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’ BLG model ABJM model Studies of these models the deeper understanding about the M-theory Dealing with Multiple membranes (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. Non-commutative geometry in the 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. About the Seiberg-Witten Map In D-brane theory, D-brane in Constant B-field D-brane in Non-commutativity These two description is physically equivalent ( Just the regularization is different) Seiberg-Witten Map (Decided from the fact the gauge transformation of one side must corresponds to the gauge trans. of the other side)

11. First order of the Seiberg-Witten Map is already proposed in Ho-Imamura-Matsuo and Shiba JHEP 0808:014,2008 M5-brane in Non-commutativity M5-brane in Constant C-field Seiberg-Witten Map in M5 Also in the ‘M5-brane theory’ in BLG model Seiberg-Witten Map is proposed ( = 0 of BLG M5)

12. Open questions (what we try to study)

13. 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 !

14. Relationship between the PST formalism Seiberg-Witten map ? Pasti-Sorokin-Tonin Phys. Lett. B 398, 41 (1997) PST Formalism (with constant C-field) M5 in BLG model ? Some limit ? We will study the relationship

15. How we study the relationship? Comparing M2-brane soliton solution on M5-brane in the both formalism M5 M2 M5 M2 PST Formalism (with constant C-field) BLG Model

16. Our Works (the study of the M2-brane soliton solution on the M5-brane )

17. Solution in BLG

18. Field content and parameter of M5 in BLG 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

19. 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

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

21. BPS equation (Chirality condition of the solution) 1. From the direction M5-brane extends 2. From the direction M2-brane extends

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

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

24. Solution in PST

25. 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 We take ( << 1) M2 M5

26. 1. We rotate the coordinate and solution by Rotate the coordinate M5 M2 M5

27. Then we obtained 2. Performing Seiberg-Witten Map If we take The solution coincides with solution in M5 in BLG!

28. Indication PST Formalism with C-field M5 in BLG model Connected by Seiberg-Witten Map! These two formalism deeply relate with each other ! The ‘M5-brane’ in BLG would be really M5-brane in M-theory! By some limit, they might translate to each other

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

30. *All order correspondence between the two actions ? by the Seiberg-Witten Map Need to *study the quantization of Nambu-Poisson bracket, (Our work might also give some hint for this direction. Our direction might be complementary to this study.) Future work