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Constructions of [L.sub.[infinity]] Algebras and Their Field Theory Realizations.

1. Introduction

Lie groups are ubiquitous in mathematics and theoretical physics as the structures formalizing the notion of continuous symmetries. Their infinitesimal objects are Lie algebras: vector spaces equipped with an antisymmetric bracket satisfying the Jacobi identity. In various contexts it is advantageous (if not strictly required) to generalize the notion of a Lie algebra so that the brackets do not satisfy the Jacobi identity. Rather, in addition to the "2-bracket," general "n-brackets" [l.sub.n] are introduced on a graded vector space for n = 1, 2, 3, ..., satisfying generalized Jacobi identities involving all brackets. Such structures, referred to as [L.sub.[infinity]] or strongly homotopy Lie algebras, first appeared in the physics literature in closed string field theory [1] and in the mathematics literature in topology [2-4]. A closely related cousin of [L.sub.[infinity]] algebras is [A.sub.[infinity]] algebras, which generalize associative algebras to structures without associativity [5, 6].

Our goal in this paper is to prove general theorems about the existence of [L.sub.[infinity]] structures for given "initial data" such as an antisymmetric bracket and to discuss their possible field theory realizations. First, as a warm-up, we answer the following natural question: Given a vector space V with an antisymmetric bracket [*,*], under which conditions can this algebra be extended to an [L.sub.[infinity]] algebra with [l.sub.2](v, w) = [v, w]? We will show that this is always possible. More specifically, we will prove the following theorem: The graded vector space X = [X.sub.1] + [X.sub.0], where [X.sub.0] = V is the space of degree zero and [X.sub.1] = [V.sup.*] is isomorphic to V and of degree one, carries a 2-term [L.sub.[infinity]] structure, meaning that the highest nontrivial product is [l.sub.3], which encodes the "Jacobiator" (i.e., the anomaly due to the failure of the original bracket to satisfy the Jacobi identity). We have been informed that this theorem is known to some experts, and one instance of it has been stated in [7], but we have not been able to find a proof in the literature. (See also [8, 9] for examples of finite-dimensional [L.sub.[infinity]] algebras.)

At first sight the above theorem may shed doubt on the usefulness of [L.sub.[infinity]] algebras, since it states that any generally non-Lie algebra can be extended to an [L.sub.[infinity]] algebra. It should be emphasized, however, that for a generic bracket the resulting structure is quite degenerate in that the 2-term [L.sub.[infinity]] algebra may not be extendable further in a nontrivial way, say by including a vector space [X.sub.-1]. Such extensions are particularly important for applications in theoretical physics as here [X.sub.-1] encodes the "space of physical fields", [X.sub.0] the space of "gauge parameters," and [X.sub.1] the space of "trivial parameters" whose action on fields vanishes [10]. Thus, if [X.sub.1] is isomorphic to [X.sub.0] there is no nontrivial action of [X.sub.0] on the physical fields and hence no genuine field theory realization of the [L.sub.[infinity]] algebra. In order to obtain nontrivial field theory realizations we will next prove a much more general theorem that covers the case of the Jacobiator being of a special form. Specifically, we will prove that if the Jacobiator takes values in the image of a linear operator that defines an ideal of the original algebra then there exists a 3-term [L.sub.[infinity]] algebra whose highest bracket in general is a nontrivial [l.sub.4]. A special case is the Courant bracket investigated by Roytenberg and Weinstein [11], for which the 4-bracket trivializes, but which is extendable and realized in string theory, in the form of double field theory [10,12,13].

We will illustrate these results with examples. Our investigation arose in fact out of the question whether the nonassociative octonions (more precisely, the 7-dimensional commutator algebra of imaginary octonions) can be viewed as part of an [L.sub.[infinity]] algebra. Our first theorem implies that the answer is affirmative, with the total graded space being 14dimensional, which we will see is minimal. However, given the theorem, the existence of this [L.sub.[infinity]] structure does not express a nontrivial fact about the octonions. Moreover, this [L.sub.[infinity]] structure is not extendable, which implies with the results of [10] that the octonions, at least when realized as a 2-term [L.sub.[infinity]] algebra, cannot realize a nontrivial gauge symmetry in field theory.

As recently discovered in [14] and further investigated in [15,16], the octonions are related to the phase space of nongeometric backgrounds in M-theory (nongeometric R-flux or non-geometric Kaluza-Klein monopoles in M-theory). Furthermore, a contraction of the octonions leads to the string theory "R-flux algebra" of [7,17-20] and also to the "magnetic monopole algebra" of [20-26]. The Jacobiator of the R-flux algebra only takes values in a one-dimensional subspace, and therefore these contracted nonassociative algebras may in fact be extendable. Here it is sufficient to take [X.sub.1] to be one-dimensional, leading to an 8-dimensional [L.sub.[infinity]] algebra. (A 14-dimensional and hence nonminimal [L.sub.[infinity]] realization of the R-flux algebra has already been given in [7].)

The remainder of this paper is organized as follows. In Section 2 we briefly review the axioms of [L.sub.[infinity]] algebras. In Section 3 we prove the theorem that for arbitrary 2-bracket as initial data there is an [L.sub.[infinity]] structure on the "doubled" vector space. This theorem will then be significantly generalized in Section 4. In Section 5 we discuss examples, such as the octonions, the "R-flux algebra," and the Courant algebroid. In the appendix we prove an analogous result for [A.sub.[infinity]] algebras.

2. Axioms of [L.sub.[infinity]] Algebras

We begin by stating the axioms of an [L.sub.[infinity]] algebra. It is defined on a graded vector space

[mathematical expression not reproducible], (1)

and we refer to elements in [X.sub.n] as having degree n. We also refer to algebras with [X.sub.n] = 0 for all n with [absolute value of n] [greater than or equal to] k as a k-term [L.sub.[infinity]] algebra. There are a potentially infinite number of generalized multilinear products or brackets [l.sub.k] having k inputs and intrinsic degree k-2, meaning that they take values in a vector space whose degree is given by

deg([l.sub.k]([x.sub.1], ..., [x.sub.k])) = k - 2 + [k.summation over (i=1)]deg ([x.sub.i]). (2)

For instance, [l.sub.1] has intrinsic degree -1, implying that it acts on the graded vector space according to

[mathematical expression not reproducible]. (3)

Moreover, the brackets are graded (anti-)commutative in that, e.g., [l.sub.2] satisfies

[mathematical expression not reproducible], (4)

and similarly for all other brackets.

The brackets have to satisfy a (potentially infinite) number of generalized Jacobi identities. In order to state these identities we have to define the Koszul sign [epsilon]([sigma]; x) for any [sigma] in the permutation group of k objects and a choice x = ([x.sub.1], ..., [x.sub.k]) of k such objects. It can be defined implicitly by considering a graded commutative algebra with

[mathematical expression not reproducible], (5)

where in exponents [x.sub.i] denotes the degree of the corresponding element. The Koszul sign is then inferred from

[x.sub.1] [conjunction] ... [conjunction] [x.sub.k] = [epsilon]([sigma]; x)[x.sub.[sigma](1)] [conjunction] ... [conjunction] [x.sub.[sigma](k)]. (6)

The [L.sub.[infinity]] relations are given by

[summation over (i+j=n+1)][(-1).sup.i(j-1)] [summation over ([sigma])][(-1).sup.[sigma]][epsilon]([sigma]; x) x [l.sub.j]([l.sub.i]([x.sub.[sigma](1)], ..., [x.sub.[sigma](i)]), [x.sub.[sigma](i+1)], ... [x.sub.a(n)]) = 0, (7)

for each n = 1, 2, 3, ..., which indicates the total number of inputs. Here [(-1).sup.[sigma]] gives a plus sign if the permutation is even and a minus sign if the permutation is odd. Moreover, the inner sum runs, for a given i, j [greater than or equal to] 1, over all permutations [sigma] of n objects whose arguments are partially ordered ("unshuffles"), satisfying

[sigma](1) [less than or equal to] ... [less than or equal to] [sigma](i), [sigma](i + 1) [less than or equal to] ... [less than or equal to] [sigma](n). (8)

We will now state these relations explicitly for the values of n relevant for our subsequent analysis. For n = 1 the identity reduces to

[l.sub.1]([l.sub.1](x)) = 0, (9)

stating that [l.sub.1] is nilpotent, so that (3) is a chain complex. For n = 2 the identity reads

[mathematical expression not reproducible], (10)

meaning that [l.sub.1] acts like a derivation on the product [l.sub.2]. For n = 3 one obtains

[mathematical expression not reproducible]. (11)

We recognize the last three lines as the usual Jacobiator. Thus, this relation encodes the failure of the 2-bracket to satisfy the Jacobi identity in terms of a 1- and 3-bracket and the failure of li to act as a derivation on [l.sub.3]. Finally, the n = 4 relations read

[mathematical expression not reproducible], (12)

where we named the l.h.s. O([x.sub.1], ..., [x.sub.4]) for later convenience. For a 2-term [L.sub.[infinity]] algebra there are no 4-brackets and hence the above right-hand side is zero. The n = 4 relation then poses a nontrivial constraint on [l.sub.2] and [l.sub.3], while all higher [L.sub.[infinity]] relations will be automatically satisfied.

3. A Warm-Up Theorem

We now prove the first theorem stated in the introduction. Consider an algebra (V, [*, *]) with bilinear antisymmetric 2-bracket, i.e.,

[v, w] = - [w, v] [for all]v, w [member of] V, (13)

but we do not assume that the bracket satisfies any further constraints. In particular, the Jacobi identity is generally not satisfied, so that the Jacobiator

Jac(u, v, w) [equivalent to] [[u, v], w] + [[v, w], u] + [[w, u], v], (14)

in general is nonzero. We then have the following.

Theorem 1. The graded vector space

X = [X.sub.1] + [X.sub.0], (15)

where [X.sub.0] = V and [X.sub.1] = [V.sup.*] with [V.sup.*] a vector space isomorphic to V, carries a 2-term [L.sub.[infinity]] structure whose nontrivial brackets are given by

[l.sub.1]([v.sup.*]) = v, (16)

[l.sub.2](v, w) = [v, w], (17)

[l.sub.2]([v.sup.*], w) = [[v, w].sup.*], (18)

[l.sub.3](u, v, w) = -Jac[(u, v, w).sup.*]. (19)

Comment. We denote the elements of [V.sup.*] by [v.sup.*], [w.sup.*], etc., and the isomorphism by

[mathematical expression not reproducible], (20)

and similarly for its inverse. For instance, if V carries a non-degenerate metric we can take [V.sup.*] to be the dual vector space of V and the isomorphism to be the canonical isomorphism. (More simply, we can think of [V.sup.*] as a second copy of V and of the isomorphism as the identity, but at least for notational reasons it is important to view V and [V.sup.*] as different objects.)

Proof. The proof proceeds straightforwardly by fixing the products so that the n= 1, 2, 3 relations are partially satisfied and then verifying that in fact all [L.sub.[infinity]] relations are satisfied. First, [l.sub.1] maps [X.sub.1] = [V.sup.*] to [X.sub.0] = V, and we take it to be given by the (inverse) isomorphism (20),

[for all][v.sup.*] [member of] [X.sub.1], v [member of] [X.sub.0] : [l.sub.1]([v.sup.*]) = v, [l.sub.1](v) = 0. (21)

The second relation in here is necessary because there is no space [X.sub.-1] in (15). The n = 1 relations [l.sup.2.sub.1] = 0 then hold trivially.

Next, we fix the [l.sub.2] product by requiring [l.sub.2](v, w) = [v, w] on [X.sub.0] = V and imposing the n = 2 relation (10). For arguments of total degree 0 this relation is trivial because of the second relation in (21). For arguments of total degree 1 we have

[l.sub.1]([l.sub.2]([v.sup.*], w)) = [l.sub.2]([l.sub.1]([v.sup.*]), w) - [l.sub.2]([v.sup.*], [l.sub.1](w)) = [l.sub.2](v, w) = [v, w], (22)

where we used (21). Using (21) on the l.h.s. we infer

[l.sub.2]([v.sup.*], w) = [[v, w].sup.*] [??] [l.sub.2](w, [v.sup.*]) = [[w, v].sup.*]. (23)

Since there is no space [X.sub.2] we have [l.sub.2]([v.sup.*], [w.sup.*]) = 0. This is consistent with the n = 2 relation (10) for arguments of total degree 2:

0 = [l.sub.1]([l.sub.2]([v.sup.*], [w.sup.*])) = [l.sub.2](v, [w.sup.*]) - [l.sub.2]([v.sup.*], w) = [[v, w].sup.*] - [[v, w].sup.*] = 0,

where we used (23). Thus, all n = 2 relations are satisfied.

Let us now consider the n = 3 relations (11). For arguments of total degree 0 (i.e., all taking values in [X.sub.0]), it reads

[mathematical expression not reproducible], (25)

from which we infer

[l.sub.3](u, v, w) = -Jac[(u, v, w).sup.*] [member of] [X.sub.1]. (26)

Due to the antisymmetry of the bracket [*,*], the Jacobiator is completely antisymmetric in all arguments, and (26) is consistent with the required graded commutativity of [l.sub.3]. Since there is no space [X.sub.2], [l.sub.3] is trivial for any arguments in [X.sub.1]. We have thus determined all nontrivial n-brackets.

So far we have verified the n = 1, 2 relations and the n = 3 relation for arguments of total degree 0. We now verify the remaining [L.sub.[infinity]] relations. The n = 3 relation for arguments of total degree 1 reads

[mathematical expression not reproducible], (27)

and is thus satisfied. The n = 3 relations for arguments of total degree larger than 1 are trivially satisfied, completing the proof of all n = 3 relations.

Finally, we have to verify the n = 4 relations. Since there is no nontrivial [l.sub.4] these require that the left-hand side of (12) vanishes identically for [l.sub.2] and [l.sub.3] defined above. This follows by a direct computation that we display in detail. First, for arguments [v.sub.1], [v.sub.2], [v.sub.3], [v.sub.4] [member of] [X.sub.0] of total degree 0 one may verify that (12) is completely antisymmetric in the four arguments. Writing [[summation].sub.anti] for the totally antisymmetrized sum (carrying 4! = 24 terms and pre-factor 1/4!) we then compute for the left-hand side of (12)

[mathematical expression not reproducible]. (28)

Here we used repeatedly the total antisymmetry in the four arguments, in particular in the last step that under the sum [[[[v.sub.3], [v.sub.4]], [[v.sub.1], [v.sub.2]]].sup.*] then vanishes. The n = 4 relations for arguments of total degree 1 or higher are trivially satisfied because they would have to take values in spaces of degree 2 or higher, which do not exist. The [L.sub.[infinity]] relations for n > 4 are trivially satisfied for the same reason. This completes the proof.

4. Main Theorem

The above theorem states that an arbitrary bracket can be extended to an [L.sub.[infinity]] algebra. For generic brackets, this [L.sub.[infinity]] structure is, however, quite degenerate in that it may not be extendable further, say by adding a further space [X.sub.-1]. Indeed, if the violation of the Jacobi identity is "maximal" and the Jacobiator takes values in all of V, the space [X.sub.1] has to be as large as V, and the image of the map [l.sub.1] : [X.sub.1] [right arrow] [X.sub.0] equals [X.sub.0] = V. Consequently, one cannot introduce a further space [X.sub.-1] together with a nontrivial [l.sub.1] : [X.sub.0] [right arrow] [X.sub.-1] satisfying [l.sup.2.sub.1] = 0. Since in physical applications [X.sub.-1] serves as the space of fields, such brackets do not lead to [L.sub.[infinity]] algebras encoding a nontrivial gauge symmetry.

More interesting situations arise when the Jacobiator takes values in a proper subspace U [subset] V, for then it is sufficient to set [X.sub.1] = U and to take [l.sub.1] = 1 to be the "inclusion" defined for any u [member of] U by [iota](u) = u, viewing u as an element of V. Indeed, it is easy to verify, provided the subspace forms an ideal (i.e., [for all]u [member of] U, v [member of] V : [u, v] [member of] U), that the above proof goes through as before. In this case, further extensions of the [L.sub.[infinity]] algebra may exist. In the following we will prove a yet more general theorem that is applicable to situations where the Jacobiator takes values in the image of a linear map that itself may have a nontrivial kernel. Then there is an extension to a 3-term [L.sub.[infinity]] algebra that generally requires a nontrivial 4-bracket.

Theorem 2. Let (V, [*, *]) be an algebra with bilinear antisymmetric 2-bracket as in Section 3, and let D : U [right arrow] V be a linear map satisfying the closure conditions

[Im (D), V] [subset] Im (D), (29)

together with the Jacobiator relation

[for all][v.sub.1], [v.sub.2], [v.sub.3] [member of] V: Jac ([v.sub.1], [v.sub.2], [v.sub.3]) [member of] Im (D), (30)

where Im(D) and ker(D) denote image and kernel of D, respectively. Then there exists a 3-term Lm structure with [l.sub.2](v, w) = [v, w] on the graded vector space with

[mathematical expression not reproducible], (31)

where [X.sub.0] = V, [X.sub.1] = U, [X.sub.2] = ker(D) and [iota] denotes the inclusion of ker(D) into U. The highest nontrivial bracket in general is given by the 4-bracket (and the complete list of nontrivial brackets is given in eq. (54) below).

Notation and Comments. We denote the elements of V by v, w, ..., the elements of U by [alpha], [beta], ... and the elements of ker(D) by c, c', ... The condition (30) implies that there is a multilinear and totally antisymmetric map f : [V.sup.[cross product]3] [right arrow] U so that

[for all][v.sub.1], [v.sub.2], [v.sub.3] [member of] V: Jac ([v.sub.1], [v.sub.2], [v.sub.3]) = Df([v.sub.1], [v.sub.2], [v.sub.3]). (32)

The condition (29) states that the bracket of an arbitrary v [member of] V with D[alpha], [alpha] [member of] U, lies in the image of D, i.e., we can write

[for all]v [member of] V, [alpha] [member of] U: [D[alpha], v] = D (v([alpha])), v([alpha]) [member of] U. (33)

We can think of the operation on the r.h.s. as defining for each v [member of] V a map on U, [alpha] [??] v([alpha]) [member of] U. This map is defined by (33) only up contributions in the kernel, as is the function f in (32), but the following construction goes through for any choice of functions satisfying (33), (32). (The algebras resulting for different choices of these functions are almost certainly equivalent under suitably defined [L.sub.[infinity]] isomorphisms; see, e.g., [27], but we leave a detailed analysis for future work.)

Proof. As for Theorem 1, the proof proceeds by determining the n-brackets from the [L.sub.[infinity]] relations as far as possible and then proving that in fact all relations are satisfied. The n = 1 relations [l.sup.2.sub.1] = 0 for [l.sub.1] defined in (31) are satisfied by definition since D([iota](c)) = 0 for all c [member of] ker(D). In the following we systematically go through all relations for n = 1, ..., 5.

n = 2 relations: The n = 2 relations are satisfied for arguments of total degree zero, since [l.sub.1] acts trivially on [X.sub.0]. For arguments [alpha] [member of] [X.sub.1], v [member of] [X.sub.0] of total degree 1 we need

[l.sub.1]([l.sub.2]([alpha], v)) = [l.sub.2]([l.sub.1]([alpha]), v) = [D[alpha], v] = D(v([alpha])), (34)

where we used (33). As the l.h.s. equals D([l.sub.2](a, v)), this relation is satisfied if we set

[l.sub.2]([alpha], v) = v([alpha]) [member of] [X.sub.1]. (35)

For arguments [alpha], [beta] [member of] [X.sub.1] of total weight 2 we compute

[mathematical expression not reproducible], (36)

using (35) in the last step. As [l.sub.1] on the l.h.s. acts by inclusion, we can satisfy this relation by setting

[l.sub.2]([alpha], [beta]) = -(D[alpha])([beta]) - (D[beta])([alpha]) [member of] ker(D), (37)

but it remains to prove that the r.h.s. indeed takes values in the kernel. This follows by setting v = D[beta] in (33):

[D[alpha], D[beta]] = D((D[beta])([alpha])) [??] D((D[alpha])([beta]) + (D[beta])([alpha])) = 0, (38)

using the fact that the bracket is antisymmetric. Note that (37) is properly symmetric in its two arguments, in agreement with the graded commutativity (4). Another choice of arguments of total degree 2 is v [member of] [X.sub.0], c [member of] [X.sub.2], for which we require

[l.sub.1]([l.sub.2](v, c)) = [l.sub.2]([l.sub.1](v), c) + [l.sub.2](v, [l.sub.1](c)) = [l.sub.2](v, [iota](c)) = -v([iota](c)), (39)

where we used (35) in the last step, recalling [iota](c) [member of] [X.sub.1]. Thus, using [l.sub.1] = [iota] on the l.h.s. together with the graded symmetry we have

[iota]([l.sub.2](c, v)) = v([iota](c)). (40)

We can also write this as (Here we employ the map on [X.sub.2] induced by v([alpha]) via v(c) := v([iota](c)), which lies in ker(D) as a consequence of Dc = 0 and (33))

[for all]c [member of] [X.sub.2], v [member of] [X.sub.0] : [l.sub.2](c, v) = v(c) [member of] [X.sub.2]. (41)

We next consider arguments c [member of] [X.sub.2], a [member of] [X.sub.1] of total degree 3, for which [l.sub.2] must vanish as there is no vector space [X.sub.3]. This leads to a constraint from the n = 2 relation:

0 = [l.sub.1]([l.sub.2](c, [alpha])) = [l.sub.2]([iota](c), [alpha]) + [l.sub.2](c, D[alpha]) = -(D[alpha])(c) + [l.sub.2](c, D[alpha]), (42)

where we used (37) and Dc = 0. This relation is satisfied for (41). Finally, the n = 2 relations are trivially satisfied for arguments of total degree 4 or higher, completing the proof of all n = 2 relations.

n = 3 relations: We now consider the n = 3 relations for arguments [v.sub.1], [v.sub.2], [v.sub.3] [member of] [X.sub.0] of total degree zero:

0 = [l.sub.1]([l.sub.3]([v.sub.1], [v.sub.2], [v.sub.3])) + Jac ([v.sub.1], [v.sub.2], [v.sub.3]). (43)

Recalling (32) and that [l.sub.1] = D when acting on [X.sub.1], we infer that this relation is satisfied for

[l.sub.3]([v.sub.1], [v.sub.2], [v.sub.3])) = -f([v.sub.1], [v.sub.2], [v.sub.3]) [member of] [X.sub.1]. (44)

Next, for arguments [alpha] [member of] [X.sub.1], [v.sub.1], [v.sub.2] [member of] [X.sub.0] of total weight 1 the n = 3 relation reads

[mathematical expression not reproducible], (45)

where we used repeatedly (35). Moreover, we used (44) and that [l.sub.3]([alpha], [v.sub.1], [v.sub.2]) [member of] [X.sub.2] on which [l.sub.1] acts as the inclusion. We will next prove that the function

g([alpha], [v.sub.1], [v.sub.2]) [equivalent to] f(D[alpha], [v.sub.1], [v.sub.2]) + [[v.sub.1], [v.sub.2]]([alpha]) + [v.sub.1]([v.sub.2]([alpha])) - [v.sub.2]([v.sub.1]([alpha])), (46)

takes values in the subspace ker(D). We have to prove that the r.h.s. is annihilated by D. To this end we compute for the first term with (32)

[mathematical expression not reproducible], (47)

where we repeatedly used (33). This show that the r.h.s. of (46) is annihilated by D, proving that g takes values in [X.sub.2] = ker(D). We can thus satisfy (45) by setting

[l.sub.3]([alpha], [v.sub.1], [v.sub.2]) = g([alpha], [v.sub.1], [v.sub.2]) [member of] [X.sub.2]. (48)

We next recall that there can be no nontrivial [l.sub.3] for arguments [[alpha].sub.1], [[alpha].sub.2] [member of] [X.sub.1], v [member of] [X.sub.0] of total degree 2. Thus, the n = 3 relation for these arguments has to be satisfied for the products already defined. We then compute from (11), noting that it is symmetric in [[alpha].sub.1], [[alpha].sub.2] and writing [[summation].sub.sym] for the symmetrized sum,

[mathematical expression not reproducible], (49)

where we used (35), (37), and, in the third equality, (48). It is now easy to see that under the symmetrized sum all terms cancel, using in particular that f is totally antisymmetric. Thus, this n = 3 relation is satisfied. Since the n = 3 relations for total degree 3 or higher are trivially satisfied, we have completed the proof of all n = 3 relations.

n = 4 relations: We consider the n = 4 relations (12) for arguments of total degree 0. Precisely as in (28) we compute

[mathematical expression not reproducible]. (50)

In contrast to (28) this is not zero in general, but we can now have a nontrivial [l.sub.4] taking values in [X.sub.2]. We next prove that the function defined by

[mathematical expression not reproducible], (51)

takes values in ker(D). To this end we have to show that it is annihilated by D:

[mathematical expression not reproducible]. (52)

Thus, the n = 4 relation can be satisfied by setting

[1.sub.4]([v.sub.1], ..., [v.sub.4]) = h([v.sub.1], ..., [v.sub.4]) [member of] [X.sub.2]. (53)

We have now determined all nontrivial brackets, which we summarize here:

[mathematical expression not reproducible], (54)

with the functions g, h defined in (46) and (51), respectively. All further [L.sub.[infinity]] relations have to be satisfied identically. Let us next consider the n = 4 relations (12) for arguments [v.sub.1], [v.sub.2], [v.sub.3] [member of] [X.sub.0], [alpha] [member of] [X.sub.1] of total degree 1. It is easy to see that (12) is then totally antisymmetric in [v.sub.1], [v.sub.2], [v.sub.3], and writing [[summation].sub.anti] for the antisymmetric sum over these three arguments we compute

[mathematical expression not reproducible], (55)

where we used the products already defined, in particular (48), and the relation (32) for the Jacobiator. We observe that various terms cancelled under the totally antisymmetric sum. In order to satisfy the n = 4 relation (12), the remaining terms need to be equal to [l.sub.4]([v.sub.1], [v.sub.2], [v.sub.3], D[alpha]). To see this note that writing (53) with an antisymmetrized sum over only the first three arguments one obtains

[mathematical expression not reproducible]. (56)

Specializing this to [l.sub.4]([v.sub.1], [v.sub.2], [v.sub.3], D[alpha]) we infer that it equals (55), completing the proof of this n = 4 relation. It is easy to see that for arguments of total degree 2 or higher the n = 4 relations are trivially satisfied. Thus, we have verified all n = 4 relations.

n = 5 relations: We have not displayed the [L.sub.[infinity]] relations in Section 2 for n [greater than or equal to] 5 explicitly as these get increasingly laborious. However, it is easy to see that here the only nontrivial n = 5 relation has arguments [v.sub.1], ..., [v.sub.5] [member of] [X.sub.0], which are of even degree so that the Koszul sign becomes [epsilon]([sigma]; v) = 1. Moreover, [l.sub.5] is trivial, and it is then easy to verify that (7) reduces to

[mathematical expression not reproducible], (57)

where the sum antisymmetrizes over all five arguments. Upon inserting the products in (54), it is a straightforward direct calculation, largely analogous to (55), to verify that this relation is identically satisfied. As these are the only nontrivial [L.sub.[infinity]] relations for n = 5 or higher, this completes the proof.

Specializations. As a special case of Theorem 2 let us assume that the Jacobiator takes values in a subspace U [subset] V, which forms an ideal of the bracket. In this case we can take D = [iota] to be the inclusion map U [right arrow] V. Since its kernel is trivial, we have [X.sub.2] = {0}, and the algebra can be reduced to a 2-term [L.sub.[infinity]] algebra. Indeed, the action of v [member of] V on U that is implicit in (33) then reduces to

u [??] V (u) [equivalent to] -[v, u] [member of] U. (58)

Using this and Jac([v.sub.1], [v.sub.2], [v.sub.3]) = f([v.sub.1], [v.sub.2], [v.sub.3]), it is straightforward to verify that all products in (54) that take values in [X.sub.2] trivialize. In particular, [l.sub.4] trivializes. Theorem 1 is contained as a special case, for which U = V.

5. Examples

We will now discuss a few examples, which get increasingly less trivial, with the goal to illustrate the scope of the above theorems.

The octonions: The seven imaginary octonions [e.sub.a], a = 1, ..., 7 satisfy the algebra

[e.sub.a][e.sub.b] = -[[delta].sub.ab]1 + [[eta].sub.abc][e.sub.c], (59)

and thus the commutation relations

[[e.sub.a], [e.sub.b]] = 2[[eta].sub.abc][e.sub.c], (60)

where the structure constants are defined as follows. Splitting the index as a = (i, [bar.i], 7), where i, [bar.i] = 1, 2, 3, [[eta].sub.abc] is the totally antisymmetric tensor defined by

[mathematical expression not reproducible], (61)

with the three-dimensional Levi-Civita symbol satisfying [[epsilon].sub.123] = 1. (This coincides with the conventions of [14].) [[eta].sub.abc] satisfy the following relations

[[eta].sub.abe][[eta].sub.cde] = 2[[delta].sub.c[a][[delta].sub.b]d] - [[THETA].sub.abcd], [[THETA].sub.abcd] [equivalent to] [1/3!][[epsilon].sub.abcdefg][[eta].sub.efg]. (63)

Using these it is straightforward to compute the Jacobiator:

Jac([e.sub.a], [e.sub.b], [e.sub.c]) = -12[[THETA].sub.abcd][e.sub.d]. (63)

It is easy to verify with this expression that each generator appears on the right-hand side, see [14]. Thus, the Jacobiator does not take values in a proper subspace, and therefore the [L.sub.[infinity]] extension requires a doubling to a 14-dimensional space (with basis {[e.sub.a], [e.sup.*.sub.a]}) as in Theorem 1, with the nontrivial brackets being given in addition to (60) by

[mathematical expression not reproducible]. (64)

There is no further nontrivial extension; in particular, this algebra cannot describe a nontrivial gauge symmetry in a field theory.

The R-flux algebra: This algebra, introduced in [17-19], is a contraction of the algebra of imaginary octonions in the following sense [14]: (As shown in [16], the algebra of octonions can be also contracted in an analogous way to the magnetic monopole algebra, which is isomorphic to the R-flux algebra upon exchange of position and momentum variables.) We decompose [e.sub.a] = ([e.sub.i], [f.sub.i], [e.sub.7]), with i = 1, 2, 3, and introduce a scaling parameter [lambda] to define

[mathematical expression not reproducible]. (65)

Expressing the algebra (60) now in terms of x, p, I and sending [lambda] [right arrow] 0 one obtains the.R-flux algebra

[mathematical expression not reproducible], (66)

where I is a central element that commutes with everything. It is easy to see that the only nonvanishing Jacobiator is

Jac([x.sup.i], [x.sup.j], [x.sup.k]) = 3[e.sup.ijk]I. (67)

Thus, the Jacobiator takes values in the one-dimensional subspace spanned by I. According to the specialization discussed after the proof of Theorem 2, we can then define an [L.sub.[infinity]] structure on [X.sub.1] + [X.sub.0], where [X.sub.0] = {[x.sup.i], [p.sub.i], I} and [X.sub.1] = {[I.sup.*]}. In addition to the 2-brackets defined by (66) we have the nontrivial products

[l.sub.i]([I.sup.*]) = I, [l.sub.3]([x.sup.i], [x.sup.j], [x.sup.k]) = -3[[epsilon].sup.ijk][I.sup.*]. (68)

The Courant algebroid: The Courant bracket of generalized geometry or the "C-bracket" of double field theory has a nonvanishing Jacobiator. Denoting the arguments of this bracket, i.e., the elements of [X.sub.0], by [[xi].sub.1], [[xi].sub.2], etc., it is given by

[mathematical expression not reproducible], (69)

where <,> denotes the O(d, d) invariant metric and D is the exterior derivative in generalized geometry or the doubled partial derivative in double field theory. The bracket satisfies for a function [chi]

[D[chi], [xi]] = -[1/2]D<D[chi], [xi]>, (70)

so that for our current notation we read off with (33)

[xi]([chi]) = -[1/2]<D[chi], [xi]>. (71)

It was established by Roytenberg and Weinstein that the Courant algebroid defines a 2-term [L.sub.[infinity]] algebra with the highest bracket being [l.sub.3], which is defined by f, and [X.sub.1] being the space of functions [11]. The space [X.sub.2] of constants (the kernel of the differential operator D) is not needed as all brackets in (54) taking values in [X.sub.2] vanish. For instance, [l.sub.2] for two functions [[chi].sub.1], [[chi].sub.2] [member of] [X.sub.1] becomes

[l.sub.2]([[chi].sub.1], [[chi].sub.2]) = -(D[[chi].sub.1]) ([[chi].sub.2]) - (D[[chi].sub.1])([[chi].sub.2]) = <D[[chi].sub.1], D[[chi].sub.2]> = 0. (72)

In double field theory language this is zero because of the "strong constraint," and it is also one of the axioms of a Courant algebroid (see definition 3.2, axiom 4 in [11]). The vanishing of all other products taking values in [X.sub.2] can be verified similarly using the relations given, for instance, in [10]. Thus, the existence of an [L.sub.[infinity]] structure on the Courant algebroid is a corollary of the more general Theorem 2.

6. Conclusions

We established general theorems about the existence of [L.sub.[infinity]] algebras for a given bracket and discussed possible field theory realizations. This includes well-known examples such as the Courant algebroid as special cases. Most importantly, it then remains to construct explicit examples of algebras that obey the conditions of Theorem 2 and that really do use the full structure possible, particularly a nontrivial 4-bracket. This may require identifying a structure that relaxes some of the axioms of a Courant algebroid.

Moreover, it is clear that there will be further generalizations of this theorem. For instance, the construction of Theorem 2 could be extended by taking the map [l.sub.1] : [X.sub.2] [right arrow] [X.sub.1] not to be the inclusion map but rather a nontrivial operator that again could have a nontrivial kernel, which in turn would necessitate a new space [X.sub.3] and higher brackets beyond a 4-bracket. These may be useful for generalizations of double and exceptional field theory [28, 29]. Indeed, it is to be expected that the gauge structure of exceptional field theory requires [L.sub.[infinity]] algebras with arbitrarily high brackets [30], as is also the case in closed string field theory [1]. Moreover, in order to obtain interesting [L.sub.[infinity]] algebras with nontrivial field theory realizations, for special cases it is instrumental to take an appropriate bracket as starting point. For instance, for the [E.sub.8(8)] theory in [31] the naive bracket does not yield a Jacobiator living in the image of an appropriate operator (or, equivalently, the naive bracket does not transform covariantly under its own "adjoint" action [32]), but rather the vector space has to be suitably enlarged from the beginning, leading to a so-called Leibniz-Loday structure [33].

Appendix

[A.sub.[infinity]] and Nonassociative Algebras

In analogy to the doubling of vector spaces introduced for the [L.sub.[infinity]] realization of Theorem 1 we will show that every nonassociative algebra has a realization as an [A.sub.[infinity]] algebra. An [A.sub.[infinity]] algebra is a graded vector space V together with a collection [[m.sub.k] | k [member of] N} of multilinear maps [m.sub.k] : [[cross product].sup.k]V [right arrow] V of internal degree k - 2 satisfying the following fundamental identity [4]

[mathematical expression not reproducible], (A.1)

for every n [member of] N. The first four equations read explicitly

(i) n = 1, deg = -2:

0 = [m.sub.1]([m.sub.1] ([a.sub.1])). (A.2)

(ii) n = 2, deg = -1:

[mathematical expression not reproducible]. (A.3)

(iii) n = 3, deg = 0:

[mathematical expression not reproducible]. (A.4)

(iv) n = 4, deg = 1:

[mathematical expression not reproducible]. (A.5)

Let (V,*) be a nonassociative algebra and [V.sup.*] a vector space isomorphic to V with the isomorphism denoted by V [contains as member of] a [??] [a.sup.*] [member of] [V.sup.*]. The graded vector space of the [A.sub.[infinity]] algebra is then defined as

[X.sub.1] = [V.sup.*], [X.sub.0] = V. (A.6)

In addition we define the following products

[m.sub.1]([a.sup.*]) = a, [m.sub.2]([a.sub.1], [a.sub.2]) = [a.sub.1] x [a.sub.2]. (A.7)

Using this construction, the n = 1 [A.sub.[infinity]] equation is trivially satisfied. For the second equation we compute

[mathematical expression not reproducible] (A.8)

= -[m.sub.1]([m.sub.2] ([a.sup.*.sub.1], [a.sub.2])) + [a.sub.1] x [a.sub.2], (A.9)

from which we conclude

[m.sub.2]([a.sup.*.sub.1], [a.sub.2]) = [([a.sub.1] x [a.sub.2]).sup.*]. (A.10)

For two arguments of degree 1 we compute

[mathematical expression not reproducible] (A.11)

= [m.sub.2]([a.sub.1], [a.sup.*.sub.2]) - [([a.sub.1] x [a.sub.2]).sup.*], (A.12)

from which we conclude

[m.sub.2]([a.sub.1], [a.sup.*.sub.2]) = [([a.sub.1] x [a.sub.2]).sup.*]. (A.13)

Note that the m-products have no a priori symmetry properties, so the [m.sub.2]-product has to be specified for each order of entries individually.

The n = 3 equations read

0 = [m.sub.1]([m.sub.3]([a.sub.1], [a.sub.2], [a.sub.3])) + [m.sub.2]([m.sub.2]([a.sub.1], [a.sub.2]), [a.sub.3]) - [m.sub.2]([a.sub.1], [m.sub.2]([a.sub.2], [a.sub.3])) (A.14)

= [m.sub.1]([m.sub.3]([a.sub.1], [a.sub.2], [a.sub.3])) + ([a.sub.1] x [a.sub.2]) x [a.sub.3] - [a.sub.1] x ([a.sub.2] x [a.sub.3]), (A.15)

from which we infer that the 3-product is defined by the associator:

[m.sub.3]([a.sub.1], [a.sub.2], [a.sub.3]) = -Ass [([a.sub.1], [a.sub.2], [a.sub.3]).sup.*]. (A.16)

Moreover, for total degree 1 we compute

[mathematical expression not reproducible] (A.17)

= -Ass[([a.sub.1], [a.sub.2], [a.sub.3]).sup.*] + [(([a.sub.1] x [a.sub.2]) * [a.sub.3]).sup.*] - ([a.sub.1] x ([a.sub.2] x [a.sub.3])), (A.18)

which is therefore satisfied.

We claim that the n = 4 equations are satisfied for [m.sub.4] = 0, which we verify by a direct computation:

[mathematical expression not reproducible]. (A.19)

The terms exactly cancel. This completes the proof that any nonassociative algebra can be embedded into an [A.sub.[infinity]] algebra.

https://doi.org/10.1155/2018/9282905

Data Availability

This work is theoretical and does not use any data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

We would like to thank Ralph Blumenhagen, Michael Fuchs, Ezra Getzler, Tom Lada, Martin Rocek, Christian Saemann, Jim Stasheff, Richard Szabo, and Barton Zwiebach for useful discussions and comments. Olaf Hohm is supported by a DFG Heisenberg Fellowship of the German Science Foundation (DFG). Vladislav Kupriyanov is supported by the Capes-Humboldt Fellowship No. 0079/16-2. The work of Dieter Lust is supported by the ERC Advanced Grant No. 320045 "Strings and Gravity."

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Olaf Hohm (iD), (1) Vladislav Kupriyanov, (2,3) Dieter Lust, (2,4) and Matthias Traube (2)

(1) Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794-3636, USA

(2) Max-Planck-Institut fur Physik, Werner-Heisenberg-Institut, Fohringer Ring 6, 80805 Miinchen, Germany

(3) Universidade Federal do ABC, Santo Andre, SP, Brazil

(4) Arnold Sommerfeld Center for Theoretical Physics, Department fir Physik, Ludwig-Maximilians-Universitat MUnchen, Theresienstrafie 37, 80333 MUnchen, Germany

Correspondence should be addressed to Olaf Hohm; ohohm@scgp.stonybrook.edu

Received 2 July 2018; Revised 3 October 2018; Accepted 14 October 2018; Published 1 November 2018

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