Post on 28-Dec-2015
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
)
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）
GSW
Grand unified 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）
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
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 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)
??
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 !! 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
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 beembedded in the M5-brane gauge symmetry inthe 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