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Strong Homotopy Lie Algebras, Generalized Nahm Equations and Multiple M2-branes

We review various generalizations of the notion of Lie algebras, in particular those appearing in the recently proposed Bagger-Lambert-Gustavsson model, and study their interrelations. We find that Filippov's n-Lie algebras are a special case of strong homotopy Lie algebras. Furthermore, we define a class of homotopy Maurer-Cartan equations, which contains both the Nahm and the Basu-Harvey equations as special cases. Finally, we show how the super Yang-Mills equations describing a Dp-brane and the Bagger-Lambert-Gustavsson equations supposedly describing M2-branes can be rewritten as homotopy Maurer-Cartan equations, as well.


Introduction and results
Since Witten's introduction of M-theory, our understanding of this proposal for a nonperturbative unification of the various superstring models has grown quite slowly. Little more than a year ago, however, a description of the effective dynamics of stacks of M2branes was proposed by Bagger, Lambert and independently by Gustavsson [1,2]. The construction of the Bagger-Lambert-Gustavsson (BLG) model relies crucially on mathematical structures known as 3-Lie algebras [3]. It was soon found that the BLG model together with the special 3-Lie algebra A 4 should be interpreted as the effective description of a stack of two M2-branes [4]. Unfortunately, subsequent analysis showed that this is the only 3-Lie algebra which can be used in the construction of the BLG theory [5]; all other 3-Lie algebras suitable for that purpose are direct sums of A 4 .
Therefore, a generalization of the BLG model is clearly needed and over the last year, three such generalizations were pursued. First, positive definiteness of the bilinear pairing of the 3-Lie algebra was relaxed. This leads to ghosts in the theory, which have to be removed. At least one of these procedures [6] yields d = 3, N = 8 super Yang-Mills (SYM) theory, which is not the desired outcome. Second, it was noticed that the BLG model can be rewritten as a gauge theory [7] having a Lie algebra as its gauge algebra. Thus, the original incarnation of the BLG model in terms of 3-Lie algebras might have been misleading and 3-Lie algebras could have entered the theory only accidentally for the case of two M2-branes. Third, one can use an extended version of 3-Lie algebras, which in general leads to theories with a smaller amount of supersymmetry than the original N = 8 of the BLG model [8,9,10,11].
In this note, we present evidence that the BLG model based on 3-Lie algebras should be reconsidered within the framework of strong homotopy Lie algebras, also known as L ∞ algebras [12,13]. These are natural generalizations of Lie algebras arising in algebraic homotopy theory and deformation theory. Just as one can consider the Maurer-Cartan equation describing deformation problems governed by a Lie algebra, an L ∞ algebra defines a 'homotopy Maurer-Cartan (hMC) equation' related to an associated deformation problem [14]. In a physics context, L ∞ algebras and homotopy Maurer-Cartan equations have appeared naturally in string field theory [15], BV quantization [16], topological open string theory and topological field theory [17,18,19,20,21,22,23] as well as gauge theory [24].
Certain classical field equations can be written in the form of homotopy Maurer-Cartan equations associated with appropriate L ∞ algebras. For example, Yang-Mills theory and its maximally supersymmetric extension were discussed in [25] and [26], the Einstein equations were rewritten in hMC form up to second order in [27]. All this, together with the results of this paper, supports the conjecture of Andrei S. Losev that for all classical field equations, an interpretation in terms of hMC equations exists.
As a side remark, we note that the twistor description provides an alternate route to reformulating field equations in hMC form. Over twistor space, solutions to classical field equations correspond to topologically trivial holomorphic vector bundles over a twistor space P. Such vector bundles can be described in terms of holomorphic connections given locally by gauge potentials A 0,1 ∈ Ω 0,1 (P, g) which are holomorphically flat, i.e. they satisfy the Maurer Cartan equation F 0,2 :=∂A 0,1 + A 0,1 ∧ A 0,1 = 0 . (1.1) Here, g is an ordinary Lie algebra, and thus the twistor description yields an equivalence between classical field equations and Maurer-Cartan equations for connection one-forms taking values in a Lie algebra. For more details and examples of field theories allowing for a twistorial description, see e.g. [28] and references therein. This approach, however, requires an auxiliary twistor space, while the above mentioned reformulations of Yang-Mills and Einstein equations in terms of hMC equations live on the same space as the original equations.
In this paper, we place the BLG model into the context of L ∞ algebras and hMC equations. First, we review the relations between Filippov's n-Lie algebras, Lie n-algebras and L ∞ algebras. We find that n-Lie algebras are a special class of ungraded L ∞ algebras. We then construct a family of models given by hMC equations and special L ∞ algebras, which correspond to Nahm-type equations. In particular, both the Nahm and the Basu-Harvey equations [29] are contained in this family as special cases. After this, we discuss L ∞ algebras whose hMC equations yield both the SYM equations 1 as well as the full supersymmetric BLG equations. We conclude with a few remarks on the reduction process from the BLG equations to the SYM equations.

Higher Lie structures
We shall need certain successive generalizations of the notion of a Lie algebra, all of which are based on promoting the Lie bracket to operations of higher arity, and on successive weakenings of the classical Jacobi identity. There are two types of generalizations, which can be classified into graded and ungraded, depending on whether the underlying vector space and operations are required or not to be graded or homogeneous 2 . The mathematical details can be found in the Appendix.

Ungraded generalizations of the notion of a Lie algebra
The first ungraded generalization useful for our purpose is provided by the notion of a n-Lie algebra, which was introduced by Filippov [3]. To define such algebras, one replaces the Lie bracket by an n-ary operation which is required to be totally antisymmetric and to obey a partial weakening of the Jacobi identity known as Filippov's fundamental identity.
A wider generalization is provided by the notion of Lie n-algebra in the sense of Hanlon and Wachs [30], which again uses a single totally antisymmetric n-operation of higher arity, this time required to obey a further weakening of the Jacobi identity which we shall call the homotopy Jacobi identity. It was shown in [31] that n-Lie algebras are Lie n-algebras, because Filippov's fundamental identity implies the homotopy Jacobi identity.
The widest generalization we shall need is the notion of a (ungraded) strong homotopy Lie (or L ∞ ) algebra, which is well-known in algebraic homotopy theory, where it originated. This is obtained by allowing for a countable family of multilinear antisymmetric operations 1 Our rewriting of this theory differs from that found in [26]. 2 More generally, each higher Lie structure can be defined as an algebra object in a certain monoidal category V. The ungraded extensions arise by taking V to be the category of vector spaces over some field k of characteristic zero, while the graded extensions arise when V is a category of graded vector spaces over k. of all arities n ≥ 1, constrained by a countable series of generalizations of the Jacobi identity known as the L ∞ identities. This notion admits specializations indexed by subsets 3 S ⊂ AE * of arities and which are defined by requiring vanishing of all products of arities not belonging to S. This leads to the notion of L S algebra. When S = {n 1 , . . . , n p } is a finite set (with n 1 < . . . < n p ), the result of this specialization is called an L n 1 ,...,np algebra, also known as an L (N ) algebra when S = {1, . . . , N }. The case S = {n}, when only a single product of arity n is non-vanishing, recovers the notion of a Lie n-algebra, i.e. all n-Lie algebras are L n algebras. All in all, we have a series of successive generalizations: Lie algebras ⊂ n-Lie algebras ⊂ Lie n-algebras L n algebras ⊂ (ungraded) L ∞ algebras We refer the reader to the Appendix for more details.

Graded versions
An obvious variant of the algebraic structures above is to consider their graded versions. In this case the underlying vector space is graded, and each operation is graded-antisymmetric as well as homogeneous of a certain degree. The various extensions of the Jacobi identity are replaced by graded variants, which now contain appropriate sign factors. In general, one can consider a grading through elements of an Abelian group G, and define the signs appearing in various equations (such as the graded antisymmetry and graded Filippov/homotopy Jacobi/L ∞ identities) by using a group morphism φ from G to 2 . This leads to the notions of (G, φ)-graded n-Lie algebras, Lie n-algebras and L ∞ algebras/L S algebras: graded Lie algebras ⊂ graded n-Lie algebras ⊂ graded Lie n-algebras graded L n algebras ⊂ graded L ∞ algebras The choice G = {0} (the trivial Abelian group) and φ = 0 recovers the ungraded theories, whose multilinear operations are antisymmetric in the usual sense rather than graded antisymmetric. In the L ∞ case, the choices G = , φ(n) = n(mod 2) and G = 2 , φ = id 2 recover the notions of -and 2 -graded L ∞ algebras as they appear in algebraic homotopy theory. More details can be found in the Appendix.
Remark. For other generalizations of 3-Lie algebras appearing in the Bagger-Lambert model as well as their relations to Lie triple systems, see [8,9,10]. There, however, the multilinear operations are not totally antisymmetric. As noted in the conclusions of [11], they might find an interpretation within a further extension of Lie algebras, the strong homotopy pre-Lie algebras.

Homotopy Maurer-Cartan equations
Given a -graded L ∞ algebra L (see Appendix), the homotopy Maurer-Cartan equations (hMC) are the following equations for an element φ of L which is weakly-homogeneous of odd parity: Here µ ℓ are the multiplications in L. These equations are invariant under the infinitesimal gauge transformations: where λ is a a weakly-homogeneous element of L of even parity, playing the role of a gauge generator. We illustrate this in some two particular cases: (1) In a differential -graded Lie algebra with µ 1 = d and µ 2 = [ , ], the hMC equations reduce to the classical Maurer-Cartan equations: (2) In an L 1,2,n algebra with µ 1 = d and µ 2 (a, b) = [a, b], they take the form:

A family of L ∞ algebras
Let V be an (n + 1)-dimensional complex vector space endowed with a nondegeneratebilinear symmetric form ( , ). Below, we consider the Clifford algebra C(V, Q), where Q is the quadratic form defined by ( , ); if there is a canonical choice for Q, we also write C(V ) as a shorthand. Picking an orthonormal basis e 1 . . . e n+1 of V , this can be presented as the unital associative algebra generated by e i with the relations: Let A m : V ×r → C(V, Q) be the antisymmetrization operator: where juxtaposition in the right hand side stands for the multiplication in C(V, Q). Since each map A m is multilinear and alternating, it factors through a mapĀ m : ∧ m V → C(V, Q). Setting ∧ • V := ⊕ n+1 m=0 ∧ m V , it is well-known that the mapĀ := n+1 m=0Ā m : ∧ • V → C(V, Q) is a linear isomorphism. We will use the standard notation e [i 1 . . . e im] for the element A m (e i 1 , . . . , e im ) of C(V, Q). Via the isomorphismĀ, the grading on ∧ • V induces a grading of the underlying vector space of C(V, Q), whose homogeneous subspaces we denote by C m (V, Q); these subspaces of the Clifford algebra are isomorphic with ∧ m V via the maps A m . Of course, this grading of C(V, Q) is not preserved by the Clifford multiplication.
Consider the element e := e 1 ...e n+1 of C n+1 (V, Q), which corresponds to the non-vanishing element e 1 ∧ . . . ∧ e n+1 of the determinant line ∧ n+1 V via the isomorphismĀ n+1 : ∧ n+1 V → C n+1 (V, Q). From (3.1), we have the following identities: Now let (L, [ , . . . , ]) be an ungraded n-Lie algebra over the complex numbers. If n > 2, then we consider the vector space: is the basic Lie algebra of L, whose natural action on L we denote by the symbol ⊲ (see Appendix). If n = 2, we set F L = L. An element a of Λ i L will be said to be of order i and we introduce the notationâ = i. To describe both the Nahm and the Basu-Harvey equations themselves, L is sufficient; however, it turns out that including the other powers allows us to also describe gauge transformations. Consider the infinite-dimensional complex vector space L : We view L as a -graded vector space whose grading is concentrated in degrees 0, 1 and 2. Also, we endow L with its natural differential 4 µ 1 (ω) := −(d ⊗ id)ω, where d is the de Rham differential on the real line Ê and ω ∈ Ω(Ê, L). A degree zero element of L is simply a g L -valued function λ ∈ Ω 0 (Ê) ⊗ g L . A degree one element φ of L decomposes as: where A is a g L -valued one-form on the real line and X is a function on the real line taking values in the vector space V n+1 ⊗ L. We view A as a connection one-form for the trivial principal bundle G over the real line whose fiber is the Lie group obtained by exponentiating g L , and treat X as a section of the trivial vector bundle over Ê associated to G via the action (induced by ⊲) of g L on the vector space V n+1 ⊗ L. Accordingly, we let ∇ s be the covariant derivation operator defined by the connection one-form A, i.e dX + A ⊲ X = ∇ s (X)ds for any function X on the real line taking values in V n+1 ⊗ L. A degree two element ρ of L is a V n+1 ⊗ L-valued one-form on the real line. Thus, a general element x ∈ L decomposes as: Besides the differential µ 1 , we introduce the following products on L: Here and in the following, we include factors in the products µ ℓ such that the normalizations in the equations of motion arising from the hMC equations match the canonical choices.
It is easy to check that the products introduced above satisfy the homotopy Jacobi identities (B.37), hence (L, µ 1 , µ 2 , µ n ) is a -graded L 1,2,n algebra, i.e. a -graded L ∞ algebra whose only non-vanishing products are µ 1 , µ 2 and µ n (see Appendix). The identity involving two copies of µ n holds because ds ∧ ds = 0 on the real line. The other non-trivial homotopy Jacobi identity involves products µ 2 (λ, ) and µ n (X, . . . , X), and holds due to relation (A.5), which in turn is a consequence of Filipov's fundamental identity (A.2) satisfied by the bracket of L (see Appendix).
The homotopy Maurer-Cartan equation (2.1) for φ = A + X: reduces to: and identity which we shall call the generalized Nahm equation. Since A is flat (we have dA = 0 for dimension reasons) and Ê is contractible, the connection A can be gauged away provided that its behavior at infinity is tame enough. The infinitesimal gauge transformations (2.2) take the form: Since the last term vanishes in our case, this is the same as: 10) or, more explicitly: where λ ∈ Ω 0 (Ê) ⊗ g L as above.
In the following, we discuss the two special cases n = 2 and n = 3, which correspond to the Nahm and Basu-Harvey equations, respectively.

The L ∞ algebra of the Nahm equation
The Nahm equation was originally introduced to extend the ADHM construction of instantons to the case of monopoles [32,33]. Later, this construction found an interpretation in terms of D-branes in superstring theory [34]. In this interpretation, one considers a stack of N D1-brane ending on a D3-brane in type IIB string theory, or, in type IIA, a stack of N D2-brane ending on a D4-brane. The D-branes extend into spacetime as follows: For simplicity, let us restrict to the D1/D3 case in the following. Furthermore, we will parameterize the real line extending in the 2-direction by the variable s and the indices i, j, k = 1, 2, 3 correspond to the three directions Ê 3 345 . The D1-brane is located at x 1 = x 2 = x 3 = 0 and the configuration is assumed to be time independent.
This D-brane configuration is a BPS configuration, and it is thus effectively described by a dimensional reduction of the self-dual Yang-Mills equations. Adopting the perspective of the D3-brane, one obtains the Bogomolny monopole equations in three dimensions. The D1-brane appears as a monopole in the effective description of the D3-brane, with a scalar field with profile φ ∼ 1 r , where r = x 2 1 + x 2 2 + x 2 3 [35,36]. Inversely, the stack of D1-branes is described by the dimensional reduction of the self-dual Yang-Mills equations to the one dimension parameterized by s, which yields the Nahm equation: differential of a flat connection A on Ê, which can be gauged away provided that its behavior at infinity is tame enough. The components X i of X are valued in the Lie algebra L := su(N ), and the prefactor ε ijk comes from the quaternionic structure underlying the Nahm equation. The D3-brane appears in this picture as the so-called fuzzy funnel solution [37]: Making the separation ansatz X i (s) = f (s)G i , we obtain the solution (3.14) The interpretation of this solution is as follows: For every s, the cross-section of the stack of D1-branes is blown up into a fuzzy sphere described by the representation of SU(2) formed by the G i . The point at s is thus increased to that of a D3-brane (with non-commutative world volume for N < ∞), and at s = 0, this D3-brane becomes flat as the radius R ∼ f (s) of the fuzzy sphere tends to infinity. Note that the profiles of both interpretations are compatible: The Nahm equation (3.13) is now a special case of the homotopy Maurer-Cartan equations we considered above. We choose n = 2 and work with the 2-Lie algebra L = su(N ), from which we construct the ungraded L ∞ algebra L : is generated by e 1 , e 2 , e 3 satisfying e i e j +e j e i = 2δ ij ; moreover, we have e = e 1 e 2 e 3 = 1. The only non-trivial products in L are µ 1 and µ 2 , where again µ 1 (x) = (d ⊗ id)x for x ∈ L and µ 2 is defined through: where the bracket [ , ] is the Lie bracket of L. The homotopy Maurer-Cartan equation takes the form: which is equivalent to the Nahm equation (3.13). The gauge transformations preserving this equation take the form: where λ ∈ Ω 0 (Ê) ⊗ L. These, of course, coincide with the homotopy gauge transformations (2.2) as they apply to our case.

The L ∞ algebra of the Basu-Harvey equation
In their paper [29], Basu and Harvey suggested a generalized Nahm equation as a description of a stack of N M2-branes ending on an M5-brane. The configuration in flat spacetime is here given by: where we parameterize again the real line extending in the 2-direction by s. The coordinates 3456 in the M5-brane, which is perpendicular to the M2-branes. From the analysis of the Abelian theory [36], we know that such a stack of M2-branes should appear in the worldvolume theory on the M5-brane as a scalar field with profile 1 r 2 , where r = x 2 1 + ... + x 2 4 . The M5-brane, in turn, should therefore appear as the scalar field in a solution to the worldvolume theory on the M2-brane with profile 1 √ s . Assuming a splitting ansatz for the scalar field X i (s) = f (s)G i , i = 1, ..., 4, one concludes that the equations should take the following form [29]: where the X i take values in a 3-Lie algebra L. Again, these equations can be recast in terms of our Maurer-Cartan equations: they correspond to the case n = 3. Given a 3-Lie algebra L, we have an associated 2-Lie algebra g L . As before, we define L : As generators for the Clifford algebra C(Ê 4 ) we take an orthonormal basis e 1 . . . e 4 of Ê 4 . In this case one has e := e 1 ...e 4 .
We endow L with the non-trivial products µ 1 (x) = −(d⊗id)x as well as µ 2 and µ 3 defined through: The gauge transformations (2.2) preserving this equation take the form:

The full theories
The Nahm equation and supposedly the Basu-Harvey equation are BPS equations in the effective description of D1-and M2-branes respectively. Hence, they partly determine the supersymmetry transformations of these effective theories, and from closure of the supersymmetry algebra, the field equations and ultimately the Lagrangian can be derived. This, in fact, is how Bagger and Lambert [1] and independently Gustavsson [2] obtained the BLG model in the first place. In this section, we show that graded homotopy Maurer-Cartan equations appear, albeit the grading in the underlying L ∞ algebra is slightly less natural than it was in the case of BPS equations.

The description of D1-and D2-branes dynamics by homotopy Maurer-Cartan equations
It is well-known that the effective dynamics of D-branes is described by maximally supersymmetric Yang-Mills (SYM) theory dimensionally reduced to their worldvolume, see [38] for details. For a stack of n flat Dp-branes, one starts with the Lagrangian of N = 1, U(n) SYM theory in ten dimensions, Here, F M N are the components of the field strength of the U(n) connection, whose covariant derivative we denote by ∇. The fermions of the theory are described by ten-dimensional Majorana-Weyl spinors ψ transforming in the adjoint representation of the gauge group, while Γ M are the Gamma matrices in the corresponding representation of the Clifford algebra C(Ê 1,9 ). The dimensional reduction of the bosonic part of the action is obtained as usual by splitting the ten-dimensional gauge potential A M into a p + 1-dimensional gauge potential A µ , µ = 0, ..., p and 9 − p scalar fields X i , i = 1, ..., 9 − p. Setting all derivatives along directions transverse to the worldvolume of the Dp-brane to zero gives: The resulting equations of motion are: Gauge transformations are generated by su(n)-valued functions λ on Ê 1,p and on infinitesimal level, the fields transform according to It has been shown in [26] that the pure Yang-Mills equations can be interpreted as homotopy Maurer-Cartan equations using a BRST complex. Earlier, in [25], the maximally supersymmetric Yang-Mills were recast into hMC form using a pure spinor formulation. Here, we will present a simple reinterpretation of the N = 8 SYM equations in two or three 5 dimensions as hMC equations without invoking more than classical structures. To be concise, we first discuss the purely bosonic part of the action in detail; the necessary modifications for the supersymmetric extension will be listed in the next section.
The main difference to the Nahm and the Basu-Harvey equations is the fact that the differential equations (4.3) are non-linear. We will therefore have to define µ 1 as a non-linear differential operator. It should be stressed, that our rewriting of the two-dimensional N = 8 SYM equations as homotopy Maurer-Cartan equations is by no means unique.
On Ê 1,p , we consider the de Rham differential d, the Hodge operator * with respect to the Minkowski metric η µν and the top form ω = dx 0 ∧ . . . ∧ dx p . Furthermore, let q = 9 − p as a shorthand and, as before, C(Ê q ) = C 0 (Ê q ) ⊕ ... ⊕ C q (Ê q ) is the Clifford algebra with generators e 1 , ..., e q and e := e 1 e 2 ...e q . The vector space underlying our L ∞ structure L for SYM theory will be the same as for the Nahm equation extended to the two-dimensional world-volume of the Dp-branes, and we employ again the 2-Lie algebra L = su(N ). The L ∞ algebra will be supported on the infinite-dimensional vector space: where: L 0 := Ω 0 (Ê 1,p ) ⊗ L , (4.6) We view L as a -graded vector space whose grading is concentrated in degrees 0,1 and 2, with L 0 , L 1 and L 2 playing the role of homogeneous subspaces of the corresponding degree.
For what follows, we consider objects: We will treat A as a connection one-form on the wordlvolume valued in the trivial principal bundle G defined by the Lie group with Lie algebra g L , X as a section of the trivial vector bundle with fiber C 1 (Ê q ) ⊗ L associated with G via the representation induced by ⊲ and λ as a generator of the group of gauge transformations of G. For these fields, we now define various higher products µ k . The modified Koszul rule of the L ∞ products gives the relations µ 2 (X, A) = µ 2 (A, X), µ 3 (X, X, A) = µ 3 (X, A, X) = µ 3 (A, X, X) and    These are the expressions appearing in equation (4.3a). The products mapping to Ω 2 (Ê 1,1 )⊗ C 1 (Ê 8 ) ⊗ g L are defined according to 10) and they contain all the terms appearing in equation (4.3b). Every bracket maps elements of degree one into elements of degree two. As all brackets containing degree two elements vanish, the homotopy Jacobi identities are all satisfied trivially. Therefore, L with the brackets above forms an L ∞ algebra.
In this form, the N = 8 SYM equations in two or three dimensions read as In the homotopy Maurer-Cartan equations, we take φ = (A, ψ,ψ, X), where ψ ∈ Γ(S) and ψ ∈ Γ(S). We therefore extend L tô which we associate with the following grading: The additional products we have to introduce are straightforwardly read off the equations of motion (4.17): (4.20) The homotopy Jacobi identities are again trivially satisfied, as a product µ k 1 having another bracket µ k 2 amongst its arguments is defined to be vanishing.
To capture gauge transformations, we additionally introduce The homotopy Jacobi identities then extend the action of gauge transformations to products µ k containing ψ in the arguments. The homotopy Maurer-Cartan equations together with the L ∞ algebra L endowed with all of these brackets reproduce both the two-dimensional N = 8 SYM equations and the associated gauge transformations of the fields.
Note that the association of degrees to the various subspaces of L seems to be rather ad-hoc. A more enlightening approach to the grading of the L ∞ algebra is obtained from considering ghost number grading of a BRST complex, as done for pure Yang-Mills theory in [26].

Review of the BLG theory
We start from a metric 3-Lie algebra (L, ( , )) with associated Lie algebra g L . The bracket and the metric on L can be used to construct an invariant bilinear form (( , )) on g L , which is not identical with the Killing form on g L , and which is not positive definite [10]: (4.22) One easily verifies symmetry of (( , )). The matter field content of the BLG theory is given by the Goldstone fields arising from the spacetime symmetries broken by the presence of the M2-branes. We have thus 8 scalar fields X I , I = 1, ..., 8 and a Majorana spinor Ψ of Spin(1, 10) as their superpartner. The matter fields take values in the 3-Lie algebra L with invariant form ( , ). In addition, we have a gauge potential A µ taking values in Ω 1 (Ê 1,2 ) ⊗ g L . As gamma matrices, we use the 11-dimensional ones, which are split according to Γ M , M = 0, ..., 10 → (Γ µ , e I ), (µ = 0, .., 2, I = 1, ..., 8). For simplicity, we introduce X = e I X I and the invariant form ( , ) is assumed to include a trace where necessary. The Lagrangian density can then be written in the form: ) .
up to translations, equations of motion and gauge transformations. On infinitesimal level, the latter are generated by λ ∈ Ω 0 (Ê 1,2 ) ⊗ g L according to

Supersymmetric extension
For the supersymmetric extension, we use the field content presented in section 4.3, that is, we work with the spinor bundle S over Ê 1,10 restricted to Ê 1,2 . The fully supersymmetric equations of motion read as 6 To incorporate the fermions, we need to extend L as followŝ (4.37) and introduce the further products (4.38) These are enough to fully capture the equations of motion, and to allow for gauge transformations, we extend the set of defined products by The homotopy Jacobi identities then define for us the action of µ 2 (λ, ) on any of the products (4.38), and thus the description of the BLG equations together with their gauge symmetry in terms of homotopy Maurer-Cartan equations is complete.

From M2-branes to D2-branes
In [4], a nice reduction mechanism was proposed to descend from the BLG model with 3-Lie algebra A 4 describing two M2-branes to d = 3, N = 8 SYM theory with gauge group U(2) describing two D2-branes. The idea behind this is to compactify one direction, say x 10 on a circle and to assume that this compactification associates an expectation value with the corresponding scalar X8 = X 48 τ 4 : Rewriting the hMC equations of the BLG model using the expectation value for X8 as well as the decomposition of the gauge potential, the equations of motion for B a µ become algebraic, and we can eliminate this field. The result are the hMC equations of d = 3 N = 8 SYM theory plus corrections in lower powers of g Y M . In a strong coupling expansion, in which the subleading terms are neglected, the corrections vanish, and the effective description of a stack of two D2-branes is reproduced [4].

Appendix: Various extensions of the notion of a Lie algebra
In this appendix, we give a brief account of the various higher Lie structures used in this paper, which in particular fixes our conventions.

A. Ungraded extensions of the notion of Lie algebra
Let k be a field of characteristic zero, for example k = Ê or k = and R be an associative and commutative unital k-algebra, for example R = Ê or R = again.

n-Lie algebras in the sense of Filippov
Definition. An n-Lie algebra over R is a (unital) R-module L endowed with an R-multili- Notice that 2-Lie algebras are ordinary Lie algebras, for which the fundamental identity (A.2) corresponds to the usual Jacobi identity. A basic example is the n-Lie algebra A n+1 over Ê whose underlying vector space is an oriented Euclidean vector space of dimension n + 1 and whose n-bracket is given by the cross product of an ordered system of n vectors: which shows thatD is a linear representation (called the canonical representation) of the basic algebra on the underlying vector space of L. The Lie algebra of inner derivations is the image of this representation, i.e. we have Inn R (L) = ImD. The action of u ∈ g L on an element x ∈ L will be denoted by: The fact that the representationD of g L on L acts through derivations of (L, [ , . . . , ]) is equivalent with the statement that the n-bracket [ , . . . , ] satisfies Filippov's fundamental identity (A.2).
The algebra F L . When L is an ordinary Lie algebra with bracket [ , ], then we have [[ , ]] = [ , ] and g L coincides with L as a Lie algebra. In this case, the representationD of g L in L is the adjoint representation ad of L in itself. The fact that ad acts in L through derivations of (L, [ , ]) is the well-known reformulation of the classical Jacobi identity. Filippov's notion of n-Lie algebra is inspired by generalizing this take on the classical Jacobi identity. Let L be an n-Lie algebra with n > 2. Since we have an action of g L on L, the R-module F L := g L ⊕L carries an algebraic structure consisting of two R-bilinear and alternating maps, namely a 2-bracket [ , ] : F L × F L → F L and an n-bracket [ , . . . , ] : F ×n L → F L , defined as follows for all u, v, u 1 . . . u n ∈ g L and all x, y, x 1 . . . x n ∈ L: We will see later that the triple (F L , [ , ], [ , . . . , ]) is an (ungraded) L 2,n algebra over R. and satisfies the homotopy Jacobi identity: 14) The definition given above coincides with that used by Hanlon and Wachs [30] but differs from that used e.g. in [39]. The following result was proved by Dzhumadil'daev 7 : Proposition.
[31] Filippov's fundamental identity (A.2) implies the homotopy Jacobi identity, i.e. any n-Lie algebra (L, [ , . . . , ]) is also a Lie n-algebra. Dzhumadil'daev also shows that the converse statement is untrue, a counterexample being provided by Jacobian algebras, which are Lie n-algebras but not n-Lie.
As we shall see below, Lie n-algebras are the same as ungraded L n algebras, a particular case of ungraded L ∞ algebras. Hence Dzhumadil'daev's result above shows that the theory of n-Lie algebras is a special case of the theory of L ∞ algebras..
Definition. An (ungraded) L ∞ (or strong homotopy Lie) algebra over R is an R-module L endowed with a family of R-multilinear maps µ n : L ×n → L (n ≥ 1) such that: (1) Each µ n is alternating, i.e.: (2) Each of the following countable tower of ungraded L ∞ identities is satisfied: for all n ≥ 1 and all x 1 , ..., x n ∈ L.
Definition. Let S ⊂ AE * be a proper subset of the set AE * of all non-vanishing natural numbers. An (ungraded) L S algebra over R is an L ∞ algebra over R whose products µ i vanish for i / ∈ S.
Thus an L S algebra has only the products µ n with n ∈ S, and some of the identities (A.16) might be trivially satisfied. Some particular cases which are common in practice are the following: (1) S = {1 . . . n} for some n ≥ 1. In this case, and L S algebra is also called an L (n) algebra. Such an L ∞ algebra has only the products µ 1 . . . µ n , and one need only consider the first 2n − 1 identities in (A.16) (those with k = 1 . . . 2n − 1) since all identities with k ≥ 2n are satisfied trivially.
Remark. It is obvious from the definitions above that a Lie n-algebra is the same as an (ungraded) L n algebra.
(2) An L (2) algebra has a linear unary product µ 1 of weak degree and bilinear binary product µ 2 , the second being alternating. Conditions (A.16) say that µ 1 squares to zero, that it is a derivation of µ 2 : .18) and that µ 2 satisfies the Jacobi identity: These constraints state that (L, µ 1 , µ 2 ) is a differential Lie algebra. An L 2 algebra is an L (2) algebra with trivial differential, i.e. an ordinary Lie algebra. (

B. The graded versions
Preparations. Let G be an Abelian group and fix a group morphism φ : G → 2 . Recall that a G-graded R-modules is an R-module V endowed with a submodule decomposition: Given such a module, its pure (or homogeneous) elements are those elements x ∈ V for which there exists some g = g x such that x ∈ V g . In this case, we let deg x := g. We also letx := φ(deg x) = φ(g) ∈ 2 , which we will call the parity of x. Thus x is even if its parity equals0 ∈ 2 and odd if its parity equals1 ∈ 2 . Notice that the sign factor (−1)g is unambiguously defined. The reduced grading of V is the 2 The tensor powers ⊗ n R V of V are defined as usual, while the (G, φ)-graded symmetric and graded exterior powers ⊙ n R V and ∧ n R V are defined with sign factors induced by the parity of homogeneous elements. All these R-modules are again G-graded.
Given a permutation σ ∈ S n , we define the Koszul sign ǫ(σ, x 1 . . . x n ) of n pure elements x 1 . . . x n ∈ V via the identity: where x 1 ⊙ . . . ⊙ x n is the pure element of ⊗ n R V defined by x 1 . . . x n . The modified Koszul sign χ(σ, x 1 . . . x n ) is defined through: and we have: Given another group morphism ψ : G → 2 and a (G, ψ)-graded R-module U , an R-linear As in the 2 -graded case, we can define the notions of (G, φ)-graded symmetric and graded antisymmetric R-multilinear maps η : V ×n → V , and find that these factor through R-linear mapsη from ⊗ n R V and ∧ n R V to V , respectively. for all homogeneous x, y ∈ L. We letD := α and say that D is even if α =1 and that D is odd if α =1. The set of weakly homogeneous derivations of L is denoted by Der R (L). It is a graded Lie algebra when endowed with the usual graded commutator of homogeneous linear operators in L, which makes it into a graded Lie subalgebra of End R (L). For any homogeneous element x ∈ L, the associated adjoint action ad x is weakly homogeneous of parityx, and ad gives an R-linear representation of the graded Lie algebra (L, [ , ]) on L, i.e. ad : L → End R (L) is a morphism of graded Lie algebras. Given this property, the graded Jacobi identity (B.22) is equivalent to the statement that ad x is a weakly homogeneous derivation for every homogeneous element x ∈ L. Notice that 2-Lie algebras are ordinary (G, φ)graded Lie algebras, for which the graded fundamental identity (B.26) becomes the usual (G, φ)-graded Jacobi identity.
Let us fix a (G, φ)-graded n-Lie algebra L. Such an algebra defines an ordinary (G, φ)graded Lie bracket [[ , ]] (called the associated basic bracket) on the graded R-module g L := ∧ n−1 R L via: The (G, φ)-graded Lie R-algebra (g L , [[ , ]]) is called the basic graded Lie algebra of L.
Definition. An R-linear map D : L → L is called a weakly homogeneous derivation of L of parity α ∈ 2 if: for any x 1 , . . . , x n ∈ L. We setD := α. which shows thatD is a linear representation (called the canonical representation) of the basic algebra on the underlying graded vector space of L; this means thatD is a morphism of graded Lie algebras from g L to End R (L). The graded Lie algebra of inner derivations is the image of this representation, i.e. we have Inn R (L) = ImD. The action of u ∈ g L on an element x ∈ L will be denoted by: u ⊲ x :=D u (x) . When S = {n}, the corresponding L S algebras are also called graded L n algebras. When S = {1, . . . , n}, they are called graded L (n) algebras.
(1) Consider a (G, φ)-graded L (1) algebra structure on a G-graded vector space L. There is just one weakly homogeneous linear map µ 1 of weak degree1, and the conditions (B.37) reduce to the single condition µ 1 • µ 1 = 0, which says that (L, µ 1 ) is a (G, φ)-graded complex. Clearly a (G, φ)-graded L 1 algebra is the same thing as an L (1) algebra. The choice (A), (B) above for (G, φ) lead to the usual notions of -graded and 2 -graded complexes respectively.
C. Lie n-algebras are L ∞ algebras Proposition. A Lie n-algebra over k is the same as a (G, φ)-graded L n algebra L over k based on choice (C) for (G, φ), i.e. G = {0} and φ(0) = 0.
Proof. Indeed, substituting this choice of (G, φ) into the definition of an L n algebra obviously reproduces the definition of a Lie n-algebra. It follows that Lie n-algebras are simply the 'ungraded version' of L n algebras. Recall that any n-Lie algebra is a symmetric and therefore an Lie n-algebra [31]. It follows that any n-Lie algebra is an L n algebra. Therefore the theory of n-Lie algebras is a special case of the theory of L ∞ algebras. Thus, quite a few extensions of the notion of a Lie algebra which have been considered in the literature are particular cases of L ∞ algebras, which, as expected from the work of Stasheff, play a unifying role.