The BLG model is a 3-dimensional SCFT involving a Chern-Simons theory coupled to matter. It is argued to be the worldvolume theory of 2 coincident M2-branes with 16 manifest supersymmetries. The generalization to an arbitrary number of M2-branes is supposed to be given by the ABJM model.
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The BLG model Lagrangian involves a trilinear operation on the scalar fields $\phi \in V$
Moreover, the supersymmetry of the Lagrangian hinges on the fact that this map satisfies a condition that has some similarity to a Jacobi identity for the binary operation on a Lie algebra.
Therefore, superficially, it looks like this might be the trinary bracket on an L-∞ algebra structure on the space $V$.
On the one hand, indeed, by the discussion at supergravity C-field , the M2-brane is charged under a circle 3-bundle with connection whose higher gauge theory is controled by Lie 3-algebras in direct analogy to how the higher gauge theory of the string is controled by gerbes/principal 2-bundles and their Lie 2-algebras and that of charged particles by ordinary Lie algebras.
Apparantly motivated by an intuition along these lines, (BaggerLambert) named $(V,[-,-,-])$ a 3-algebra. This terminology was picked up by many authors In the process, it transmuted sometimes to “3-Lie algebra” and sometimes even to “Lie 3-algebra”.
Unfortunately, the Bagger-Lambert “3-algebra” is not a Lie 3-algebra in the established sense of an L-∞ algebra structure on a graded vector space $V$ concentrated in the lowest three degrees. At least not without some modifications in the interpretation of the map $[-,-,-]$.
The reason is that for the notion of an L-∞ algebra (as discussed there) it is crucial that $V$ is a $\mathbb{N}$-graded (or $\mathbb{Z}$-graded) vector space and that the $n$-ary brackets respect the degree in a certain way. But in the BaggerLambert-proposal, $V$ is all concentrated in a single degree (is regarded as ungraded). One immediately finds that in this case the $L_\infty$-respect of $[-,-,-]$ for the grading would imply that $V$ is taken to be in degree $1/2$. Since this is not in $\mathbb{N}$, it does not yield an $L_\infty$-algebra.
Notice that the $\mathbb{N}$-grading (or $\mathbb{Z}$-grading) of $L_\infty$-algebras is crucial for the homotopy theoretic interpretation of L-∞ algebras as higher Lie algebras. None of the good theory of $L_\infty$-algebras survives when this grading is dropped. This grading has its origin in the Dold-Kan correspondence, which establishes integral graded homological structures as models for structures in higher category theory. Notably, a higher Lie algebra is supposed to have a Lie integration to a smooth $n$-groupoid. Under this process, the elements in degree $k$ of the higher Lie algebra become tangents to the space of k-morphisms of this smooth $n$-groupoid. Clearly, here only integer $k$ do make sense.
On the other hand, it is of course possible to consider the structure on “$L_\infty$-algebras without grading”, even if these will not have a good theory. This notion has once been introduced by Filippov (Sib. Math. Zh. No 6 126–140 (195)) under the term n-Lie algebra .
Beware, therefore, that the innocent-looking difference between the terms
corresponds, unfortunately, to a major difference in the behaviour of the concepts behind these terms.
In conclusion, it is clear that 2-brane physics is governed by Lie 3-algebraic structures, but it is not yet clear how the trinary operation highlighted by BaggerLambert would be an example.
In view of this it might be noteworthy that the equivalent reformulation and generalization of the BLG model by the ABJM model does not involve any “3-algebras” at all. In fact at least most of the “3-algebras” appearing in the membrane literature may be understood as being data of a plain Lie algebra with an invariant product and a representation (MFMR 08). These authors summarize the state of affiars on p. 3 as
All this prompts one to question whether the 3-algebras appearing in the constructions [1–3,10,11] play a fundamental role in M-theory or, at least insofar as the effective field theory is concerned, are largely superfluous. The equivalence of [10] and [6] and the abundance of new theories (dual to known M-theory backgrounds) which seem not to involve a 3-algebra might suggest the latter. Nonetheless, given our lack of understanding of how to incorporate in Lie-algebraic terms the expected properties of M-theoretic degrees of freedom, like the entropy scaling laws for M2- and M5-brane condensates, it may be useful to understand the precise relation between the 3-algebras appearing in the recent literature on superconformal Chern–Simons theory and Lie algebras.
The article (MFMR 08) provides this relation and under this relation the “3-algebraic” BLG model has then been understood as a special case of the ordinary Lie algebraic ABJM theory. A review is in (Bagger-Lambert 12).
It has also been suggested that “3-algebras” are to be interpreted in 2-plectic structure (Saemann-Szabo).
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
The original articles are
Jonathan Bagger, Neil Lambert, Modeling Multiple M2’s, Phys. Rev. D75, 045020 (2007). (hep-th/0611108).
Jonathan Bagger, Neil Lambert, Phys. Rev. D77, 065008 (2008). (arXiv:0711.0955).
and concerning the “3-algebra”-structure also
A comprehensive review and survey is in
Discussion in Horava-Witten theory reducing M2-branes to heterotic strings is in
The interpretation of at least most of the “3-algebra” appearing in the membrane literature in terms of plain Lie algebras is due to
The suggestion that BGL “3-algebras” are to be interpreted in 2-plectic geometry appears in