string solitons in the m5-brane worldvolume with a nambu-poisson structure and seiberg-witten map...
TRANSCRIPT
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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]
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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
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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)
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BLG model.
N=8 3-d SUSY ‘gauge’ model described by the Lie 3-algebra
Originally Formulation of Multiple M2-branes
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Action
Multiple M2-branes in (1+10) dimensions.
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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
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Study of ‘the M5-brane’
(Our main purpose)
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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
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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 reductionto IIA string theory
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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)
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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)
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Open questions (what we try to
study)
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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 !
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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
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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
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Our Works
(the study of the M2-brane soliton solution on the M5-brane )
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Solution in BLG
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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
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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
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SUSY transformation law of fermion
BPS equation
gives BPS equation
= 0
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BPS equation (Chirality condition of the solution)1. From the direction M5-brane extends
2. From the direction M2-brane extends
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Final form of the BPS equation describing an M2-brane on M5-brane where
We solved the equation by the perturbation of
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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
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Solution in PST
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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
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1. We rotate the coordinate and solution by
Rotate the coordinate
M5
M2
M2
M5
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Then we obtained
2. Performing Seiberg-Witten Map
If we take The solution
coincides with solution in M5 in BLG!
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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 bereally M5-brane in M-theory!
By some limit, they might translate to each other
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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
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*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