Weakly Constrained Double Field Theory as the Double Copy of Yang-Mills Theory

Weakly constrained double field theory, in the sense of Hull and Zwiebach, captures the subsector of string theory on toroidal backgrounds that includes gravity, $B$-field and dilaton together with all of their massive Kaluza-Klein and winding modes, which are encoded in doubled coordinates subject to the `weak constraint'. Due to the complications of the weak constraint, this theory was only known to cubic order. Here we construct the quartic interactions for the case that all dimensions are toroidal and doubled. Starting from the kinematic $C_{\infty}$ algebra ${\cal K}$ of pure Yang-Mills theory and its hidden Lie-type algebra, we construct the $L_{\infty}$ algebra of weakly constrained double field theory on a subspace of the `double copied' tensor product space ${\cal K}\otimes\bar{\cal K}$, by doing homotopy transfer to the weakly constrained subspace and performing a non-local shift that is well-defined on the torus. We test the resulting three-brackets, and establish their uniqueness up to cohomologically trivial terms, by verifying the Jacobi identities up to homotopy for the gauge sector.


Introduction
String theory is often thought of as something quite different from quantum field theory.There are, however, formulations of string theory as an 'ordinary' field theory, known as string field theory, with the only somewhat unusual feature being that it carries an infinite number of component fields (see [1,2] for modern reviews).These component fields include familiar fields such as vector spin-1 gauge fields (in open string theory) or the tensor spin-2 fluctuations of gravity (in closed string field theory).Since string theory is UV-finite, we thus have with closed string field theory a quantum field theory of gravity that is at least perturbatively well-defined.
However, string theory and string field theory are technically infamously involved and also include numerous exotic ingredients such as extra dimensions and infinite towers of massive fields of ever-increasing spin.Undoubtedly, this state of affairs is part of the reason that so far no compelling scenario has emerged of how to connect string theory to real-world observations.At the same time, general qualitative aspects of the real-world physics encoded in the standard model of particle physics are naturally found in string theory.Yang-Mills theory, for instance, which governs all interactions in the realm of particle physics, can be obtained from open string field theory by eliminating (or rather integrating out) all massive string modes.Since Yang-Mills theory defines a perfectly good quantum field theory, without any need to pass to the full open string field theory, one may thus wonder, by analogy, whether there are consistent theories of quantum gravity that are 'smaller' than the full closed string field theory, perhaps consistent duality-invariant subsectors of string theory that include only some of the massive string modes.(Since general relativity and the low-energy supergravity actions of string theory are certainly non-renormalizable and are unlikely to be UV-finite it is clear that, in contrast to Yang-Mills theory, any putative quantum gravity theory has to include some, and most likely infinite towers of, extra states in order to improve the UV-behavior.) In this paper, we explicitly construct a gravity theory, known as weakly constrained double field theory [3], that includes some infinite towers of massive string modes, which provide a promising subsector for two reasons: First, at the level of scattering amplitudes there are deep relations between open and closed string theory, the Kawai-Lewellen-Tye (KLT) relations [4], which have been shown by Bern, Carrasco and Johansson (BCJ) to have field theory analogues [5,6].Such so-called 'double copy' constructions relate pure Yang-Mills theory to 'N = 0 supergravity', i.e.Einstein-Hilbert gravity coupled to a two-form (B-field) and a scalar (dilaton).In view of the double copy this theory is most efficiently formulated as a (strongly constrained) double field theory (DFT) [7][8][9][10].Second, DFT is believed to exist also in a weakly constrained version that features genuine massive string modes and is expected to exhibit an improved UV behavior.Concretely, weakly constrained DFT is defined on toroidal backgrounds and includes the massless fields of N = 0 supergravity together with all of their massive Kaluza-Klein and winding modes.Such a theory can in principle be derived from the full closed string field theory by integrating out all fields that do not belong to the DFT sector [11][12][13].As argued by Sen, the weakly constrained DFT so obtained would inherit the UV finiteness of string theory [11].Therefore, 'bootstrapping' such a theory directly from Yang-Mills theory via double copy, plausibly upon also including α ′ corrections [14,15], appears to be a promising path towards quantum gravity.
The construction of weakly constrained DFT to be presented here, which was announced and outlined in [16], is based on homotopy algebras such as homotopy Lie or L ∞ algebras.In theoretical physics, such structures were first discovered in string field theory [17] and only later realized to govern also conventional field theories such as Yang-Mills theory, see in particular the early work of A. Zeitlin [18,19] (which remarkably anticipated already aspects of double copy [20,21]).Homotopy algebras also play a role in the formulation of quantum field theory due to Costello [22] and Gwilliam-Costello [23].We refer to [24] for a self-contained introduction to L ∞ algebras and the general dictionary between field theories and L ∞ algebras.
Using this framework one may start from Yang-Mills theory, viewed as an L ∞ algebra, and give a perfectly precise meaning to the notion of 'stripping off' color factors.While there is no such thing as a field theory of 'color-stripped Yang-Mills fields', there is a homotopy algebra of such 'fields', a C ∞ rather than an L ∞ algebra.Specifically, the vector space X YM , on which the L ∞ algebra of Yang-Mills theory is defined, can be decomposed as the tensor product X YM = K ⊗ g, where g is the 'color' Lie algebra of the gauge group, and the C ∞ algebra K encodes the 'kinematics' of Yang-Mills theory [19].C ∞ algebras are homotopy versions of commutative associative algebras, which means that the (graded commutative) product is only associative 'up to homotopy', with the failure of associativity being governed by a '3-product'.The kinematic algebra K of Yang-Mills theory is thus an 'associative-type' algebra.However, the study of scattering amplitudes underlying the double copy indicates that there is also a hidden 'Lie-type' algebra.At the level of any local off-shell Lagrangian formulation there is no such Lie algebra in the strict sense, not even 'up to homotopy', but a further relaxation of the Lie algebra axioms was proposed by Reiterer in [25] and proved to be realized in Yang-Mills theory in four (Euclidean) dimensions (with complexified fields and momenta).This algebra goes under the forbidding yet fitting name BV ∞ and is a generalization of a Batalin-Vilkovisky (BV) algebra.Here refers to the flat space wave operator that, being of second order, is the origin of the obstructions that prevent K from carrying a homotopy Lie algebra.Nevertheless, the BV ∞ algebra seems to be at the core of the so-called color-kinematics duality of Yang-Mills scattering amplitudes.More recently, for Yang-Mills theory in arbitrary dimensions (and spacetime signature), we displayed this algebra up to trilinear maps [26].
In order to double copy Yang-Mills theory one considers the tensor product K ⊗ K with a second copy K of the kinematic algebra.This total space consists of functions of doubled (unconstrained) coordinates (coordinates x associated to K and coordinates x associated to K).This space inherits an algebra of precisely the same kind: a BV ∆ ∞ algebra, where ∆ = 1 2 ( − ¯ ) is the difference of the respective wave operators.This unconstrained space does not carry an unobstructed L ∞ algebra, and hence no consistent field theory, due to ∆ being second order, but by restricting to a suitable subspace one can eliminate the obstructions.Identifying coordinates x with coordinates x implies ∆ ≡ 0, which yields the strongly constrained DFT that can be viewed as a duality invariant formulation of N = 0 supergravity.More precisely, so far this was established to quartic order in fields [26].(See [27] for a quick and polemic introduction, [28][29][30][31][32][33] for DFT and double copy, and [34][35][36][37][38][39] for homotopy algebras and double copy.) In this paper we will show how to construct weakly constrained DFT for toroidal and hence Euclidean backgrounds in which ∆ = 0 is imposed as a constraint on fields, which then still genuinely depend on doubled coordinates and hence encode both physical winding and momentum modes.(Since all dimensions are toroidal and doubled this theory does not yet include a non-compact time direction, which at least in conventional thinking should remain undoubled.We leave the construction of the full Lorentzian theory for future work.)To this end, one performs homotopy transfer to the subspace with ∆ = 0 (see, e.g., [12] for a self-contained introduction to homotopy transfer).This still does not give an unobstructed L ∞ algebra, but by further imposing an algebraic constraint known from the level-matching constraints of string theory one can redefine the desired 3-bracket by a non-local but perfectly well-defined shift so that one obtains a genuine L ∞ algebra to the order relevant for the quartic theory.This solves a problem that was outstanding since the modern inception of DFT by Hull and Zwiebach in 2009 [3].It should be emphasized that this solution of the problem is unique, up to cohomologically trivial redefinitions, given the 'initial data' of the differential B 1 and 2-bracket B 2 of weakly constrained DFT encoded in the cubic theory of [3].
What is perhaps the most striking aspect of this solution of the long-open problem of constructing weakly constrained DFT is that it relates to, and in some ways is almost identical with, deep hidden structures that are present in Yang-Mills theory proper, without any reference to gravity.While the conventional Lagrangian formulation of Yang-Mills theory relies only on the color Lie algebra g and the kinematic C ∞ algebra, the computation of scattering amplitudes requires more structures, as for instance exhibited in gauge conditions.Given these extra structures, the kinematic vector space comes close to be a (homotopy) Lie algebra, but this is obstructed by the wave operator being of second order.When computing scattering amplitudes one goes on-shell, so that gives zero when acting on single fields (polarization vectors), but even then the algebraic structure is obstructed since the product of two on-shell fields is generally not on-shell.Thus, even on the subspace with = 0 the BV ∞ algebra does not yield an unobstructed homotopy Lie algebra.In the amplitude literature it has been shown how to shift the kinematic numerators so that these obey Jacobi-type identities, a property known as color-kinematic duality.The problem of constructing weakly constrained DFT is therefore technically analogous to the problem of making color-kinematics manifest in Yang-Mills theory proper, just with BV ∆ ∞ instead of BV ∞ .We hope to further explore this intriguing connection in the future.
The remainder of this paper is organized as follows.In sec. 2 we introduce the C ∞ algebra of the kinematic space K of Yang-Mills theory, to the order of tri-linear maps, and we introduce the BV ∞ algebra.While to a large part this is a review of results presented in [26], we also introduce a more streamlined notation for objects of K and its multilinear maps, which is instrumental in order to efficiently compute the double copied maps in later sections.These results are useful additions to [26], even just for strongly constrained DFT.In sec. 3 we prove, again to the order of tri-linear maps relevant for the quartic theory, that the (unconstrained) doubled space K ⊗ K carries a BV ∆ ∞ algebra.Finally, in sec.4 we construct weakly constrained DFT to quartic order, by first doing homotopy transfer to the subspace with ∆ = 0 and then performing a non-local but well-defined shift.We verify the inevitability of this non-local shift by computing the 3brackets of the gauge sector and by verifying the generalized Jacobi identities.We close with a summary and outlook in sec. 5. where we discuss possible applications and generalizations.In two appendices we collect all maps for Yang-Mills theory and we give some of the technically challenging proofs.

The kinematic algebra of Yang-Mills
Here we start by reviewing the BV ∞ algebra of Yang-Mills theory, up to its trilinear maps.In doing so we will introduce the necessary formalism and fix our conventions and notation.We follow closely the discussion in [26,40], although with some differences in the notation that, we believe, lead to a more streamlined treatment.
We employ a formulation of Yang-Mills theory with an auxiliary scalar field ϕ, whose action is given by [40] where f abc are the structure constants of the color Lie algebra g.The cubic and quartic vertices are standard, and one recovers the usual action upon integrating out ϕ.In Yang-Mills theory all objects including gauge parameters, fields, field equations, etc., take values in the Lie algebra g of the gauge group.It is thus natural to view the space of Yang-Mills theory as the tensor product K ⊗ g, where elements of K are color-stripped local spacetime fields, which we still refer to as gauge parameters, fields and so on.

The graded vector space K
Let us describe in more detail the structure of the kinematic vector space K.It is a graded vector space given by the direct sum of subspaces K i of homogenous degree: K = 3 i=0 K i .Elements in each K i are identified as gauge parameters λ, fields A, field equations E and Noether identities N according to the following diagram: We take the field A to contain both the (color-stripped) gauge vector field A µ and the scalar ϕ, and similarly the equations of motion E have a vector and a scalar component.
In order to display explicitly its degree structure, we find it useful to view K as the tensor product of a finite-dimensional graded vector space Z = 3 i=0 Z i with the space of smooth spacetime functions of degree zero: K = Z ⊗C ∞ (M).Here M is flat d−dimensional Minkowski spacetime, but the signature is immaterial for the following discussion.The vector space Z is defined by giving a basis.To this end let us introduce a (d + 2)-component graded vector θ M = (θ + , θ µ , θ − ), where µ = 0, 1, . . ., d − 1 is a Lorentz vector index.The degrees of the components are given by |θ A basis of Z is then given by The above characterization of Z exhibits the manifest Z 2 symmetry that exchanges θ M and c θ M .This isomorphism between the subspaces generated by θ M and c θ M , respectively, can be implemented by nilpotent operators b and c defined by their action on the basis Z A : (2.6) The degrees of b and c are thus fixed to be |c| = +1 and |b| = −1, and from their definition one can see that they obey the algebra The basis elements of Z can be displayed according to their degree in a way that emphasizes the Z 2 symmetry: where we have indicated the action of b (c acts by reversing the arrows).
In addition to the above Z 2 symmetry, Z can be equipped with an odd symplectic bilinear form1 ω of degree |ω| = −3, satisfying which is symmetric since it always pairs odd with even elements.We specify ω by giving its components ω(Z A , Z B ) in the above basis: where η µν is the d−dimensional Minkowski metric, and all other pairings vanish.
Upon tensoring Z with smooth functions, we obtain the kinematic space K of Yang-Mills theory.The degree in K coincides with the one in Z, meaning that for an homogeneous element ψ = Z f (x) one has |ψ| = |Z|.An arbitrary element in K can thus be expanded as (2.11) Comparing the degree structure (2.2) of K with (2.8), one infers that the Yang-Mills fields, parameters and so on are given by the following vectors in K with homogeneous degrees: (2.12) The Z 2 structure and the action of b and c are inherited from Z.One can indeed draw the same diagram (2.8) in K to display this: where we omitted the Z A and only wrote the component fields.The odd symplectic pairing ω induces a degree −3 inner product , in K, defined by More specifically, using (2.10) one can see that the non-vanishing pairings are between fields A and field equations E: and between gauge parameters λ and Noether identities N : 2.2 C ∞ algebra on K Having described K as a graded vector space, we now turn to reviewing the algebraic structures that can be defined on it.The consistency of Yang-Mills theory as a field theory (this includes, for instance, gauge covariance of the field equations and closure of the gauge algebra) is encoded, upon factoring out color, by a C ∞ algebra structure on K [19,35,40].This is a homotopy generalization of a commutative associative algebra where a graded vector space (K in the case at hand), is equipped with multilinear maps or products m n obeying a set of quadratic relations.
For the case of Yang-Mills theory, the only non-vanishing maps are an operator m 1 of degree +1, a bilinear product m 2 of degree zero and a trilinear product m 3 of degree −1, summarized as The nontrivial C ∞ relations to be satisfied then consist of stating that m 1 is a differential, which makes K into a chain complex.Physically, m 2 1 = 0 encodes, in particular, gauge invariance of the free theory under linearized gauge transformations.
• The differential m 1 acts as a derivation on m 2 (Leibniz rule): where ψ i in exponents always denotes the degree |ψ i |.This requirement ensures, upon tensoring with color, consistency of Yang-Mills theory up to cubic order.
• The product m 2 is associative up to homotopy: which is responsible for consistency of the theory up to quartic order and thus fully, given that Yang-Mills theory has no higher vertices.
In a C ∞ algebra, the products m n have to further obey symmetry constraints under permutations of arguments.Specifically, the m n have to vanish under so-called shuffle permutations.For our purposes, the relevant symmetry properties are which, for m 2 , is the same as graded symmetry.For the following discussion we find it more convenient to work with a different representation of m 3 , which we denoted m 3h in [26], defined as This is just a redefinition of m 3 , not a projection, as it can be inverted explicitly by using the above formula.The redefined m 3h is graded symmetric in its first two arguments and vanishes upon total graded symmetrization.Otherwise stated, it is a graded hook representation in terms of Young diagrams.
Given the tensor product structure of K = Z ⊗C ∞ (M) and the expansion (2.11) of arbitrary vectors, the m n products act on elements of K as follows: where the operations on the right are defined as follows: First, the operators mn (Z A 1 , . . ., Z An ) are Z−valued multidifferential operators acting on the component fields as (2.23) Second, µ just denotes the local pointwise product: To clarify this notation, let us give some explicit examples (the explicit form of all m n products can be found in [40]).Acting on the basis vectors of Z 1 (corresponding to fields), we have (2.25) Using (2.22) one computes the action of the differential m 1 on a field where we omitted the explicit spacetime dependence of the component fields, and denoted the contraction of Lorentz indices with a dot.One can see that setting m 1 (A) = 0 corresponds to the free Maxwell equations upon solving for ϕ.
For the next example, the non-vanishing part of m 2 between fields ( The pointwise multiplication implemented by µ then yields (with a dot denoting contraction of Lorentz indices) which gives the color-stripped cubic vertex of Yang-Mills as A 3 , m 2 (A 1 , A 2 ) .Similarly, the only non-vanishing component of m 3h comes from the operator yielding corresponding to the color-stripped quartic vertex.A complete list of the operators mn can be found in appendix A.

BV ∞ algebra on K
While it is rather straightforward to determine that K carries a C ∞ structure (this is 'just' rephrasing its usual consistency conditions), the next algebraic layer on K is highly nontrivial and plays a crucial role in the double copy construction.To see how this deeper structure arises on K, one has to look at the interplay of the C ∞ algebra, given by the products m n , with the b operator introduced before.From its definition in (2.6) and the expression of the differential m 1 one may verify that it obeys where = ∂ µ ∂ µ is the wave operator.It turns out that (2.32) should be viewed as the general defining property of b, with our realization (2.6), (2.13) being a particular case. 2 Although b does not play a role in the consistency relations encoded in the C ∞ algebra of the theory, it can be viewed as providing a gauge fixing condition as b(A) = 0, as well as the related propagator as b acting on the space of equations.The peculiar property of our realization (2.6) is that it implying in particular that it is local and does not contain spacetime derivatives.This will be instrumental in order to construct a local theory from double copy.
With this second differential at our disposal, one can study its compatibility with the C ∞ products.Its graded commutator with m 1 is given in (2.32).Going one step further, b does not act as a derivation on m 2 .Rather, the failure to do so defines a bracket b 2 : on which b acts as a derivation by construction.Given a product m 2 and a bracket b 2 , one can ask if they are mutually compatible, i.e. if they obey the graded Poisson identity If this were the case (which also requires m 2 to be associative), b 2 would be a graded Lie bracket, and the triplet (b, m 2 , b 2 ) would form a BV algebra.While this happens for Chern-Simons theory [36,45], it is not the case for Yang-Mills theory (at least in standard formulations).The compatibility (2.35) holds only up to an homotopy θ 3 and further deformations originating from (2.32).This prompts a cascade of higher relations, defined as a BV ∞ algebra in [25].
In order to give all the relevant relations of the resulting BV ∞ algebra (up to trilinear maps), we shall review a convenient input-free notation introduced in [26], which will allow us to establish the results of the forthcoming sections.In this part we will denote by O any linear operator in K, of degree |O|.Generic bilinear and trilinear maps will be denoted by M and T , respectively, with arbitrary intrinsic degrees |M| and |T |.Similarly to (2.22), these generic maps act on elements of K as where Ô, M and T are Z−valued multidifferential operators acting on the component functions.We define the graded commutator of operators where every symbol in exponents refers to the degree of a map or element.The commutators of an operator O with bilinear and trilinear maps M and T are the bilinear map [O, M] and trilinear map [O, T ] given by (2.38) The action of an operator O on a map (be it another operator, a bilinear or trilinear map) gives a map of the same kind, e.g.
Finally, composition of bilinear maps is defined from the left and denoted by juxtaposition: This is sufficient for our purposes, since all bilinear maps involved are graded symmetric.With this notation one can check that [O, −] is a derivation on commutators and compositions, in the sense that it obeys (2.41) We now turn to discuss the symmetry properties of trilinear maps T .Since they are all graded symmetric in the first two arguments (this is the reason we chose to work with m 3h rather than m 3 ), they can be decomposed into a totally graded symmmetric part, which we denote by T s , and a graded hook part T h := T − T s .In terms of T , the symmetrized map T s acts on three inputs as (2.42) In line with (2.36) we want to associate a multidifferential operator Ts to T s , such that To do so, we start by introducing a permutation operator Σ, which acts on trilinear operators as and thus obeys and Σ 3 = 1.We then use this to define a projector π, obeying π 2 = π, so that the symmetrized and hook operators Ts and Th are defined via In terms of the permutation operator Σ, π is explicitly given by which reproduces, upon using (2.43), the expression (2.42) for the map T s .We will then use interchangeably the notation T s ≡ T π for the symmetrized map as well.The operator Ts obeys the graded symmetry property (2.48) which implies the standard graded symmetry of the map T s (ψ 1 , ψ 2 , ψ 3 ) upon permutations of the inputs.
Let us illustrate the action of π with a concrete example.We consider the trilinear map T associated with the operator ) and evaluating the pointwise product one obtains where we abbreviated A i = θ µ A µ i .According to the definition (2.47), the symmetrized operator T π is given by yielding the symmetrized map (2.52) From the definition (2.38) of the commutator [O, T ], one can check that the action of O preserves the symmetry property of the map T in the sense that We conclude this review of the input-free formulation by focusing on the possible obstructions.Since we are working on flat spacetime, the wave operator commutes with all the multidifferential operators Ô, M and T in (2.36).Its commutators with the maps O, M and T are thus entirely determined by the commutator of on the pointwise product of functions.We thus define the following operators, acting on three local functions: The subscript in d s alludes to the Mandelstam variable s, and should not be confused with the symmetrization T s .One can compose a Z−valued tri-differential operator T with d s and d , which we denote by juxtaposition: These are also Z−valued tri-differential operators which generate the corresponding maps T d s and T d .For instance, one has ) and so on.Under projection by π, d s and d obey The d operator is always related to a total commutator with , in the sense that With this notation at hand, we can summarize all the relevant BV ∞ relations up to trilinear maps [26]: two-bracket and deformed Leibniz, (2.59) The explicit maps for m n and θ 3 can be found in the appendix of [26], while b 2 and b 3 are easily derived from these by taking b−commutators.The corresponding differential operators mn and θ3 are listed in appendix A.
From the above table one can see that the only consistent subsector is given by the original C ∞ algebra (m 1 , m 2 , m 3h ).On the other hand, the brackets (b 1 ≡ m 1 , b 2 , b 3 ) form an L ∞ algebra only up to −deformations, governed at this level by m 2 and θ 3 .Armed with this structure on K, in the next section we will show that a natural BV ∆ ∞ algebra exists on the tensor product K ⊗ K of two copies of K.

BV ∆
∞ algebra on K ⊗ K In this section we will consider two copies of the kinematic algebra K and show that the respective BV ∞ algebras give rise to a BV ∆ ∞ algebra on the tensor product X := K ⊗ K, where ∆ := 1 2 ( − ¯ ).This will be used in the next sections to derive the three-brackets of double field theory (DFT), both on arbitrary flat backgrounds in the strongly constrained sense and on a torus in the weakly constrained sense.At this point we should mention that the space X is not the L ∞ graded vector space of DFT, which we refer to as V DFT and which is a linear subspace of X to be described below.

Grading and maps on K ⊗ K
We start by spelling out the structure of the tensor product X = K ⊗ K as a graded vector space.From now on, we will denote all elements and maps of K with a bar on the same symbols used for K. Recalling that K = Z ⊗ C ∞ (M), one obtains that X similarly factorizes as a finite-dimensional graded vector space tensored with functions on a doubled spacetime: , which under a certain topological completion holds for the tensor product.Throughout this section, we will take M and M to be d−dimensional flat spaces with unspecified signature, but later on we will specialize to Euclidean signature.Given the structure (3.1) of X and two copies of the finite-dimensional basis, i.e. {Z A } of Z and { Z Ā} of Z, we can expand an arbitrary element Ψ ∈ X as where we denote coordinates of the doubled space by (x µ , xμ ), and from now on we will use capital Ψ for elements in X .The degree in X is defined as the sum of the degrees in K and K, with an additional shift by 2. Specifically, for a homogenous element Ψ = Z Z F (x, x) we set where we recall that degrees in Z (the same for Z) are displayed in (2.8).The shift in degree is by an even amount, so it is immaterial for sign factors like (−1) |Ψ| and thus strictly not necessary.However, the definition (3.3) complies with standard L ∞ degrees in the resulting double field theory.
Given the definition (2.36) and the expansion (3.2) we now proceed to lift the action of operators O : K → K and Ō : K → K to X by defining where the differential operators Ô and Ô act on functions of x and x by taking ∂ µ = ∂ ∂x µ and ∂μ = ∂ ∂ xμ derivatives, respectively.This allows us to sum operators from K and K and yield well-defined operators on X , such as O + Ō.Similarly, tensor products of bilinear maps M and M are defined to act on elements of X as follows: with a completely analogous expression for T ⊗ T (Ψ 1 , Ψ 2 , Ψ 3 ).With these definitions we can extend the input-free notation of the previous section to X .It turns out that operators O and Ō commute (in the graded sense).To show this we compute where we omitted the explicit dependence on (x, x) in intermediate steps.This can be written as the input-free relation [O, Ō] = 0, where the graded commutator is defined as in (2.37), albeit acting on elements of K ⊗ K.A similar computation using the definition (3.5) shows that operators of K commute with bilinear and trilinear maps of K and viceversa, in the sense with analogous formulas for commutators with T ⊗ T .Nesting of bilinear maps can also be extended naturally by defining where the composition M 1 M 2 (same for the barred ones) is defined by (2.40).
Finally, one can introduce on X a symmetric projector Π, obeying Π 2 = Π, via where the global phase is In terms of the map T ⊗ T , this results in which makes the graded symmetry manifest.One can lift the definition of the single copy π or π to a trilinear map T ⊗ T on X by and using (2.43), (2.47) for the single copy symmetrized maps.From this it follows that πΠ = πΠ = ππ, which further implies the decomposition (3.12)

BV ∆ ∞ algebra on X
We are now ready to show that, given the BV ∞ algebras on K and K, a natural BV ∆ ∞ algebra arises on X .As for the single copies, for the moment we will work this out up to trilinear maps.The starting point is the C ∞ sector of (2.59).With the differentials m 1 , m1 and the two-products m 2 and m2 we define a differential M 1 and two-product M 2 on X by (3.7) it is immediate to see that M 1 is nilpotent and acts as a derivation on M 2 : We next study the associativity of M 2 by computing its associator.Due to the graded symmetry of M 2 , the latter is equivalent to the hook projection M 2 M 2 (1 − Π).Using the definition (3.13), the property (3.12) of projectors and the homotopy associativity of m 2 and m2 we can compute where in the last step we used [m 1 , m 2 m 2 ] = 0 (and the barred relation) to extract a total differential M 1 .At this stage an ambiguity arises on how to treat the last term, since for arbitrary ξ.Keeping ξ arbitrary leads to a one-parameter family of three-products, differing by an M 1 −exact term (which for maps means a total M 1 commutator).This is expected, since in a homotopy associative algebra the three-product is defined only up to an M 1 −closed quantity.For simplicity we choose ξ = 0 and obtain obeying homotopy associativity in the form Even though the original C ∞ algebras on K and K have no higher products than m 3h , the tensor algebra X are expected to have infinitely many higher M n , which we will not explore further.
In order to go beyond the C ∞ structure, one has to identify the analogue of the b differential on the tensor space.Two natural candidates are the linear combinations For establishing a BV ∆ ∞ algebra on X , both choices b ± for the second differential are equivalent.Our choice is dictated by the goal of constructing double field theory on a suitable subspace of X which, in particular, requires an unobstructed L ∞ algebra.In view of the fact that the subspace V DFT is partly determined by constraining Since ∆ arises as the above commutator, it is guaranteed to commute with both M 1 and b With this choice one can construct a degree +1 two-bracket B 2 , in perfect analogy with the single copy version (2.34): where in the second line we emphasized its tensor product structure that is of the schematic form "Lie ⊗ Commutative".While b − is trivially a derivation for B 2 , M 1 is not.The obstruction is easily computed and takes the same form as in (2.59): where we used [M 1 , M 2 ] = 0.For the ∆−obstructions on trilinear maps we define D ∆ := Given the definition of D s and D ∆ and the projector (3.9), they obey similarly to (2.56) for d s and d .
Given the product M 2 and the bracket B 2 = [b − , M 2 ], the Poisson compatibility condition (defined as in (2.35) upon replacing m 2 → M 2 and b 2 → B 2 ) can be formulated as The first form of (3.25) emphasizes the Poisson relation between B 2 and M 2 , while the second form shows that this is equivalent to b − being second order with respect to M 2 .Since the single copy b 2 and m 2 obey one does not expect (3.25) to vanish, but rather to obey a similar relation in terms of M 3h and a Θ 3 yet to be determined.To show that this is indeed the case, it is convenient to split the computation of (3.25) into its totally symmetric and hook parts.We start from the hook, which is simple to determine: where we identified One can see that (3.27) has the same form as the hook projection of (3.26), and the relation between Θ 3h and M 3h is the same as in (2.59) for compatibility of the homotopies.Computing the symmetric projection of (3.25) is considerably more involved.We spell out the computation in detail in appendix B, which results in where Θ 3s is determined in terms of Yang-Mills maps as Given the Poisson relation, one can determine the jacobiator 3 B 2 B 2 Π of the two bracket B 2 by taking a b − commutator.This is also presented in appendix B and yields the deformed where the three-bracket is given by This concludes the BV ∆ ∞ relations up to trilinear maps, which we summarize in the following table, analogous to the single copy one (2.59): two-bracket and deformed Leibniz, (3.31)In particular, notice that the brackets of the L ∞ sector (albeit obstructed) are all determined in terms of other structures as . This is in close analogy with closed string field theory, where all the string brackets, apart from B 1 , are b − −exact [17].
We conclude this section by collecting the expressions of the relevant maps in terms of Yang-Mills building blocks: 4 Weakly constrained double field theory In this section we will start from the BV ∆ ∞ algebra on X and construct the L ∞ algebra of weakly constrained DFT on a spatial torus, up to and including the three-bracket, which encodes all the quartic structures of the theory.The obvious issue is that the L ∞ sector of (3.31) is obstructed on X , due to ∆.The idea is that these obstructions should be milder when considering the relevant subspace of weakly constrained fields, obeying ∆Ψ = 0.
Let us start by giving the precise definition of the graded vector space V DFT carrying the L ∞ algebra of double field theory.This is given by the following linear subspace of X = K ⊗ K: Given that an arbitrary element of X can be expanded as in (3.2), with the graded vectors Z A = (θ M , c θ M ) and Z Ā = ( θ M , c θ M ), one can explicitly characterize the elements of V DFT as where c + := c + c, so that c From the degree assignment (3.3), one can see that the degrees in V DFT are given by where we recall that we associate degrees (+1, 0, −1) to M = (+, µ, −).One thus finds, for instance, the DFT gauge parameters in degree −1 to be given by which thus consist of two vector parameters, related to generalized diffeomorphisms, and a Stückelberg scalar parameter.Similarly, one has the fields in degree zero: comprising the tensor fluctuation e µν , two scalars (one combination is related to the dilaton, the other is pure gauge), and two auxiliary vectors.This is precisely the field content of DFT as first introduced in [3].
In order to construct an L ∞ algebra on V DFT , we will proceed in two steps: we will first transport the BV ∆ ∞ structure of X to the subspace X := ker∆ via homotopy transfer, to be discussed momentarily, and in the second step we will restrict the maps to act on kerb − .In this last step the BV ∆ ∞ structure will be lost, leaving an unobstructed L ∞ algebra on V DFT .

Homotopy transfer to ker∆
In order to perform homotopy transfer, we shall find a suitable projector P ∆ to ker∆, together with an homotopy operator h of degree |h| = −1, obeying Since X = ker∆ ⊂ X is a subspace of X , we consider the projector P ∆ : X → X as an operator in X , implicitly assuming a trivial inclusion map ι : X → X (see e.g.[12] for an introduction to homotopy transfer).The projector has to be properly normalized and is required to be a chain map, meaning that it should obey We further require that the homotopy h obeys the so-called side conditions: In order to define P ∆ , we specialize the signature and topology of our underlying doubled space to be a doubled Euclidean square torus.In particular, we have two copies of the Euclidean metric δ µν and δ μν and we identify coordinates with periodicity 2π.Any function on the doubled torus can be expanded in discrete Fourier modes as with an unconstrained sum over discrete momenta (k µ , kμ ) ∈ Z 2d .The obstruction ∆ acts on (4.9) as This allows us to write the projectors P ∆ and (1 − P ∆ ) explicitly in terms of the Fourier expansion by inserting suitable Kronecker deltas: This operator clearly projects to ker∆, since ∆ P ∆ = P ∆ ∆ = 0 , (4.12) and squares to itself: P 2 ∆ = P ∆ .Moreover, being a function of ∆, it commutes with M 1 , thus complying with the properties (4.7).To construct the homotopy operator, we shall first introduce the "propagator" G. Given a function orthogonal to ker∆, meaning it obeys P ∆ f = 0, one can invert ∆ by means of the propagator G, defined as Such operator is clearly not defined on ker∆.Nevertheless, the following operator relations hold on the full space X : as one may quickly verify.Since ∆ = [M 1 , b − ], it is now straightforward to find the homotopy: which indeed obeys the fundamental relation (4.6) and the side conditions (4.8).To verify this one uses that b − is nilpotent and commutes with P ∆ and G.
Equipped with the projector and homotopy maps we are now ready to transport the BV ∆ ∞ structure to X .Homotopy transfer is well-established for L ∞ and A ∞ algebras but, to the best of our knowledge, it has not been discussed for BV ∞ or BV ∆ ∞ algebras.We will thus proceed step by step in a constructive way.From now on, we will denote all transferred maps in X with an overline, which should not be confused with the second copy K, since the underlying Yang-Mills maps will not play a role anymore.Here all inputs are intended as Ψ i , living in ker∆.Since we do not display any input, we shall denote by M| X the restriction of any multilinear map M (including operators) to act on elements of the subspace X ⊂ X .
We begin with the differential, which is unchanged thanks to [M 1 , P ∆ ] = 0: Next, the transferred 2-product M 2 is simply obtained by restricting the map to the subspace and projecting the output back to the subspace: which obeys ∆M 2 = 0 by definition.Similarly, B 2 is given by projection, and it retains its BV relation with M 2 : The original failure of M 1 to be a derivation of B 2 is now cured for B 2 : where we used the fact that acting on elements of ker∆ one has [∆, M 2 ] = ∆M 2 = 0.
We move on to the first trilinear map, which requires the homotopy operator.In order to find M 3h , we compute the associator of M 2 (we leave the restriction ( with the transferred three-product We now define the Poisson compatibility of the transferred M 2 and B 2 as the one in X (see (3.25)) by Since now both M 2 and B 2 commute with M 1 , one immediately has that the expression (4.23) is M 1 −closed.Its hook part is also exact, since for The ∆−obstruction above vanishes, since [∆, M 3h ] = ∆M 3h = 0 when acting on inputs in X .Computing the symmetric part of (4.23) is, as usual, more complicated.
We obtain Now we can use in order to pull out some M 1 −commutators: where we used P ∆ [∆, M 2 hM 2 ]| X = P ∆ ∆M 2 hM 2 | X = 0.The last term above can be further manipulated as follows: where we used (3.24) and the vanishing of total ∆−commutators under P ∆ .We can now use the homotopy Poisson relation (B.13) on X to rewrite (4.27) as with the transferred Θ 3 given by The full homotopy Poisson relation thus reads This is telling us that the BV ∆ ∞ algebra is well-behaved under homotopy transfer, but this is not enough to completely remove the obstructions.The reason is that ∆ is not zero as an operator on X .Rather, only total ∆−commutators are transferred to zero.
The main difference of (4.31) compared to the identity (B.13) is that both sides of (4.31) are M 1 −closed.This is obvious for the left-hand side, since M 1 commutes with both M 2 and B 2 , while checking it for the right-hand side requires some computation: including the deformation.It is interesting to note that only the hook part of Θ 3 (and thus M 3h ) contributes to (4.33), since The transferred three-bracket is given by as dictated by BV ∆ ∞ .Upon using the expression (4.30), one can see that B 3 has also the standard form in terms of homotopy transfer of L ∞ algebras: In the following subsection we will study the obstruction to the homotopy Jacobi identity encoded in the last term of (4.33) upon restricting the inputs to kerb − .Only the L ∞ brackets B n will restrict to V DFT , and we will show that a well-defined albeit non-local deformation of B 3 can be used to remove the obstruction.

Restriction to ker b − and three-bracket
In the last section we have shown that the space X of weakly constrained fields still carries a BV ∆ ∞ structure with milder, but non-vanishing, ∆ deformations.The DFT vector space (4.1) can be viewed, in terms of X , as the subspace V DFT = kerb − ⊂ X .In the following we will denote the restriction of maps M (which already act on inputs in X ) to act on elements in kerb − by M| kerb − .Let us stress that here we are merely restricting the inputs to lie in kerb − , but no homotopy transfer is involved.While generic operators and maps of the BV ∆ ∞ algebra on X are not well-defined on V DFT upon restricting the inputs, the brackets of the L ∞ sector are, as we will show in a moment.
From now on we will focus on the (obstructed) L ∞ sector on X , with brackets B n given by obeying the following quadratic relations: Before studying the obstruction of the homotopy Jacobi identity, we recall that restricting the inputs to kerb − gives us a well-defined differential and two-bracket on V DFT [40], which we denote by B 1 and B 2 , respectively: As we have mentioned, B 1 and B 2 are well-defined on kerb − , since they obey b − B i = 0.While it is trivial to see this for B 2 from (4.38), for the differential it can be shown by computing DFT → V DFT restrict correctly and obey nilpotency and the Leibniz relation, thus defining a consistent field theory (DFT) to cubic order.
We now move on to the Jacobi identity of B 2 , which is given by restriction of the corresponding relation (4.37): with the obstruction denoted by O. Taking into account the restriction to kerb − , one can express the obstruction as where we used that [b − , T ]| kerb − = b − T | kerb − .This expression can be further manipulated by using the definition (4.21) of the transferred M 3h : where we used The homotopy (4.15) obeys ∆h = b − (1 − P ∆ ) and, for inputs in kerb − , we can also write yielding a simpler expression for the obstruction: which we will use to determine whether it can be removed.
First of all, the obstruction is closed: [B 1 , O] = 0, as can be seen by taking a B 1 −commutator of (4.40).However, O is given by projection P ∆ of an otherwise not closed quantity: with explicit O and W given by Since O is not closed, it certainly cannot be exact.It is thus hard to expect that one can extract a B 1 −commutator from O in a simple way.
In order to prove that O is, in fact, exact, we shall consider the Laplacian corresponding to Euclidean signature: which acts on the Fourier expansion (4.9) as Since the metric in both k 2 and k2 is Euclidean, ∆ + is almost invertible.The only solution to ∆ + f (x, x) = 0 is the doubled zero mode f (0, 0), which is allowed on the doubled torus due to its nontrivial topology.We can thus associate a zero mode projector P 0 to ker∆ + , with a corresponding homotopy At this stage, it is important to notice that nonlinear combinations of fields of the form do not contain zero modes.This is easily seen from the fact that in the sum above (k µ , kμ ) = −(l µ , lμ ), given that k 2 = k2 , while l 2 = l2 .The total momentum above is thus (k µ + l µ , kμ + lμ ) = (0, 0).On such combinations one has (1 In order to see that we can apply this argument to our obstruction, let us act with O in (4.44) on three arbitrary inputs ( where by (123) we denote graded symmetrization in the labels.The B 2 B 2 term above has momenta of the form (4.50), given the explicit projector (1 − P ∆ ) and recalling The M 3h term falls in the same category, since P ∆ only acts on input functions, and one has for weakly constrained functions F i (x, x) obeying ∆F i = 0.
Having shown that ∆ + is invertible on O, we can prove that O is exact: where we used [B 1 , O] = 0.
Since we have shown that the obstruction is exact, we can shift the original where B 1 and B 2 are given by (4.38), and the final three-bracket reads Notice that the standard homotopy part B 2 hB 2 in the definition (4.35) of the transported B 3 drops on kerb − , due to h ∝ b − and B 2 ∝ b − .We have thus succeeded in constructing the threebracket of weakly constrained DFT on a purely spatial torus.Since the whole construction is fairly abstract and intricate, in the next section we will provide an explicit check of the above results by computing the gauge algebra.

Gauge algebra
We now compute explicitly a consistent subsector of the gauge algebra of weakly constrained DFT, which is encoded in the homotopy Jacobi relation with the Jacobiator Jac(Λ 1 , Λ 2 , Λ 3 ) defined as The input labels inside of the square brackets [123] on top of the the last equal sign denote antisymmetrization of the labels.In the above equation, as in the remainder of the paper, we employ the convention where we used the antisymmetry of B 2 when acting on gauge parameters: In order to check the identity (4.57), it will be convenient to rewrite the individual terms of the Jacobiator in a different but equivalent way.One can rewrite the inner projector of the nested brackets in the Jacobiator as P ∆ = 1 − (1 − P ∆ ) while keeping the external projector untouched.Doing so yields Jac(Λ 1 , Λ 2 , Λ 3 ) where here and in what follows it is understood that all the maps are acting on kerb − ∩ ker∆ and in order to simplify our notation we introduced the perpendicular projector B 2 ⊥ in the second term in the last line which denotes (1 − P ∆ ) B 2 .The above split will be useful once we compute the part of the homotopy Jacobi relation that contains the three-bracket B 3 because it contains a term with As presented in equation (4.4), a generic gauge parameter in double field theory has three components: two vector components and one scalar component.However, in order to simplify the computation we will restrict our attention to vanishing λν and η while only keeping λ µ .For this reason from now on we consider the consistent subsector of the gauge algebra with parameters of the form Λ = −θ µ θ+ λ µ .(4.61) The homotopy Jacobi relation (4.57) takes values in the space of gauge parameters, and hence consists of three components.For this reason we will check the gauge algebra explicitly displaying the basis elements of the DFT space Z A Z B .This will allow us to keep track of the different components of the relation.
We now turn to finding the two-bracket between gauge parameters using the technology developed in section 3.For gauge parameters defined as in (4.61), we have and we used the component form of m2 (θ µ , θ ν ) and m2 ( θ+ , θ+ ), which can be found in the appendix in equation (A.4).Using the above expression for the two-bracket B 2 we obtain the following Jacobiator: where we use P ∆ ✷ = P ∆ ∆ + .
Having the Jacobiator (4.63) at our disposal, in order to verify the homotopy Jacobi relation (4.57) we need the following components of B 3 : first, B 3 on three gauge parameters, whose only non-trivial part can be found with the following computation: where we used the component form θ3s (θ µ , θ ν , θ ρ ) shown in equation (A.12).Second, we need to find B 3 (Λ 1 , Λ 2 , Ψ).From a computation analogous to the above, we find From this expression one infers by inspection of the third to fifth line that the non-locality inherent in 1 ∆ + is unavoidable: there is no overall ∆ + that can be factored out to cancel it, as ∂ν is contracted with e µν and not with a derivative.This changes after replacing the field in (4.65) by B 1 (Λ), which is the next step in order to verify the homotopy Jacobi relation.For instance, in the last line in equation (4.65) one obtains where the equality follows using the Leibniz rule and the antisymmetry of the labels.Under the projector P ∆ we can then use the weak constraint ∂ν ∂ν ≡ ¯ = together with P ∆ = P ∆ ∆ + to cancel 1 ∆ + .Doing so for the other terms and appropriate antisymmetrizations of the inputs leads to which has no non-localities.

Conclusions and Outlook
In this paper we have explicitly constructed weakly constrained double field theory to quartic order in fields, encoded in the three-brackets of the corresponding L ∞ algebra.Due to the 'weak constraint' originating from the level-matching constraints of string theory, the construction of such a theory is a highly non-trivial problem and requires an essential non-locality, which is also present in the full string theory.Specifically, the weak constraint requires that all fields are annihilated by ∆, the second-order Laplacian w.r.t. the flat metric of signature (d, d).It is precisely the second-order character of ∆ that complicates the construction of an algebra of fields, since the product of two fields satisfying the weak constraint in general does not satisfy the weak constraint.Rather, the naive product has to be modified by projecting the output to the ∆ = 0 subspace, an operation that singles out certain Fourier modes and is hence non-local.Consequently, the resulting product is non-associative.
It is relatively straightforward to solve the resulting consistency problems to cubic order [3,46], which is essentially due to a 'kinematical accident', but to quartic and higher order it is highly non-trivial to construct a consistent field theory (an L ∞ algebra).In this paper we give the corresponding L ∞ algebra up to and including three-brackets, constructed via a double copy procedure from Yang-Mills theory.Apart from the non-locality inherent in the weak constraint we found the need for additional non-localities in the form of inverses of ∆ + , the positive definite flat-space Laplacian, but they only show up in terms where they are fully well-defined on the torus.We verified that these new non-localities are inevitable given the problem we set out to solve: finding the three-brackets B 3 so that the Jacobiator relation involving B 2 B 2 is obeyed.Since B 1 and B 2 are fixed from the cubic theory of Hull and Zwiebach in [3], 3 which agrees with our double copy [40], the only freedom is the definition of B 3 (which is only well-defined up to cohomologically trivial contributions that drop out from [B 1 , B 3 ]).We have verified for the gauge sector that the inverses of ∆ + are essential.
The research presented here should be generalized in many directions, which include: • So far we have given the three-brackets only in the case that all dimensions are toroidal and hence Euclidean, with all coordinates being doubled.It remains to include an undoubled time coordinate or, more generally, an arbitrary number of dimensions for the 'external' or non-compact space.Thus, the theory presented here should be thought of as the 'internal' sector of a split (or Hamiltonian-type) formulation as in [50,51] for double field theory (or in [52][53][54] for the closely related U-duality invariant 'exceptional field theory').It would be interesting to see whether such split formulations can be interpreted as a tensor product between 'internal' and 'external' algebras along the lines of [55,56].
• One of the potentially most important applications, and one of strongest original motivations, of a weakly constrained double field theory is in the realm of cosmology.One may imagine massive string modes being excited in the very early universe that leave an imprint on the cosmic microwave background (CMB).For instance, the string gas cosmology proposal of Brandenberger-Vafa invokes the winding modes that must be present if some of the spatial dimensions in cosmology are toroidal [57], see [58] for a recent review.Generalizing the previous item, it remains to find a weakly constrained double field theory on time-dependent Friedmann-Robertson-Walker backgrounds, generalizing the cubic perturbation theory of [46] to quartic and higher order.
• Independent of the inclusion of non-compact dimensions, the arguably most important outstanding problem is to generalize the construction to higher order in fields, even just for the 'internal' or compact dimensions.Since the quartic theory exists, it is virtually certain that the theory exists to all orders, but since the detailed construction is already quite involved for the three-brackets we need a more efficient formulation for the kinematic BV ∞ structures that are present in Yang-Mills theory proper in order to display the algebra and its double copy to all orders. 4It is intriguing that this is a problem already in pure Yang-Mills theory, which thus displays a complexity comparable to that of gravity.
• Our double copy procedure developed in [26,40], which is based on the additional structures involving the 'b-ghost', may appear rather special and only applicable to peculiar formulations, but this is not so.We hope to be able to illustrate this with further examples in the future and to develop the general theory further.Notably, in this paper we have been cavalier about the cyclic structure of the L ∞ algebra, which is needed in order to write an action.Thus, the results presented here are strictly applicable to the equations of motion only.We leave the detailed construction of the cyclic L ∞ brackets, which might differ from the ones presented here by cohomologically trivial shifts (that, however, in the language of BV, are not symplectomorphisms) to future work.
• The weakly constrained double field theory constructed here to quartic order is quite complicated and non-local.While above we have emphasized that the non-localities are inevitable given the fixed starting point encoded in B 1 and B 2 of the cubic theory of Hull and Zwiebach, it is conceivable that there are simpler versions that carry more propagating fields, which would manifest themselves already to quadratic order, and that are effectively integrated out in the theory encountered here.One may wonder if there are versions with weaker constraints or perhaps even no level-matching constraints, as recently explored in string field theory [59,60].
• It would be very interesting to generalize weakly constrained double field theory to theories including massive 'M-theory states' of the kind required by U-duality invariance.The strongly constrained versions are known as exceptional field theory (see, e.g., [52-54, 61, 62]).One of the challenges here is that there is no immediate analogue of the double copy construction from Yang-Mills theory, but one may speculate that there are exotic field theories waiting to be constructed who could serve as similar building blocks [63].
This ensures the graded symmetry of the map θ 3 (ψ 1 , ψ 2 , ψ 3 ) in the first two arguments.Since all the maps associated with (A.10) are totally graded symmetric, the corresponding operators obey θ3 = θ3s = θ 3 π.The remaining permutations of the arguments (Z A , Z B , Z C ) can then be recovered from (2.48).The next θ3 operators have both a totally graded symmetric part θ3s = θ 3 π and a hook part θ3h = θ3 − θ 3 π, which we give separately: and one can see that they are antisymmetric in the simultaneous exchange of µ ↔ ν and The last group of non-vanishing θ3 also has totally graded symmetric and hook components, given by In this appendix we compute the symmetric projection of the Poisson relation (3.25), in order to determine the symmetric part of the homotopy Θ 3 .We then use this to compute the jacobiator of the bracket B 2 , yielding the deformed homotopy Jacobi identity.
We begin by writing the maps in terms of their Yang-Mills building blocks: ) which we sometimes summarize by writing |θ M | = 1 − M , where M means +1, 0 or −1, depending on the index.Next, we take a second copy of these vectors with degrees shifted by one, which we denote by c θ M , with |c θ M | = 2 − M , or

( 3 . 13 )
From the graded symmetry of m 2 and m2 one can easily show that M 2 is graded symmetric in X .Due to the degree shift (3.3), one has |M 1 | = +1 and |M 2 | = +2.Upon using (3.6) and

. 18 ) 1 2
Both b ± are nilpotent and (anti)commute with each other.Their commutators with M 1 , which determine the obstruction ∆, are given by [M 1 , b ± ] =

1 2
(d − d¯ ) and D s := 1 2 (d s − ds ) which act on three functions F i (x, x) as

(4. 32 )
We thus see that the ∆−obstruction in (4.31) is M 1 −closed.If it were exact, one could shift Θ 3 to obtain a genuine homotopy Poisson identity.If this is not the case, on the other hand, M 3h D s Π would be a genuine cohomological obstruction.For the last step, let us determine the homotopy Jacobi relation.Due to B 2 = [b − , M 2 ], the Jacobiator is given by taking a b − −commutator of the left-hand side of (4.31).The computation is entirely analogous to the one leading to (B.15), and we obtain