m-theory studies with lie-3 algebra tomohisa takimi (ntu) 2010 3/4 @ ncku

M-theory studies with Lie-3 algebra

Tomohisa Takimi (NTU)2010 3/4 @ NCKU

1. String theory and the M-theory.

Unifying Forces1. Electro-magnetic

force2. Weak force

3. Strong force

4. Gravity force

( 電磁気力） (QED)

（弱い力）

（強い力）(QCD)

（重力）

Glashow-Weinberg-Salam theory (1967

)


force2. Weak force

3. Strong force

4. Gravity force


（弱い力）


（重力）

GSW

Grand unified theory ?


force2. Weak force

3. Strong force

4. Gravity force


（弱い力）


（重力）

GSW

GUT

String Theory

Strings

Most Fundamental object would be strings !

String theory

Open string

Gauge bosonElectro-magnetic,

weak, strong forces

Assuming the fundamental object as 1-dimensionally extending object

Closed string

Gravitational force

String theoryConsistent only in the 10-dimensional space.

How do we get 4-dimensional space?Compactifying the extra

dimensions

In larger scale

Compactified direction can not be seen.

(low energy) 虫

小

姐

Problem of the string theory.

1. Defined only in perturbatively(not fundamental so)

2. There are two many theories(type-IIA, type-IIB, type-I, SO(32), E8 X E8)

Which is the real model describing nature ?

Breakthrough(1)

Discovery of the D(p)-brane

End of open string can move only

on the D-branes.

Breakthrough(1)

Non-perturbative (p)-dimensional object which the open string can end on.

Remarkable fact found out.

T-duality

Spectrum does not changed under this transformation Symmetry of the

theory!!Due to the D-brane, such a transformation can be done also in open strings.

Hint to solve the problem.

It is found out that a part of different theories are connected via T-duality

IIA IIB

D-brane itself is non-perturbative object.

To connect the all theory

M-theory is suggested !!

What is M-theory ?

String theory

Defined in the perturbative theory at small string coupling region

(Non-perturbative effect)

M-theory 1.‘non-perturbative fundamental theory of

superstring’2. 11-dimensional space-time theory

M-theory string-theory

3. 5-superstring theories are connected by M-theory

MIIA

IIB E8XE8

SO(32)

T-dual

S1 compact

T-dual

S1/Z2

IS-dual

M-theory

Theory with extended objects giving the degree of freedom in superstring theory.

‘M5-brane’ S1

M-theory

(ex

‘D4-brane’

‘M2-brane’

‘D2-brane’‘M2-brane’

‘Fundamental string’

Type IIA superstring

Fundamental degrees of freedom in M-theory have not been understood enough !

M-theory is not understood at all…

We need further investigation still now !

Recent progress in the ‘M-theory’

BLG model

Studies of this models

the deeper understanding about the fundamental degree of freedom in M-theory

First model dealing with Multiple membranes

(Bagger-Lambert, Phys. Rev. D 77, 065008 (2008))

(A. Gustavsson, Nucl.Phys.B811:66-76,2009)

(M5-brane is also included)

Special character of BLG model.

N=8 3-d SUSY ‘gauge’ model described by the Lie 3-algebra

Usual quantum field theory: described by Lie-algebra

What is Lie 3-algebra ? •Extension of the Lie algebra

(1)Lie algebra (Lie ring):

Product is defined with 2-components of the set (2)Lie 3-algebra (not ring):

Product is defined with 3-components not 2 !

Product (commutation

relation):

‘3-

Product’:

Properties

(1)Lie algebra VS (2)3-algebra(1)Lie

algebra 1. Metric (inner product):

2. Gauge transformation law

In compact Lie algebra, it is positive definite

symmetric

3. Fundamental identity (Leibniz rule)

For center elements

(2)3-algebra

symmetric

1. Metric (inner product):

2. Gauge transformation law:

3. Fundamental identity (Leibniz rule)

For center elements

Among the finite algebras, only A_4 can satisfy (1)positive definite metric (2) fundamental identity(3)gauge invariance of metric

Different from Lie algebra ex)

4. Gauge invariance of the metric

Structure constant becomes antisymmetric

What is BLG model ?

(1+2)-d gauge theory based on 3-algebra.

:8 component of the scalar fields.

:16 components Majorana spinor with SO(8) Chirality condition

:(1+2)-d gauge field with 2 gauge indices.

Field contents of the theory.

:8 transverse directions in 11-d target space :gauge indices

Action

There is not kinetic term for gauge field, only the Chern-Simon term exists. (due to SUSY)

Multiple M2-branes in (1+10) dimensions.

The symmetries of the theory

* Gauge symmetry

* Supersymmetry

Lie 3-algebra essentially enters above sym. transf.

After we perform the S1 compactification,(1+2)-dimensional U(N) supersymmetric gauge theory shows up.N D2-brane system in

string theory

Strong evidence that it is a multiple membrane system in the M-theory.

# of multiple membrane = dimension of Lie 3-algebra

N Dp-brane effective theory = p-dimensional U(N) gauge theory

Open string Behave as gauge

fields or matter with adjoint representation

A M5-brane in the BLG model

BLG model involve not only multiple membrane but also Single M5-brane.

M5-brane:

(1+5) dimensional extended object allowed to exist in the M-theory.

Ho-Matsuo (JHEP 0806:105,2008)

Ho-Imamura-Matsuo-Shiba(JHEP 0808:014,2008)

Nambu-Poisson structure in the Lie 3-algebra*dimension of Lie 3-algebra =Lie 3-algebra can be written by

the Fourier modes along the internal 3-directions

(as well as) (something like

root)

We denote the basis of Lie 3-algebra with Fourier mode as

*3-product becomes Nambu-Poisson bracket.

*Inner product is

Field variable

It seems to enhance (1+2) dimensional

world-volume (1+(2+3)) dimensional world-volume But the kinetic term along the 3

enhanced direction is still lacking.

How the internal 3-direction shows up as real world-volume by inducing the kinetic term along the directions ?A. Expanding the field around the background

Then the potential term serves the kinetic term

(1+2) dimensional theory is really enhanced to (1+5) dimensional theory.

Action

Nambu-Poisson structure

Serves kinetic terms

Trace changed to integration over enhanced direction

M5-brane action shows up !!

Meaning of background C-

field 3 dimensional version of the B-field in the non-commutative gauge theory

* Enhancement of the world-volume is caused by the C-field !!

* Moreover the enhanced direction has ‘non-commutative property’

In string theory

D0-brane

B-field

A D2-brane with NC-geometry

In BLG model

Membranes

C-field

A M5-brane with ‘NC-geometry’ (3-dimensional

version)

Is it really the M5-brane in the M-theory ? Does degrees of freedom in the string

theory show up after the S1 compactification ?

*Doubling dimensional reduction

‘M5-brane’ in 11-dimensions

D4-brane in string

theory ? Yes !!

M5-brane theory with Nambu-Poisson bracket Double dimensional

reduction D4-brane theory with Poisson bracket. (1+4) dimensional non-commutative Supersymmetric gauge theory in

Picking up only up to Poisson limit

M5-brane theory in BLG model by Nambu-Poisson bracket

Poisson limit of NC gauge theory

Double dimensional reduction


??????? What theory with NP-structure ??

NC gauge theory in full order of



Natural question

This question

Quantization of Nambu-Poisson bracket

Quantization of Poisson bracket

NC gauge theory with Moyal product

Pursuing the M5-brane theory with NP-structure reproducing the NC gauge theory

What is NC gauge theory and the Moyal product.

B場が入った弦理論から正準量子化で、非可換。また、弦の散乱振幅計算すると、Moyal積の構造が出てくる。






Natural question

Key viewpointPoisson limit of the NC gauge

transformation

These are actually the volume preserving diffeormorphism in 2-dimensional space.

with closed Lie-algebra structure

Full order of NC gauge transformation Deformation of the volume

preserving diffeomorphism in Poisson limit

With the deformation of the algebra

Full order of NC gauge transformation Poisson limit of the NC gauge transformation

vs

(1) The gauge transformation in both cases are generated by the same Lie-algebra.

generated by

(2) But the following are modified by parameter *Representation

space *Product (Lie bracket)

*Representation space

Poisson limit

Full order

Seiberg-Witten Map

Poisson limit

*Algebra

Full order

Similar treatment might be applicable to the M5-brane theory with NP-structure.

Gauge symmetry in BLG model with NP-bracket

Actually these can be regarded as the volume preserving diffeomorphism in the enhanced 3-dimensions.

with

Similar treatment

Can we deform the gauge transformation

(1) With keeping the same set

(2) Deforming the *Representation

space *Product (Lie bracket)

??





Deformation of gauge symmetry O.K ???

Deformation of gauge symmetry

???

Very Bad News

Actually,

No !!

No-go theorem

There is no non-trivial deformation of the volume preserving diffeomorphism in

By Lecomte & Roger (1996, French paper)

Proof Let us consider the deformation of the product

If the deformation is absorbed by the linear transformation of the algebra, it is not real non-trivial deformation

Change of the algebra by the linear transformation

If The deformation is not real non-trivial deformation

Note !! Operator

has nilpotency

If the deformed product satisfied the Jacobi identity,

So the non-trivial deformation with Jacobi indentity must be non-zero element of the cohomology of Must be non-trivial in

But ! It was shown that

is trivial in

(1)

(2)

is one dimension in d = 3 But contradict with the next

leading order of the Jacobi identity

Can not be satisified !!

There is no non-trivial satisfying Jacobi identity in

There is no non-trivial deformation of the volume preserving diffeomorphism in





Deformation of gauge symmetry O.K ???

Deformation of gauge symmetry

???

Is there alternative candidate ?

How about it ?





Another way of double dimensional reduction

Actually






Why ?If the NC gauge theory with full order is realized by the different double dimensional reduction from M5-brane theory in BLG model,

NC gauge transformation must be able to beembedded in the M5-brane gauge symmetry inthe BLG model !!But such embedding is

impossible !!

証明をここに入れる

環の構造を持ってないところが、ネックになっていそうとコメント（ associativeじゃないのがネック）

Summary

* Lie 3-algebra and BLG model gives us a new description of the M-theory.

*Lie 3-algebra involves the theory with Nambu-Poisson bracket, which gives us the new description of another degree of freedom in M-theory, moreover the M-theory counter part of the ‘NC-geometry’

*This kind of study serves us interesting both mathematical and physical subjects

#Quantization of Nambu-Poisson bracket, #non-associative algebra,#M-theory origin of the gauge symmetry ?#Other new degree of freedom ?

#Origin of the Chan-Paton factor in string ?

It is very worthy to consider about it.

Although we can not define the gauge invariant inner product for the background,

The gauge symmetry of the action is kept as unbroken.

Because the background shows up only through the NP-bracket, After go through the NP-bracket it changed to trace element

m-theory studies with lie-3 algebra tomohisa takimi (ntu) 2010 3/4 @ ncku

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