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3d CFT and Multi M2-brane Theory on

M. Ali-AkbariSchool of physics, IPMarXiv:0902.2869 [hep-th]

JHEP 0903:148,2009

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Outline

1. Mini-review of BLG theory1-1. 3-algebra1-2. 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 theoryJ. 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 :

Bosonic fields :

Fermionic field :

As scalar fields in representation of SO(8)(corresponding to the eight directions transverse to M2-branes).

Non-propagating gauge fields.

in representation of SO(8).

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 .

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3-algebra1. 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

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The BLG Lagrangian is :

where

Indices take the values with being the dimension of 3-algebra.

Supersymmetry variations are :

Superalgebra closes up to a gauge transformation on shell.

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Some properties of BLG 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]]

1.Euclidean signature which are gauge theory

Original BLG :

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 :

Metric is positive definite.Structure constant is totally antisymmetric and real.

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5. According to AdS/CFT and holographic principle this model lives on the boundary of which is .

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.

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]]

ABJM model[O. Aharony, O. Bergman, D. Louis Jafferis and J. Maldacena; arXiv:0806.1218[hep-th]]

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]]

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Killing spinor

Metric of

Killing spinor of

Killing spinors on can be found in following way by using Killing spinors on .

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Relation betweenand

Killing spinor of

Then we have

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New BLG theory

where .

SUSY variations

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

BPS equation

By definition a BPS configuration is a state which is invariant under some Specific supersymmetry transformations.

where

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BPS equation

In order to solve above equation we introduced

where

Then BPS equation leads

,( is the SO(4) chirality)

which has a solution if . These solution are exactly fuzzy three spherewith SO(4) symmetry. Above equation shows that our solutions are ¼ BPS.

, is a dimensional constant,

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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 casesor

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Basu-Harvey configuration

“Basu-Harvey limit”

and then