ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

On Multiple M2-brane Model(s) and Its N =8 Superspace
Formulation(s)

I. A. Bandos

Abstract

Wegive abrief review ofBagger-Lambert-Gustavsson (BLG)model, with emphasis on its version invariant under thevolume
preserving diffeomorphisms (SDiff3) symmetry. We describe the on-shell superfield formulation of this SDiff3 BLG model
in standard N = 8, d = 3 superspace, as well as its superfield action in the pure spinor N = 8 superspace. We also
briefly address the Aharony-Bergman-Jafferis-Maldacena (ABJM/ABJ) model invariant under SU(M)k × SU(N)−k gauge
symmetry, and discuss the possible form of their N = 6 and, for the case of Chern-Simons level k = 1,2, N = 8 superfield
equations.

1 Introduction
In the fall of 2007, motivated by the search for a low-
energy description of the multiple M2-brane system,
Bagger, Lambert and Gustavsson [1, 2, 3] proposed a
N = 8 supersymmetric superconformal d = 3 model
based on Filippov three algebra [4] instead of Lie al-
gebra.

1.1 3-algebras

Lie algebras are defined with the use of antisym-
metric brackets [X, Y ] = −[Y, X] of two elements,
X =

∑
a

X aTa and Y =
∑

a

Y aTa, called Lie brack-

ets or commutator. The brackets of two Lie algebra
generators, [Ta, Tb] = fab

cTc, are characterized by an-
tisymmetric structure constants fab

c = −fabc = f[ab]c
which obey the Jacobi identity f[ab

dfc]d
e = 0 ⇔

[Ta, [Tb, Tc]]+ [Tc, [Ta, Tb]]+ [Tb, [Tc, Ta]] = 0.
In contrast, the general Filippov 3-algebra is de-

fined by 3-brackets

{Ta, Tb, Tc} = fabcd Td , fabcd = f[abc]d (1)

which are antisymmetric and obey the so-called ‘fun-
damental identity’

{Ta, Tb, {Tc1, Tc2, Tc3}} =
3{{Ta, Tb, T[c1}, Tc2, Tc3]}} .

(2)

Towrite an action for some 3-algebra valued field the-
ory, one needs as well to introduce an invariant inner
product or metric

hab =< Ta, Tb > . (3)

Then for the metric 3-algebra the structure constants
obey fabcd := fabc

ehed = f[abcd].

An example of infinite dimensional 3-algebra is de-
fined by the Nambu brackets (NB) [5] of functions on
a 3-dimensional manifold M3

{Φ,Ξ,Ω} = �ijk ∂iΦ ∂jΞ ∂kΩ ,
∂i := ∂/∂y

i , i =1,2,3 .
(4)

Here yi = (y1, y2, y3) are local coordinates on M3,
Φ = Φ(y), Ξ = Ξ(y) and Ω = Ω(y) are functions
on M3, and �ijk is the Levi-Cevita symbol (it is con-
venient to define NB using a constant scalar density e
[6], but this is not important for our present discussion
here and we simplify the notation by setting e = 1).
These brackets are invariant with respect to the vol-
ume preserving diffeomorphisms of M3, which we call
SDiff3 transformations. In practical applications one
needs to assume compactness of M3. For our discus-
sion here it is sufficient to assume that M3 has the
topology of sphere S3.
Another example of 3-algebra, which was present

already in the first paper by Bagger and Lambert [1]
is A4 realized by generators Ta, a =1,2,3,4 obeying

{Ta, Tb, Tc} = �abcd Td , a, b, c, d =1,2,3,4 . (5)

These are related to the 6 generators Mab of SO(4)
as Euclidean d = 4 Dirac matrices are related to
the Spin(4) = SU(2)× SU(2) generators, Ta ↔ γa,
Mab ↔ 1/2γab := 1/4(γaγb − γbγa).
A more general type of 3-algebras with not com-

pletely antisymmetric structure constants were dis-
cussed e.g. in [7], [8] and [9]. In particular, as
it was shown in [8], the Aharony-Bergman-Jafferis-
Maldacena (ABJM) model [10] is based on a partic-
ular ’hermitian 3-algebra’ the 3-brackets of which can
be defined on two M × N (complex) matrices Zi, Zj
and an N × M (complex) matrix Z†k by [8]

[Zi, Zj;Z†k]
M×N

= ZiZ†kZ
j − Zj Z†kZ

i . (6)

Contribution to the Selected Topics in Mathematical and Particle Physics Prague 2009. Procs. of the Conference on the occasion
of Prof. Jiri Niederle’s 70th birthday.

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Acta Polytechnica Vol. 50 No. 3/2010

1.2 BLG action

The BLG model on general 3-algebra is described in
terms of an octet of 3-algebra valued scalar fields in
vector representation of SO(8), φI(x) = φIa(x)Ta, an
octet of 3-algebravalued spinor fields in spinor (say, s-
spinor) representationofSO(8), ψαA(x)= ψαA

a(x)Ta,
and the vector gauge field Aabμ in the bi-fundamental
representation of the 3-algebra. The BLG Lagrangian
reads

LBLG = T r
[
−
1
2
|Dφ|2 −

g2

12

{
φI , φJ , φK

}2 − (7)
i

2
ψ̄γμDμψ +

ig

4

{
φI , φJ , ψ̄

}
ρIJ ψ

]
+
1
2g
LCS ,

I =1, . . . ,8 .

where g is a real dimensionless parameter, LCS is
the Chern-Simons (CS term) for the gauge potential
Aμb

a = Acdμ fdcb
a which is also used to define the co-

variant derivatives of the scalar and spinor fields. The
Spin(8) indices are suppressed in (7); ρI := ρI

AḂ
are

the 8 × 8 Spin(8) ‘sigma’ matrices (Klebsh-Gordan
coefficients relating the vector 8v and two spinor,
8s and 8c, representations of SO(8)). These obey
ρI ρ̃J + ρI ρ̃J =2δIJ I with their transpose ρ̃I := ρ̃I

ȦB
;

notice that ρIJ := (ρ[I ρ̃J])AB and ρ̃
IJ := (ρ̃[I ρJ])ȦḂ

are antisymmetric in their spinor indices.
This model possesses N = 8 supersymmetry and

superconformal symmetries the set of which includes
8 special conformal supersymmetries. Hence the total
number of supersymmetryparameters is 2×8+2×8=
32. This coincides with the number of supersymme-
tries possessed by M2-brane [11] and the conformal
symmetry was expected for infrared fixed point (low
energy approximation) of the multiple M2-brane sys-
tem [12]. Thus, action (7) was expected to play for
the multiple M2-brane system the same rôle as it is
played by the U(N) SYM action for the multiple Dp-
brane system [13] (with N Dp-branes).
However, if thiswere the case, thenumber of gener-

ators of the Filippov 3-algebrawould be related some-
how to the number of M2-branes composing the sys-
tem the low energy limit of which is described by the
action (7). This expectation enters in conflict with
the relatively poor structure of the set of finite dimen-
sional Filippov 3-algebraswith positively definitemet-
ric (3): this set was proved to contain the direct sums
of A4 and trivial one-dimensional 3-algebras only (see
[14, 15] as well as [16] and refs therein).
A veryuseful rôle in searching for resolution of this

paradoxwasplayedbytheanalysisbyRaamsdock [17],
who reformulated the A4 BLG model in matrix nota-
tion. ThiswasusedbyAharony,Bergman, Jafferis and
Maldacena [10] to formulate an SU(N)k × SU(N)−k
and then [26] SU(M)k × SU(N)−k gauge invariant
CS plusmattermodels, which are believed to describe
the low energymultipleM2-branedynamics. The sub-

script k denotes the so-calledCS level, this is to saythe
integer coefficient in front of theCS term in the action
of the CS plus matter models. In the dual description
of theABJMmodel byM-theory on the AdS4×S7/Zk
[10] the same integer k characterizes thequotientof the
7-sphere.
TheABJM/ABJmodel possessesonlyN =6man-

ifest supersymmetries, which is natural for k > 2,
as the AdS4 × S7/Zk backgrounds with k > 2 pre-
serve only 24 of 32M-theory supersymmetries in these
cases. The nonperturbative restoration of N = 8 su-
persymmetry for k = 1,2 cases was conjectured al-
ready in [10]. Recently this enhancement of super-
symmetry was studied in [9], where its relation with
some special ‘identities’ (whichwepropose to callGR-
identities orGustavsson-Rey identities) conjectured to
be true due to the properties of monopole operators
specific for k = 1,2 is proposed. We shortly discuss
the ABJM/ABJ model in the concluding part of this
paper.

1.3 NB BLG action

Coming back to the 3-algebraBLG models, we notice
that inside their set there are clear candidates for the
N → ∞ limit of the multiple M2-brane system, which
one can view as describing possible ‘condensates’ of
coincident planarM2-branes. These are the BLG the-
ories in which the Filippov 3-algebra is realized by
the Nambu-bracket (4) of functions defined on some
3-manifold M3. This model was conjectured [18, 19]
to be related with the M5-brane [20, 21, 22] wrapped
over M3 (see [6] and recent [23] for further studyof this
proposal) and was put in a general context of SDiff3
gauge theories in [24].
It is described in terms of Spin(8) 8v-plet of real

scalar fields φI (I = 1, . . .8), and a Spin(8) 8s-
plet of Majorana anticommuting Sl(2;R) spinor fields
ψA (A = 1, . . . ,8), both on the Cartesian product
of 3-dimensional Minkowski spacetime with some 3-
dimensional closed manifold without boundary, M3.
These fields transforms as scalars with respect to
SDiff3: δξφ = −ξi∂iφ , δξψ = −ξi∂iψ, where ξi = ξi(y)
is a divergenceless SDiff3 parameter.
The action of thisNambubracket realizationof the

Bagger-Lambert-Gustavssonmodel (NB BLG model)
is

LN B BLG =
∮
d3y

[
−
1
2
e |Dφ|2 −

i

2
e ψ̄γμDμψ +

ig

4
εijk∂iφ

I ∂j φ
J
(
∂kψ̄ρ

IJ ψ
)
− (8)

g2

12
e
{
φI , φJ , φK

}2]
+
1
2g
LCS

In (8) the trace T r of (7) is replaced by integral
∮

d3y

over M3 and LCS is the CS-like term involving the
SDiff3 gauge potential si and gauge pre-potential Ai

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Acta Polytechnica Vol. 50 No. 3/2010

[24]. The gauge potential si = dxμsiμ transforms
under the local SDiff3 with ξ

i = ξi(x, y) as δξs
i =

dξi − ξj ∂j si + sj ∂j ξi and is used to construct SDiff3
covariant derivatives of scalar and spinor fields

Dφ = dφ + si∂iφ , Dψ = dψ + si∂iψ . (9)

As the gauge field takes values in the Lie algebra of
the Lie group of gauge symmetries, and this is as-
sociated with volume preserving diffeomorphisms the
infinitesimal parameter of which is a divergenceless
three-vector ξi(x, y), ∂iξ

i = 0, the SDiff3 gauge field
si =dxμsiμ(x, y) obeys

∂is
i ≡ 0 ⇔ ∂isiμ ≡ 0 (10)

which implies the possibility to express it, at least lo-
cally, in terms of gauge pre-potential one-form Ai =
dxμAμi(x),

si = �ijk∂j Ak ⇔ siμ = �
ijk∂j Aμk . (11)

Also the covariant field strength

F i =dsi + sj∂j s
i =
1
2
dxμ ∧dxν F iνμ . (12)

satisfies the additional identity

∂iF
i ≡ 0 ⇔ ∂iF iμν ≡ 0 (13)

and can be expressed (locally) in terms of pre-field
strength,

F i = εijk∂j Gk ⇔ F iμν = ε
ijk∂j Gμν k , (14)

Gi =dAi + s
j ∂[j Ai] =

1
2
dxμ ∧dxν Gνμi . (15)

TheCS-like term in (8) is expressed through the gauge
potential and pre-potential by

LCS =
∮
d3y �μνρ

[(
∂μs

i
ν

)
Aρ i −

1
3
�ijks

i
μs

j
ν s

k
ρ

]
, (16)

or, in terms of differential forms, by LCS =∮
d3y

[
dsi ∧ Ai −

1
3
�ijks

i ∧ sj ∧ sk
]
. The formal ex-

terior derivative of LCS can be expressed through the
field strength and pre-field strength by

dLCS =
∮
d3y F i ∧ Gi . (17)

The Lagrangian density (8) varies into a total
spacetime derivative under the following infinitesimal
supersymmetry transformations with 8c-plet constant
anticommuting spinor parameter �α

Ȧ
(Ȧ =1, . . . ,8):

δφI = i�ρ̃I ψ , δAμi = −ig
(
�γμρ̃

I ψ
)
∂iφ

I , (18)

δψ =
[
γμρIDμφI −

g

6

{
φI , φJ , φK

}
ρIJK

]
� .

The BLG equations of motion are

DμDμφI =
ig

2
εijk∂iφ

J ∂j ψ̄ρ
IJ ∂kψ −

g2

2

{{
φI , φJ , φK

}
, φJ , φK

}
,

γμDμψ = −
g

2
ρIJ

{
φI , φJ , ψ

}
, (19)

F iμν = −g εμνρε
ijk

[
∂j φ

IDρ∂kφI −
i

2
∂j ψγ

ρ∂kψ

]
.

2 NB BLG in N =8
superfields

The NB BLG equations of motion can be obtained
from the set of superfield equations in N = 8 super-
space [30]. Wewill reviewthis approach in this section.
Let us introduce 8v-plet of scalar, and SDiff3-

scalar, superfields φI, the lowest component of which
(also denoted by φI) may be identified with the
BLG scalar fields, and impose on it the following
superembedding-like equation [30]1

DαȦφ
I = iρ̃I

ȦB
ψαB. (20)

The SDiff3-covariant spinorial derivatives on N =8
superspace, entering (20),

DαȦ = DαȦ + ςαȦ
i∂i , (21)

are constrained to obey the following algebra [30]

[DαȦ, DβḂ]+ = 2iδȦḂ(Cγ
μ)αβDμ + (22)

2i�αβWȦḂ
i ∂i ,

whereDμ = ∂μ+isiμ∂i is the 3-vector covariantderiva-
tive which obeys

[
DαȦ, Dμ

]
= FαȦ μ

i∂i , [Dμ, Dν] = Fμν i ∂i . (23)

Eqs. (22), (23) are equivalent to the Ricci iden-
tity DD = F i∂i for the covariant exterior derivative
D := d+si∂i = EαȦDαȦ +E

μDμ , plus the constraint
F i

αȦ βḂ
=2iCαβ WȦḂ

i.

The basic SDiff3 gauge superfield strength WȦḂ
i is

antisymmetric on c-spinor indices (this is to say WȦḂ
i

is in the 28 of SO(8)); it is also divergence-free, so

WȦḂ
i = −WḂȦ

i , ∂iWȦḂ
i =0 . (24)

Using the Bianchi identity DF i =0, one finds that

1The name comes from the observation that (20) can be obtained from the superembedding equation for a single M2-brane [25] by
first linearizing with respect to the dynamical fields in the static gauge, and then covariantizing the result with respect to SDiff3.

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Acta Polytechnica Vol. 50 No. 3/2010

FαȦ μ
i = i

(
γμWȦ

i
)
α

, WαḂ
i :=

i

7
DαȦWȦḂ

i , (25)

Fμν
i =

1
16

�μνρDȦγ
ρWȦ

i ,

and that

Dα(ȦWḂ)Ċ
i = iWαḊ

i
(
δḊ(ȦδḂ)Ċ − δḊĊ δȦḂ

)
, (26)

DȦαWβḂ
i = (Cγμ)αβ

(
DμWȦḂ

i −4δȦḂWμ
i
)
. (27)

We see that the SDiff field strength supermultiplet in-
cludes a scalar 28 (WȦḂ

i), a spinor 8c (WαȦ
i) and

a singlet divergence-free vector (W μi = DȦγ
ρWȦ

i).
There are many other independent components, but
these become dependent on-shell as far as we are
searching for a description of Chern-Simons (CS)
rather than the Yang-Mills one. The relevant super-
Chern-Simons (super-CS) system superfield equation
in the absence of ‘matter’ supermutiplets is obviously
WȦḂ

i = 0, since this sets to zero all SDiff3 field
strengths; in particular it implies F iμν = 0. In the
presence of matter, the super-CS equation may get a
nonvanishing right hand side.
Indeed, acting on the superembedding-like equa-

tion (20) with an SDiff3-covariant spinor derivative,
and making use of the anticommutation relation (22),
one finds that Dα[Ȧρ̃

I
Ḃ]C ψ

α
C = 2WȦḂ

i∂iφ
I which is

solved by the ‘super-CS’ equation [30]

WȦḂ
i =2gεijk∂iφ

I ∂j φ
J ρ̃IJ

ȦḂ
. (28)

It was shown in [30] that the two N = 8 super-
field equations (20) and (28) imply theNambu-bracket
BLG equations (19).

3 NB BLG in pure-spinor
superspace

An N = 8 superfield action for the abstract BLG
model, i.e. for theBLGmodel basedonafinitedimen-
sional 3-algebra, which in practical terms implies A4
or the direct sum of several A4 and trivial 3-algebras,
was proposed by Cederwall [28]. Its generalization for
the case of NB BLGmodel invariant under infinite di-
mensional SDiff3 gauge symmetry, constructed in [24],
will be reviewed in this section.
The pure-spinor superspace of [28] is parametrized

by the standard N =8 D =3 superspace coordinates
(xμ, θα

Ȧ
) together with additional pure spinor coordi-

nates λα
Ȧ
. These are described by the 8c-plet of com-

plex commuting D = 3 spinors satisfying the ‘pure
spinor’ constraint

λγμλ := λα
Ȧ
γ

μ
αβ λ

β

Ȧ
=0 . (29)

This is a variant of the D =10 pure-spinor super-
space first proposed by Howe [31] (see [32] for earlier
attempt to use pure spinors in the SYM and super-
gravity context). From a more general perspective,

the approach of [28] can be considered as a realiza-
tion of the harmonic superspace programme of [33]
(although one cannot state that the algebra of all the
symmetries of the superfield action of [28] are closed
off shell, i.e. without the use of equations of mo-
tion). The D = 10 pure spinors are also the cen-
tral element of the Berkovits approach to covariant
description of quantum superstring [34]. In this ap-
proach thepure spinors are considered tobe theghosts
of a local fermionic gauge symmetry related to the κ-
symmetry of the standardGreen-Schwarz formulation.
This ‘ghost nature’ may be considered as a justifica-
tion for that the pure-spinor superfields are assumed
(in [28, 24] andhere) to be analytic functions of λ that
can be expanded as a Taylor series in powers of λ. To
discuss the BLG model, we allow all the pure spinor
superfields to depend also on the local coordinates yi

of the auxiliary compact 3-dimensional manifold M3.
Following [28], we define the BRST-type operator

(cf. [34])

Q := λα
Ȧ
DαȦ , (30)

which satisfies Q2 ≡ 0 as a consequence of the pure
spinor constraint (29). We now introduce the 8v-plet
of complex scalar N =8 ‘matter’ superfields ΦI, with
SDiff3 transformation

δΦI =Ξi∂iΦ
I (31)

characterized by the commuting M3-vector parameter
Ξi =Ξi(y).
We allow these superfields to be complex because

they may depend on the complex pure-spinor λ but,
to make contact with the spacetime BLG model, we
assume that the leading term in its decomposition in
power series on complex λ

ΦI = φI +O (λ) , (32)

is given by a real 8v-plet of ‘standard’ N = 8 scalar
superfields, like the basic objects in Sec. 2.
Let us consider (complex and anticommuting) La-

grangian density

L
0
mat =

1
2
MIJ

∮
d3y eΦI QΦJ , (33)

where MIJ = λ
α
Ȧ

ρ̃IJ
ȦḂ

λαḂ is one of the two nonvanish-
ing analytic pure spinor bilinears

MIJ := λ
α ρ̃IJ λα , N

μ
IJKL := λ γ

μρ̃IJKLλ . (34)

It is important that, due to (29), these obey the iden-
tities (see [24] for a detailed proof)

MIJ ρ̃
J λ ≡ 0 , M[IJ MKL] =0 , (35)
NP Q[IJ · NKL]P Q ≡ 0 .

To construct the N = 8 supersymmetric action
with the use of the Lagrangian (33) one needs to spec-
ify an adequate superspace integration measure. We

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Acta Polytechnica Vol. 50 No. 3/2010

refer to [29] for details on such a measure, which has
the crucial property of allowing us to discard aBRST-
exact terms when varying with respect ΦI. Then, as
a consequence of this and also of the identities (35),
the action is invariant under the gauge symmetries
δΦI = λα

Ȧ
ρ̃I

ȦB
ζαB + QK

I for arbitrary pure-spinor-

superfield parameters ζα and K
I .

The variation with respect to ΦI yields the super-
field equation

MIJ QΦ
J =0 , (36)

which implies, as a consequence of the pure-spinor
identities, that

QΦI = λρ̃IΘ (37)

for some 8s-plet of complex spinor superfields ΘαȦ.
The first nontrivial (∼ λ) term in the λ-expansion of
this equation is precisely the free field limit of the on-
shell superspace constraint (20), DαȦφ

I = iρ̃I ȦBψαB,
with ψ =Θ|λ=0. 2 In the light of the results of Sec. 2,
this implies that the free field (g �→ 0) limit of the NB
BLG field equations (19) can be obtained from the
pure spinor superspace action (33).
Now, as the free field limit is reproduced, to con-

struct the pure spinor superspace description of the
NB BLG system we need to describe its gauge field
(Chern-Simons) sector and to use it to gauge the
SDiff3 invariance. To this end, we introduce an M3-
vector-valued complex anticommuting scalar Ψi with
the SDiff3 gauge transformations

δΨi = QΞi +Ψj∂j Ξ
i −Ξj ∂jΨi , ∂iΞi =0 (38)

involving the commuting M3-vector parameter Ξ
i =

Ξi(x, θ, λ;yj) and its derivatives. In the present con-
text,Ψi will play the role of the SDiff3 gaugepotential.
We require that ∂iΨ

i =0 so that, locally on M3,

Ψi = εijk∂j Πk , (39)

where Πi is the complex anticommuting, and space-
time scalar, pre-gauge potential of this formalism.
Using Ψi we can define an SDiff3-covariant exten-

sion of QΦI by

QΦI := QΦI +Ψi∂iΦ
I (40)

and construct the generalization of (33) invariant un-
der local SDiff3 symmetry (31), (38):

Lmat =
1
2
MIJ

∮
d3y eΦI QΦJ , (41)

MIJ = λ ρ̃
IJ �λ .

Next we have to construct the (complex and
fermionic) Lagrangian density LCS describing the

(Chern-Simons) dynamics of the gauge potential Ψi.
To this end we introduce the field-strength superfield

Fi := QΨi +Ψj∂jΨi = εijk∂jGk , (42)

where the last equality is valid locally on M3 and

Gi := QΠi +Ψj∂jΨi (43)

is the pre-field-strength superfield of this formalism.
Both Fi and Gi are SDiff3 covariant, so FiGi is an
SDiff3 scalar. Furthermore, the integral of this den-
sity over M3 is Q-exact, in the sense that∫

d3y e FiGi = Q LCS , (44)

where

LCS =
∫
d3σ e

(
Πi QΨ

i −
1
3
�ijkΨ

iΨjΨk
)
(45)

is the complex and anti-commuting CS-type La-
grangiandensity [24] which canbe used, togetherwith
Lmat of (41), to construct the candidate Lagrangian
density of the NB BLG model,

L = Lmat −
1
g

LCS . (46)

The Πi equation of motion of this combined La-
grangian is

Fi =
g

2e
MIJ �

ijk∂jΦ
I ∂kΦ

J . (47)

At this stage it is important to assume that Ψi has
‘ghostnumberone’ [28],whichmeans that it is apower
series in λ with vanishing zeroth order term (and sim-
ilarly for its pre-potential Πi). In other words

Ψi = λα
Ȧ
ςi
αȦ

, (48)

where ςi is an M3-vector-valued 8c-plet of arbitrary
anticommuting spinors. Its zerothcomponent in the λ-
expansion is the fermionic SDiff3 potential introduced,
with the same symbol, in (21). With this ‘ghost num-
ber’ assumption, (47) produces at lowest nontrivial
order (∼ λ2) the superspace constraints (22) for the
‘ghost number zero’ contribution ςi|λ=0 to the pure
spinor superfield ςi in (48), accompanied by the su-
per CS equation (28) for the field strength WȦḂ con-
structed from this potential.
Anheuristic justificationof the assumption (48), so

crucial to obtain the correct super-CS equations, can
be found in thatwith this formofΨi the covariantized
BRSToperator in (40) doesnot containa contribution

2Notice that the above mentioned gauge symmetry δΦI = λα
Ȧ

ρ̃I
ȦB

ζαB of the action (33) contributes to δ(QΦ
I) the terms of at least

the second order in λ. Then the induced transformation of the pure spinor superfield Θ
αȦ
in (37) is of the first order in λ so that

ψ
αȦ
= Θ

αȦ
|λ=0, entering the superembedding-like equation (20), is inert under those transformations.

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Acta Polytechnica Vol. 50 No. 3/2010

of ghost number zero, i.e. it has the form of (30),
Q = λȦ

α
DαȦ, but with the SDiff3 covariant Grass-

mann derivative DαȦ = DαȦ + ξ
i
αȦ

∂i.

Varying the interacting action with respect to ΦI

results in SDiff3 gauge invariant generalization of Eqs.
(36),

MIJ QΦ
J =0 , (49)

which contains, as the first nontrivial (∼ (λ)3) term
in the λ-expansion, precisely the superembedding-like
equation (20) with ψ =Θ|λ=0.
We have now shown, following [24], how the on-

shell N = 8 superfield formulation of Sec. 2, and
hence all BLG field equations (19), may be extracted
from the equations of motion derived from the pure
spinor superspace action (46). Of course, the field
content and equations of motion should be analyzed
at all higher-orders in the λ-expansion. To this end,
one must take into account the existence of additional
gauge invariance [28, 29]

δΦI = λ̄ρ̃I ζα +(Q +Ψ
j ∂j)K

I , (50)

δΠi = K
I MIJ ∂iΦ

J ,

for arbitrarypure-spinor-superfieldparameters ζα and
KI .
What one can certainly state, even without a de-

tailed analysis of these symmetries, is that, if addi-
tional fields are present inside the pure spinor super-
fields of the model (46), they are decoupled from the
BLGfields in the sense that theydonotenter the equa-
tions of motion of the BLG fields which are obtained
from the pure spinor superspace equations. This al-
lowed us [24], following the terminology of [28], to call
(46) the N = 8 superfield action for the NB BLG
model.

4 Remarks on ABJM/ABJ
model

The N = 6 pure spinor superspace action for the
ABJMmodel [10] invariantunder SU(N)k×SU(N)−k
gauge symmetry, was proposed in [29]3. One can ex-
tract the standard (not pure spinor)N =6superspace
equation by varying the action of [29] and fixing its
gauge symmetries. It is also instructive (and probably
simpler) to develop independently the on-shell N =6
superspace formalism for the ABJM as well as for the
ABJ [26] model invariant under SU(M)k × SU(N)−k
symmetry [37].
For anyvalue of theCS-level k the startingpoint of

the on-shell N = 6 superfield formalism could be the

following (superembedding-like) superspace equation
for complex M × N matrix superfield Zi [37]4

D
I
αZ

i = γ̃Iij ψαj , (51)

I =1,2, . . . ,6, i, j =1,2,3,4 .

Here γ̃Iij =
1
2
�ijklγIkl = −(γ

I
ij)

∗ and γIij = −γ
I
ji are

SO(6) Klebsh-Gordan coefficients (generalized Pauli
matrices), which obey γI γ̃J + γJ γ̃I = δIJ. The ma-
trix superfield Zi carries (M,N̄) representation of the
SU(M)×SU(N)gaugegroup. Itshermitianconjugate
Z
†
i is N × M matrix carrying (M̄,N) representation
and obeying DIαZ

†
i = γ

I
ij ψ

†j
α . Note that, although in

the original ABJM model [10] M = N, the N × N
matrix superfields Zi and Z†i carry different represen-
tation of SU(N) × SU(N): (N,N̄) and (N̄,N), re-
spectively. Here we speak in terms of the case with
M �= N, which is terminologically simpler, but all our
arguments clearly also apply for M = N.
The Grassmann spinorial covariant derivatives DIα

in (51) includes the gaugegroup SU(M)×SU(N) con-
nection and obey the algebra

{DIα, D
J
β} = iγ

a
αβ δIJDa + i�αβW IJ . (52)

This algebra involves the 15-plet of the basic field
strength superfields W IJ = −W JI which can be ex-
pressedthroughthematter superfieldsbythe following
N =6 super-CS equation [37]

W IJSU(M) = iZ
i
Z
†
j γ

IJ
i
j , W IJSU(N) = iZ

†
jZ

iγIJ i
j . (53)

Here W IJSU(M) and W
IJ
SU(N) are the basic field strength

corresponding to SU(M) and SU(N) subgroups of the
gauge group SU(M)k×SU(N)−k. One can check that
the consistency conditions for Eqs. (51) and (53) are
satisfied if the matter superfield obeys the superfield
equation of motion

γJij D
β(I D

J)
β Z

j +4iγJij[Z
j , Zk;Z†k]+ (54)

4iγJjk[Z
j, Zk;Z†i] ,

where [Zj , Zk;Z†k] are hermitian 3-brackets (6).
This superfield equation implies, in particular, the
fermionic equations of motion [37]

γaαβ Daψ
β
i = −

2
3
[ψαj , Z

j;Z†i]+
1
6
[ψαi, Z

j;Z†j]+(55)

1
2
�ijkl[Z

j , Zk;ψ†lα ] .

Werefer to [37] for further details on theN =6 super-
space formalism of the ABJM/ABJ model, including

3Note the existence of the off-shell N = 3 superfield formalism for the ABJM model [35] which was used to develop the quantum
calculation technique in [36]
4Here and below we use the Latin symbols from the middle of the alphabet, i, j, . . ., to denote the four-valued SU(4) index,

i, j, . . . = 1,2,3,4; we hope that this will not produce confusion with real 3-valued vector indices of M3, see secs. 1.3, 2 and 3, as far
as we do not use these in the present discussion.

19



Acta Polytechnica Vol. 50 No. 3/2010

for the explicit form of the bosonic equations of mo-
tion.
Searching for an N = 8 superfield formulation for

the ABJM/ABJ models with CS levels k = 1,2 it is
natural to assume that the universal N = 6 sector
is present as a part of N = 8 superspace formalism
and, to describe two additional fermionic directions of
N = 8 superspace, introduce, in addition to six DIα,
one complex spinor Grassmann derivative Dα, and its
conjugate (Dα)

† = −D̄α obeying

{Dα, D̄β} = iγaαβDa + i�αβ W , (56)
{Dα, Dβ} =0 , {D̄α, D̄β} =0 ,

{Dα, DJβ} = i�αβ W
J , (57)

{D̄α, DJβ} = i�αβW̄
J .

The structure of additional N = 2 supersymmetries
proposed in [9] suggests to impose on the basic N =8
superfields the chirality condition in the new fermionic
directions [37],

D̄αZ
i =0 , DαZ

†
i =0 . (58)

While the natural candidate for the super-CS equation
for the SO(6) scalar superfield strength W is

W = ZiZ†i , (59)

to write a possibly consistent super-CS equation for
6 complex field strength W J, which has to be chiral,
DαW

J = 0 = D̄αW̄
J, to provide the consistency of

the constraints (56), (57) and (52),

W̄ JSU(M) =∝ Z
iγJij Z̃

j , W JSU(M) =∝ Z̃
†
i γ̃

J ij
Z
†
j , (60)

one needs to involve “non-ABJM superfields”, the
leading components of which are the “non-ABJM

fields” of [9]. These are N × M matrix Z̃
i
and M × N

matrix Z̃†i which obey

D̄αZ̃
i
=0 , DαZ̃

†
i =0 (61)

and must be related with ABJM superfields Zi, Z†i
by using the suitable monopole operators (converting
(M̄,N) representation into (M,N̄))whichexist for the
case ofCS levels k =1,2only [9]. According to [9], the
existence of these monopole operators is reflected by
the ‘identities’ between hermitain three brackets (6)
of the ABJM and non-ABJM (super)fields. The set of
these ‘GR-identities’ includes

[(. . .), Z̃†i ; Z̃
i] = −[(. . .), Zi ;Z†i] . (62)

The consistency of the system of N =8 superfield
equations (51)–(60) and the set of GR-identities nec-
essary for that are presently under investigation [37].

Acknowledgement

The author thanks José de Azcárraga, Warren Siegel,
Dmitri Sorokin, Paul Townsend and Linus Wulff for

useful discussions. This work was partially supported
by the Spanish MICINN under the project FIS2008-
1980 and by the Ukrainian National Academy of Sci-
ences and Russian RFFI grant 38/50–2008.

Notice added: After this manuscript has been
finished, a paper [38] devoted to N = 8 superspace
formulations of d = 3 gauge theories appeared on the
net. It contains a detailed description of the on-shell
N = 8 superspace formulation of the BLG model for
finite dimensional three algebras, similar to the for-
mulation of the SDiff3 invariant Nambu bracket BLG
model in [30], and of its derivation starting from the
gauge theory constraints and Bianchi identities. Also
the component field content of the SYMmodel defined
by the constraints (22) and its finite-3-algebra coun-
terpart is discussed there.

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Igor A. Bandos
E-mail: igor bandos@ehu.es
Ikerbasque, the Basque Foundation for Science and
Department of Theoretical Physics
University of the Basque Country
P.O. Box 644, 48080 Bilbao, Spain

22