NB BLG model in N=8 superfields

We develop the N=8 superfield description of the Bagger-Lambert-Gustavsson (BLG) model in its Nambu bracket (NB) realization. The basic ingredient is the octet of scalar d=3 N=8 superfields depending also on the coordinates of a compact three dimensional space M. It is restricted by the 'superembedding-like' equation, which can be treated as covariantization of the linearized superembedding equation for supermembrane (M2-brane) with respect volume preserving diffeomorphisms SDiff3 of M. The curvatures of SDiff3 connection are expressed through the scalar superfields by the N=8 superfield generalization of the Chern-Simons equation (super-CS equation). We show how the dynamical BLG equations appear when studying consistency of these basic equations.


Introduction
Recently, motivated by the search for the Lagrangian of multiple M2-brane system, Bagger, Lambert [1] and Gustavsson [2] proposed the d = 3 N = 8 supersymmetric action based on Filippov 3-algebra instead of Lie algebra. A particular infinite dimensional 3-algebra related with three dimensional volume preserving diffeomorphism group SDiff 3 is given in terms of Nambu brackets (Nambu-Poisson brackets) [3] on a three dimensional compact manifold M 3 . For three functions on M 3 , Φ(y i ), Ξ(y i ), Ω(y i ) (i = 1, 2, 3), the Nambu brackets are defined by (see [4,5] and refs. therein; here, following [6,7], we have introduced a fixed M 3 densityē =ē(y) in the definition of Nambu brackets).
As it was stressed in [4], Bagger-Lambert-Gustavsson model with Nambu bracket realization of the 3-algebra (NB BLG model) can be treated as 6-dimensional field theory. The BLG gauge fields become the gauge fields for the 3-volume preserving diffeomorphisms (see [8] as well as [7] and refs. therein). The SDiff 3 gauge potential is given by the 1-form (see [7] for more details). Furthermore, the authors of [4] proposed the identification of the NB BLG model with M-theory 5-brane (M5-brane, see [9] for equations of motion and [10] for the covariant action) with the worldvolume chosen to be R 1+2 ⊗M 3 . However, an attempt to obtain the NB BLG model from light-cone M5-brane [6] has resulted only in reproducing the Carrollian limit of the NB BLG model. This suggests to study the NB realization of BLG model separately, and this was the subject of [5,6,7,11] and also of the original papers [4].
In this letter we present N=8 superfield description of the NB BLG model. It is the on-shell superfield description which does not allow for constructing the action, but reproduces equations of motion as the selfconsistency conditions of the basic equations.
2 Basic superfield equations 2.1 Superembedding-like equation for octet of d=3 N=8 scalar superfields The complete on-shell N=8 superfield description of the Nambu bracket realization of the Bagger-Lambert-Gustavsson model (NB BLG model) is provided by the octet of scalar d=3, N=8 superfields, φ I = φ I (x µ , θα ; y i ), depending on additional coordinates y i (i = 1, 2, 3) of a compact space M 3 , which obeys the following basic equation Here and below α, β, γ = 1, 2 are spinorial and a, b, c = 0, 1, 2 are vector indices of SO(1, 2),γ IȦ B := γ I BȦ are the SO(8) Klebsh-Gordan coefficients relating 8 v , 8 s and 8 c representation, which obey γ (IγJ) = δ IJ I s ,γ (I γ J) = δ IJ I c 1 , and ψ αB is a fermionic superfield which is expressed through φ I by the γ I -trace part of Eq. (2.1). Finally D αȦ is the covariant Grassmann derivative. It is covariant with respect to d = 3, N = 8 supersymmetry and under the volume preserving diffeomorphisms of M 3 (SDiff 3 group). Hence it involves a fermionic SDiff 3 connection ς i αȦ and, when act on SDiff 3 scalars (like φ I and ψ αB ), reads where D µ is the vector covariant derivative, which reads as D µ = ∂ µ + is i µ ∂ i when acting on SDiff 3 scalars. It involves the vector SDiff 3 gauge potential s i µ defined by The matrices γ µ αβ in (2.2) are real and symmetric; they obey Finally, WȦḂ i is the basic superfield strength of the SDiff 3 gauge supermultiplet. This carries the indices of 28 representation of SO(8), i.e. WȦḂ i = −WḂȦ i , and is a vector field with respect to SDiff 3 gauge group.

N=8 superfield generalization of the Chern-Simons gauge field equation
We impose on WḂȦ i the superfield generalization of the Chern-Simons field equation. This reads 3 Notice that WȦḂ i in (2.3) automatically satisfies the condition ∂ i (ēWȦḂ i ) = 0 necessary for any SDiff 3 field strength [7] (see (1.2)). 1 The SO (8)   As far asγ IJ AḂ form the complete basis in the space of antisymmetric 8×8 matrices, an equivalent form of the super Chern-Simons equation (2.3) is given by and, secondly, that tensorial gauge field strength ( In (3.6) one recognizes the Chern-Simons type gauge field equations which can be obtained from the BLG Lagrangian of [1]. This expresses the tensorial gauge field strength through the matter (super)fields. The dynamical bosonic and fermionic equations of motion of the NB BLG model follow from the superembedding-like equation (2.1) and the super-CS equation (2.3). Indeed, with the use of (2.2), one finds that the selfconsistency condition for Eq. (2.1) gives the expression for Grassmann covariant derivative of the fermionic superfield ψ βB in (2.1), Next stage is to study the selfconsistency conditions for Eq.  5 These equations can be also obtained from consistency of (2.1) with the use of (2.3) (see below).
Chern-Simons equation (3.6)) 6 . Taking this into account in the SO(1,2) vector -SO(8) scalar (∝ δȦḂγ µ αβ ) irreducible part we obtain the BLG Dirac equation which can be equivalently written in the following compact form As usually, the bosonic equations of motion can be obtained by taking the covariant spinorial derivative of the fermionic ones. Acting by the covariant (SDiff 3 and SUSY covariant) spinor derivatives on (3.8), and extracting the ∝ ǫ αβ γ I CȦ irreducible part one finds The ∝ γ µ αβ γ I CȦ irreducible part of the same relation can be used to obtain the bosonic Chern-Simons equation (3.6), while the ∝ γ IJK CȦ irreducible parts vanish identically 7 . To conclude, the superembedding-like equation (2.1), supplemented by the covariant derivative algebra (2.2) with the composite scalar field strength (2.3), restricts field content of the basic octet of d=3, N=8 scalar superfields φ I , depending in addition on three coordinates of a compact space M 3 , to the NB BLG supermultiplet, and, furthermore, accumulates all the equations of motion of the NB BLG model.

Conclusions
In this letter we presented the N=8 superfield description of the Nambu bracket (NB) realization of the Bagger-Lambert-Gustavsson (BLG) model. It is given by an octet of scalar N=8, d=3 superfields φ I which, in addition, depend on the three coordinates y i of compact space M 3 . This octet of superfields is restricted by Eq. (2.1), which, as we have shown, contains all the equations of motion of the NB BLG model when supplemented (at least, when supplemented) by super-Chern-Simons equation (2.3) (or, equivalently, (2.4)).
We call the basic Eq. (2.1) superembedding-like equation because of its relation with the superembedding equation describing one M2-brane in the d=3 N=8 worldvolume superspace which is as follows. To obtain (2.1), one has first to linearize the 6 On the way of such a derivation of (3.5) one should use the requirement of that the dependence of M 3 coordinates should not be restricted, i.e. no additional conditions on ∂ i φ I may occur. Then, coming to the equation (W i A − . . .)∂ i φ I = κ B γ I BȦ , one concludes that κ B = 0 and that W i A = . . . where multidots denote the r.h.s of Eq. (3.5). 7 To prove this, one has to use the consequences {φ L , φ [I , {φ J , φ K] , φ L }} = 0 and ǫ IJKLMN P Q {φ L , φ M , {φ N , φ P , φ Q }} = 0 of the so-called fundamental identity {φ L , φ M ,ē{φ N , φ P , φ Q }} = 3{ē{φ L , φ M φ [N }, φ P , φ Q] }} (the presence of the densityē(y) in the fundamental identity and its absence in its consequences above is not occasional). supermembrane superembedding equation [12], see [13], and to fix the so-called static gauge on the worldvolume superspace, arriving at the equation D αȦ X I = 2iγ IȦ B Ψ αB . Then one replaces the octet of d=3, N=8 superfields X I (x, θ) by the octet of superfields depending also on coordinates of M 3 , X I (x, θ) → φ I (x, θ, y) (this automatically produces Ψ αB (x, θ) → ψ αB (x, θ, y)) and covariantize the result with respect to the volume preserving diffeomorphisms of M 3 (D αȦ → D αȦ = D αȦ + ς i αȦ ∂ i ). We hope that our superfield description will be useful in studying the properties of the NB BLG model and in understanding its physical meaning.
Actually, such a way of passing from the complete nonlinear description of one M2-brane in the frame of superembedding approach [12] to the NB BLG modelnamely first linearization, and than obtaining nonlinearities by a separate covariantization with respect to SDiff 3 ,-suggests that NB BLG model may be not a description of multiple M2-brane, but rather an independent-and without any doubt very interesting-d = 3, N = 8 supersymmetric dynamical system.
Actually, a search for alternative candidates on the rôle of multiple M2-brane action can be witnessed. A very incomplete list includes the N = 6 supersymmetric model of [14], d=3, N=2 supersymmetric models of [15], as well as very recent construction of a candidate multiple M2-brane bosonic action, similar to the Myers action for the coincident bosonic D-branes, in [16].
The generalization of our on-shell N=8 superstring description for the case of arbitrary 3-algebra seems to be possible 8 and, in many respects, looks interesting to develop in details.