Octonions, G_2 and generalized Lie 3-algebras

We construct an explicit example of a generalized Lie 3-algebra from the octonions. In combination with the result of arXiv:0807.0808, this gives rise to a three-dimensional N=2 Chern-Simons-matter theory with exceptional gauge group G_2 and with global symmetry SU(4)\times U(1). This gives a possible candidate for the theory on multiple M2-branes with G_2 gauge symmetry.


Generalized Lie 3-algebras
Let us first give the definition of a generalized metric Lie 3-algebra following [1] 3 .
We can rewrite these conditions in terms of structure constants. Structure constants f abc d are introduced just as in the case of Lie 3-algebras: [e a , e b , e c ] = f abc d e d . (2.4) where e a 's are basis of A and h ab = (e a , e b ).
The conditions we imposed above on generalized metric Lie 3-algebras can be reformulated using the structure constants. The fundamental identity reads as (2.5) and the remaining conditions are captured by the symmetry properties 4 Given a generalized Lie 3-algebra, we can write down the Lagrangian of the theory, using four sets . . , 4) of chiral superfields and a vector superfield V . Here we do not write down the explicit form of the action; see [1] for details. Since the action is written in N = 2 superfield formalism, it is clear that we have N = 2 supersymmetry. Global symmetry SU (4) × U (1) is also manifest from the form of the action.
At this point some readers might wonder whether N = 2 supersymmetry in three dimensions is enough to determine the form of the Lagrangian completely. In fact, the answer is definitely no. The possible ambiguity resides in the form of the superpotential W(Φ), which should be a polynomial in Φ and should be constructed from the triple bracket and the metric. Possible forms of W include: The first term is similar to the potential term of the BLG theory, whereas the second term is absent in the ordinary (non-generalized) Lie 3-algebras. In this sense, the theories of [1] are different from BLG(-type) theories constructed from ordinary Lie 3-algebras.

An example of a generalized Lie 3-algebra from octonions
In the previous section, we explained general formalism applicable to arbitrary generalized metric Lie 3-algebras. However, we still have to find explicit examples of generalized Lie 3-algebras to complete the story. In [1], the only example discussed in [1] is the algebra C 2d , and we are definitely in need of more explicit examples of generalized Lie 3-algebras. In this section, we thus give an explicit construction of a generalized Lie 3-algebra using the octonions, which is one of the most famous non-associative algebras in the literature 5 .
Octonions O are one of the four normed division algebras (the other three are R, C and H). It is an important example of alternative algebra (see Appendix for definition of alternative algebras), which is a special class of non-associative algebra. It is spanned by 8 basis e a (a = 0, 1, . . . , 7), whose multiplication table is given in Table 1. Here we are taking e 0 to be an identity.  It is easy to check that this algebra is nonassociative; for example, (e 1 e 2 )e 3 = e 4 e 3 = −e 6 but e 1 (e 2 e 3 ) = e 1 e 5 = e 6 .
5 See [14] for examples of old discussions of octonions in the context of membranes and [15] for more recent discussions in the context of BLG theories. However, these works use the so-called "octonionic structure constants" φ ijk defined by 1) or its dual φ ijkl in seven dimensions which is different from our structure constants f abcd . If we use the notation shown below (3.19), then the antisymmetric part Φ F of f abcd corresponds to φ ijkl .
Define D x,y by where bracket [−, −] is the commutator as usual. More explicitly, From this definition, we have Now the important fact is that this D x,y is actually a derivation [16]. In order words, for all x, y, z, w ∈ A. In fact, it is known [16] that any derivation D can be written as the sum of D x,y : for some constants c x,y .
Let us now define our 3-bracket by 6 [x, y, z] := D x,y (z). (3.9) By explicit computations, it is again easy to check that this 3-bracket is not totally antisymmetric, although it is antisymmetric with respect to the interchange of x and y due to (3.6).
Having defined our 3-bracket, we now have to verify that our 3-bracket satisfies the fundamental identity. The key to prove this is (3.7).
, and if we use the explicit expression, D a,b (c) is a sum of products of a, b and c. Now D x,y acts as a derivation for each of these terms, and thus on the whole D a,b (c). This proves the fundamental identity.
Of course, we have to define a metric on O before verifying other axioms. Since O is a normed algebra, we have a natural metric. For an element x = x 0 e 0 + . . . x 7 e 7 ∈ O, define its conjugate x * by where Re is defined by taking the e 0 component: x a e a = x 0 . (3.11) In particular, for basis e i , e j , we have (e i , e j ) = δ ij . (3.12) and in general x a y a (3.13) This metric is clearly symmetric in x and y. Moreover, since the metric is diagonal in our basis, we will hereafter not worry about the differences of upper and lower indices.
Now the metric defined above satisfies (xy, z) = (x, yz) (3.14) for all x, y, z ∈ O. To prove this, it suffices to show that (e i e j , e k ) = (e i , e j e k ). (3.15) We only need to verify this when (e i e j , e k ) = 0, namely e i e j = ±e k . From the multiplication  and it is clear from the final expression that the result is antisymmetric in z, w. Note that in the final line, we have used the identity (xy, zw) = (yx, wz), (3.17) which can again be verified by similar considerations as in (3.14).
It is another straightforward exercise of octonions to verity that the remaining axiom of generalized Lie 3-algebras: Summarizing, if we define 3-bracket by (3.9) and metric by (3.10), then all the conditions of generalized metric 3-Lie algebra is satisfied. In the next section we will see that this algebra gives an interesting theory when applied to the formalism of [1].
Before finishing this section, it is probably instructive to comment on the relation of our 3-bracket with previous approaches. Since the 3-bracket [x, y, z] is antisymmetric in first two indices, the 3-bracket defines a linear map Φ : Φ is broken down into two components Φ F and Φ L [9]: When Φ = Φ F (i.e. Φ L = 0), [x, y, z] is totally antisymmetric and we have an ordinary Lie 3-algebra as discussed in [2]. When Φ = Φ L (i.e. Φ F = 0), the algebraic structure satisfying (3.20) is called Lie triple systems (see [18] for a recent discussion). For our 3-bracket [x, y, z], Φ F = 3 x, y, z = 0 (here we used (A.5) in Appendix), where x, y, z is the so-called associator which is defined by x, y, z = x(yz) − (xy)z. (3.21) Now recall that the Bagger-Lambert paper [2] constructs 3-brackets from the nonassociative algebras by [x, y, z] BL := x, y, z ± (perm.). (3.22) Since associator of octonions is antisymmetric with respect to its three arguments (see Appendix), we learn that our Φ F (x, y, z) and [x, y, z] BL coincide. However, we have another piece Φ L (x, y, z) = [x, y, z]− 3 x, y, z = 0, and our 3-bracket is a mixture of the above two. The fundamental identity is not satisfied by either Φ F and Φ L , and only by their sum Φ. This explains the similarities and differences of our approach and the approach taken by [2].

M2-branes with gauge group G 2 ?
Having obtained an example of generalized 3-Lie algebra, we turn to the physical implication of this result.
Let us study the gauge symmetry of our theory. For that purpose, we need the gauge transformation of gauge fields: This is the usual gauge transformation with parameter λ b a . Just as in BLG theory, structure constants are antisymmetric when a and b are exchanged, and thus the gauge group is the subgroup of SO(O) ≃ SO (8). But we can indeed say more than that.
Of course, it is easy to notice that the gauge group is a subgroup of SO (7). This is because our 3-bracket is zero whenever we have an identity e 0 in one of its arguments: [x, y, e 0 ] = [x, e 0 , y] = [e 0 , x, y] = 0, (4.2) or in terms of structure constants, Does this mean that the gauge group is SO (7)? The answer is no; explicit computations by Mathematica tells us that the dimension of the gauge group is 14.
In order to determine the gauge group, recall that [x, y, −] acts as a derivation: This means that, if we define λ : A → A by λ · e a = λ cd f cdb a e b , we have λ(zw) = (λz)w + z(λw) (4.5) This means that λ is a derivation. Namely, gauge transformations are contained in the set of derivations of O. Now it is known since long ago [19] that derivation of O is nothing but the exceptional Lie algebra (8). (4.6) This means that the gauge group of our theory is G 2 , as claimed above.
We have constructed a three-dimensional theory with N = 2 supersymmetry and with SU (4) × U (1) global symmetry. What is the physical meaning of this fact? First, this is inconsistent (at least in our example) with statement in [1] that we have SU (4) × U (1) R-symmetry. Indeed, the classification of N = 6 theories in [10] tells us that theories with N = 6 should have gauge group SU (n) × U (1), Sp(n) × U (1), SU (n)×SU (n) and SU (n)×SU (m)×U (1) with possibly additional U (1)'s, and no G 2 gauge groups are allowed. Thus SU (4) × U (1) should be considered as a global symmetry, rather than a R-symmetry.
Second, the appearance of nonassociative algebras and generalized Lie 3-algebras strongly suggest the connection to membrane physics. Unfortunately, the connection of generalized Lie 3-algebras and worldvolume theories on M2-branes are currently not known, but our theory is certainly the possible candidate for the worldvolume of M2-branes with gauge group G 2 . If this is indeed the case, this would be a novel way to realize exceptional gauge symmetry in M-theory 7 .
In this paper, we gave an explicit example of a generalized Lie 3-algebra, using one of the most famous example of non-associative algebras, namely the octonions. When combined with the result of [1], we have three-dimensional Chern-Simons-matter theories with gauge group G 2 and with global symmetry SU (4)×U (1). The appearance of generalized Lie 3-algebra suggests that this theory is a possible candidate for the theory on M2-branes with exceptional gauge group G 2 .
This raises many question which needs further exploration. First, it would be interesting to study the moduli space of this theory. Just as in the BLG/ABJM theories and their variants [7,20], this will give invaluable information about the physical interpretation of our theory. Related to this question is the reduction to type IIA theory (probably along the lines of [21]). The G 2 gauge symmetry should be broken into a smaller symmetry group in this process.
We can also envisage possible generalizations. In this paper, we have taken octonions as an example, but our strategy should be much more general. As written in Appendix, much of our construction applies to a wider class of non-associative algebras known as alternative algebras, and it would be interesting to consider generalizations to other alternative algebras. Aside from alternative algebras, another interesting class of nonassociative algebras is Jordan algebras. Interestingly, other exceptional gauge groups F 4 , E 6 , E 7 and E 8 are related to Jordan algebras in interesting way [22] (for example, F 4 is equal to Isom(OP 2 ), where OP 2 is an octonionic projective plane), we have the hope of constructing theories with these exceptional gauge groups.
Finally, in BLG theory, we have obtained a class of theories by relaxing the condition of positivity of the metric [4]. The same strategy should also work of our case as well. For example, we have an algebra split-octonion, which has signature (4,4), as contrast to the ordinary octonions whose signature is (8,0).
Although the issue of unitarity is subtle, these algebras might has some role to play in physics.

Note Added
After the completion of version 1 of this paper, we received a paper [9] which discusses general theory of generalized Lie 3-algebras. In particular, they show that there exists a one-to-one correspondence between a generalized metric Lie 3-algebra and a pair consisting of a Lie algebra and its faithful orthogonal representation. In this general framework, our example is constructed from a Lie algebra G 2 and the octonions as its eight-dimensional representation, which in turn decomposes as 8 = 7 + 1 as can be seen in (4.3). Still, it still seems highly non-trivial to check explicitly that the 3-brackets defined from equation (9) of [9] matches with our definitions of 3-brackets. We thank José Figueroa-O'Farrill for valuable comments on this point.