Constructions of L ∞ algebras and their field theory realizations

We construct L∞ 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 non-trivial 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. ar X iv :1 70 9. 10 00 4v 3 [ m at hph ] 2 5 O ct 2 01 8


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 'nbrackets' 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 ∞ 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][3][4].A closely related cousin of L ∞ algebras are A ∞ algebras, which generalize associative algebras to structures without associativity [5].
Our goal in this paper is to prove general theorems about the existence of L ∞ 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 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 1 + X 0 , where X 0 = V is the space of degree zero and X 1 = V * is isomorphic to V and of degree one, carries a 2-term L ∞ structure, meaning that the highest non-trivial product is 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 [6], but we have not been able to find a proof in the literature.(See also [7,8] for examples of finite-dimensional L ∞ algebras).
At first sight the above theorem may shed doubt on the usefulness of L ∞ algebras, since it states that any 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 non-trivial 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', X 0 the space of 'gauge parameters' and X 1 the space of 'trivial parameters' whose action on fields vanishes [9].Thus, if X 1 is isomorphic to X 0 there is no non-trivial action of X 0 on the physical fields and hence no genuine field theory realization of the L ∞ algebra.In order to obtain non-trivial 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 non-trivial 4 .A special case is the Courant bracket investigated by Roytenberg and Weinstein [10], for which the 4-bracket trivializes, but which is extendable and realized in string theory, in the form of double field theory [9,11,12].
We will illustrate these results with examples.Our investigation arose in fact out of the question whether the non-associative octonions (more precisely, the 7-dimensional commutator algebra of imaginary octonions) can be viewed as part of an L ∞ 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 non-trivial fact about the octonions.Moreover, this L ∞ structure is not extendable, which implies with the results of [9] that the octonions, at least when realized as a 2-term L ∞ algebra, cannot realize a non-trivial gauge symmetry in field theory.
As recently discovered in [13] and further investigated in [14,15], the octonions are related to the phase space of non-geometric backgrounds in M-theory (non-geometric R-flux or nongeometric Kaluza-Klein monopoles in M-theory).Furthermore, a contraction of the octonions leads to the string theory 'R-flux algebra' of [6,[16][17][18][19] and also to the 'magnetic monopole algebra' of [19][20][21][22][23][24][25].The Jacobiator of the R-flux algebra only takes values in a one-dimensional subspace, and therefore these contracted non-associative algebras may in fact be extendable.Here it is sufficient to take X 1 to be one-dimensional, leading to an 8-dimensional L ∞ algebra.(A 14-dimensional and hence non-minimal L ∞ realization of the R-flux algebra has already been given in [6].) The remainder of this paper is organized as follows.In sec. 2 we briefly review the axioms of L ∞ algebras.In sec. 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 sec.4. In sec.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.

Axioms of L ∞ algebras
We begin by stating the axioms of an L ∞ algebra.It is defined on a graded vector space and we refer to elements in X n as having degree n.We also refer to algebras with X n = 0 for all n with |n| ≥ k as a k-term L ∞ algebra.There are a potentially infinite number of generalized multi-linear products or brackets k having k inputs and intrinsic degree k − 2, meaning that they take values in a vector space whose degree is given by For instance, 1 has intrinsic degree −1, implying that it acts on the graded vector space according to Moreover, the brackets are graded (anti-)commutative in that, e.g., 2 satisfies 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 (σ; x) for any σ in the permutation group of k objects and a choice x = (x 1 , . . ., x k ) of k such objects.It can be defined implicitly by considering a graded commutative algebra with where in exponents x i denotes the degree of the corresponding element.The Koszul sign is then inferred from x 1 ∧ . . .∧ x k = (σ; x) x σ(1) ∧ . . .∧ x σ(k) . (2.6) The L ∞ relations are given by i+j=n+1 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 We will now state these relations explicitly for the values of n relevant for our subsequent analysis.For n = 1 the identity reduces to meaning that 1 acts like a derivation on the product 2 .For n = 3 one obtains We recognize the last line 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 1 to act as a derivation on 3 .Finally, the n = 4 relations read where we named the l.h.s.O(x 1 , . . ., x 4 ) 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 non-trivial constraint on 2 and 3 , while all higher L ∞ relations will be automatically satisfied.

A warm-up theorem
We now prove the first theorem stated in the introduction.Consider an algebra (V, [ 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 in general is non-zero.We then have the following Theorem 1: The graded vector space where X 0 = V and X 1 = V * with V * a vector space isomorphic to V , carries a 2-term L ∞ structure whose non-trivial brackets are given by ) Comment: We denote the elements of V * by v * , w * , etc., and the isomorphism by and similarly for its inverse.For instance, if V carries a non-degenerate 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 the n = 1, 2, 3 relations are partially satisfied and then verifying that in fact all L ∞ relations are satisfied.First, 1 maps X 1 = V * to X 0 = V , and we take it to be given by the (inverse) isomorphism (3.8), The second relation in here is necessary because there is no space X −1 in (3.3).The n = 1 relations 2 1 = 0 then hold trivially.Next, we fix the 2 product by requiring 2 (v, w) = [v, w] on X 0 = V and imposing the n = 2 relation (2.10).For arguments of total degree 0 this relation is trivial because of the second relation in (3.9).For arguments of total degree 1 we have where we used (3.9).Using (3.9) on the l.h.s.we infer Since there is no space X 2 we have 2 (v * , w * ) = 0.This is consistent with the n = 2 relation (2.10) for arguments of total degree 2: where we used (3.11).Thus, all n = 2 relations are satisfied.
Let us now consider the n = 3 relations (2.11).For arguments of total degree 0 (i.e., all taking values in X 0 ), it reads from which we infer Due to the antisymmetry of the bracket [• , •], the Jacobiator is completely antisymmetric in all arguments, and (3.14) is consistent with the required graded commutativity of 3 .Since there is no space X 2 , 3 is trivial for any arguments in X 1 .We have thus determined all non-trivial 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 ∞ relations.The n = 3 relation for arguments of total degree 1 reads: 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 non-trivial 4 these require that the left-hand side of (2.12) vanishes identically for 2 and 3 defined above.This follows by a direct computation that we display in detail.First, for arguments v 1 , v 2 , v 3 , v 4 ∈ X 0 of total degree 0 one may verify that (2.12) 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 (2.12) Here we used repeatedly the total antisymmetry in the four arguments, in particular in the last step that under the sum ]] * 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.

Main theorem
The above theorem states that an arbitrary bracket can be extended to an L ∞ 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 X 1 has to be as large as V , and the image of the map 1 : X 1 → X 0 equals X 0 = V .Consequently, one cannot introduce a further space X −1 together with a non-trivial 1 : X 0 → X −1 satisfying 2 1 = 0. Since in physical applications X −1 serves as the space of fields, such brackets do not lead to L ∞ algebras encoding a non-trivial gauge symmetry.
More interesting situations arise when the Jacobiator takes values in a proper subspace U ⊂ V , for then it is sufficient to set X 1 = U and to take 1 = 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 non-trivial kernel.Then there is an extension to a 3-term L ∞ algebra that generally requires a non-trivial 4-bracket: Let (V, [ •, • ]) be an algebra with bilinear antisymmetric 2-bracket as in sec.3, and let D : U → V be a linear map satisfying the closure conditions together with the Jacobiator relation where Im(D) and Ker(D) denote image and kernel of D, respectively.Then there exists a 3-term L ∞ structure with 2 (v, w) = [v, w] on the graded vector space with where X 0 = V , X 1 = U , X 2 = Ker(D) and ι denotes the inclusion of Ker(D) into U .The highest non-trivial bracket in general is given by the 4-bracket (and the complete list of non-trivial brackets is given in eq.(4.26) 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 (4.2) implies that there is a multi-linear and totally antisymmetric map f : The condition (4.1) states that the bracket of an arbitrary v ∈ V with Dα, α ∈ U , lies in the image of D, i.e., we can write 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 (4.5)only up contributions in the kernel, as is the function f in (4.4), but the following construction goes through for any choice of functions satisfying (4.5), (4.4). 1 1 The algebras resulting for different choices of theses functions are almost certainly equivalent under suitably defined L∞ isomorphisms, see, e.g., [26], 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 ∞ relations as far as possible and then proving that in fact all relations are satisfied.The n = 1 relations 2 1 = 0 for 1 defined in (4.3) 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 1 acts trivially on X 0 .For arguments α ∈ X 1 , v ∈ X 0 of total degree 1 we need where we used (4.5).As the l.h.s equals D( 2 (α, v)), this relation is satisfied if we set For arguments α, β ∈ X 1 of total weight 2 we compute using (4.7) in the last step.As 1 on the l.h.s acts by inclusion, we can satisfy this relation by setting but it remains to prove that the r. using that the bracket is antisymmetric.Note that (4.9) is properly symmetric in its two arguments, in agreement with the graded commutativity (2.4).Another choice of arguments of total degree 2 is v ∈ X 0 , c ∈ X 2 , for which we require where we used (4.7) in the last step, recalling ι(c) ∈ X 1 .Thus, using 1 = ι on the l.h.s.together with the graded symmetry we have We can also write this as We next consider arguments c ∈ X 2 , α ∈ X 1 of total degree 3, for which 2 must vanish as there is no vector space X 3 .This leads to a constraint from the n = 2 relation: where we used (4.9) and Dc = 0.This relation is satisfied for (4.13).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 1 , v 2 , v 3 ∈ X 0 of total degree zero: Recalling (4.4) and that 1 = D when acting on X 1 , we infer that this relation is satisfied for where we used repeatedly (4.7).Moreover, we used (4.16) and that 3 (α, v 1 , v 2 ) ∈ X 2 on which 1 acts as the inclusion.We will next prove that the function 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 (4.4) where we repeatedly used (4.5).This show that the r.h.s. of (4.18) is annihilated by D, proving that g takes values in X 2 = Ker(D).We can thus satisfy (4.17) by setting We next recall that there can be no non-trivial 3 for arguments α 1 , α 2 ∈ X 1 , v ∈ X 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 (2.11), noting that it is symmetric in α 1 , α 2 and writing sym for the symmetrized sum, where we used (4.7), (4.9) and, in the third equality, (4.20).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 (2.12) for arguments of total degree 0. Precisely as in (3.16) we compute In contrast to (3.16) this is not zero in general, but we can now have a non-trivial 4 taking values in X 2 .We next prove that the function defined by takes values in Ker(D).To this end we have to show that it is annihilated by D: Thus, the n = 4 relation can be satisfied by setting We have now determined all non-trivial brackets, which we summarize here: with the functions g, h defined in (4.18) and (4.23), respectively.All further L ∞ relations have to be satisfied identically.Let us next consider the n = 4 relations (2.12) for arguments v 1 , v 2 , v 3 ∈

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 X 2 = {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 (4.5) then reduces to Using this and Jac(v ), it is straightforward to verify that all products in (4.26) that take values in X 2 trivialize.In particular, 4 trivializes.Theorem 1 is contained as a special case, for which U = V .

Examples
We will now discuss a few examples, which get increasingly less trivial, with the goal to illustrate the scope of the above theorems.
where the structure constants are defined as follows.Splitting the index as a = (i, ī, 7), where i, ī = 1, 2, 3, η abc is the totally antisymmetric tensor defined by with the three-dimensional Levi-Civita symbol satisfying 123 = 1.(This coincides with the conventions of [13].)The η abc satisfy the following relations Using these it is straightforward to compute the Jacobiator: Jac(e a , e b , e c ) = −12 Θ abcd e d . (5.5) It is easy to verify with this expression that each generator appears on the right-hand side, see [13].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 {e a , e * a }) as in Theorem 1, with the non-trivial brackets being given in addition to (

5.2) by
There is no further non-trivial extension; in particular, this algebra cannot describe a non-trivial gauge symmetry in a field theory.
The R-flux algebra: This algebra, introduced in [16][17][18], is a contraction of the algebra of imaginary octonions in the following sense [13]: 3 We decompose e a = (e i , f i , e 7 ), with i = 1, 2, 3, and introduce a scaling parameter λ to define (5.7) Expressing the algebra (5.2) now in terms of x, p, I and sending λ → 0 one obtains the R-flux algebra where I is a central element that commutes with everything.It is easy to see that the only non-vanishing Jacobiator is Jac(x i , x j , x k ) = 3 ijk I . (5.9) 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 X 1 + X 0 , where X 0 = {x i , p i , I} and X 1 = {I * }.In addition to the 2-brackets defined by (5.8) we have the non-trivial products (5.10) The Courant algebroid: The Courant bracket of generalized geometry or the 'C-bracket' of double field theory have a non-vanishing Jacobiator.Denoting the arguments of this bracket, i.e., the elements of X 0 , by ξ 1 , ξ 2 , etc., it is given by Jac 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 so that for our current notation we read off with (4.5) It was established by Roytenberg and Weinstein that the Courant algebroid defines a 2-term L ∞ algebra with the highest bracket being 3 , which is defined by f , and X 1 being the space of functions [10].The space X 2 of constants (the kernel of the differential operator D) is not needed as all brackets in (4.26) taking values in X 2 vanish.For instance, 2 for two functions χ 1 , χ 2 ∈ X 1 becomes 2 (χ 1 , χ 2 ) = −(Dχ 1 )(χ 2 ) − (Dχ 2 )(χ 1 ) = Dχ 1 , Dχ 2 = 0 . (5.14) 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 [10]).The vanishing of all other products taking values in X 2 can be verified similarly using the relations given, for instance, in [9].Thus, the existence of an L ∞ structure on the Courant algebroid is a corollary of the more general Theorem 2.

Conclusions
We established general theorems about the existence of L ∞ 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 non-trivial 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 1 : X 2 → X 1 not to be the inclusion map but rather a non-trivial operator that again could have a non-trivial kernel, which in turn would necessitate a new space X 3 and higher brackets beyond a 4-bracket.These may be useful for generalizations of double and exceptional field theory [27].Indeed, it is to be expected that the gauge structure of exceptional field theory requires L ∞ algebras with arbitrarily high brackets [28], as is also the case in closed string field theory [1].Moreover, in order to obtain interesting L ∞ algebras with non-trivial field theory realizations, for special cases it is instrumental to take an appropriate bracket as starting point.For instance, for the E 8 (8) theory in [29] 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 [30]), but rather the vector space has to be suitably enlarged from the beginning, leading to a so-called Leibniz-Loday structure [31].