =0, the ‘M5-brane theory’ would become the theory in the constant C-field M5-brane in Non-commutativity M5-brane in Constant C-field Seiberg-Witten Map

11. Open questions (what we try to study)

12. Although there are several evidences that the BLG would capture the M5-brane information, we are not still sure that ‘the M5-brane in the BLG’ is exactly M5-brane in the M-theory ! We should clarify the relationship between already established M5-brane formulations !

13. Relationship between the ‘Ordinary description’ (PST formalism) of the M5-brane Seiberg-Witten map ? Pasti-Sorokin-Tonin Phys. Lett. B 398, 41 (1997) PST Formalism (with constant C-field) M5 in BLG model ? Some limit ? Here we study the relationship between the PST formalism

14. Our Aim We want to clarify what ‘the M5-brane’ in the BLG model is. We would investigate it, by studying the relationship between PST M5-brane (with constant C-field) via the study of the M2-brane soliton solution on the M5-brane from BLG.

15. Our Works (the study of the M2-brane soliton solution on the M5- brane from BLG. )

16. Field content of and M5-brane theory 2,3,4,5 direction 1-direction M5 :two-form tensor fields :bosonic fields transverse to M5-brane :fermion :M5-brane world-volume coordinate :SUSY parameter Direction related with Nambu-poisson bracket

17. Anzats for the BPS configuration of M2 on M5 Among the transverse fields only the is turn on 2,3,4,5 direction 6-direction 1-direction M5 M2

18. SUSY transformation law of fermion BPS equation gives BPS equation = 0

19. BPS equation (Chirality condition of the solution) SUSY in the M5-brane theory from BLG is originally imposed M2 brane ending on the M5-brane imposes

20. Final form of the BPS equation describing an M2-brane on M5-brane where We solved the equation by the perturbation of

21. BPS solution up to O(g) Let us compare with this solution with the solution in the PST M5-brane formulation with the constant C-field background

22. Relation ship between the solution in the constant C-field in PST formalism Solution in the constant C-field M2-brane is tilted by for the 2-direction due to the effect of C-field along 3,4,5-direction (residing in 3,4,5-direction ) Youm, Phys Rev D63 (2001) 045004,Michishita, JHEP 09 (2000) 036

23. 1. We resolve the tilt by rotating the coordinate and solution by Here is a real direction of the M2-brane soliton transverse to M5-brane M5 (Proper representation of the solution)

24. Then we obtained 2. We performed Seiberg-Witten Map (We expect this map connect with the C-field solution and our solution) If we take This solution coincides with our solution!

25. Indication PST Formalism M5 in BLG model Connected by Seiberg-Witten Map! These two formalism should deeply relates with each other ! The ‘M5-brane’ in BLG would be really M5-brane ! By some limit, they translate to each other

26. Summary We clarified that at least order O(g), M2-soliton on the M5-brane in the Nambu- Poisson bracket are derived from M2-brane soliton on M5-brane in constant C-field in PST formalism by Seiberg-Witten Map

27. *All order correspondence by the Seiberg-Witten Map between the two actions ? Motivate to *The study of the quantization of Nambu- Poisson bracket, (We might be able to construct the M-theory counter part of the non-commutative gauge theory ?) Future work