3d CFT and Multi M2-brane Theory on 2/19/2019 3d CFT and Multi M2-brane Theory on M. Ali-Akbari School of physics, IPM arXiv:0902.2869 [hep-th] JHEP 0903:148,2009 1

Outline Mini-review of BLG theory 1-1. 3-algebra 2/19/2019 Outline 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 2

Mini-review of BLG theory 2/19/2019 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 : 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

2/19/2019 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

Indices take the values with being the dimension of 3-algebra. 2/19/2019 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

Some properties of BLG theory 2/19/2019 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 : 6

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: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 .

2/19/2019 Killing spinor Killing spinors on can be found in following way by using Killing spinors on . Metric of Killing spinor of 8

Then we have Relation between and Killing spinor of

New BLG theory SUSY variations where . 2/19/2019 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

Closure of supersymmetry over the fermionic field leads to 2/19/2019 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

By tacking super variations of the fermion eqution of motion we have : 2/19/2019 By tacking super variations of the fermion eqution of motion we have : Finally BLG theory action is where 12

2. Parity invariance( ) is 2/19/2019 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

2/19/2019 BPS configuration By definition a BPS configuration is a state which is invariant under some Specific supersymmetry transformations. BPS equation where 14

,( is the SO(4) chirality) 2/19/2019 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

One find another solution when is not constant and BPS equation 2/19/2019 One find another solution when is not constant and BPS equation leads to and then that we have used Two different cases or 16

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