An N=1 Superfield Action for M2 branes

We present an octonionic N=1 superfield action that reproduces in components the action of Bagger and Lambert for M2 branes. By giving an expectation value to one of the scalars we obtain the maximally supersymmetric superfield action for D2 branes.

In this short note, we present an N = 1 superfield action whose component expansion gives the BL theory for a 3-algebra with totally antisymmetric structure constants f abcd .We use real three-dimensional superfields both for the matter as well as for the Chern-Simons part of the action.The crucial point is the use of the octonionic self-dual tensor in the construction of the real superpotential.In this way, the superpotential is only manifestly SO(7) invariant.However, for specially chosen couplings, the component action coincides with the BL action, and hence full SO(8) symmetry is restored.We believe that octonions will play a fundamental role in future studies of AdS 4 /CFT 3 .
Our motivation comes in part from corresponding studies in AdS 5 /CFT 4 where the N = 1 formulation of N = 4 SYM has been an extremely efficient tool for studies of anomalous dimensions, non-renormalization properties and integrability.We believe that our N = 1 action will be similarly useful in this case too.
As a simple test for our action we follow [7] and demonstrate that giving an expectation value in one of the scalar supefields, our action yields the maximally supersymmetric YM theory in 2+1 dimensions, as it should.

The N = 1 superfield action
We consider eight real N = 1 superfields 2 as where a denotes the index of the three-algebra algebra with structure constants f abcd .We use the SO(8) triality tensor Γ I A Ȧ, I, Ȧ, A = 1, 2, .., 8.In the representation where ΓI 8A = −δ I A (see appendix B), choosing the superspace coordinate to point in the 8 direction we have essentially made equivalent the vector and one of the two spinorial representations of SO (8).We use the notation of [46] summarized along with other useful relations and conventions in the Appendix A.
Our superfield action couples the matter superfields to a Chern-Simons gauge superfield Γ α ab in the Wess-Zumino gauge, that carries two gauge group indices a, b Apart from the overall normalization, the only adjustable parameter is the coupling constant of the real superpotential.Nevertheless, we keep all coefficients α, β, γ, k arbitrary having in mind possible generalizations of the action (2).The crucial point is the use of the self-dual 3 eight dimensional tensor C IJKL , I, J, K, L = 1, 2, ..8 that describes the embeddings of SO (7) into SO(8) [47,48,49,50].Its properties are briefly recalled in the Appendix B. Hence, the presence of the superpotential implies that (2) has only SO(7) manifest global symmetry.Our strategy is to fix the coefficients in (2) by comparing the resulting component action with the one of BL.In this way, the SO(7) symmetry is enhanced to SO(8) and hence we achieve maximal supersymmetry.
We detail next the various projections.A) Scalar kinetic terms Earlier works on the superfield formulation of Chern-Simons theories coupled to matter include [44,45].
3 Similar result can be obtained using the anti self-dual tensor. where B) Chern-Simons terms: The equations of motion for the auxiliary fields read Using ( 7) and ( 8), the terms contributing to the potential from A), B) and C) that involve the auxiliary fields give To arrive at the last line we have used the contraction of the self-dual form [47,48,50] It is also crucial that the third line in (9) vanishes.This can be shown for a general 3-algebra with totally antisymmetric structure constants f abcd .Indeed, using the fundamental identity in the relevant part of the third line of ( 9) we obtain after some relabeling where we have also used the antisymmetry of C KLOM .Hence this term vanishes.
Putting everything together we arrive at the component action Now, in our representation we have so that for the spinor kinetic term we have Then, choosing we obtain the action This coincides exactly with the Bagger and Lambert action given in [3].To see that, notice that in our notations we use the purely imaginary charge conjugation matrix C to raise and lower spinor indices.Therefore for a real Majorana spinor ψ we have the identifications ( ψ and also to match the fermion kinetic term that requires (note the position of the γ-matrix indices) Our final superspace action is then written as

M2 to D2
As a simple test for our action (21) we give an expectation value to one of the scalars.Following [7] we expect that the resulting action will be the maximally supersymmetric YM theory in 2+1 dimensions for the gauge group SU(2).We split the I = 1, .., 8 index of the scalar superfield as I → (i, 8) with i = 1, ..., 7 and the SO(4) index â = 1, ..., 4 as â → (a, x) with a = 1, ..., 3. Then we give an expectation value to the scalar superfield which we identify with the dimensionful coupling constant of the 2+1 SYM as For the spinor superfield we define We rewrite the superspace action in terms of the new fields and indices.
i) Kinetic terms: where we defined the gauge covariant derivative ii) Superpotential term: where the dots indicate subleading terms in the large g Y M limit which we have discarded.
iii) Chern-Simons terms: To arrive in the the last line we have used the fact that in the Wess-Zumino gauge D α Γ α is vanishing and D α Γ β can be symmetrized.Next, we derive the equations of motions for the B auxiliary superfield neglecting the terms cubic in B and also terms of the form B 2 Φ 2 , as in [7].We thus obtain where we defined Inserting B into the action and collecting all the terms we get where the dots stand for subleading contributions and we substituded the values in ( 16), so that now Rescaling the gauge superfield as in [7] A → 1/2A we see that W α d W d α gives the right SYM kinetic term in the Wess-Zumino gauge and the covariant derivative in (26) assumes its standard form.Finally, ignoring the contribution of Φ i x that has completely decoupled, we arrive at This is the superfield Langangian for maximally supersymmetric YM in 2+1 dimensions for SU (2).It is intriguing to notice that the Lagrangian ( 33) is remarkably similar to the octonionic N = (1, 1) sigma model (in two dimensions) of [51].

Conclusions
We have presented an N = 1 superfield action in three dimensions that in components gives the Bagger-Lambert action for a general 3-algebra with totally antisymmetric structure constants f abcd .Crucial in our construction were the self-dual octonionic tensors C IJKL .Although the tensors are SO(7) invariant, we have shown that a special choice of the parameters in the action enhances the global symmetry to SO (8).We have demonstrated that a superhiggs mechanism yields the maximally supersymemtric 2+1 YM theory on D2 branes, curiously in a formalism resembling a two-dimensional sigma model.We hope that our superfield action and its generalizations can be used in N = 1 superfield calculations that should shed more light into the AdS 4 /CFT 3 correspondence.

A Superspace Notations
In this appendix we collect the useful identities for our superfields and gamma matrices.We follow Superspace [46].The component field definitions are as In the analog of the Wess-Zumino procedure we can gauge away B and χ component fields.Our spacetime signature is (−, +, +).The purely immaginary totally antisymmetric symbol C αβ is used to raise and lower spinor indexes according to the ց convention We represent vectors in spinor notation as symmetric matrices: Then the following relations hold: We conclude with some useful relations for three dimensional D-algebra computations:

B Octonionic conventions
In the octonion algebra we can choose a basis of elements We can also introduce the seven dimensional dual of the structure constants: Combining these two objects one can construct an SO(7) invariant tensor C IJKL I,J,K,L=1,...,8 which is self dual in 8 dimensions, by taking: Octonionic structure constants can be used to construct SO(8) gamma matrices.For instance a suitable representation of the triality tensor that enters BL susy transformations is given by: A, Ȧ = 1, ..., 8 Defining ΓI ȦA = Γ T I ȦA it is easy to see that Γ I ΓJ + Γ J ΓI = 2δ IJ and therefore we can write down 16x16 matrices satisfying the clifford algebra With the above definitions it can be shown that so that the antisymmetrized product of Γ's of the scalar-fermion interaction can be written by means of the C IJKL tensor.