1. 3d CFT and Multi M2-brane Theory on
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3d CFT and Multi M2-brane Theory on
M. Ali-Akbari
School of physics, IPM
arXiv:  [hep-th]
JHEP 0903:148,2009 1

2. Outline Mini-review of BLG theory 1-1. 3-algebra
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Outline
Mini-review of BLG theory
algebra
1-2. some properties of BLG theory
2. BLG theory on
2-1. Killing spinor on
3. BPS configuration 2

3. Mini-review of BLG theory
2/19/2019
Mini-review of BLG theory
J. Bagger and N. Lambert; arXiv: [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 :
It describes the worldvolume of membranes at low energy.
It is an example of the
As scalar fields in representation of SO(8)
(corresponding to the eight directions transverse to M2-branes).
Non-propagating gauge fields.
Bosonic fields :
Fermionic field :
in representation of SO(8). 3

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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

5. Indices take the values with being the dimension of 3-algebra.
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Indices take the values with being the dimension of 3-algebra.
Supersymmetry variations are :
Superalgebra closes up to a gauge transformation on shell.
The BLG Lagrangian is :
where 5

6. Some properties of BLG theory
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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: [hep-th]]
Then the theory has been written as an ordinary gauge theory with gauge
group as [Mark Van Raamsdonk; arXiv: [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/ ]
3. Party invariance : 6

7. Algebra and another approach is ABJM model.
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: [hep-th]]
ABJM model
[O. Aharony, O. Bergman, D. Louis Jafferis and J. Maldacena; arXiv: [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: [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 can be found in following way by using Killing spinors on
Metric of
Killing spinor of 8

9. Then we have
Relation between
and
Killing spinor of

10. New BLG theory SUSY variations where .
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New BLG theory
SUSY variations
where
Closure of scalar field leads to :
where ,
and
We didn’t need equation of motion for scalar fields. 10

11. Closure of supersymmetry over the fermionic field leads to
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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 11

12. By tacking super variations of the fermion eqution of motion we have :
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By tacking super variations of the fermion eqution of motion we have :
Finally BLG theory action is
where 12

13. 2. Parity invariance( ) is
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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 13

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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 14

15. ,( is the SO(4) chirality)
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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. 15

16. One find another solution when is not constant and BPS equation
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One find another solution when is not constant and BPS equation
leads to
and then
that we have used
Two different cases or 16

17. Basu-Harvey configuration
“Basu-Harvey limit”
and then