On multiple M2-brane model(s) and its N=8 superspace formulation(s)

We give a brief review of Bagger-Lambert-Gustavsson (BLG) model, with emphasis on its version invariant under the volume 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 $SU(M)_{k}\times SU(N')_{-k}$ 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.


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 algebra. In contrast, the general Filippov 3-algebra is defined by 3-brackets which are antisymmetric and obey the so-called 'fundamental identity' To write an action for some 3-algebra valued field theory, one needs as well to introduce an invariant inner product or metric Then for the metric 3-algebra the structure constants obey f abcd := f abc e h ed = f [abcd] . An example of infinite dimensional 3-algebra is defined by the Nambu brackets (NB) [5] of functions on a 3-dimensional manifold M 3 {Φ, Ξ, Ω} = ǫ ijk ∂ i Φ ∂ j Ξ ∂ k Ω , ∂ i := ∂/∂y i , i = 1, 2, 3 .
Here y i = (y 1 , y 2 , y 3 ) are local coordinates on M 3 , Φ = Φ(y), Ξ = Ξ(y) and Ω = Ω(y) are functions on M 3 , and ǫ ijk is the Levi-Cevita symbol (it is convenient 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 volume preserving diffeomorphisms of M 3 , which we call SDiff 3 transformations. In practical applications one needs to assume compactness of M 3 . For our discussion here it is sufficient to assume that M 3 has the topology of sphere S 3 . Another example of 3-algebra, which was present already in the first paper by Bagger and Lambert [1] is A 4 realized by generators T a , a = 1, 2, 3, 4 obeying These are related to the 6 generators M ab of SO(4) as Euclidean d = 4 Dirac matrices are related to the A more general type of 3-algebras with not completely antisymmetric structure constants were discussed 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 particular 'hermitian 3-algebra' the 3-brackets of which can be defined on two M × N (complex) matrices Z i , Z j and an N × M (complex) matrix Z † k by [8] [ 1.2. BLG action. The BLG model on general 3-algebra is described in terms of an octet of 3algebra valued scalar fields in vector representation of SO(8), φ I (x) = φ Ia (x)T a , an octet of 3-algebra valued spinor fields in spinor (say, s-spinor) representation of SO(8), ψ αA (x) = ψ αA a (x)T a , and the vector gauge field A ab µ in the bi-fundamental representation of the 3-algebra. The BLG Lagrangian reads where g is a real dimensionless parameter, L CS is the Chern-Simons (CS term) for the gauge potential A µb a = A cd µ f dcb a which is also used to define the covariant derivatives of the scalar and spinor fields. The Spin (8) indices are suppressed in (7); ρ I := ρ I AḂ are the 8 × 8 Spin(8) 'sigma' matrices (Klebsh-Gordan coefficients relating the vector 8 v and two spinor, 8 s and 8 c , representations of SO (8)). These obey ρ IρJ +ρ IρJ = 2δ IJ I with their transposeρ I :=ρ IȦ B ; notice that ρ IJ := (ρ [IρJ] ) AB andρ IJ := (ρ [I ρ J] )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 supersymmetry parameters is 2×8+2×8=32. This coincides with the number of supersymmetries possessed by M2-brane [11] and the conformal symmetry was expected for infrared fixed point (low energy approximation) of the multiple M2-brane system [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 this were the case, the number of generators of the Filippov 3-algebra would be related somehow to the number of M2-branes composing the system 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 dimensional Filippov 3-algebras with positively definite metric (3): this set was proved to contain the direct sums of A 4 and trivial one-dimensional 3-algebras only (see [14,15] as well as [16] and refs therein).
A very useful rôle in searching for resolution of this paradox was played by the analysis by Raamsdock [17], who reformulated the A 4 BLG model in matrix notation. This was used by Aharony, 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 plus matter models, which are believed to describe the low energy multiple M2-brane dynamics. The subscript k denotes the so-called CS level, this is to say the integer coefficient in front of the CS term in the action of the CS plus matter models. In the dual description of the ABJM model by M-theory on the AdS 4 × S 7 /Z k [10] the same integer k characterizes the quotient of the 7-sphere.
The ABJM/ABJ model possesses only N = 6 manifest supersymmetries, which is natural for k > 2, as the AdS 4 × S 7 /Z k backgrounds with k > 2 preserve only 24 of 32 M-theory supersymmetries in these cases. The nonperturbative restoration of N = 8 supersymmetry for k = 1, 2 cases was conjectured already in [10]. Recently this enhancement of supersymmetry was studied in [9], where its relation with some special 'identities' (which we propose to call GR-identities or Gustavsson-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-algebra BLG 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 planar M2-branes. These are the BLG theories in which the Filippov 3-algebra is realized by the Nambu-bracket (4) of functions defined on some 3-manifold M 3 . This model was conjectured [18,19] to be related with the M5-brane [20,21,22] wrapped over M 3 (see [6] and recent [23] for further study of this proposal) and was put in a general context of SDiff 3 gauge theories in [24].
It is described in terms of Spin (8)  These fields transforms as scalars with respect to SDiff 3 : The action of this Nambu bracket realization of the Bagger-Lambert-Gustavsson model (NB BLG model) is In (8) the trace T r of (7) is replaced by integral d 3 y over M 3 and L CS is the CS-like term involving the SDiff 3 gauge potential s i and gauge pre-potential A i [24]. The gauge potential s i = dx µ s i µ transforms under the local SDiff 3 with ξ i = ξ i (x, y) as δ ξ s i = dξ i − ξ j ∂ j s i + s j ∂ j ξ i and is used to construct SDiff 3 covariant derivatives of scalar and spinor fields As the gauge field takes values in the Lie algebra of the Lie group of gauge symmetries, and this is associated with volume preserving diffeomorphisms the infinitesimal parameter of which is a divergenceless three-vector ξ i (x, y), ∂ i ξ i = 0, the SDiff 3 gauge field s i = dx µ s i µ (x, y) obeys which implies the possibility to express it, at least locally, in terms of gauge pre-potential one-form Also the covariant field strength satisfies the additional identity and can be expressed (locally) in terms of pre-field strength, The CS-like term in (8) is expressed through the gauge potential and pre-potential by or, in terms of differential forms, by The formal exterior derivative of L CS can be expressed through the field strength and pre-field strength by The Lagrangian density (8) varies into a total spacetime derivative under the following infinitesimal supersymmetry transformations with 8 c -plet constant anticommuting spinor parameter ǫ α A (Ȧ = 1, . . . , 8): The BLG equations of motion are The NB BLG equations of motion can be obtained from the set of superfield equations in N = 8 superspace [30]. We will review this approach in this section. Let us introduce 8 v -plet of scalar, and SDiff 3 -scalar, superfields φ I , the lowest component of which (also denoted by φ I ) may be identified with the BLG scalar fields, and impose on it the following superembedding-like equation [30] 3 The SDiff 3 -covariant spinorial derivatives on N = 8 superspace, entering (20), are constrained to obey the following algebra [30] [D αȦ , where Eqs. (22), (23) are equivalent to the Ricci identity DD = F i ∂ i for the covariant exterior derivative The basic SDiff 3 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 Using the Bianchi identity DF i = 0, one finds that and that We see that the SDiff field strength supermultiplet includes a scalar 28 (WȦḂ i ), a spinor 8 c (W αȦ i ) and a singlet divergence-free vector (W µi = DȦγ ρ WȦ 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 SDiff 3 field strengths; in particular it implies F i µν = 0. In the presence of matter, the super-CS equation may get a nonvanishing right hand side.

NB BLG in pure-spinor superspace
An N = 8 superfield action for the abstract BLG model, i.e. for the BLG model based on a finite dimensional 3-algebra, which in practical terms implies A 4 or the direct sum of several A 4 and trivial 3-algebras, was proposed by Cederwall [28]. Its generalization for the case of NB BLG model invariant under infinite dimensional SDiff 3 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 (x µ , θ α A ) together with additional pure spinor coordinates λ α A . These are described by the 8 c -plet of complex commuting D = 3 spinors satisfying the 'pure spinor' constraint This is a variant of the D = 10 pure-spinor superspace first proposed by Howe [31] (see [32] for earlier attempt to use pure spinors in the SYM and supergravity context). From a more general perspective, the approach of [28] can be considered as a realization 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 motion). The D = 10 pure spinors are also the central element of the Berkovits approach to covariant description of quantum superstring [34]. In this approach the pure spinors are considered to be the ghosts of a local fermionic gauge symmetry related to the κ-symmetry of the standard Green-Schwarz formulation. This 'ghost nature' may be considered as a justification for that the pure-spinor superfields are assumed (in [28,24] and here) 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 y i of the auxiliary compact 3-dimensional manifold M 3 .
Following [28], we define the BRST-type operator (cf. [34]) which satisfies Q 2 ≡ 0 as a consequence of the pure spinor constraint (29). We now introduce the 8 v -plet of complex scalar N = 8 'matter' superfields Φ I , with SDiff 3 transformation characterized by the commuting M 3 -vector parameter Ξ i = Ξ i (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 is given by a real 8 v -plet of 'standard' N = 8 scalar superfields, like the basic objects in Sec. 2.
Let us consider (complex and anticommuting) Lagrangian density where M IJ = λ α Aρ IJ AḂ λ αḂ is one of the two nonvanishing analytic pure spinor bilinears It is important that, due to (29), these obey the identities (see [24] for a detailed proof) To construct the N = 8 supersymmetric action with the use of the Lagrangian (33) one needs to specify an adequate superspace integration measure. We refer to [29] for details on such a measure, which has the crucial property of allowing us to discard a BRST-exact terms when varying with respect Φ I . Then, as a consequence of this and also of the identities (35), the action is invariant under the gauge symmetries δΦ I = λ α Aρ IȦ B ζ αB + QK I for arbitrary pure-spinor-superfield parameters ζ α and K I . The variation with respect to Φ I yields the superfield equation which implies, as a consequence of the pure-spinor identities, that for some 8 s -plet of complex spinor superfields Θ αȦ . 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), D αȦ φ I = iρ IȦ B ψ αB , with ψ = Θ| λ=0 . 4 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 construct 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 SDiff 3 invariance. To this end, we introduce an M 3 -vector-valued complex anticommuting scalar Ψ i with the SDiff 3 gauge transformations involving the commuting M 3 -vector parameter Ξ i = Ξ i (x, θ, λ; y j ) and its derivatives. In the present context, Ψ i will play the role of the SDiff 3 gauge potential. We require that ∂ i Ψ i = 0 so that, locally on M 3 , where Π i is the complex anticommuting, and spacetime scalar, pre-gauge potential of this formalism. Using Ψ i we can define an SDiff 3 -covariant extension of QΦ I by and construct the generalization of (33) invariant under local SDiff 3 symmetry (31), (38): Next we have to construct the (complex and fermionic) Lagrangian density L CS describing the (Chern-Simons) dynamics of the gauge potential Ψ i . To this end we introduce the field-strength superfield 4 Notice that the above mentioned gauge symmetry δΦ I = λ α Aρ IȦ B ζ αB 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.
where the last equality is valid locally on M 3 and is the pre-field-strength superfield of this formalism. Both F i and G i are SDiff 3 covariant, so F i G i is an SDiff 3 scalar. Furthermore, the integral of this density over M 3 is Q-exact, in the sense that where is the complex and anti-commuting CS-type Lagrangian density [24] which can be used, together with L mat of (41), to construct the candidate Lagrangian density of the NB BLG model, The Π i equation of motion of this combined Lagrangian is At this stage it is important to assume that Ψ i has 'ghost number one' [28], which means that it is a power series in λ with vanishing zeroth order term (and similarly for its pre-potential Π i ). In other words where ς i is an M 3 -vector-valued 8 c -plet of arbitrary anticommuting spinors. Its zeroth component in the λ-expansion is the fermionic SDiff 3 potential introduced, with the same symbol, in (21). With this 'ghost number' assumption, (47) produces at lowest nontrivial order (∼ λ 2 ) the superspace constraints (22) for the 'ghost number zero' contribution ς i | λ=0 to the pure spinor superfield ς i in (48), accompanied by the super CS equation (28) for the field strength WȦḂ constructed from this potential. An heuristic justification of the assumption (48), so crucial to obtain the correct super-CS equations, can be found in that with this form of Ψ i the covariantized BRST operator in (40) does not contain a contribution of ghost number zero, i.e. it has the form of (30), Q = λȦ α D αȦ , but with the SDiff 3 covariant Grassmann derivative D αȦ = D αȦ + ξ i αȦ ∂ i . Varying the interacting action with respect to Φ I results in SDiff 3 gauge invariant generalization of Eqs. (36), 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] δΦ I =λρ I ζ α + (Q + Ψ j ∂ j )K I , for arbitrary pure-spinor-superfield parameters ζ α and K I . What one can certainly state, even without a detailed analysis of these symmetries, is that, if additional fields are present inside the pure spinor superfields of the model (46), they are decoupled from the BLG fields in the sense that they do not enter the equations of motion of the BLG fields which are obtained from the pure spinor superspace equations. This allowed us [24], following the terminology of [28], to call (46) the N=8 superfield action for the NB BLG model.

Remarks on ABJM/ABJ model
The N = 6 pure spinor superspace action for the ABJM model [10] invariant under SU (N ) k × SU (N ) −k gauge symmetry, was proposed in [29] 5 . One can extract the standard (not pure spinor) N = 6 superspace 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].
where [Z j , Z k ; Z † k ] are hermitian 3-brackets (6). This superfield equation implies, in particular, the fermionic equations of motion [37] γ a αβ D a ψ β We refer to [37] for further details on the N = 6 superspace formalism of the ABJM/ABJ model, including for the explicit form of the bosonic equations of motion. 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 D I α , one complex spinor Grassmann derivative D α , and its conjugate (D α ) † = −D α obeying {D α ,D β } = iγ a αβ D a + iǫ αβ W , 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],