**Dp-branes, NS5-branes and U-duality from ** **nonabelian (2,0) theory with Lie 3-algebra**

**Yoshinori Honma ** (SOKENDAI, KEK)

### in collaboration with M. Ogawa and S. Shiba arXiv: 1103.XXXX

seminar@NTU, Mar. 4, 2011

( 本間 良則 )

### Introduction

During the past three years, there has been a lot of works about the action of 3D Chern-Simons-matter theory

They have arisen from searching the low energy effective action of multiple M2-branes Novelty is the appearance of new algebraic structure,

Lie 3-algebra

[Bagger-Lambert, Gustavsson]

However, the structure constant must satisfy the following Fundamental idenity (generalization of Jacobi identity)

for the closure of gauge symmetry

This identity is highly restrictive and a few examples are known in maximaly SUSY case

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### Introduction

3D Finite dim.

positive norm

negative norm [Ho-Imamura-Matsuo][Gomis et al.] [Benvenuti et al.]

**Lorentzian BLG**

component associated to Lorentzian generator becomes ghosts

But are Lagrange multipliers and these can be integrated out constraint equation

constant solution (VEV)

3d N=8 SYM (D2-brane) (novel Higgs mechanism) _{2/24}

### Introduction

N M2-branes on an orbifold

Lorentzian BLG theory can be derived from ABJM theory [Y.H.-Iso-Sumitomo-Zhang ‘08]

U(N)×U(N) (or SU(N)×SU(N)) Chern-Simons-matter theory

On the other hand, from the brane construction, the low energy effective action of arbitrarly of M2-branes is proposed♯

(from an analysis of moduli space) ABJM theory [Aharony-Bergman-Jafferis-Maldacena]

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Gauge tr. of bifundamental matter field

### Introduction

Take a linear combination of generators

Gauge structure of L-BLG

SU(N)×trans.

( in N=2, ISO(3) )

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### Introduction

M2

L-BLG action ABJM action

scaling limit

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### Introduction

What about M5-brane?

World volume theory of M2-brane (BLG theory, ABJM theory and their relationship) has been intensely studied and generate many interesting development (AdS/CMP, integrability, …)

low energy dynamics of M5-brane is thought to be described by a 6D theory which has

► supersymmetry

► Conformal symmetry

► SO(5)R symmetry

field contents are 5 scalars, a self dual 2-form and fermions [(2,0) tensor multiplet]

Covariant description of self dual form is not so easy and only the abelian (single M5-brane) case is known [Aganagic et.al.][Bandos et.al.]

But recently, a new approach toward the nonabelianization is proposed and our work is exploration of its properties

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**Outline**

### Introduction

### (2,0) SUSY in 6D (review) Dp&NS5 from (2,0) theory Aspects of U-duality

### Conclusion and Discussion

### (2,0) SUSY in 6D (review)

### (2,0) SUSY in 6D

Abelian (2,0) theory

Recently, N. Lambert and C. Papageorgakis generalize this to non-Abelian case with

Guiding principle is the emergence of the 5D SYM SUSY transformaion under the reducion

linear SUSY transformations are

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### (2,0) SUSY in 6D

Lie 3-algebra naturally appear once again Introduce a new (auxiliarly) field

And the proposed SUSY transformations of non-Abelian (2,0) theory are

In the following discussions, we treat to be totally antisymetric This SUSY trans. respects SO(5)R and dilatation symmetry

(appropriate as the M5-brane theory)

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### Non-Abelian (2,0) theory

proposed non-Abelian (2,0) SUSY transformation closes under the following EOM and constraints

We can recover 5D SYM(D4-brane) by taking a VEV

KK-tower along the M-direction doesn’t appear [Lambert-Papageorgakis-Schmidt Sommerfeld ‘10]

Absence of

( contradicts the Jacobi identity)

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### Dp&NS5 from (2,0) theory

### Dp-branes from (2,0) theory

Now we start with

generalized loop algebra

This can be regarded as the original Lorentzian Lie 3-algebra including loop algebra ( in d=1, Kac-Moody algebra )

[Ho-Matsuo-Shiba ‘09][Kobo-Matsuo-Shiba ‘09]

This central extension is crucial to realize the torus compactification This type of BLG theory can be obtained by the scaling limit of the orbifolded ABJM theory [Y.H.-Zhang]

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### Dp-branes from (2,0) theory

first we expand the fields as

we apply this algebra to the nonabelian (2,0) theory with Lie 3-alebra

Higgs ghost

scalar field

(and gauge field )

auxiliarly field auxiliarly field gauge field

component

EOM ：

preserve all the SUSY

We choose VEV’s as

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### Dp-branes from (2,0) theory

M5-brane

Physical meaning of setting the VEV as

D4-brane Dp-brane (p>4)

radius of M-circle + moduli of

( 5： M-direction)

It is convenient to use the projection operator which determine how to decompose into and

bocomes fiber direction of Dp-brane w.v.

torus compactification along the I directions

These VEVs corresponds to the moduli parameter of torus compactification

and the metric of torus is determined by

Later we will see that is actually compactified on

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### Dp-branes from (2,0) theory

component

► field and constraints

dimensional reduction of M-direction (M5→D4)

► spinor field

► scalar field

Using projection operator , we decompose the scalar fields as

(gauge field of fiber direction)

Then we obtain the kinetic part of gauge field (of fiber direction) as well as the scalar field

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### Kaluza-Klein mass by Higgs mechanism

In this stage, we can see how the higher dim. (p>4) Dp-brane theory arise

► In D4-brane perspective, this theory has mass term

similar mass terms exist for all the fields with index

### KK-tower

Finally we obtain D(d+4)-brane whose worldvolume is

► If we define gamma matrices of new direction as , they satisfy and

Therefore, if we do a Fourier transformation, we obtain

Same procedure works out and we can construct higher dimensional fields defined by

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### Dp-branes from (2,0) theory

► gauge field

► self dual 2-form

We substitute the EOM of gauge field and the self duality conditon into the EOM of self dual 2-form

Then we obtain the EOM of the Yang-Mills gauge field

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### Dp-branes from (2,0) theory

We derive the equations of motion of Dp-brane whose world volume is from nonabelian (2,0) theory with Lie 3-algebra

We finally obtain the following EOM

These are precisely the EOM of (5+d)D SYM !!

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### NS5-branes from (2,0) theory

Type IIA NS5-brane is obtained by choosing VEV’s as

► However, it only provides the copies of the free (2,0) tensor multiplet and no proper interaction-like term seems to exist

For the Type IIB NS5-brane, the dimensional reduction occurs but

another direction of world volume appears and resulting theory becomes (1+5)D Moreover, in this case, we can read the string coupling from the gauge field and this enables us to check the S-duality between NS5-brane and D5-brane

► In this case dimensional reduction caused by doesn’t occur

because of the absence of the VEV of

So far, we consider only the reduction to the Dp-brane

So the world volume remains to be (1+5)D

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### IIB NS5-branes from (2,0) theory

This is because, in order to obtain IIB NS5-brane, we interchange the M-direction and T-duality direction in the D5-brane case

And we choose the VEV as

and reformulate the fields in a slightly different way from the previous case as We start with Lorentzian Lie 3-algebra with Kac-Moody algebra

M5-brane

IIA NS5-brane IIB NS5-brane

( 10： M-direction) compactification along 5 direction

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### IIB NS5-branes from (2,0) theory

For example, EOM of scalar field of 10 direction is VEV ：

This was an auxiliarly field on the Dp-brane but now this becomes a gauge field on the IIB NS5-brane

► Together with the identification ,

We finally obtain the expected EOM of extra gauge field

Similarly, other EOM’s are easily obtained and they are all consistent with the (1,1) vector multiplet of IIB NS5-brane

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### Aspects of U-duality

### D5-brane on S^1

VEV corresponds with the compactification radius of M-direction as and the radius of transverse direction T-duality acts is

T-duality

On the other hand, we have obtained the D5-brane action given by

and these are consitent with the expected T-duality relation

First we consider the simplest case, D5-branes on ( M-theory compactified on ) In this case, the U-duality group is

(note that the world volume of fiber direction of D5-brane is a dual circle)

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### D5-brane on S^1

for the IIB NS5-brane, we can read the string coupling from the coefficient of the kimetic term of gauge field

This is the inverse of the string coupling in D5-brane theory and we see that the S-transformation is represented by the rotation of VEV

T-transformation

Therefore, we can realize the SL(2,Z) transformaion as a rotation of the VEV, as expected

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In general, the U-duality group is

### Dp-brane on T^(p-4)

Then we consider general case, Dp-branes on ( M-theory compactified on ) In this case, we can realize the moduli parameter as

and part of it can be realized by the transformaion of VEV’s as

However, we cannnot reproduce all the moduli parameters, at least in our set up

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d Background fields Parameter sp.

D5 1 D6 2 D7 3 D8 4 D9 5

: NS-NS 2-form

: R-R form field

deformation of 3-algebra

?? Nambu-Poisson like bracket?

### Dp-brane on T^(p-4)

Realization of the moduli parameter

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We derive Dp&NS5 from nonabelian (2,0) theory

As a consistency check, we see that the expected U-duality relations are co rrectly reproduced

In paricular, we realize the S-duality between IIB NS5 ⇔ D5

It is known that the Lorentzian BLG theory are derived from the scaling limi t of the ABJM theory and it is just conceivable that certain quiver gauge th eory has a origin of nonabelian (2,0) theory with Lie 3-algebra (but in gene ral the inverse process of scaling limit is not so easy)

## Conclusion and Discussion

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### equation

(pure gauge)

gauge field