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]
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
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)
BLG model.
N=8 3-d SUSY ‘gauge’ model described by the Lie 3-algebra
Originally Formulation of Multiple M2-branes
Action
Multiple M2-branes in (1+10) dimensions.
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
Study of ‘the M5-brane’
(Our main purpose)
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
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 reductionto IIA string theory
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
Open questions (what we try to study)
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 relationshipbetween already established M5-brane formulations !
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
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.
Our Works (the study of the M2-brane soliton solution on the M5-brane from BLG. )
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
Anzats for the BPS configuration of M2 on M5
Among the transverse fieldsonly the
is turn on
2,3,4,5 direction
6-direction
1-direction
M5
M2
SUSY transformation law of fermion
BPS equation
gives BPS equation
= 0
BPS equation (Chirality condition of the solution)SUSY in the M5-brane theory from BLGis originally imposed
M2 brane ending on the M5-brane imposes
Final form of the BPS equation describing an M2-brane on M5-brane where
We solved the equation by the perturbation of
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
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
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)
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!
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 bereally M5-brane !
By some limit, they translate to each other
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
*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