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’

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)

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 bracket3

. BLG can be a ‘M5-brane world-volume’ theory

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

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)

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

modelSeiberg-Witten Map is proposed

( = 0 of BLG M5)

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 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

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

Our Works

(the study of the M2-brane soliton solution on the M5-brane )

Solution in BLG

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

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)1. From the direction M5-brane extends

2. From the direction M2-brane extends

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 this solution

with the solution in the PST M5-brane in the constant C-field background

Solution in PST

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

1. We rotate the coordinate and solution by

Rotate the coordinate

M5

M2

M2

M5

Then we obtained

2. Performing Seiberg-Witten Map

If we take The solution

coincides with solution in M5 in BLG!

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

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

*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