3d CFT and Multi M2-brane Theory
on
M. Ali-Akbari
School of physics, IPM
arXiv:0902.2869 [hep-th]
JHEP 0903:148,2009
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Outline
1. Mini-review of BLG theory
1-1. 3-algebra
1-2. some properties of BLG theory
2. BLG theory on
2-1. Killing spinor on
3. BPS configuration
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Mini-review of BLG theory
J. Bagger and N. Lambert; arXiv: 0711.0955[hep-th]
As a three domensional superconformal field theory with OSp(8|4) superalgebra.
The bosonic part of the superalgebra is :
SO(8)xSO(3,2)
R-symmetry
Conformal symmetry
Motivation to study 3d CFT :
1. It describes the worldvolume of membranes at low energy.
2. It is an example of the
.
Bosonic fields :
As scalar fields in representation of SO(8)
(corresponding to the eight directions transverse to M2-branes).
Non-propagating gauge fields.
Fermionic field :
in
representation of SO(8).
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3-algebra
1. Totally antisymmetric 3-bracket over three 3-algebra generators :
2. Trace over the 3-algebra indices :
3. Fundamental identity (It is essential for closuer of gauge fields) :
4. Gauge invarivace :
or
4
Supersymmetry variations are :
Indices take the values
with
being the dimension of 3-algebra.
Superalgebra closes up to a gauge transformation on shell.
The BLG Lagrangian is :
where
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Some properties of BLG theory
1.Euclidean signature which are
gauge theory
It was proven that since the metric is positive definite the theory has solution
which is
.
[J.P. Gauntlett and J.B. Gutowski; arXiv:0804.3078[hep-th]]
Then the theory has been written as an ordinary gauge theory with gauge
group as
.
[Mark Van Raamsdonk; arXiv:0803.3803[hep-th]]
Original BLG :
Metric is positive definite.
Structure constant is totally antisymmetric and real.
2. The low energy limit of multiple M2-branes theory is expected to be an
interacting 2+1 dimensional superconformal(Osp(8|4)superalgebra) field
theory with eight transverse scalar fields as its bosonic content .
[J. H. Schwarz; arXiv:hep-th/0411077]
3. Party invariance :
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4.There are two different approach to account for an arbitrary number of membrans.
One approach is Lorentzian signature which are
theories based on any Lie
Algebra and another approach is ABJM model.
Lorentzian signature
Metric is not positive definite.
Structure constant is totally antisymmetric and real.
[S.Benvenuti, D. Rodriguez-Gomez, E. Tonni and H. Verlinde; arXiv:0805.1087[hep-th]]
ABJM model
[O. Aharony, O. Bergman, D. Louis Jafferis and J. Maldacena; arXiv:0806.1218[hep-th]]
ABJM theories have been obtained by relaxing the condition on 3-bracket so
that it is no longer real and antisymmetric in all three indices but the metric is
positive definite yet.
[J. Bagger and N. Lambert; arXiv: 0807.0163[hep-th]]
5. According to AdS/CFT and holographic principle this model lives on the
boundary of
which is
.
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Killing spinor
Killing spinors on
on
.
can be found in following way by using Killing spinors
Metric of
Killing spinor of
8
Then we have
Relation between
and
Killing spinor of
9
New BLG theory
SUSY variations
where
.
Closure of scalar field leads to :
where
,
and
We didn’t need equation of motion for scalar fields.
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Closure of supersymmetry over the fermionic field leads to
where the equation of motion is
and
The last closure is
with the following equation of motion
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By tacking super variations of the fermion eqution of motion we have :
Finally BLG theory action is
where
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1. For positive definite metric the above theory can be written as SU(2)xSU(2)
gauge theory.
2. Parity invariance(
) is
3. It is easy to check that ABJM model can be written in the same way
if one adds an appropriate term in variation of fermionic field which is
where
is in 6 of SU(4) and raised A index indicates
that the field is in 4 of SU(4).
4. Superalgebra
where
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BPS configuration
By definition a BPS configuration is a state which is invariant under some
Specific supersymmetry transformations.
BPS equation
where
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BPS equation
In order to solve above equation we introduced
,
is a dimensional constant,
where
Then BPS equation leads
,(
is the SO(4) chirality)
which has a solution if
. These solution are exactly fuzzy three sphere
with SO(4) symmetry. Above equation shows that our solutions are ¼ BPS.
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One find another solution when
leads to
is not constant and BPS equation
and then
that we have used
Two different cases
or
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Basu-Harvey configuration
“Basu-Harvey limit”
and then
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