Contents

# Contents

## Idea

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 black M2-branes with 16 manifest supersymmetries. The generalization to an arbitrary number of M2-branes is supposed to be given by the ABJM model.

$N$ Killing spinors on
spherical space form $S^7/\widehat{G}$
$\phantom{AA}\widehat{G} =$spin-lift of subgroup of
isometry group of 7-sphere
3d superconformal gauge field theory
on back M2-branes
with near horizon geometry $AdS_4 \times S^7/\widehat{G}$
$\phantom{AA}N = 8\phantom{AA}$$\phantom{AA}\mathbb{Z}_2$cyclic group of order 2BLG model
$\phantom{AA}N = 7\phantom{AA}$
$\phantom{AA}N = 6\phantom{AA}$$\phantom{AA}\mathbb{Z}_{k\gt 2}$cyclic groupABJM model
$\phantom{AA}N = 5\phantom{AA}$$\phantom{AA}2 D_{k+2}$
$2 T$, $2 O$, $2 I$
binary dihedral group,
binary tetrahedral group,
binary octahedral group,
binary icosahedral group
$\phantom{AA}N = 4\phantom{AA}$$\phantom{A}2 D_{k+2}$
$2 O$, $2 I$
binary dihedral group,
binary octahedral group,
binary icosahedral group
(HLLLP 08b, Chen-Wu 10)

## Properties

(…)

### Boundary conditions

Discussion of boundary conditions of the BLG model, leading to brane intersection with M-wave, M5-brane and MO9-brane is in (Chu-Smith 09, BPST 09).

### The “$3$-algebra” structure

The BLG model Lagrangian involves a trilinear operation on the scalar fields $\phi \in V$

$[-,-,-] : V^{\otimes 3} \to V \,,$

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 controlled 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.

Motivated by attempts in Basu-Harvey 04 to generalize Nahm's equations for fuzzy funnels of D2-D4-brane intersections to M2-M5-brane intersections, Bagger-Lambert 06 introduced the algebraic structure that came to be known the M2-brane 3-algebra, further highlighted by Gustavsson 07. 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”.

Beware that the Bagger-Lambert “M2-brane 3-algebra” is not- aLie 3-algebra_ in the established sense of an L-∞ algebra structure on a graded vector space $V$ concentrated in the lowest three degrees.

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 [13,10,11] 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).

$d$$N$superconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
$\phantom{A}3\phantom{A}$$\phantom{A}2k+1\phantom{A}$$\phantom{A}B(k,2) \simeq$ osp$(2k+1 \vert 4)\phantom{A}$$\phantom{A}SO(2k+1)\phantom{A}$
$\phantom{A}3\phantom{A}$$\phantom{A}2k\phantom{A}$$\phantom{A}D(k,2)\simeq$ osp$(2k \vert 4)\phantom{A}$$\phantom{A}SO(2k)\phantom{A}$M2-brane
D=3 SYM
BLG model
ABJM model
$\phantom{A}4\phantom{A}$$\phantom{A}k+1\phantom{A}$$\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}$$\phantom{A}U(k+1)\phantom{A}$D3-brane
D=4 N=4 SYM
D=4 N=2 SYM
D=4 N=1 SYM
$\phantom{A}5\phantom{A}$$\phantom{A}1\phantom{A}$$\phantom{A}F(4)\phantom{A}$$\phantom{A}SO(3)\phantom{A}$D4-brane
D=5 SYM
$\phantom{A}6\phantom{A}$$\phantom{A}k\phantom{A}$$\phantom{A}D(4,k) \simeq$ osp$(8 \vert 2k)\phantom{A}$$\phantom{A}Sp(k)\phantom{A}$M5-brane
D=6 N=(2,0) SCFT
D=6 N=(1,0) SCFT

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

## References

### Precursors

The lift of Dp-D(p+2)-brane bound states in string theory to M2-M5-brane bound states/E-strings in M-theory, under duality between M-theory and type IIA string theory+T-duality, via generalization of Nahm's equation (this eventually motivated the BLG-model/ABJM model):

### General

The original articles are

and with special emphasis on the “M2-brane 3-algebra”-structure also:

The interpretation in terms of branes at a $\mathbb{Z}/2$-ADE-singularities with discrete torsion in the supergravity C-field is due to

Review:

Discussion of boundary conditions leading to brane intersection with M-wave, M5-brane and MO9-brane is in

Discussion in Horava-Witten theory reducing M2-branes to heterotic strings is in

The interpretation of at least most of the “M2-brane 3-algebra” appearing in the membrane literature in terms of plain metric Lie algebras is due to