M-theory studies with Lie-3 algebra Tomohisa Takimi (NTU) 2010 NCKU

1. String theory and the M-theory.

Unifying Forces 1. Electro-magnetic force 2. 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)

Unifying Forces 1. Electro-magnetic force 2. Weak force 3. Strong force 4. Gravity force ( 電磁気力） (QED) （弱い力） （強い力） (QCD) （重力） GSW Grand unified theory ?

Unifying Forces 1. Electro-magnetic force 2. Weak force 3. Strong force 4. Gravity force ( 電磁気力） (QED) （弱い力） （強い力） (QCD) （重力） GSW GUT String Theory

Strings Most Fundamental object would be strings !

String theory Open string Gauge boson Electro-magnetic, weak, strong forces Assuming the fundamental object as 1-dimensionally extending object Closed string Gravitational force

String theory Consistent 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 IIAIIB D-brane itself is non-perturbative object.

To connect the all theory M-theory is suggested !!

What is M-theory ? String theoryDefined in the perturbative theory at small string coupling region (Non-perturbative effect) M-theory 1.‘non-perturbative fundamental theory of superstring’ dimensional space-time theory M-theory string-theory

3. 5-superstring theories are connected by M-theory M IIA IIB E8XE8 SO(32) T-dual S1 compact T-dual S1/Z2 I S-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, (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

M5-brane theory in BLG model by Nambu- Poisson bracket ??????? What theory with NP- structure ?? NC gauge theory in full order of Poisson limit of NC gauge theory Double dimensional reduction 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。

M5-brane theory in BLG model by Nambu- Poisson bracket ??????? What theory with NP- structure ?? NC gauge theory in full order of Poisson limit of NC gauge theory Double dimensional reduction Natural question

Key viewpoint Poisson 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 limitFull 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) ??

M5-brane theory in BLG model by Nambu-Poisson bracket ??????? What theory with NP- structure ?? NC gauge theory in full order of Poisson limit of NC gauge theory 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 !! Operatorhas 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

M5-brane theory in BLG model by Nambu-Poisson bracket ??????? What theory with NP- structure ?? NC gauge theory in full order of Poisson limit of NC gauge theory Deformation of gauge symmetry O.K ??? Deformation of gauge symmetry ???

M5-brane theory in BLG model by Nambu-Poisson bracket ??????? What theory with NP- structure ?? NC gauge theory in full order of Poisson limit of NC gauge theory Deformation of gauge symmetry O.K ??? Deformation of gauge symmetry ???

Is there alternative candidate ?

How about it ? M5-brane theory in BLG model by Nambu-Poisson bracket NC gauge theory in full order of Poisson limit of NC gauge theory Double dimensional reduction Another way of double dimensional reduction

Actually M5-brane theory in BLG model by Nambu-Poisson bracket NC gauge theory in full order of Poisson limit of NC gauge theory Double dimensional reduction Another way of double dimensional reduction

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 be embedded in the M5-brane gauge symmetry in the BLG model !! But such embedding is impossible !!

証明をここに入れる

M5-brane theory in BLG model by Nambu-Poisson bracket NC gauge theory in full order of Poisson limit of NC gauge theory Double dimensional reduction Another way of double dimensional reduction

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.

完

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