non-anticommutative abj theory
TRANSCRIPT
Available online at www.sciencedirect.com
Nuclear Physics B 869 [PM] (2013) 598–607
www.elsevier.com/locate/nuclphysb
Non-anticommutative ABJ theory
Mir Faizal
Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom
Received 5 December 2012; accepted 24 December 2012
Available online 8 January 2013
Abstract
In this paper we will discuss non-anticommutative deformations of the harmonic superspace. We willanalyse the non-anticommutative deformation of the superspace that break the supersymmetry from N = 3supersymmetry to N = 2 supersymmetry. We will then study the ABJ theory in this non-anticommutativesuperspace. This deformed ABJ theory will be shown to posses N = 5 supersymmetry.© 2012 Elsevier B.V. All rights reserved.
1. Introduction
In four dimensions N = 2 supersymmetry has been studied in harmonic superspace [1,2], andthis has been adopted for analysing N = 3 supersymmetry in three dimensions [3–5]. The har-monic superspace variable parameterise the coset SU(2)/U(1) and are well suited for analysingtheories with hight amount of supersymmetry. Thus, harmonic superspace has been used foranalysing the ABJM theory [6], which is a superconformal Chern–Simons-matter theory withmanifest N = 6 supersymmetry [6–11]. This theory is thought to be a low energy descriptionof N M2-branes on C4/Zk orbifold because it coincides with the BLG theory for the onlyknown example of a Lie 3-algebra [12–15]. So, the supersymmetry of the ABJM theory is ex-pected to be enhanced to full N = 8 supersymmetry for k = 1,2 [16,17]. The gauge fields inthe ABJM theory are governed by the Chern–Simons action and the matter fields live in the bi-fundamental representation of the gauge group U(N) × U(N). A generalisation of the ABJMtheory to the case where the matter fields live in the bifundamental representation of gauge groupU(M) × U(N) with M �= N has been made [18–21]. This theory is called the ABJ theory andit also has N = 6 supersymmetry. However, unlike the ABJM theory, non-planar corrections to
E-mail address: [email protected].
0550-3213/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nuclphysb.2012.12.018
M. Faizal / Nuclear Physics B 869 [PM] (2013) 598–607 599
the two-loop dilatation generator of ABJ theory mix states with positive and negative parity, andthis mixing is proportional to M − N [22]. So, for M = N , when the ABJ theory reduces to theABJM theory, there is no mixing.
In string theory the NS background causes a noncommutative deformation between the space-time coordinates [23–26], and a gravitino background causes a noncommutative deformationbetween the spacetime and Grassmann coordinates [27–31]. All these deformations preserve thesupersymmetry of the theory. However, the presence of a RR background causes a deformationbetween the Grassmann coordinates and thus breaks the a certain amount of the supersymmetryof the theory [32–40]. In four dimensions this can give rise to a fractional amount of supersym-metry like N = 1/2 supersymmetry. Non-anticommutative deformation of harmonic superspacehas also been analysed [39,41–43].
As M-theory is dual to type II string theory, a deformation of the string theory side willalso generate a deformation on the M-theory side. Thus, a noncommutative deformation of theM-theory will be dual to a noncommutative deformation of type II string theory caused by NSbackground. Similarly, a non-anticommutative deformation of the M-theory will be dual to a non-anticommutative deformation of type II string theory caused by RR background. In fact, thisduality can be explicitly verified by using the novel Higgs mechanism [44–47]. Thus, if weperform the higgsing of non-anticommutative M2-branes we will obtain non-anticommutativeD2-branes. So, the non-anticommutative ABJ theory is dual to type II string theory deformed bya RR background. Noncommutative deformation of the M2-branes in N = 1 superspace havebeen already studied [11,49]. However, non-anticommutative deformation of the M2-branes hasnot been studied.
This non-anticommutative deformation of the M2-branes can occur due to the presence ofa curved three form field Cμντ . This is because the ABJM theory is the boundary gauge theorydual to the eleven-dimensional supergravity on AdS4 × S7/Zk . A deformation of this eleven-dimensional supergravity on AdS4 ×S7/Zk can be caused by a three form field Cμντ . A constantthree form field Cμντ is only expected to change the gauge group of the theory without breakingany supersymmetry. Thus, the ABJ theory can be viewed as a deformation of the ABJM theory.However, a deformation of the eleven-dimensional supergravity on AdS4 × S7/Zk by a curvedthree form field Cμντ will change the geometry considerably and is expected to partially breakthe supersymmetry. The boundary theory dual to this deformed eleven-dimensional supergravitywill be a non-anticommutative ABJ theory.
Now, if Hμντρ the field strength of this three form field, then we expect a non-anticommutativedeformation proportional to {θa, θb} ∼ (γ μγ νγ τ γ ρHμντρ)ab to occur. However, it was not possi-ble to study the non-anticommutative deformations of the ABJM theory in the N = 1 superspaceas that would break all the manifest supersymmetry of the superspace. It would thus be inter-esting to analyse the non-anticommutative deformations of this theory in superspace with higheramount of manifest supersymmetry. So, in this paper we analyse the non-anticommutative defor-mation of the ABJ theory in harmonic superspace. Non-anticommutative deformations break thetotal supersymmetry of this theory from N = 6 supersymmetry to N = 5 supersymmetry.
2. Harmonic superspace
In this section we will review harmonic superspace in three dimensions [3–5,50]. These har-monic variable are parameterise by the coset SU(2)/U(1). So, the harmonic variables u± aresubjected to the constraints u+iu−
i = 1, u+iu+i = u−iu−
i = 0, and the superspace coordinates aregiven by
600 M. Faizal / Nuclear Physics B 869 [PM] (2013) 598–607
z = (xab, θ++
a , θ−−a , θ0
a , u±i
), (1)
where θ±a = θ
ija u±
i u±j and θ0
a = θija u+
i u−j . In the harmonic superspace the following derivatives
are constructed
∂++ = u+i
∂
∂u−i
, ∂−− = u−i
∂
∂u+i
,
∂0 = u+i
∂
∂u+i
− u−i
∂
∂u−i
. (2)
These derivatives are used to define the following derivatives
D−−a = ∂
∂θ++a+ 2iθ−−b∂A
ab, D0a = −1
2
∂
∂θ0a+ iθ0b∂A
ab,
D++a = ∂
∂θ−−a. (3)
Apart from these derivatives, the following derivatives are also constructed
D++ = ∂++ + 2iθ++aθ0b∂Aab + θ++a ∂
∂θ0a+ 2θ0a ∂
∂θ−−a,
D−− = ∂−− − 2iθ−−aθ0b∂Aab + θ−−a ∂
∂θ0a+ 2θ0a ∂
∂θ++a,
D0 = ∂0 + 2θ++a ∂
∂θ++a− 2θ−−a ∂
∂θ−−a. (4)
The superalgebra satisfied by these derivatives is given by{D++
a ,D−−b
} = 2i∂Aab,
{D0
a,D0b
} = −i∂Aab,[
D∓∓,D±±a
] = 2D0a,
[D0,D±±
a
] = ±2D±±a ,
∂0 = [∂++, ∂−−]
,[D++,D−−] = D0,{
D±±a ,D0
b
} = 0,[D±±,D0
a
] = D±±a . (5)
The harmonic superspace has N = 3 supersymmetry in three dimensions. The generators ofthis N = 3 supersymmetry in three dimensions are given by
Q++a = u+
i u+j Q
ija , Q−−
a = u−i u−
j Qija ,
Q0a = u+
i u−j Q
ija , (6)
where
Qija = ∂
∂θaij
− θijb∂ab. (7)
In harmonic superspace, the superfields which are independent of the θ−−a are called analytic
superfields. Thus, these analytic superfields satisfy
D++a ΦA = 0 ⇒ ΦA = ΦA(ζA), (8)
where the coordinates parameterising the analytic subspace are given by
ζA = (xab, θ++
a , θ0a , u±)
. (9)
A i
M. Faizal / Nuclear Physics B 869 [PM] (2013) 598–607 601
Here xabA is given by
xabA = (γm)abxm
A = xab + i(θ++aθ−−b + θ++bθ−−a
). (10)
It is convenient to define
d9z = − 1
16d3x
(D++)2(
D−−)2(D0)2
,
dζ (−4) = 1
4d3xA du
(D−−)2(
D0)2. (11)
Furthermore, a conjugation in this superspace is defined by(u±
i
) = u±i ,(xmA
) = xmA ,(
θ±±a
) = θ±±a ,
(θ0a
) = θ0a . (12)
The analytic superspace measure is real ˜dζ (−4) = dζ (−4) and the full superspace measure isimaginary d9z = −d9z because this conjugation is squared to −1 on the harmonics and to 1on xm
A and Grassmann coordinates.
3. Deformation of harmonic superspace
In this section we will analyse non-anticommutative deformation of the harmonic super-space. Such deformation occurs due to a RR background in the string theory [32,33]. Unlikethe noncommutative deformation caused by a NS background or gravitino background, the non-anticommutative deformation a certain amount of the breaks the supersymmetry of the theory.In four dimensions this deformation can be used to break half the supersymmetry of a N = 1supersymmetry theory to obtain a theory with N = 1/2 supersymmetry. This is because in fourdimensions the supersymmetric generator QA can be split into Qa and Qa . So, it is possible tobreak the supersymmetry corresponding to one of these generators without breaking the otherone. This is thus done by imposing the following anticommutator
{θa, θb} = Cab. (13)
This will break the supersymmetry corresponding to Qa without breaking the supersymmetrycorresponding to Qa . We could also break the supersymmetry corresponding to Qa withoutbreaking the supersymmetry corresponding to Qa . Now if we project the undeformed N = 1supersymmetric in four dimensions to three dimensions, then it will have N = 2 supersymmetry.This is because in three dimensions both Qa and Qa act as separate supercharges. Now theN = 1/2 supersymmetry in four dimensions corresponded to N = (1,0) or N = (0,1) in threedimensions. It is not possible to obtain a N = 1/2 supersymmetry in three dimensions from thisas there are enough degrees of freedom to do so.
The N = 2 supersymmetric theory in four dimensions is generated by four supercharges. It ispossible to break the supersymmetry with respect to any number of these supercharges. Thus,if the supercharges are denoted by Q±
a and Qa . Then it is possible to obtain a N = 1/2 theoryby breaking the supersymmetry with respect to three of these supercharges by imposing thefollowing anti-commutators{
θ+a , θ+
b
} = C+aa ,
{θ−a , θ+
a
} = C−aa ,{
θ+, θ+} = C+ . (14)
a b ab
602 M. Faizal / Nuclear Physics B 869 [PM] (2013) 598–607
From a three-dimensional perspective this corresponded to breaking a N = 4 supersymmetrictheory to N = ((1,0), (0,0)). We could obtain similar deformations, N = ((0,1), (0,0)), N =((0,0), (1,0)) and N = ((0,0), (0,1)), depending upon which supercharges are left undeformed.It is also be possible to deform the four-dimensional theory with N = 2 supersymmetry as{
θ+a , θ+
b
} = C+aa ,
{θ+a , θ+
b
} = C+ab. (15)
This will corresponding to N = ((1,0), (1,0)) in three dimensions. Now we can use generateother similar deformations like N = ((0,1), (1,0)), N = ((0,1), (0,1)), N = ((1,0), (0,1)).Finally, we can break only one of the supercharges in the four-dimensional theory and obtaina N = 2/3 supersymmetric theory. Thus, if we impose{
θ+a , θ+
b
} = C+aa , (16)
then we obtain a N = 2/3 in four dimensions. This corresponded to N = ((1,0), (1,1)) inthree dimensions. Similarly, we can obtain N = ((0,1), (1,1)), N = ((1,1), (0,1)) and N =((1,1), (1,0)), in three dimensions.
We could also start from the harmonic superspace in three dimensions and impose the follow-ing deformation{
θ++a , θ++
a
} = C++ab . (17)
This will break the supersymmetry corresponding to Q++a without breaking the supersymmetry
corresponding to Q−−a and Q0
a . Thus, we will obtain N = 2 supersymmetry in three dimen-sions. This we could have similarly broken the supersymmetry with respect to Q−−
a or Q0a and
left the remaining two intact to obtain N = 2 supersymmetry in three dimensions. We can alsobreak the supersymmetry with respect to any two supercharges say Q++
a and Q−−a and leave the
supersymmetry with respect to Q0a intact by imposing the following deformations{
θ++a , θ++
a
} = C++ab ,
{θ−−a , θ−−
a
} = C−−ab . (18)
This way we will obtain a theory with N = 1 supersymmetric. We could have similarly haveleft either Q++
a or Q−−a intact to obtain a theory with N = 1 supersymmetric. It may be noted
that N = 3 supersymmetry in three dimensions corresponded to N = 3 supersymmetry in twodimensions. This is because each of the supercharges splits into two independent superchargesQ++± , Q−−± or Q0±. Thus, N = 2 supersymmetry in three dimensions will correspond to one ofthe following, ((1,1), (1,1), (0,0)), ((0,0), (1,1), (1,1)) and ((1,1), (0,0), (1,1)), in two di-mensions. Similarly, N = 1 supersymmetry in three dimensions will correspond to one of thefollowing, ((1,1), (0,0), (0,0)), ((0,0), (1,1), (0,0)) and ((0,0), (0,0), (1,1)), in two dimen-sions.
4. Deformed ABJ theory
In this section we will analyse non-anticommutative deformation of the ABJ theory in theharmonic superspace with one of generators of the supersymmetry broken due to the deforma-tion. Other deformations can be analysed in a similar way. So, to start with defining a vectorfield V ++ in the harmonic superspace. We now deform the harmonic superspace by breaking thesupersymmetry generated by Q++
a by imposing the following relations,{θ++a , θ++
a
} = C++. (19)
ab
M. Faizal / Nuclear Physics B 869 [PM] (2013) 598–607 603
We now use Weyl ordering and express the Fourier transformation of a superfield on this de-formed superspace as
V ++(z) =∫
dp V ++(p) exp(ipz), (20)
where
exp(ipz) = exp(−ikx − π++aθ++
a − π−−aθ−−a − π0aθ0
a
),
dp = d3k d2π++ d2π−− d2π0,
V ++(p) = V ++(k,π++,π−−,π0, u±)
. (21)
Thus, we obtain a one to one map between a function of z to a function of ordinary superspacecoordinates z via
V ++(z) =∫
dp V ++(p) exp(ipz). (22)
We can express the product of two fields V ++(z)V ++(z) on this deformed superspace as
ˆV ++(z)V ++(z) =∫
dp1 dp2 exp[i((p1 + p2)z
)]exp(i�)V ++(p1)V
++(p2), (23)
where
exp(i�) = exp
[−1
2
(C++abθ++2
a θ++1b
)]. (24)
This motivates the definition of the star product between ordinary vector fields, which is nowdefined as
V ++(z) � V ++(z) = exp
[−1
2
(C++ab∂++2
a ∂++1b
)]V ++(z1)V
++(z2)
∣∣∣∣z1=z2=z
. (25)
If we impose the following deformation{θ++a , θ++
a
} = C++ab ,
{θ−−a , θ−−
a
} = C−−ab , (26)
and proceed in a similar way, we obtain the following definition of the star product betweenordinary vector fields
V ++(z) � V ++(z) = exp
[−1
2
(C++ab∂++2
a ∂++1b + C−−ab∂−−2
a ∂−−1b
)]× V ++(z1)V
++(z2)∣∣z1=z2=z
. (27)
However, we will only analyse the deformation corresponding to Eq. (19) in this paper. So, in thispaper the deformed ABJ model will have manifest N = 2 supersymmetry. It is now possible towrite V −− in-terms of V ++ as
V −−(z, u) =∞∑
n=1
(−1)n∫
du1 · · ·dun E++, (28)
where
E++ = V ++(z, u1) � V ++(z, u2) � · · · � V ++(z, un)
(u+u+)(u+u+) · · · (u+u+). (29)
1 1 2 n
604 M. Faizal / Nuclear Physics B 869 [PM] (2013) 598–607
Now we can write the action for the deformed ABJ theory. This theory is invariant under thegauge group U(N) × U(M). In this theory the matter fields are denoted by (q+)
B
A and (q+)AB
and the gauge fields corresponding to U(M) and U(N) are denoted by (V ++L )AB and (V ++
R )A
B ,respectively. Here the underlined indices refer to the right U(M) gauge group. The covariantderivatives for the matter fields in the deformed ABJ theory can be written as
∇++q+ = D++q+ + V ++L � q+ − q+ � V ++
R ,
∇++q+ = D++q+ − q+ � V ++L + V ++
R � q+. (30)
We can write the action for the ABJ theory in the deformed harmonic superspace as
S = SCS,k
[V ++
L
]�+ SCS,−k
[V ++
R
]�+ SM
[q+, q+]
�, (31)
where
SCS,k
[V ++
L
]�= ik
4πtr
∞∑n=2
(−1)n
n
∫d3x d6θ du1 · · ·dun H++
L ,
SCS,−k
[V ++
R
]�= − ik
4πtr
∞∑n=2
(−1)n
n
∫d3x d6θ du1 · · ·dun H++
R ,
SM
[q+, q+]
�= tr
∫d3x dζ (−4) q+ � ∇++ � q+, (32)
where
H++L = V ++(z, u1)L � V ++(z, u2)L � · · · � V ++(z, un)L
(u+1 u+
2 ) · · · (u+n u+
1 ),
H++R = V ++(z, u1)R � V ++(z, u2)R � · · · � V ++(z, un)R
(u+1 u+
2 ) · · · (u+n u+
1 ). (33)
This theory is invariant under the following infinitesimal gauge transformations
δq+ = ΛL � q+ − q+ � ΛR,
δq+ = ΛR � q+ − q+ � ΛL,
δV ++L = −D++ΛL − [
V ++L ,ΛL
]�,
δV ++R = −D++ΛR − [
ΛR,V ++R
]�. (34)
The deformation of the ABJ theory breaks the supersymmetry from N = 3 supersymmetry toN = 2 supersymmetry. However, the original ABJ theory had N = 6 supersymmetry. We havebroken manifest the supersymmetry corresponding to Q++
a , so we should still be left with N = 5supersymmetry. Now as we have manifest N = 2 supersymmetry generated by to the super-charges Q−−
a and Q0a , we should have an additional N = 3 supersymmetry to generate N = 5
supersymmetry. This is achieved by the following supersymmetric transformations,
δεq+ = iεa∇0
a � q+,
δεq+ = iεa∇0
a � q+,
δεV++L = 8π
kεaθ0
a � q+ � q+,
δεV++R = 8π
εaθ0a � q+ � q+, (35)
k
M. Faizal / Nuclear Physics B 869 [PM] (2013) 598–607 605
where
∇0a � q+ = ∇0
a � q+ + θ−−a
(W++
L � q+ − q+ � W++R
),
∇0a � q+ = D0
aq+ + V 0
La � q+ − q+ � V 0Ra,
V 0La = −1
2D++
a V −−L ,
V 0Ra = −1
2D++
a V −−R . (36)
Here ∇0a q+ and ∇0
a q+ are obtained via conjugation and the field strengths W++R and W++
L aredefined by
W++L = −1
4D++aD++
a V −−L ,
W++R = −1
4D++aD++
a V −−R . (37)
These field strengths satisfies
D++W++L + [
V ++L ,W++
L
]�= 0,
D++W++R + [
V ++R ,W++
R
]�= 0. (38)
Now using Fierz rearrangement, we get −δεSM [q+, q+]� = δεSCS,k[V ++L ]� + δεSCS,−k[V ++
R ]�,and so we have
δεS = 0, (39)
and thus the action is invariant under N = 5 supersymmetry. Thus, unlike the undeformed ABJtheory which is invariant under N = 5 supersymmetry, the deformed ABJ theory is only invariantunder N = 5 supersymmetry.
5. Conclusion
In this paper we analysed the non-anticommutative deformation of the ABJ theory in har-monic superspace. This theory is dual to multiple D2-brane in RR background. The full multipleD2-brane action includes couplings to the background fields of type II string theory. For a singlebrane this would be the pull back of Cμντ to the world volume but for the non-Abelian mul-tiple D2-brane action it must include further dielectric couplings to all of the RR form fields.M-theory contains a background three form field and its dual field. These should reduce to theRR to fields of string theory. The D-branes have been studied in various backgrounds. So, it willbe interesting to analyse the coupling of the three form field explicitly to the multiple M2-braneaction in harmonic superspace.
As the RR background partially break the supersymmetry in type II string theory, the dualdeformations of it will also partially break the supersymmetry on the M-theory side. We analysethe M2-branes in harmonic superspace. This harmonic superspace initially had manifest N = 3supersymmetry. However, after the non-anticommutative deformation, it only had N = 2 su-persymmetry. This was because the supersymmetry corresponding to Q++
a was broken by theimposition of the non-anticommutative deformation of the superspace. Thus, the total supersym-metry of the ABJ theory was reduced from N = 6 to N = 5 supersymmetry. We also discussed
606 M. Faizal / Nuclear Physics B 869 [PM] (2013) 598–607
the deformations of the harmonic superspace that break the supersymmetry corresponding totwo of the supercharges, namely Q++
a and Q−−a , respectively. This deformation only leaves
manifest N = 1 supersymmetry unbroken. Thus a similar analysis of the ABJ theory in thissuperspace will only have N = 4 supersymmetry. There are other type of deformations that oc-cur in superspace. These occur due to non-vanishing values of commutators between spacetimeand Grassmann coordinates and physically correspond to a deformation generated by a gravitinobackground. It will be interesting to study the ABJ theory with this kind of deformations in har-monic superspace. The interesting thing about these deformations is that they do not break anyamount of supersymmetry. Thus, the ABJ theory with deformed superspace, where the deforma-tions are caused by a gravitino background will preserve all of the N = 6 supersymmetry.
Chern–Simons-matter theories also have important applications in condensed matter physics.This is because of their relevance to the fractional quantum Hall effect, which is based on theconcept of statistical transmutation [51–54]. Recently, supersymmetric generalisation of the frac-tional quantum Hall effect has also been investigated [55–58]. In particular, physical propertiesof the topological excitations in the supersymmetric quantum Hall liquid were discussed in adual supersymmetric Chern–Simons theory [59]. Furthermore, a close connection between thefractional quantum Hall noncommutativity of the spacetime has been discovered [60–63]. Thus,the results of this paper can have interesting condensed matter applications. This is becausewe can analyse the non-anticommutative deformation of the supersymmetric fractional quantumHall effect. This can change the behaviour of fractional condensates and thus have importantconsequences for the transport properties in supersymmetric quantum hall systems.
It may be noted that the BRST symmetry of the ABJM theory has been analysed in deformedN = 1 superspace [10,11,48]. So, it will be interesting to analyse the BRST symmetry of ABJtheory in deformed harmonic superspace. We can also use the BRST symmetry of this theory toshow the unitarity of the S-matrix. It is possible to reduce the ABJM action to a N = 8, super-Yang–Mills theory describing N D2-branes by using the novel higgsing mechanism [44–47].In this higgsing mechanism the gauge group of the ABJM theory is spontaneously broken downto its diagonal subgroup. This analysis has also been performed in N = 1 superspace [11,49],and it will interesting to repeat this analysis in harmonic superspace. By doing that we will beable to analyse this higgsing mechanics for the ABJM theory with non-anticommutative defor-mations. This will give us a better understanding of the existence of these non-anticommutativedeformation in the M-theory, as we will be able to relate it to the familiar objects in the stringtheory.
References
[1] A.S. Galperin, E.A. Ivanov, V.I. Ogievetsky, E.S. Sokatchev, Harmonic Superspace, Cambridge University Press,2001.
[2] A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky, E. Sokatchev, Class. Qant. Grav. 1 (1984) 469.[3] A. Galperin, E. Ivanov, V. Ogievetsky, E. Sokatchev, JETP 40 (1984) 912.[4] B.M. Zupnik, Supersymmetries and quantum symmetries, in: J. Wess, E. Ivanov (Eds.), Lecture Notes in Physics,
vol. 524, Springer, 1998, p. 116.[5] B.M. Zupnik, D.V. Khetselius, Sov. J. Nucl. Phys. 47 (1988) 730.[6] I.L. Buchbinder, E.A. Ivanov, O. Lechtenfeld, N.G. Pletnev, I.B. Samsonov, B.M. Zupnik, JHEP 0903 (2009) 096.[7] O. Aharony, O. Bergman, D.L. Jafferis, J. Maldacena, JHEP 0810 (2008) 091.[8] M. Naghdi, Int. J. Mod. Phys. A 26 (2011) 3259.[9] A. Gustavsson, JHEP 1101 (2011) 037.
[10] M. Faizal, Comm. Theor. Phys. 57 (2012) 637.[11] M. Faizal, Phys. Rev. D 84 (2011) 106011.
M. Faizal / Nuclear Physics B 869 [PM] (2013) 598–607 607
[12] J. Bagger, N. Lambert, Phys. Rev. D 75 (2007) 045020.[13] J. Bagger, N. Lambert, Phys. Rev. D 77 (2008) 065008.[14] J. Bagger, N. Lambert, JHEP 0802 (2008) 105.[15] A. Gustavsson, Nucl. Phys. B 811 (2009) 66.[16] O-Kab Kwon, P. Oh, J. Sohn, JHEP 0908 (2009) 093.[17] H. Samtleben, R. Wimmer, JHEP 1010 (2010) 080.[18] O. Aharony, O. Bergman, D.L. Jafferis, JHEP 0811 (2008) 043.[19] S. Cremonesi, JHEP 1101 (2011) 076.[20] J. Evslin, S. Kuperstein, JHEP 0912 (2009) 016.[21] J.A. Minahan, O. Ohlsson Sax, C. Sieg, J. Phys. A 43 (2010) 275402.[22] P. Caputa, C. Kristjansen, K. Zoubos, Phys. Lett. B 677 (2009) 197.[23] N. Seiberg, E. Witten, JHEP 9909 (1999) 032.[24] M.R. Douglas, N.A. Nekrasov, Rev. Mod. Phys. 73 (2001) 977.[25] S. Doplicher, K. Fredenhagen, J.E. Roberts, Commun. Math. Phys. 172 (1995) 187.[26] A. Connes, Noncommutative Geometry, Academic Press, London, 1990.[27] J. de Boer, P.A. Grassi, P. van Nieuwenhuizen, Phys. Lett. B 574 (2003) 98.[28] K. Ito, S. Sasaki, JHEP 0611 (2006) 004.[29] K. Ito, Y. Kobayashi, S. Sasaki, JHEP 0704 (2007) 011.[30] P. Meessen, T. Ortin, Nucl. Phys. B 684 (2004) 235.[31] N. Berkovits, N. Seiberg, JHEP 0307 (2003) 010.[32] H. Ooguri, C. Vafa, Adv. Theor. Math. Phys. 7 (2003) 53.[33] N. Seiberg, JHEP 0306 (2003) 010.[34] E. Chang-Young, H. Kim, H. Nakajima, JHEP 0804 (2008) 004.[35] K. Araki, T. Inami, H. Nakajima, Y. Saito, JHEP 0601 (2006) 109.[36] J.S. Cook, J. Math. Phys. 47 (2006) 012304.[37] Y. Kobayashi, S. Sasaki, Phys. Rev. D 72 (2005) 065015.[38] N. Berkovits, N. Seiberg, JHEP 0307 (2003) 010.[39] B. Safarzadeh, Phys. Lett. B 601 (2004) 81.[40] A.F. Ferrari, M. Gomes, J.R. Nascimento, A.Yu. Petrov, A.J. da Silva, Phys. Rev. D 74 (2006) 125016.[41] I.L. Buchbinder, E.A. Ivanov, O. Lechtenfeld, I.B. Samsonov, B.M. Zupnik, Phys. Part. Nucl. 39 (2008) 759.[42] E. Ivanov, O. Lechtenfeld, B. Zupnik, Nucl. Phys. B 707 (2005) 69.[43] A. De Castro, E. Ivanov, O. Lechtenfeld, L. Quevedo, Nucl. Phys. B 747 (2006) 1.[44] S. Mukhi, C. Papageorgakis, JHEP 0805 (2008) 085.[45] Y. Pang, T. Wang, Phys. Rev. D 78 (2008) 125007.[46] T. Li, Y. Liu, D. Xie, Int. J. Mod. Phys. A 24 (2009) 3039.[47] J.P. Allen, D.J. Smith, JHEP 1108 (2011) 078.[48] M. Faizal, Europhys. Lett. 98 (2012) 31003.[49] S.V. Ketov, S. Kobayashi, Phys. Rev. D 83 (2011) 045003.[50] M. Faizal, Mod. Phys. Lett. A 27 (2012) 1250147.[51] D. Orgad, S. Levit, Phys. Rev. B 53 (1996) 7964.[52] A. Lopez, E. Fradkin, Phys. Rev. B 69 (2004) 155322.[53] B. Skoric, A.M.M. Pruisken, Nucl. Phys. B 559 (1999) 637.[54] I. Ichinose, A. Sekiguchi, Nucl. Phys. B 493 (1997) 683.[55] K. Hasebe, Phys. Rev. D 72 (2005) 105017.[56] K. Hasebe, Phys. Rev. Lett. 94 (2005) 206802.[57] K. Hasebe, Phys. Lett. A 372 (2008) 1516.[58] E. Ivanov, L. Mezincescu, P.K. Townsend, JHEP 0601 (2006) 143.[59] K. Hasebe, Phys. Rev. D 74 (2006) 045026.[60] A. Jellal, Mod. Phys. Lett. A 18 (2003) 1473.[61] A. Pinzul, A. Stern, Mod. Phys. Lett. A 18 (2003) 1215.[62] O.F. Dayi, A. Jellal, J. Math. Phys. 43 (2002) 4592.[63] M. Daoud, A. Hamama, Int. J. Mod. Phys. A 23 (2008) 2591.