# Constructions ofL∞ Algebras and Their Field Theory Realizations

Authors: Olaf Hohm; Vladislav Kupriyanov; Dieter Lüst; Matthias Traube
Journal: Advances in Mathematical Physics
 (2018)

---

## Abstract

We constructL∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these L∞ algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define L∞ algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term L∞ algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid.


---

## Body

## 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” ln 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∞ 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∞ algebras is A∞ algebras, which generalize associative algebras to structures without associativity [5, 6].Our goal in this paper is to prove general theorems about the existence ofL∞ 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∞ algebra with l2(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=X1+X0, where X0=V is the space of degree zero and X1=V∗ is isomorphic to V and of degree one, carries a 2-term L∞ structure, meaning that the highest nontrivial product is l3, 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∞ algebras.)At first sight the above theorem may shed doubt on the usefulness ofL∞ algebras, since it states thatany generally non-Lie algebra can be extended to an L∞ algebra. It should be emphasized, however, that for a generic bracket the resulting structure is quite degenerate in that the 2-term L∞ algebra may not be extendable further in a nontrivial way, say by including a vector space X-1. Such extensions are particularly important for applications in theoretical physics as here X-1 encodes the “space of physical fields”, X0 the space of “gauge parameters,” and X1 the space of “trivial parameters” whose action on fields vanishes [10]. Thus, if X1 is isomorphic to X0 there is no nontrivial action of X0 on the physical fields and hence no genuine field theory realization of the L∞ 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∞ algebra whose highest bracket in general is a nontrivial l4. 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 anL∞ algebra. Our first theorem implies that the answer is affirmative, with the total graded space being 14-dimensional, which we will see is minimal. However, given the theorem, the existence of this L∞ structure does not express a nontrivial fact about the octonions. Moreover, this L∞ structure is not extendable, which implies with the results of [10] that the octonions, at least when realized as a 2-term L∞ 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 non-geometric 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 X1 to be one-dimensional, leading to an 8-dimensional L∞ algebra. (A 14-dimensional and hence nonminimal L∞ realization of the R-flux algebra has already been given in [7].)The remainder of this paper is organized as follows. In Section2 we briefly review the axioms of L∞ algebras. In Section 3 we prove the theorem that for arbitrary 2-bracket as initial data there is an L∞ 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∞ algebras.


## 2. Axioms ofL∞ Algebras

We begin by stating the axioms of anL∞ algebra. It is defined on a graded vector space(1)X=⨁n∈ZXn,and we refer to elements in Xn as having degree n. We also refer to algebras with Xn=0 for all n with n≥k as a k-term L∞ algebra. There are a potentially infinite number of generalized multilinear products or brackets lk having k inputs and intrinsic degree k-2, meaning that they take values in a vector space whose degree is given by(2)deg⁡lkx1,…,xk=k-2+∑i=1kdeg⁡xi.For instance, l1 has intrinsic degree -1, implying that it acts on the graded vector space according to(3)⋯→X1→l1X0→l1X-1→⋯Moreover, the brackets aregraded (anti-)commutative in that, e.g., l2 satisfies(4)l2x1,x2=-11+x1x2l2x2,x1,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 signϵ(σ;x) for any σ in the permutation group of k objects and a choice x=(x1,…,xk) of k such objects. It can be defined implicitly by considering a graded commutative algebra with(5)xi∧xj=-1xixjxj∧xi,∀i,j,where in exponents xi denotes the degree of the corresponding element. The Koszul sign is then inferred from(6)x1∧…∧xk=ϵσ;xxσ1∧…∧xσk.The L∞ relations are given by(7)∑i+j=n+1-1ij-1∑σ-1σϵσ;xljlixσ1,…,xσi,xσi+1,…xσn=0,for each n=1,2,3,…, which indicates the total number of inputs. Here (-1)σ 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≥1, over all permutations σ of n objects whose arguments are partially ordered (“unshuffles”), satisfying(8)σ1≤⋯≤σi,σi+1≤⋯≤σn.We will now state these relations explicitly for the values ofn relevant for our subsequent analysis. For n=1 the identity reduces to(9)l1l1x=0,stating that l1 is nilpotent, so that (3) is a chain complex. For n=2 the identity reads(10)l1l2x1,x2=l2l1x1,x2+-1x1l2x1,l1x2,meaning that l1 acts like a derivation on the product l2. For n=3 one obtains(11)0=l1l3x1,x2,x3+l3l1x1,x2,x3+-1x1l3x1,l1x2,x3+-1x1+x2l3x1,x2,l1x3+l2l2x1,x2,x3+-1x1+x2x3l2l2x3,x1,x2+-1x2+x3x1l2l2x2,x3,x1.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 l1 to act as a derivation on l3. Finally, the n=4 relations read(12)Ox1,…,x4≡-l2l3x1,x2,x3,x4+-1x3x4l2l3x1,x2,x4,x3+-11+x1x2l2x2,l3x1,x3,x4--1x1l2x1,l3x2,x3,x4+l3l2x1,x2,x3,x4+-11+x2x3l3l2x1,x3,x2,x4+-1x4x2+x3l3l2x1,x4,x2,x3-l3x1,l2x2,x3,x4+-1x3x4l3x1,l2x2,x4,x3+l3x1,x2,l2x3,x4=-l1l4x1,x2,x3,x4+l4l1x1,x2,x3,x4+-1x1l4x1,l1x2,x3,x4+-1x1+x2l4x1,x2,l1x3,x4+-1x1+x2+x3l4x1,x2,x3,l1x4,where we named the l.h.s. O(x1,…,x4) for later convenience. For a 2-term L∞ 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 l2 and l3, while all higher L∞ 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.,(13)v,w=-w,v∀v,w∈V,but we donot assume that the bracket satisfies any further constraints. In particular, the Jacobi identity is generally not satisfied, so that theJacobiator(14)Jacu,v,w≡u,v,w+v,w,u+w,u,v,in general is nonzero. We then have the following.Theorem 1.
The graded vector space(15)X=X1+X0,where X0=V and X1=V∗ with V∗ a vector space isomorphic to V, carries a 2-term L∞ structure whose nontrivial brackets are given by (16)l1v∗=v,(17)l2v,w=v,w,(18)l2v∗,w=v,w∗,(19)l3u,v,w=-Jacu,v,w∗.Comment. We denote the elements of V∗ by v∗,w∗, etc., and the isomorphism by(20)∗:V→V∗,v↦v∗,and similarly for its inverse. For instance, if V carries a nondegenerate metric we can take V∗ to be the dual vector space of V and the isomorphism to be the canonical isomorphism. (More simply, we can think of V∗ 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∗ as different objects.)Proof.
The proof proceeds straightforwardly by fixing the products so that then=1,2,3 relations are partially satisfied and then verifying that in fact all L∞ relations are satisfied. First, l1 maps X1=V∗ to X0=V, and we take it to be given by the (inverse) isomorphism (20),(21)∀v∗∈X1,v∈X0:l1v∗=v,l1v=0.The second relation in here is necessary because there is no space X-1 in (15). The n=1 relations l12=0 then hold trivially.
Next, we fix thel2 product by requiring l2(v,w)=[v,w] on X0=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(22)l1l2v∗,w=l2l1v∗,w-l2v∗,l1w=l2v,w=v,w,where we used (21). Using (21) on the l.h.s. we infer(23)l2v∗,w=v,w∗⇔l2w,v∗=w,v∗.Since there is no space X2 we have l2(v∗,w∗)=0. This is consistent with the n=2 relation (10) for arguments of total degree 2:(24)0=l1l2v∗,w∗=l2v,w∗-l2v∗,w=v,w∗-v,w∗=0,where we used (23). Thus, all n=2 relations are satisfied.
Let us now consider then=3 relations (11). For arguments of total degree 0 (i.e., all taking values in X0), it reads(25)0=l1l3u,v,w+l2l2u,v,w+l2l2v,w,u+l2l2w,u,v=l1l3u,v,w+Jacu,v,w, from which we infer(26)l3u,v,w=-Jacu,v,w∗∈X1.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 l3. Since there is no space X2, l3 is trivial for any arguments in X1. We have thus determined all nontrivial n-brackets.
So far we have verified then=1,2 relations and the n=3 relation for arguments of total degree 0. We now verify the remaining L∞ relations. The n=3 relation for arguments of total degree 1 reads(27)0=l3l1u∗,v,w+l2l2u∗,v,w+l2l2w,u∗,v+l2l2v,w,u∗=-Jacu,v,w∗+Jacu,v,w∗,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 then=4 relations. Since there is no nontrivial l4 these require that the left-hand side of (12) vanishes identically for l2 and l3 defined above. This follows by a direct computation that we display in detail. First, for arguments v1,v2,v3,v4∈X0 of total degree 0 one may verify that (12) is completely antisymmetric in the four arguments. Writing ∑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)(28)Ov1,…,v4=∑anti-4l2l3v1,v2,v3,v4+6l3l2v1,v2,v3,v4=∑anti4Jacv1,v2,v3,v4∗-6Jacv1,v2,v3,v4∗=∑anti12v1,v2,v3,v4∗-62v1,v2,v3,v4∗+v3,v4,v1,v2∗=0.Here we used repeatedly the total antisymmetry in the four arguments, in particular in the last step that under the sum [[v3,v4],[v1,v2]]∗ 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∞ 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 anL∞ algebra. For generic brackets, this L∞ structure is, however, quite degenerate in that it may not be extendable further, say by adding a further space X-1. Indeed, if the violation of the Jacobi identity is “maximal” and the Jacobiator takes values in all of V, the space X1 has to be as large as V, and the image of the map l1:X1→X0 equals X0=V. Consequently, one cannot introduce a further space X-1 together with a nontrivial l1:X0→X-1 satisfying l12=0. Since in physical applications X-1 serves as the space of fields, such brackets do not lead to L∞ algebras encoding a nontrivial gauge symmetry.More interesting situations arise when the Jacobiator takes values in a proper subspaceU⊂V, for then it is sufficient to set X1=U and to take l1=ι to be the “inclusion” defined for any u∈U by ι(u)=u, viewing u as an element of V. Indeed, it is easy to verify, provided the subspace forms an ideal (i.e., ∀u∈U,v∈V:[u,v]∈U), that the above proof goes through as before. In this case, further extensions of the L∞ 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∞ 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→V be a linear map satisfying the closure conditions(29)ImD,V⊂ImD,together with the Jacobiator relation(30)∀v1,v2,v3∈V:Jacv1,v2,v3∈ImD,where Im(D) and ker(D) denote image and kernel of D, respectively. Then there exists a 3-term L∞ structure with l2(v,w)=[v,w] on the graded vector space with(31)X2→l1=ιX1→l1=DX0,where X0=V, X1=U, X2=ker(D) and ι 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 α,β,… and the elements of ker(D) by c,c′,… The condition (30) implies that there is a multilinear and totally antisymmetric map f:V⊗3→U so that(32)∀v1,v2,v3∈V:Jacv1,v2,v3=Dfv1,v2,v3.The condition (29) states that the bracket of an arbitrary v∈V with Dα, α∈U, lies in the image of D, i.e., we can write(33)∀v∈V,α∈U:Dα,v=Dvα,vα∈U.We can think of the operation on the r.h.s. as defining for each v∈V a map on U, α↦v(α)∈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∞ isomorphisms; see, e.g., [27], but we leave a detailed analysis for future work.)Proof.
As for Theorem1, the proof proceeds by determining the n-brackets from the L∞ relations as far as possible and then proving that in fact all relations are satisfied. The n=1 relations l12=0 for l1 defined in (31) are satisfied by definition since D(ι(c))=0 for all c∈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 l1 acts trivially on X0. For arguments α∈X1, v∈X0 of total degree 1 we need(34)l1l2α,v=l2l1α,v=Dα,v=Dvα,where we used (33). As the l.h.s. equals D(l2(α,v)), this relation is satisfied if we set(35)l2α,v=vα∈X1.For arguments α,β∈X1 of total weight 2 we compute(36)l1l2α,β=l2l1α,β-l2α,l1β=-l2β,Dα-l2α,Dβ=-Dαβ-Dβα,using (35) in the last step. As l1 on the l.h.s. acts by inclusion, we can satisfy this relation by setting(37)l2α,β=-Dαβ-Dβα∈kerD,but it remains to prove that the r.h.s. indeed takes values in the kernel. This follows by setting v=Dβ in (33):(38)Dα,Dβ=DDβα⇒DDαβ+Dβα=0,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∈X0,c∈X2, for which we require(39)l1l2v,c=l2l1v,c+l2v,l1c=l2v,ιc=-vιc,where we used (35) in the last step, recalling ι(c)∈X1. Thus, using l1=ι on the l.h.s. together with the graded symmetry we have(40)ιl2c,v=vιc.We can also write this as (Here we employ the map on X2 induced by v(α) via v(c):=v(ι(c)), which lies in ker(D) as a consequence of Dc=0 and (33))(41)∀c∈X2,v∈X0:l2c,v=vc∈X2.We next consider arguments c∈X2,α∈X1 of total degree 3, for which l2 must vanish as there is no vector space X3. This leads to a constraint from the n=2 relation:(42)0=l1l2c,α=l2ιc,α+l2c,Dα=-Dαc+l2c,Dα,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 v1,v2,v3∈X0 of total degree zero:(43)0=l1l3v1,v2,v3+Jacv1,v2,v3.Recalling (32) and that l1=D when acting on X1, we infer that this relation is satisfied for(44)l3v1,v2,v3=-fv1,v2,v3∈X1.Next, for arguments α∈X1,v1,v2∈X0 of total weight 1 the n=3 relation reads(45)0=l1l3α,v1,v2+l3l1α,v1,v2+l2l2α,v1,v2+l2l2v2,α,v1+l2l2v1,v2,α=ιl3α,v1,v2-fDα,v1,v2+v2v1α-v1v2α-v1,v2α,where we used repeatedly (35). Moreover, we used (44) and that l3(α,v1,v2)∈X2 on which l1 acts as the inclusion. We will next prove that the function(46)gα,v1,v2≡fDα,v1,v2+v1,v2α+v1v2α-v2v1α,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)(47)DfDα,v1,v2=JacDα,v1,v2=Dα,v1,v2+v2,Dα,v1+v1,v2,Dα=Dv1α,v2-Dv2α,v1-Dv1,v2α=Dv2v1α-v1v2α-v1,v2α,where we repeatedly used (33). This show that the r.h.s. of (46) is annihilated by D, proving that g takes values in X2=ker(D). We can thus satisfy (45) by setting(48)l3α,v1,v2=gα,v1,v2∈X2.
We next recall that there can be no nontriviall3 for arguments α1,α2∈X1,v∈X0 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 α1,α2 and writing ∑sym‍ for the symmetrized sum,(49)0=∑sym2l3l1α1,α2,v+l2l2α1,α2,v+2l2l2v,α1,α2=∑sym-2l3α1,Dα2,v-2l2Dα1α2,v-2l2vα1,α2=∑sym-2fDα1,Dα2,v-2Dα2,vα1-2Dα2vα1+2vDα2α1-2vDα1α2+2Dvα1α2+2Dα2vα1,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(50)Ov1,…,v4=∑anti-4l2l3v1,v2,v3,v4+6l3l2v1,v2,v3,v4=∑anti-4v1fv2,v3,v4-6fv1,v2,v3,v4.In contrast to (28) this is not zero in general, but we can now have a nontrivial l4 taking values in X2. We next prove that the function defined by(51)hv1,…,v4≡∑anti4v1fv2,v3,v4+6fv1,v2,v3,v4,takes values in ker(D). To this end we have to show that it is annihilated by D:(52)Dhv1,…,v4=∑anti4Dv1fv2,v3,v4+6Dfv1,v2,v3,v4=∑anti4Jacv2,v3,v4,v1+6Jacv1,v2,v3,v4=0. Thus, the n=4 relation can be satisfied by setting(53)l4v1,…,v4=hv1,…,v4∈X2.
We have now determined all nontrivial brackets, which we summarize here:(54)c∈X2:l1c=ιc=c∈X1,α∈X1:l1α=Dα∈X0,v,w∈X0:l2v,w=v,w∈X0,α∈X1,v∈X0:l2α,v=vα∈X1,c∈X2,v∈X0:l2c,v=vc∈X2,α,β∈X1:l2α,β=-Dαβ-Dβα∈X2,v1,v2,v3∈X0:l3v1,v2,v3=-fv1,v2,v3∈X1,α∈X1,v1,v2∈X0:l3α,v1,v2=gα,v1,v2∈X2,v1,…,v4∈X0:l4v1,…,v4=hv1,…,v4∈X2,with the functions g,h defined in (46) and (51), respectively. All further L∞ relations have to be satisfied identically. Let us next consider the n=4 relations (12) for arguments v1,v2,v3∈X0,α∈X1 of total degree 1. It is easy to see that (12) is then totally antisymmetric in v1,v2,v3, and writing ∑anti‍ for the antisymmetric sum over these three arguments we compute(55)Ov1,v2,v3,α=∑anti-l2l3v1,v2,v3,α+3l2l3α,v1,v2,v3+3l3l2v1,v2,v3,α+3l3v1,v2,l2v3,α=∑antil2fv1,v2,v3,α+3v3l3α,v1,v2+3l3v1,v2,v3,α-3l3v1,v2,v3α=∑anti-Dfv1,v2,v3α-Dαfv1,v2,v3+3v3fDα,v1,v2+v1,v2α+2v1v2α+3fDα,v1,v2,v3+v1,v2,v3α+v1,v2v3α-v3v1,v2α-3fDv3α,v1,v2+v1,v2v3α+2v1v2v3α=∑anti-Dαfv1,v2,v3+3v3fDα,v1,v2+3fDα,v1,v2,v3-3fDα,v3,v1,v2,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 l4(v1,v2,v3,Dα). To see this note that writing (53) with an antisymmetrized sum over only the first three arguments one obtains(56)l4v1,…,v4=∑antiv1,v2,v33v1fv2,v3,v4-v4fv1,v2,v3+3fv1,v2,v3,v4-3fv4,v1,v2,v3.Specializing this to l4(v1,v2,v3,Dα) 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∞ relations in Section 2 for n≥5 explicitly as these get increasingly laborious. However, it is easy to see that here the only nontrivial n=5 relation has arguments v1,…,v5∈X0, which are of even degree so that the Koszul sign becomes ϵ(σ;v)=1. Moreover, l5 is trivial, and it is then easy to verify that (7) reduces to(57)∑anti10l4l2v1,v2,v3,v4,v5+5l2l4v1,v2,v3,v4,v5+10l3l3v1,v2,v3,v4,v5=0,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∞ 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⊂V, which forms an ideal of the bracket. In this case we can take D=ι to be the inclusion map U→V. Since its kernel is trivial, we have X2={0}, and the algebra can be reduced to a 2-term L∞ algebra. Indeed, the action of v∈V on U that is implicit in (33) then reduces to(58)u↦vu≡-v,u∈U.Using this and Jac(v1,v2,v3)=f(v1,v2,v3), it is straightforward to verify that all products in (54) that take values in X2 trivialize. In particular, l4 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 ea, a=1,…,7 satisfy the algebra(59)eaeb=-δab1+ηabcec,and thus the commutation relations(60)ea,eb=2ηabcec,where the structure constants are defined as follows. Splitting the index as a=(i,i¯,7), where i,i¯=1,2,3, ηabc is the totally antisymmetric tensor defined by(61)ηijk=ϵijk,ηij¯k¯=-ϵijk,η7ij¯=δij,with the three-dimensional Levi-Civita symbol satisfying ϵ123=1. (This coincides with the conventions of [14].) ηabc satisfy the following relations(62)ηabeηcde=2δcaδbd-Θabcd,Θabcd≡13!ϵabcdefgηefg.Using these it is straightforward to compute the Jacobiator:(63)Jacea,eb,ec=-12Θabcded.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∞ extension requires a doubling to a 14-dimensional space (with basis {ea,ea∗}) as in Theorem 1, with the nontrivial brackets being given in addition to (60) by(64)l1ea∗=ea,l2ea∗,eb=2ηabcec∗,l3ea,eb,ec=12Θabcded∗.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 ea=(ei,fi,e7), with i=1,2,3, and introduce a scaling parameter λ to define(65)pi≡-12λiei,xi≡12λifi,I≡12iλ3/2e7.Expressing the algebra (60) now in terms of x,p,I and sending λ→0 one obtains the R-flux algebra(66)xi,pj=iδijI,xi,xj=iϵijkpk,pi,pj=0,where I is a central element that commutes with everything. It is easy to see that the only nonvanishing Jacobiator is(67)Jacxi,xj,xk=3ϵijkI.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∞ structure on X1+X0, where X0={xi,pi,I} and X1={I∗}. In addition to the 2-brackets defined by (66) we have the nontrivial products(68)l1I∗=I,l3xi,xj,xk=-3ϵijkI∗.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 X0, by ξ1,ξ2, etc., it is given by(69)Jacξ1,ξ2,ξ3=Dfξ1,ξ2,ξ3,fξ1,ξ2,ξ3≡12∑antiξ1,ξ2,ξ3,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 χ(70)Dχ,ξ=-12DDχ,ξ,so that for our current notation we read off with (33)(71)ξχ=-12Dχ,ξ.It was established by Roytenberg and Weinstein that the Courant algebroid defines a 2-term L∞ algebra with the highest bracket being l3, which is defined by f, and X1 being the space of functions [11]. The space X2 of constants (the kernel of the differential operator D) is not needed as all brackets in (54) taking values in X2 vanish. For instance, l2 for two functions χ1,χ2∈X1 becomes(72)l2χ1,χ2=-Dχ1χ2-Dχ2χ1=Dχ1,Dχ2=0.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 X2 can be verified similarly using the relations given, for instance, in [10]. Thus, the existence of an L∞ structure on the Courant algebroid is a corollary of the more general Theorem 2.


## 6. Conclusions

We established general theorems about the existence ofL∞ 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 Theorem2 could be extended by taking the map l1:X2→X1 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 X3 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∞ algebras with arbitrarily high brackets [30], as is also the case in closed string field theory [1]. Moreover, in order to obtain interesting L∞ algebras with nontrivial field theory realizations, for special cases it is instrumental to take an appropriate bracket as starting point. For instance, for the E8(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].



---

*Source: 9282905-2018-11-01.xml*