Triality and Bagger-Lambert Theory

We present two alternative field contents for Bagger-Lambert theory, based on the triality of SO(8). The first content is (\varphi_{A a}, \chi_{\dot A a} ; A_\m{}^{a b}), where the bosonic field \varphi is in the 8_S of SO(8) instead of the 8_V as in the original Bagger-Lambert formulation. The second field content is (\varphi_{\dot A a}, \chi^I{}_a ; A_\m{}^{a b}), where the bosonic field \varphi and the fermionic field \chi are respectively in the 8_C and 8_V of SO(8). In both of these field contents, the bosonic potentials are positive definite, as desired. Moreover, these bosonic potentials can be unified by the triality of SO(8). To this end, we see a special constant matrix as a product of two SO(8) generators playing an important role, relating the 8_V, 8_S and 8_C of SO(8) for the triality. As an important application, we give the supersymmetry transformation rule for N=6 superconformal Chern-Simons theory with the supersymmetry parameter in the 6 of SO(6), obtained by the truncation of our first field content.


Introduction
It has been recently pointed out by Bagger and Lambert (BL) [1] [2] that the totally antisymmetric triple brackets or 3-Lie algebras [3] [4] X I , X J , X K ≡ 1 3! X I , X J , X K ± (cyclic perms.) (1.1) for the element X I of non-associative algebra play a crucial role in the context of coincident M2-brane which in turn is one of the important aspects of M-theory [5] [6]. In [1] [2], an explicit lagrangian in three-dimensions (3D) with global N = 8 supersymmetry has been given with SO(4) local × SO(8) global symmetry and a Chern-Simons (CS) term.
There have been further generalizations to arbitrary non-compact Lie algebras [14] [32] whose ghost problem has been overcome by spontaneous conformal symmetry breaking [33].
However, the uniqueness of the gauge group SO(4) local has been confirmed in [34] at least for compact gauge groups. In any case, due to the tight N = 8 system [1][2] strictly constraining the field content, together with the uniqueness of SO(4) local [34], it seems extremely difficult to generalize or change the basic field content of the original BL theory [1] [2].
In this paper, we address the last question, i.e., whether the basic field content of BL theory [1][2] can be changed, or whether there is any alternative field content. Here by 'the field content of the original BL formulation', we mean the case when the SO(4) local gauge group is specified with the bosonic field X I a and its fermionic partner ψ Aa as in [2]. As explicit examples, we provide two alternative field contents to the original BL formulation [1]. Our first alternative field content is ϕ Aa , χ .

Aa
; A µ ab , where the boson ϕ Aa is in the , while the fermion χ is in the of the 8 S in the original BL formulation [2]. Our second field content is ϕ .
where the boson ϕ and fermion χ are respectively in the 8 C and 8 V of SO (8).
Correspondingly, the spinor charge Q αA is in the 8 S of SO (8). These replacements are possible thanks to the triality among 8 V , 8 S and 8 C of SO (8). We also show that our first field content with the supercharge in the 8 V of SO(8) has a direct link with N = 6 CS-matter theory [20][ 35], in which the supercharge is in the 6 of SO(6).

First Field Content
Our first field content is ϕ Aa , χ . Our total action I 1 ≡ d 3 x L 1 for the first field content has the lagrangian 3) Since the bosonic field ϕ is in the 8 S of SO(8), we use the expressions, such as the last line, e.g., The SO(4) -covariant derivative D µ acts on the ϕ's and χ's as In the last term in (2.1), the 'square' implies all the free indices a, B, C and D in one pair of the parentheses are contracted. This gives the manifestly positive-definite bosonic potential This potential has an alternative expression given in (2.14). Compared with [2], our CS term is exactly the same as that in [2], and so is the positive definiteness of the bosonic potential [2], while the χ 2 ϕ 2 term has the same magnitude as that in [2].
2, ···, 128) in the 128 (or 128) of SO(16). In our notation, we do not need the imaginary unit 'i' in front of the fermionic kinetic term, except that needed due to the signature (+, −, −) in [37]. Due to the Clifford algebra structures repeated at every eight space-time dimensions [38], the SO(8) spinorial structures of our system must be parallel to the case of SO (16) in [37]. From this viewpoint, we adopt the notation with no imaginary unit in front of the χ -kinetic term. Accordingly, we need no imaginary unit in front of the ϕ -kinetic term, either. The consistency of our notation will be seen as the emergence of the positive-definite potential (2.14a).
Our total action I is invariant under the SO(4) local symmetry and global N = 8 supersymmetry Since ϕ is in the 8 S of SO(8), we frequently use the expressions, e.g., AB ϕ Bb . The structure of supersymmetry transformation (2.7) is parallel to that in the original formulation [1] [2], such as the Dϕ or ϕ 3 -term in δ Q χ, and χϕ -term in δ Q A µ .
However, the great difference is that now the supersymmetry parameter ǫ I is in the 8 V of The closure of two supersymmetries works just as in the original formulation [2]. In fact, at the linear order, we have Compared with the original formulation [2], due to the supersymmetry parameter ǫ I in the 8 V of SO (8), the explicit The positive definite potential V 1 and the ϕ 3 -term in δ Q χ can be re-expressed in terms of the generalized 'superpotential' W ABCD as On the RHS of (2.9b), the index A is contracted within the parentheses, while the indices a, B, C, D are contracted, when the pair of parentheses is squared.
The positive definiteness of our potential is a non-trivial conclusion. Because it is the reflection of the total consistency of our system, such as the usage of our notation, in which both the fermionic and bosonic inner products do not have any imaginary unit 'i' in front.
This convention has been already used in N = 16 supergravity [37].
The confirmation of supersymmetry δ Q I 1 = 0 is more involved than the original formulation [2]. However, the basic cancellation in each sectors is parallel to [2]. In fact, the confirmation works as follows. At the quadratic order, the computation is routine. At the cubic order, we have only the χF ϕ -terms, which are parallel to [2].
At the quartic order, we have two sectors of terms: (i) (Dχ)ϕ 3 and (ii) χ 3 ϕ. For the sector (i), we need the identity for any antisymmetric tensor A BC = −A CB . It turns out that all the terms have only two The conditions of vanishing of these two kinds of terms determine the coefficients of the χ 2 ϕ 2 -term in the lagrangian and of the ϕ 3 -terms in δ Q χ.
In the sector (ii) χ 3 ϕ, we have three different structures of terms: 4) However, as the Fierzing of each of (A), (B) and (C) reveals, there are two relationships among them: Thus, all the terms no more than the (B) -terms, and their cancellation uniquely fixes the coefficient of the χ 2 ϕ 2 -term in the lagrangian.
At the quintic order, there is no term arising as in [2]. However, at the final sextic order, there is one sector of the type χϕ 5 . The analysis of this sector needs special care. First, we note that the ϕ 6 -term in L 1 can be re-expressed as an alternative form Second, it turns out that all the terms in the sextic order fall in one of the following four structures (1P), (1Q), (3P) and (5P) defined by where the terms (ξ), (η), (ζ) and (κ) are defined by The lemmas in (2.15) can be easily obtained by Fierzing. The second expressions in (2.15a) and (2.15d) are straightforward, but those in (2.15b) and (2.15c) are non-trivial to get. The expressions in terms of (ξ), (η), (ζ) and (κ) are convenient to integrate to compare δ Q L 1,ϕ 6 . In particular, the coefficient of the terms (η) and (κ) out of δ Q L 1,χ 2 ϕ 2 should be the same for them to be cancelled by δ Q L 1,ϕ 6 .

Second Field Content
Our second field content is ϕ .
Aa , χ I a ; A µ ab . Other than the representational difference of fields, the index convention is exactly the same as in section 2, e.g., ϕ in the 8 C and χ in the 8 V of SO (8). The lagrangian for our total action I 2 ≡ d 3 x L 2 is Since the ϕ's is in the 8 C of SO (8) Here again, we are using the notations, such as ( Bb . The supersymmetry parameter ǫ A is now in the 8 S of SO (8).
The closure of supersymmetries works just as in our first field content and the original formulation [2] as well. At the linear order, we have with ξ µ 3 ≡ +2(ǫ 1 γ µ ǫ 2 ) for the translation δ P , and α ab 3 ≡ −ξ µ A µ ab for the SO(4) local transformation δ G . The supersymmetry parameter ǫ A now is in the 8 S of SO (8), so that the index A is suppressed in ξ µ 3 .
Also in our second field content, its bosonic potential V 2 ≡ −L 2,ϕ 6 is positive definite: The coefficient 4c 2 /3 is the same as in the original formulation [1]. The bosonic potential V 2 and the ϕ 3 -term in δ Q χ can be re-expressed in terms of the generalized superpotential W .
These structures are parallel to the first field content case in (2.9).
The invariance confirmation δ Q I 2 = 0 is very parallel to δ Q I 1 = 0. Even the lemmas in (2.15) are parallel. For example, (2.15a) is simply replaced by whose final form is eventually the same as in (2.15a), despite the different index assignments on the ǫ's, χ's and ϕ's. Due to this parallel-ness, the confirmation of δ Q I 2 = 0 is greatly simplified.
Once we start performing the confirmation δ Q I 2 = 0, we see that the computation for the second field content is much easier than the first one. This is caused by the fact that the fermion χ I a is no longer in the 8 C , but in the 8 V of SO (8), so that necessary Fierzings are simpler.

Unification by Triality of SO(8)
We mention how the triality of SO(8) works for the three formulations, i.e., the original formulation in [2], and our first and second field contents.
First of all, we define the following constant N-matrices as products of two SO(8) generators: These constant matrices play a central role in demonstrating the triality of SO (8). For example, this constant matrix satisfies the (anti)self-duality conditions with clear symmetries among these relationships, reflecting the triality between the The proof of (4.2c) and (4.2d) can be simplified, if we use (4.3b) and (4.3d) by expressing the epsilon tensor in terms of the products of Γ -matrices. To our knowledge, these relationships associated with the triality of SO(8) have never been explicitly given in the past.
If we compare the three potentials, i.e., that in the original [2] and ours V 1 and V 2 , they reveal the symmetric expressions for these three potentials: Here V 0 is the bosonic potential in [2], and ϕ I a is their X I a in our notation. In (4.4), all the un-contracted indices within the pair of parentheses should be contracted when the 5) Here we do not use the combination of the superscripts and subscripts for the contracted indices, because it is better to keep the order of 8 V superscripts and 8 S or 8 C subscripts for the matrix N . Also, for the products of Kronecker's deltas, we use the mixed indices for an obvious reason. pair of parentheses is squared. For example in (4.4b), the indices d, I, J, K, L and D are contracted, when the pair of parentheses is squared. Due to the second terms in (4.3), these give the desired symmetric expressions in the last sides of (4.4). In other words, we have a unified expression for (4.4) as 5) where N stands for one of the three N's in (4.4), depending on the representations of ϕ a .
For example, N XY ZU X ′ Y ′ Z ′ U ′ implies N IJKL ABCD for ϕ a in the 8 S of SO(8).

Relationships with N = 6 Superconformal Chern-Simons Theory
As an important application of our first field content, we obtain the transformation rule for N = 6 superconformal Chern-Simons theory [20] [35]. 6) The importance of this relationship stems from the fact that the supersymmetry param- The basic reduction rules are represented by the Γ i 's in (5.1). The Γ 7 is defined by Γ 7 ≡ +iΓ 1 Γ 2 · · · Γ 6 , controlling the chirality for SO (6). Due to the peculiar structure of SO(6) ≈ SU(4), the subscript α and the superscript α respectively correspond to the positive and negative chiralities 6) The special feature of N = 6 was pointed out also in locally superconformal theory [39]. under Γ 7 , and they are complex conjugations to each other. Accordingly, the chirality for SO (8) corresponds to the eigen-states of the σ 3 -matrix: We also truncate ǫ 8 = ǫ 9 = 0, while maintaining our first field content with the original 32+32 degrees of freedom. Note that the symmetries of the both sides in (5.1) are consistent, because Γ i and Γ 7 are all antisymmetric.
Following this basic truncation rule, we can get the N = 6 transformation rule consistent with [20][35] We are using the notations, such as (ϕ * to save space. The on-shell closure of gauge algebra is confirmed as with the respective parameters ξ µ and Λ ab for the translation and SO(6) local symmetry.
Up to the groups SU(N ) × SU(N ) [20] and U(N ) × U(N ) [35], which are replaced by SO (4) was not the case in the original [2] and our first field content.
Our scalar potentials in both formulations are positive definite, reflecting the total consistency of our system, such as the notation with the absence of the imaginary unit 'i' in front of both fermionic and bosonic spinorial inner product. This convention has been already used in N = 16 supergravity [37]. Reflecting the triality of SO (8), the bosonic potentials V 0 , V 1 and V 2 share exactly the same positive constant 4c 2 /3.
As has been mentioned in the Introduction, BL theory [1] [2] can be obtained as the conformal limit of gauged supergravity [9]. From this viewpoint, our first content is the conformal limit of N = 8 gauged supergravity with the physical fields ϕ Aa , χ . We have so far the three distinct formulations: the original BL theory with ϕ I a , χ .

Aa
; ǫ I and the second one with ϕ .
Aa , χ I a ; ǫ A , where the ǫ's are supersymmetry parameters. Strictly speaking, there are three other formulations with ϕ .
A and ϕ I a , χ Aa ; ǫ .
A . However, the latter and the former are related through 'chirality-flip' conjugations with no essential differences.
Even though our field contents are natural consequences of SO (8)  This work is supported in part by NSF Grant # 0652996. We are indebted to the referee of this paper for the suggestion of giving an explicit connection between our first field content and N = 6 theory [20] [35].