The Gauge Structure of Double Field Theory follows from Yang-Mills Theory

We show that to cubic order double field theory is encoded in Yang-Mills theory. To this end we use algebraic structures from string field theory as follows: The $L_{\infty}$-algebra of Yang-Mills theory is the tensor product ${\cal K}\otimes \mathfrak{g}$ of the Lie algebra $\mathfrak{g}$ of the gauge group and a `kinematic algebra' ${\cal K}$ that is a $C_{\infty}$-algebra. This structure induces a cubic truncation of an $L_{\infty}$-algebra on the subspace of level-matched states of the tensor product ${\cal K}\otimes \bar{\cal K}$ of two copies of the kinematic algebra. This $L_{\infty}$-algebra encodes double field theory. More precisely, this construction relies on a particular form of the Yang-Mills $L_{\infty}$-algebra following from string field theory or from the quantization of a suitable worldline theory.


Introduction
In this paper we show that, at least to cubic order, double field theory can be derived from Yang-Mills theory through an off-shell and gauge invariant 'double copy' construction. Double copy is a powerful tool in constructing gravity scattering amplitudes from Yang-Mills or more general gauge theory amplitudes [1][2][3]. It is a central pillar of the modern amplitude program that in turn deemphasizes Lagrangians, off-shell states and gauge redundancies. As such it is often considered to be beside the point to seek a Lagrangian understanding of double copy. Nevertheless, there have been numerous attempts to establish double copy relations at the level of a Lagrangian, see e.g. [4][5][6][7][8][9][10][11][12]. Recently, together with Plefka, two of us established in [13] a close double copy relation to cubic order between the Lagrangians of Yang-Mills theory and double field theory (DFT) [14][15][16][17][18], which is a formulation of the string target space theory for graviton, B-field and dilaton that is T-duality invariant thanks to doubled coordinates. (See [19][20][21][22][23][24] for reviews and earlier work and [25][26][27] for previous work on double copy and DFT.) The construction of [13] requires integrating out the DFT dilaton and picking Siegel gauge at cubic order. Here we generalize these results by showing that the full gauge invariant DFT to cubic order, including all dilaton couplings, follows from Yang-Mills theory.
The approach employed in this paper is algebraic, based on strongly homotopy algebras such as L ∞ -algebras [28][29][30], which in turn are closely related to the Batalin-Vilkovisky (BV) formalism. L ∞ -algebras are generalizations of Lie algebras, defined on an integer graded vector space X = i∈Z X i that encodes the space of fields, the space of gauge parameters, etc., and a potentially infinite series of graded symmetric maps or brackets b n , n = 1, 2, 3, . . ., with n inputs obeying quadratic generalized Jacobi identities. When equipped with a graded symmetric inner product the action of the theory is encoded in the L ∞ brackets via where A stands for all fields and the ellipsis indicates higher order terms. For the case of Yang-Mills theory in the standard formulation (i.e. with at most quartic vertices) there are no higher brackets than 3-brackets. Similarly, gauge transformations, gauge algebra, Noether identities, etc., are all encoded in the L ∞ structure (see [31] for the general dictionary between L ∞ -algebras and field theory).
It was shown by Zeitlin that the L ∞ -algebra of Yang-Mills theory can be viewed as a tensor product of the Lie algebra g of the gauge group and a so-called C ∞ -algebra K [32]: (1. 2) A C ∞ -algebra is a homotopy version of a differential graded commutative and associative algebra. It must be emphasized that (1.2) is in no way a symbolic relation but rather a completely precise statement about the tensor product of certain algebras. As the algebra K is obtained from the Yang-Mills L ∞ -algebra by 'stripping off' color factors we will refer to it as the kinematic algebra of Yang-Mills theory, but we should point out that it does not immediately encode the relations that in the amplitude community are referred to as kinematic algebra (e.g. [33][34][35]). In any case, double copy suggests that the tensor product K ⊗K encodes a gravity theory.
In the following we will show, to cubic order in fields, that this progam can indeed be realized, with the gravity theory being DFT. This requires working with a particular formulation of Yang-Mills theory and a subspace of K ⊗K corresponding to states satisfying the level-matching constraints of closed string theory. These features are directly motivated from the string theory origin of the double copy structure: the KLT relations between open string and closed string scattering amplitudes. As such, it is natural to suspect that string field theory (SFT) is the appropriate framework to make double copy manifest. Concretely, we will use an SFT inspired formulation of Yang-Mills theory with auxiliary fields so that all kinetic terms always come with a , and gauge fixing amounts to setting some auxiliary fields to zero. In BV language this formulation is known as non-minimal and can also be derived from the quantization of a worldline theory. We will then show that the C ∞ -algebra structure induces the (cubic truncation of the) L ∞ -algebra of DFT: where the notation indicates the subspace of level-matched states. 1 As a vector space, (1.3) contains precisely the required objects to encode DFT. In particular, the doubling of coordinates of DFT is automatic and not imposed: given an algebra K of functions of x and an independent algebraK of functions ofx the tensor product is a space of functions of (x,x). This is a special case of the general relation that the tensor product of It should be noted that establishing a homotopy algebra structure on a tensor product of such algebras in general is quite subtle. While algebras with no higher than 2-products or bracket, which are known as strict, behave nicely under tensor products (so that, for instance, the tensor product of two strict C ∞ -algebras canonically yields a strict C ∞ -algebra), for general homotopy algebras it is more difficult to display a homotopy algebra structure on a tensor product. (See e.g. [39] for the case of homotopy associative or A ∞ -algebras.) For a cubic truncation the L ∞ structure requires only a 2-bracket with the correct graded symmetry properties obeying Leibniz relations with the differential. For both of these properties the truncation to the levelmatched subspace in (1.3) is instrumental. We do not yet know of a construction of the complete L ∞ -algebra on this space, as would be required in order to realize double copy to all orders.
In order to appreciate the kind of detailed relations between Yang-Mills theory and DFT that are encoded in (1.3) let us mention a particularly striking example: the DFT gauge transformations that are linear in the field e µμ (encoding metric and Kalb-Ramond fluctuations) are directly obtained from the 3-vertex of Yang-Mills theory! To see this note that according to (1.1) the cubic term of Yang-Mills theory is encoded in the 2-bracket on g-valued one-forms where f a bc are the structure constants of the Lie algebra g (the 'color factors') while • is a product on one-forms defined by where the Minkowski metric is used to raise and lower indices. (Of course, given this metric, we might as well view this as a bracket of vectors rather than one-forms.) While the cubic term in the Yang-Mills action may be re-written in various equivalent ways, say by integrations by part, 2 the corresponding L ∞ 2-bracket (1.5) is unambiguous: it takes, according to (1.6), the form of an (antisymmetrized) generalized Lie derivative of DFT (with density weight one), just with the Minkowski metric instead of the O(d, d) metric. However, in the perturbative formulation of DFT this is precisely the gauge transformation of e µμ with respect to the gauge parameter λ µ [15]: δ up to terms involving auxiliary fields of the SFT like formulation. In (1.7) the indexμ is viewed as inert, and there is an analogous gauge transformation for a gauge parameterλμ, with all derivatives and indices in (1.6) replaced by barred objects and now with the index µ being inert. Note that the above relates a 2-bracket between fields on the Yang-Mills side to a 2bracket between field and gauge parameter on the DFT side, and we will see that this precisely originates from the tensor product (1.3) as does the full cubic truncation of the L ∞ -algebra of DFT. Thus, this gauge invariant double copy is properly viewed as a map between the complete gauge theories as encoded in the corresponding L ∞ -algebras, as opposed to a simple redefinition between classical fields or their action.
The rest of this paper is organized as follows. In sec. 2 we write down the L ∞ -algebra encoding Yang-Mills theory with a particular auxiliary field and identify the C ∞ 'kinematic algebra' K. While this formulation was inspired by open string field theory and was in fact originally constructed by a related worldline quantization, we present our results without explicit reference to the SFT or worldline formulation in order to keep this section self-contained. In sec. 3 we consider the subspace of K ⊗K of level-matched states and show that it inherits the cubic truncation of an L ∞ -algebra that encodes DFT. We close with a brief conclusion and outlook in sec. 4. For completeness, and since it is of interest in its own right, we discuss the worldline approach in an appendix.

The Kinematic Algebra of Yang-Mills
In this section we present the L ∞ -algebra for Yang-Mills theory encoding its gauge structure, classical field equations and Noether identities. We use a non-standard formulation that can be motivated by open string field theory or a worldline quantization. We then explain how 'stripping off' the color part from the brackets one obtains a C ∞ -algebra, i.e. a graded commutative algebra obeying associativity up to homotopy. This 'kinematic algebra' will be used in the next section to derive double field theory to cubic order.

L ∞ -algebra of Yang-Mills
Let us consider the following form of the Yang-Mills action that includes an auxiliary scalar ϕ in the free Lagrangian: where all fields are Lie algebra valued, e.g. A µ = A a µ t a , with Lie bracket [·, ·] and generators normalized as Tr t a t b = δ ab . Upon integrating out ϕ one recovers the standard Yang-Mills action. This ensures that the standard cubic and quartic vertices in (2.1) are consistent with gauge invariance, provided that A µ has the usual transformation rule and that ϕ transforms as its on-shell value ϕ = ∂ µ A µ , i.e.
As we have mentioned, this non-standard form of the free action arises naturally from the BRST quantization of the N = 2 spinning particle (see appendix A), or from string field theory.
We now describe the L ∞ structure of Yang-Mills in this formulation. An L ∞ algebra is a graded vector space X = i X i endowed with multilinear maps b n of intrinsic degree −1, which obey quadratic Jacobi-like identities. Since Yang-Mills theory has at most quartic vertices the non-vanishing brackets are b 1 , b 2 and b 3 . The L ∞ relations in this case are: • Nilpotency of the differential • b 1 acts as a derivation on b 2 (Leibniz rule) • Homotopy Jacobi identity Here x, y, z in exponents denote the L ∞ degree of the corresponding element.
Notice that, even if higher brackets vanish, one has higher relations involving b 2 b 3 and b 3 b 2 which we do not display here. The sign conventions, referred to as the b−picture, are somewhat unconventional but are more convenient for field theory applications and also standard in string field theory [28]. In the b-picture the brackets are graded symmetric: Given this general structure, we now focus on Yang-Mills theory defined by the action (2.1). The corresponding L ∞ complex is the graded vector space with differential b 1 of degree −1. The space of highest degree X 1 contains gauge parameters λ, while the spaces of lower degrees are identified as the space of fields, equations of motion and Bianchi/Noether identities, as shown in the following diagram: The spaces of gauge parameters and Noether identities consist of scalars λ and N , respectively, while the space of fields A and the space of field equations E consist of doublets: A = (A µ , ϕ) and E = (E µ , E).
Free theory and the differential b 1 One can read off the differential b 1 from the linearized field equations and gauge transformations of (2.1): to give the action of the differential on X 1 and X 0 : where here and in the following we use the notation ∂ · A = ∂ µ A µ . Nilpotency of b 1 acting on λ encodes gauge invariance of the free field equations, b 2 1 λ = b 1 δA = 0. We define the differential b 1 acting on E as (2.11) in order to ensure that the free Noether identity is encoded in b 2 1 A = b 1 E = 0. There is no further nontrivial realization of b 2 1 = 0. As we will discuss below, the differential b 1 coincides with the first-quantized BRST operator Q of the associated worldline theory.
Interacting theory and the brackets b 2 and b 3 The field equations of the full theory can be written in L ∞ form as Since the auxiliary ϕ does not enter interactions the two and three-brackets between fields take the form for which one finds from the general dictionary between L ∞ -algebras and field theory [31]: , (2.14) with the abelian field strength f µν = ∂ µ A ν − ∂ ν A µ . The field equations (2.12) are covariant under the deformed gauge transformation which, comparing with (2.2), fixes the two bracket between a field and a gauge parameter to be Demanding that the Leibniz property (2.4) holds for all the allowed combinations of inputs, one finds the following list of non-vanishing brackets: (2.17) The homotopy Jacobi identities (2.5) are satisfied with the only non-vanishing three-bracket b µ 3 (A 1 , A 2 , A 3 ) given in (2.14).
Inner product and action We conclude the discussion of the L ∞ algebra of Yang-Mills by giving the inner product, which allows us to write the action in the standard L ∞ form [28,31]. Specifically, the inner product of X is a degree +1 pairing, including a map between fields and field equations: Given a field A and a field equation E with components we define their inner product by the spacetime integral with a similar pairing between gauge parameters λ and Noether identities N . Using the form (2.10) and (2.14) for the brackets one may verify that the Yang-Mills action (2.1) can be written in terms of the inner product as The Euler-Lagrange equations of this action indeed take the form (2.12).

The Kinematic Algebra
Having presented the L ∞ algebra of Yang-Mills, our next goal will be to disentangle the color degrees of freedom from the elements of X and from the brackets. This defines the vector space K of the kinematic algebra, whose elements are spacetime fields with no color dependence. Similarly, we will define purely kinematic products m n acting on the kinematic vector space.
Let us start with the vector space itself. Since all elements of the L ∞ algebra X are Lie algebra-valued fields, they can be written as x = x a t a ∈ X in terms of generators t a ∈ g, where g is the Lie algebra of the gauge group. This shows that the L ∞ complex X has the structure of a tensor product space X = K ⊗ g, where g is endowed with the standard Lie bracket, and K is the vector space of the kinematic algebra, such that t a ∈ g and x a ∈ K. If one considers the Lie algebra g as a special case of an L ∞ -algebra concentrated in degree +1 (in the b−picture), the generators t a have degree +1, while the Lie bracket [ , ] has intrinsic degree −1. Next, we shall define the degrees of the kinematic algebra by declaring that the degrees of elements and maps of K and g are additive, so that x = u ⊗ t , u ∈ K , t ∈ g → |x| X = |u| K + |t| g = |u| K + 1 .
(2. 22) This implies that the kinematic algebra K is given by the direct sum Although elements of K are spacetime fields with no color degrees of freedom left, informally we still refer to K 0 as the space of gauge parameters λ, K −1 as the space of fields A = (A µ , ϕ), etc., i.e. we keep the same symbols for elements of K in order to avoid burdening the notation. We shall now define the multilinear maps m n on K.
Differential m 1 Since the Lie algebra g plays no role in defining the free field equations, the differential m 1 is the same as the L ∞ differential b 1 or, more precisely, where the degree of m 1 can be inferred from the definition (2.24) and |b 1 | = −1. This yields the explicit realization Two-product m 2 We shall now define a degree zero graded commutative product m 2 on K, which thus obeys which serves as an implicit definition of m 2 . The degree The sign factor (−1) x 1 has been chosen so that (2.27) is also compatible with the symmetry property of b 2 . This can be checked by computing (2.28) Let us point out that the definition (2.27) does not imply the graded symmetry of m 2 . Rather, it implies that any part of m 2 which is not graded symmetric is projected out from the resulting b 2 . The symmetry property (2.26) has thus to be seen as part of the definition of m 2 .
Let us now use the definition (2.27) to give the explicit form of the Yang-Mills two-products m 2 . It is convenient to use the decomposition along a basis t a of g, such that x = x a ⊗ t a . In this case (2.27) reduces to in terms of component fields with adjoint color indices. Using (2.29) and the brackets (2.17) one finds that the non-vanishing products are given by where the antisymmetric product • between vector fields, already defined in the introduction, is given by The non-diagonal products with a different order of inputs, e.g. m 2 λ, A or m 2 E, A , are defined by the symmetry property (2.26).
Three-product m 3 The only non-vanishing three-bracket of Yang-Mills theory acts on three degree zero elements, which are the vector components of three fields: For this reason, the only non-vanishing kinematic three-product m 3 (u 1 , u 2 , u 3 ) acts on three elements (2.32) Upon inspection one may verify that this obeys In the language of Young tableaux this means that m 3 has the symmetry property of a (2, 1) 'hook' Young tableau. Consider now three degree zero elements x 1 , x 2 , x 3 ∈ X 0 in the L ∞ algebra X . Taking them to be of the form x i = u i ⊗ t i , with t i ∈ g and u i ∈ K −1 , the three-bracket (2.14) can be written as which guarantees total symmetry in the inputs x 1 , x 2 and x 3 thanks to (2.33) and the symmetry properties of the nested Lie algebra bracket. In order to recognize the previous expression (2.14), it is convenient to use the standard basis decomposition x = x a ⊗ t a , which yields One can use (2.32) in the expression above to immediately recover (2.14) .
In the last part of this section we will prove that the vector space K endowed with the products m 1 , m 2 and m 3 given by (2.25), (2.30) and (2.32) defines a C ∞ -algebra: a graded vector space endowed with multilinear products m n of degree |m n | = n − 2 obeying certain symmetry properties and associativity up to homotopy, see e.g. [41][42][43]. Specifically, C ∞ -algebras are A ∞ -algebras with products that obey additional symmetry properties. In particular, the two and three-products m 2 and m 3 obey which is the case for the Yang-Mills products (2.30) and (2.32), recalling that m 3 is nonvanishing only for |u 1 | = |u 2 | = |u 3 | = −1. The quadratic relations for the m n are as follows: First, the differential m 1 is nilpotent: and acts as a derivation with respect to m 2 (Leibniz rule): Second, one has associativity of m 2 up to homotopy, which is expressed as where m 3 has degree +1. Even though there are no higher products, m 2 and m 3 have to satisfy [36] in order to be consistent with the absence of m 4 . The last possible relation, involving m 3 m 3 , vanishes identically for degree reasons. We will now show that the differential (2.25), two and three-products (2.30) and (2.32) obey the graded Leibniz and homotopy associativity relations. Finally, we will establish (2.40), which concludes the proof of the quadratic relations.
Nilpotency of the differential m 1 follows immediately as above. For the Leibniz property we shall proceed in order, starting from the highest degree: Degree −1 The only Leibniz relation at degree −1 involves the product of two gauge parameters λ 1 and λ 2 , and reads as can be seen from (2.25) and (2.30). In the L ∞ -algebra associated to K by tensoring with g, this relation encodes closure of the gauge transformations of Yang-Mills.
Degree −2 Also in degree −2 one has only one relation, between a gauge parameter λ and a field A: which corresponds to gauge invariance of the field equations in X .
Degree −3 The relations in lowest degree can either take two fields A 1 and A 2 , thus expressing the deformation of the Noether identity N , or a gauge parameter λ and a field equation E, which corresponds to the consistency of N = 0 with gauge symmetries. The first one yields where we point out that, given a field equation The last Leibniz relation, with λ and E as inputs, gives which concludes the proof of the Leibniz property (2.38).
We now turn to the proof of the homotopy associativity relations (2.39). Since the only nonzero m 3 involves three vector fields, most of the m 2 products obey strict associativity. As we have done for the Leibniz relations, we shall proceed in order from the highest degree.
Degree 0 In degree zero one can only take three gauge parameters λ 1 , λ 2 and λ 3 . In this case the relation is trivial to prove, due to m 2 being an associative pointwise product: (2.45) Degree −1 In this case the only possibility is to act on two gauge parameters λ 1 , λ 2 and a field A, yielding which is also strictly associative.
Degree −2 There are now two possibilities: one takes as inputs two parameters λ 1 , λ 2 and an equation of motion E, and this is associative as well: (2.47) The second possibility is to take one parameter λ and two fields A 1 , A 2 , which is the first case to require a three product: where we used the definition (2.32). Since the only nonzero m 3 has three degree −1 inputs, the single term appearing above is the only one required to satisfy (2.39).
Degree −3 One has three cases in lowest degree: the inputs can in fact be ( The first two cases are strictly associative: (2.49) The last relation, instead, involves the three-product and is obtained as in agreement with (2.39).
In order to complete the proof that K is a C ∞ -algebra without higher products, we are left to prove the consistency condition (2.40). Since m 3 can only act on three vectors of degree −1, the only nontrivial relations are in degree −2 and −3.
Degree -2 In this case one can have three fields A i and one gauge parameter λ, giving rise to two possible relations: (2.51) The first one is easily established by computing (2.52) and the second one follows immediately from the line above.
Degree -3 One can only act on four fields, yielding which concludes the proof.

Z 2 Grading
In the last part of this section we will show that both the Yang-Mills L ∞ -algebra X and the C ∞ -algebra K admit a further Z 2 grading. This additional grading is crucial in constructing double field theory from doubling. To this end it is moreover convenient to write the kinematic algebra in terms of graded basis vectors for K, which have a natural interpretation as oscillators and ghosts of an underlying N = 2 particle or open string theory.
Let us start by recalling the component structure of the kinematic algebra K (the same applies to the L ∞ -algebra X upon tensoring K with the Lie algebra g and shifting the degrees accordingly). Specifically, we recall that both fields and field equations are split into doublets, and we group the objects of the chain complex as follows This displays the decomposition of K w.r.t. a new Z 2 degree with values 0, 1, which we name c−degree, 3 along the vertical direction above. Put differently, we assign c−degree zero to (λ, A µ , E) and c−degree 1 to (ϕ, E µ , N ), such that For the following discussions it will be useful to split the full complex K according to the c−degree alone, writing It is important to note that K (0) and K (1) are isomorphic as vector spaces, while the C ∞ degrees between the two are shifted by −1. The vectors of K (0) and K (1) then have components  thus making the isomorphism apparent. The full algebra K can thus be split as K = K (0) ⊕K (1) , which will be the most useful form to construct the DFT complex in the next section.
In order to simplify the subsequent treatment, in particular the doubling procedure, we will reformulate the above in a form which is more akin to the first-quantized description of the field theory. To this end, we introduce a basis for the triplet (2.57) in K (0) consisting of graded vectors where M = (+, µ, −) count as (+1, 0, −1) in determining the C ∞ degree. Given the isomorphism between the components of K (0) and K (1) , we shall take into account the Z 2 split due to the c−degree by tensoring the above basis with a two-dimensional Grassmann algebra generated by an odd nilpotent element c, obeying This element is nothing but the reparametrization ghost of the first-quantized theory and allows one to introduce a basis |c θ M for the second triplet (2.57) in K (1) : (2.60) In this formulation, spacetime fields are components of intrinsic vectors of K along the basis |θ M , |c θ M . As such, component fields are taken to have zero degree and the C ∞ degree is entirely carried by the basis elements |θ M and |c θ M , according to (2.58) and (2.60). An arbitrary vector u ∈ K can thus be expanded as according to the decomposition K = K (0) ⊕ K (1) . Homogeneous vectors of the different spaces K i , decomposed according to (2.55) and (2.57), can then be written as to which we will sometimes refer as classical 'string fields', gauge parameters and so on.
The same formalism can be applied at the level of the L ∞ -algebra X , by tensoring the graded vectors |θ M , |c θ M with the Lie algebra generators t a , yielding as basis elements for X (0) and X (1) . However, for the remainder of this section we will focus on the kinematic algebra K.
In order to reformulate the differential m 1 and the product m 2 in this formalism, it is useful to introduce basis vectors for the dual space K * . To do this, we shall first introduce another odd nilpotent element b that is conjugate to c and obeys b 2 = 0 , |b| K = +1 , bc + cb = 1 . (2.64) The action of c and b on the basis |θ M , |c θ M realizes the Z 2 isomorphism between K (0) and K (1) as follows: (2.65) We can now introduce a basis θ M * |, θ M * b| for the dual spaces K (0) * and K (1) * , respectively. The natural pairing with the basis of K is given by (2.67) Moreover, this basis allows to write the resolution of the identity in K as We will now show how to reformulate the differential m 1 and the product m 2 in this formalism. We start from the differential, which is defined as a degree −1 map m 1 : K → K and which in the following we will sometimes also denote by Q since it has the interpretation of the BRST operator of a worldline theory. We decompose Q as (2.69) This notation should be understood as follows: on a general vector (2.61) in K the bra vectors act in the standard fashion on ket vectors, while the spacetime derivatives act on the component spacetime fields, i.e., for (2.61) they act on u M (x) and v M (x). So defined, Q is a map Q : K → K, and one finds that it indeed reproduces the differential m 1 defined above. For instance, taking u = A ∈ K −1 to be a field one has (2.70) Another important property, which can be checked by using (2.65) and (2.67), is that which will be crucial in proving consistency of the doubling in the next section.
After discussing the realization of the differential Q (or m 1 ), we now turn to the twoproducts (2.30). To this end it is convenient to realize the degree zero map m 2 : K × K → K in terms of an element M ∈ K ⊗ K * ⊗ K * , so that The action of such an element M on u 1 ⊗ u 2 ∈ K ⊗ K is defined, given a vector u ∈ K and two dual vectors U 1 , U 2 ∈ K * , by where in the exponent we denote u 1 ≡ |u 1 | K and so on. This is a well-defined formula for finitedimensional vector spaces, but in our context the components are spacetime fields so that the algebras are infinite-dimensional and we have to be more precise about this action. Specifically, we define M by the expansion and we claim that the m 2 above is recovered upon setting M 0 00 = |θ + θ + * | θ + * | + |θ µ θ µ * | θ + * | + θ + * | θ µ * | , where we omitted the tensor product symbol between the basis elements. The action of the basis elements in here is given by (2.73), while the spacetime derivatives act on the component fields, where by ∂ 1µ and ∂ 2µ we indicate that the spacetime derivative acts on the left (respectively, right) factor of the tensor product u 1 ⊗ u 2 .
As an example of how to use these formulas to compute the products (2.30), let us compute the two-product m 2 (Λ, A) between a gauge parameter and a field: where we reinstated the ⊗ symbol to better visualize the pairing between K * ⊗ K * and K ⊗ K. We see that (2.76) reproduces the corresponding product of (2.30). Finally, one may check that (2.74) with components (2.75) gives the correct symmetry property of m 2 : We are now ready to reformulate the quadratic relations of the C ∞ -algebra in terms of operator equations. As we have anticipated, from now on we will only be concerned with the cubic theory, whose consistency relies purely on nilpotency of the differential and the Leibniz property (2.38). Nilpotency of m 1 or Q and the graded Leibniz rule (2.38) can be expressed as the operator equations where the right-hand side of the second equation defines the action of Q on the tensor product K ⊗ K, namely The Leibniz relations have been proved in section 2.2 in component form, which guarantees that (2.78) holds, but one can prove it directly in operator form. For instance, one can focus on the part of (2.78) proportional to |θ µ θ + * | θ + * | and compute where we wrote the total derivative in Q M as ∂ µ 1 + ∂ µ 2 acting on K ⊗ K. Similarly, the corresponding part of the right-hand side of (2.78) yields

(2.81)
This proves the |θ µ θ + * | θ + * | 'component' of (2.78), and all the other components can be proved in the same way. In the next section we will use two copies of the vector space K, each one endowed with its Q and M operators, to construct double field theory at cubic order. Consistency of the latter will be a direct consequence of the fundamental relations (2.78).

Cubic Double Field Theory from Doubling
We start this section by describing the L ∞ complex V for double field theory (DFT), as given in the original formulation of Hull and Zwiebach [15], and show that V is a subspace of the tensor product of two copies of the C ∞ complex K of Yang-Mills theory. We then define the DFT differential Q and two-bracket B 2 in terms of two copies of the Yang-Mills operators Q and M and prove that Q is nilpotent and acts as a derivation of B 2 . This implies consistency of DFT at cubic order. We apply these results by giving the explicit expressions for the deformed gauge transformations and the cubic action, which yields significant simplifications of the results in [15].

The Double Field Theory Complex
Double field theory is defined on a doubled spacetime, which we take to have coordinates (x µ ,xμ). These correspond to the left and right-moving parts of the closed string center of mass, while the coordinates more often used in the literature, (X µ , X µ ), correspond to the standard and dual frames with respect to T −duality. In addition, all fields and parameters in DFT are subject to the weak constraint, corresponding to the level matching constraint in string theory. In the coordinate system (x µ ,xμ) this is expressed as [15] where = η µν ∂ µ ∂ ν and = ημν∂μ∂ν are defined in terms of two copies of the Minkowski metric η and the derivatives 4 The standard supergravity solution of the weak constraint is to identify the two sets of coordinates, declaring ∂ µ =∂μ on all fields and parameters, which eliminates the dependence on the dual coordinates X µ .
We will now proceed to describe the field content of double field theory. In the formulation of [15], which arises from closed string field theory, the fields consist of a tensor e µν with no symmetry between the indices, a couple of scalar fields e andē, and a couple of vectors f µ and fμ. The tensor e µν contains the graviton and the B−field as its symmetric and antisymmetric parts, respectively. The two scalars account for the dilaton d = 1 2 (e −ē) and a pure gauge scalar ρ = 1 2 (e +ē), while the vectors f µ andfμ are auxiliary fields, similar to the scalar ϕ of Yang-Mills. Associated to the tensor e µν one has two independent vector gauge symmetries, with parameters λ µ andλμ, which are related to metric diffeomorphisms and B−field gauge transformations. Due to the presence of auxiliaries and of the pure gauge scalar ρ, we also have a scalar Stückelberg symmetry with parameter η. Finally, the presence of a two-form inside e µν implies that the gauge transformations are reducible, with a gauge-for-gauge parameter χ.
In L ∞ language, these fields, gauge parameters, field equations and so on are elements of a graded vector space V which decomposes, in the b−picture, as the direct sum In non-negative degree it contains the gauge-for-gauge parameter χ and the multiplets of gauge parameters Λ and fields ψ: The negative degree spaces consist of the field equations F (dual to fields), Noether identities N and the Noether-for-Noether identity R dual to reducibility: (3.5) A nilpotent differential b 1 , which will be discussed in the following, maps between these spaces, making V into a chain complex: We conclude this presentation by introducing a further Z 2 grading of V, which we name c + −degree and which is analogous to the one introduced for K in Yang-Mills. The vector space V can thus be split into two components according to their c + −degree: where we assign c + −degree zero and one to the component fields as  Notice that fields in (3.8) are organized vertically by decreasing L ∞ degree and, as in the Yang-Mills case, one can see that V (0) and V (1) are isomorphic as vector spaces, with L ∞ degrees shifted by −1 between the two.
We are now in the position to show that the complex V of double field theory is a subspace of the tensor product K ⊗ K of two C ∞ -algebras of Yang-Mills. To this end, let us consider two copies K and K of the Yang-Mills kinematic algebras. As we have seen in section 2.3, these split according to their respective c−degrees as From the decomposition (2.57) of Yang-Mills theory, one can see that the L ∞ vector space of DFT given by the direct sum V = V (0) ⊕ V (1) can be accommodated in the tensor products In particular, we take the tensor products of spaces of functions of x and functions ofx to give functions of (x,x), as in (1.4).
The tensor product requires a degree shift, which we describe in the following. To this end it is convenient to introduce a basis. We take the basis vectors for the two copies K and K to be |θ M , |c θ M and |θM , |cθM , respectively, according to our discussion in section 2.3. Similarly, we introduce two copies of the dual vectors: θ M * |, θ M * b| and θM * |, θM * b | , each obeying the relations (2.66) and (2.67) with the respective (b, c) and (b,c) operators. We introduce now the linear combinations defined as above on K ⊗ K. They obey 12) and allow us to write down a basis for V (0) and V (1) as 13) in terms of the natural basis |θ M , |c θ M ⊗ |θN , |cθN of K⊗K. The isomorphism between V (0) and V (1) is given by the action of (b + , c + ) as (3.14) An arbitrary element Ψ ∈ V can thus be expanded as where both φ and χ depend on doubled coordinates (x,x). In order to match the L ∞ degrees of the V complex, one has to shift by two the C ∞ degrees of the tensor product: given u ∈ K andū ∈ K, one defines which implies the following assignment for the basis vectors and the (b ± , c ± ) operators: Using this basis one can rewrite the L ∞ elements (3.4) and (3.5) of homogeneous degree as χ = |θ +θ+ χ ∈ V 2 , Λ = |θ +θμ λμ − |θ µθ+ λ µ − 2 |c + θ +θ+ η ∈ V 1 , ψ = |θ µθν e µν + 2 |θ +θ− ē + 2 |θ −θ+ e + 2 |c + θ +θμ fμ + 2 |c + θ µθ+ f µ ∈ V 0 , where the normalizations have been chosen to match the ones of [15].
We conclude this subsection by giving an alternative and more useful characterization of the vector space V as a subspace of the full tensor product K ⊗ K. Taking linear combinations of the natural basis vectors |θ M ⊗ |θN , |c θ M ⊗ |θN , |θ M ⊗ |cθN and |c θ M ⊗ |cθN of K ⊗ K we define the new basis where the action of (b ± , c ± ) follows with the algebra (3.12). This shows that vectors of V that have components only along |θ MθN and |c + θ MθN are the vectors Ψ of K⊗K obeying b − Ψ = 0. This is analogous to the constraint usually imposed in closed string field theory [28]. Recalling that all fields in V are subject to the weak constraint (3.1), the DFT complex V can be defined as the subspace

L ∞ -algebra of Double Field Theory to Cubic Order
We will now construct cubic double field theory by identifying its differential Q and two-bracket B 2 . It will be shown that these obey the quadratic relations which ensure consistency up to cubic order. In order to prove nilpotency of Q and the Leibniz property, it will be instrumental to consider V as the subspace (3.20) and work on K ⊗ K. We start from the differential, which defines the free theory. Let us consider two copies Q and Q of the Yang-Mills differential (2.69), obeying and define their sum Q + Q , acting on K ⊗ K as (3.23) As defined, this operator is not a map from V to V, but rather from the full K ⊗ K to itself. We will now prove, however, that Q + Q is well-defined on V, meaning that The first requirement for this is that Q + Q preserves the weak constraint. This is trivial, since ∆ in (3.1) commutes with both Q and Q, as is evident from (2.69). The second requirement is to preserve the space ker(b − ), according to (3.20). In order to see this, let us compute Q + Q b − : where signs are determined by all operators having odd degree. One can compute b − Q + Q in the same way and obtain the anticommutator where we used (2.71). It is now easy to see that Q + Q is well-defined on V: (3.27) Since Q + Q preserves the V subspace of K ⊗ K, we shall define the DFT differential Q as the restriction Nilpotency or Q 2 = 0 then follows immediately from Q + Q 2 = 0, which in turn follows from (3.23) together with Q 2 = Q 2 = 0 and Q and Q being odd. According to the degree assignment (3.16) and the definition (3.23), the L ∞ degree is given by Q V = Q + Q V = −1, as it should.
Having determined the DFT differential, the next goal is to find the two-bracket B 2 starting from two copies of the kinematic products M and M. The main idea comes from the fact that the separate copies of M obey the Leibniz relations in the operator form (2.78), as discussed in section 2.3. In this formalism, the two-bracket of double field theory can similarly be written as an element B ∈ V ⊗ V * ⊗ V * acting on the tensor product of two vectors in V: where the action on the tensor product V ⊗ V is defined analogously to (2.73). The Leibniz relation in (3.21) can be expressed as where the second term defines the action of Q on the tensor product V ⊗ V, namely Let us now discuss the ansatz for the two-bracket. The naive guess B ∼ M⊗M cannot work for two reasons. First of all, this naive ansatz does not respect the weak constraint, i.e. generally M ⊗ M Ψ 1 ⊗ Ψ 2 / ∈ V for Ψ 1 , Ψ 2 ∈ V. Moreover, the degree is wrong: the operator M carries degree |M| K = 0 in K, same for M in K. According to the degree shift (3.16), one has M ⊗ M V = −2, while B V = −1. The simplest ansatz which maintains the structure of a tensor product and solves both issues is given by where the overall normalization − 1 2 can be chosen at will and has been fixed to make contact with the literature. In (3.36) we have introduced the projector P ∆ to ker(∆), satisfying The ansatz (3.36) automatically respects both the weak and the algebraic constraints, in that b − B = ∆ B = 0 thanks to the explicit projection. Given two vectors Ψ 1 , Ψ 2 ∈ V of the form Ψ i = u i ⊗ū i , with u i ∈ K andū i ∈ K, the action of B is defined by where we denotedū 1 = |ū 1 | K , u 2 = |u 2 | K in the exponent, and we recall that the action of b − on K ⊗ K is given by (3.11).
We start checking the consistency of the construction by verifying the symmetry property of the two-bracket B. Taking two elements Ψ i = u i ⊗ū i with i = 1, 2 we compute which is the expected graded symmetry of the two-bracket in the b−picture. In order to obtain (3.39) we have used the symmetry property (2.77) and (3.16), namely that |Ψ| V = |u| K +|ū| K +2 for Ψ = u ⊗ū. We can now prove that the two-bracket B satisfies the Leibniz relation in the form (3.34). To do so, it is important to recall that Q = Q + Q when acting on V, as well as the identity (3.26), which allow us to write where the first equality comes from the fact that P ∆ b − Ψ ∈ V for any Ψ ∈ K ⊗ K, while the second one holds under the projector P ∆ thanks to (3.26). The last ingredient to prove the Leibniz property is that Q and Q 'commute' with M and M, respectively, in the tensor product, i.e.
We start by computing the first term of (3.34): where we used (3.40) and (3.41) to move the factors of Q and Q in the tensor product. The expression above is meant to act on V ⊗ V ⊂ K ⊗ K ⊗ K ⊗ K , with the unbarred and barred operators in (3.42) acting on the K ⊗ K and K ⊗ K subspaces, respectively. One can now use the Yang-Mills Leibniz relation (2.78) and write (3.42) as where we used that Q B is meant to act on V ⊗ V, ensuring that Q + Q = Q in the last line above.

Action and Gauge Transformations
Having proved that the two-bracket B 2 obtained by doubling obeys the Leibniz rule with respect to Q, the resulting DFT is guaranteed to be consistent at cubic order and so must be equivalent to the formulation by Hull and Zwiebach [15]. As a consistency check we explicitly compute the entire cubic action, as well as the gauge transformations of the classical fields, and show that they coincide with the original form in [15] up to field and parameter redefinitions.
One can use the gauge transformations (3.46) to derive the gauge brackets between parameters, upon taking two successive transformations. Since the operator (3.36) already contains all the DFT brackets, one can simply compute the gauge algebra by applying (3.36) where the projector P ∆ is left implicit. Let us notice that this basis of parameters allows one (to this order) to consistently setλμ and η to zero, in which case (3.47) The gauge transformations (3.46) do not coincide yet with the ones given in [15]. In order to match the latter, one has to redefine the gauge parameters as Having discussed the gauge transformations, we can now turn to the action. Given a classical string field ψ as in (3.44) and a dual field equation the inner product can be defined by the pairing The action, up to cubic order, is given in L ∞ language by where Q(ψ) and the two-bracket B(ψ ⊗ ψ) can be computed by using Let us mention that every component of B(ψ ⊗ ψ) has a projector P ∆ , which we assume to be self-adjoint 5 , thus yielding the identity when acting on the weakly constrained field ψ. In order to match the action of Hull and Zwiebach [15], one has to perform the field redefinition (3.49), together with (3.54) We stress that the form (3.53) of the cubic DFT action is a significant simplification of the one given in [15].

Conclusions and Outlook
In this paper we strengthened the recent results in [13] according to which a natural Lagrangian implementation of double copy applied to Yang-Mills theory yields, at least to cubic order, double field theory. More precisely, in [13] this was shown upon integrating out the DFT dilaton and, for the cubic couplings, upon choosing Siegel gauge. Here we generalized these results by providing a gauge invariant and off-shell double copy procedure that yields the complete DFT to cubic order, including all dilaton couplings. To this end we employed a formulation in terms of strongly homotopy Lie or L ∞ -algebras, which encode the complete data of a classical field theory. Our results highlight the usefulness of homotopy algebras for a first-principle understanding of double copy. (This point was also made at some length in [36].) We close this section with a brief list of open questions and possible follow-up projects: • It would be interesting to generalize our construction to other gauge and DFT-type theories, for instance to supersymmetric Yang-Mills theory which should be related to supersymmetric DFT [44,45].
• Given the relation between the L ∞ -algebras of Yang-Mills theory and DFT at least to cubic order there should be an intimate relation between the classical (perturbative) solutions of both theories. In the homotopy algebra formulation these solutions are given by the Maurer-Cartan elements corresponding to the L ∞ structure. It would be interesting to investigate this in detail and to compare with existing attempts in the literature on establishing double copy relations at the level of classical solutions [46][47][48].
• Arguably the most important open problem is that of extending this gauge invariant and off-shell double copy procedure to quartic and ultimately to all orders. This appears to be a hard problem for the following reason: The 2-bracket of DFT is defined by B = − 1 2 P ∆ b − M ⊗ M in terms of the 2-product M of the kinematic algebra of Yang-Mills theory, but it also involves projectors onto the subspace of level-matched states. This projection seems to be indispensable for various reasons, notably for degree reasons, and it leads to significant technical complications when trying to establish the next L ∞ relation involving the Jacobiator of the 2-bracket and the 3-bracket encoding the quartic couplings. Intriguingly, however, solving this problem would also amount to constructing a weakly constrained DFT which so far has only been possible to cubic order (due to exactly the same technical challenges). Such a weakly constrained DFT is guaranteed to exist, since in principle it is derivable from the full closed string field theory by integrating out all string modes except the ones of the DFT sector [49][50][51]. Due to the complications of closed string field theory it would, however, be extremely challenging to do so explicitly, and it is an enticing prospect that double copy might provide a short cut.

A BRST Quantization of Worldline Theory
In this appendix we show how to derive the formulation of Yang-Mills theory used in section 2 by a worldline quantization, using the N = 2 spinning particle [52][53][54][55] and its BRST quantization.
We begin by reviewing the classical worldline theory and then proceed to its BRST quantization. The fundamental worldline fields constitute a graded phase space which consists of the bosonic canonical pair (x µ , p µ ), representing target space coordinates and momenta, together with a fermionic canonical pair (θ µ , θ * µ ), which is associated with spin degrees of freedom in spacetime. The fermions θ µ and θ * µ are related by complex conjugation, i.e. (θ µ ) * = θ * µ , where we raise and lower spacetime indices with the Minkowski metric. The rigid model is invariant under global supersymmetries, as well as global time translations, generated by supercharges and the free Hamiltonian, respectively: The N = 2 spinning particle is constructed by gauging worldline supersymmetries and translations, thus turning them into local symmetries. This is achieved by means of complex worldline gravitini χ and χ * and an einbein e, respectively, leading to the action The action (A.2) is invariant under time reparametrizations τ → τ − ξ(τ ) and local supersymmetries ǫ(τ ), ǫ * (τ ) , with transformations laws δx µ = ξ p µ + i ǫ θ µ * + i ǫ * θ µ , δp µ = 0 , where both ǫ * and χ * are related to ǫ and χ by complex conjugation. Canonical quantization of the graded symplectic structure of (A.2) gives rise to the (anti)-commutation relations where we denoted the quantum operators with the same symbols as the classical variables, since from now on we will only work with the quantum theory. The generators of local (super)symmetries (A.1) turn into quantum first-class constraints obeying the superalgebra The worldline model, as it stands, describes abelian massless p−forms of arbitrary degree. In order to accommodate color degrees of freedom, we shall extend the graded phase space by a further fermionic canonical pair (t a , t a * ), as originally introduced in [56], and supplement the action (A.2) by the term where a = 1, · · · , dim g is an adjoint index of a Lie algebra g. Canonical quantization of these fermions leads to the anticommutator and we shall lower and raise adjoint indices with the Cartan-Killing metric κ ab = δ ab and its inverse. Rather than studying the physical state conditions of the theory in this form, we shall proceed to its BRST quantization, which makes the spacetime gauge structure manifest.
The space of smooth functions of x is then tensored with the Fock space of matter (the θ µ and t a sectors) and ghost oscillators. We choose the Fock vacuum to be annihilated by b and all starred operators: b, θ * µ , t a * , γ * , β * |0 = 0 . (A.11) A basis of the Fock space is thus given by arbitrary monomials in all the creation operators acting on the vacuum |0 , which we assume to be bosonic and of ghost number zero: |i, j, k| µ 1 ···µp a 1 ···aq = (γ) i (β) j (c) k (θ µ 1 · · · θ µp ) (t a 1 · · · t aq ) |0 . where the Ψ on the right-hand side is taken to be an operator-valued function acting on the vacuum. This correspondence is analogous to the operator-state correspondence in string theory and allows one to identify the state |ψ itself with the operator Ψ. In the following, we will refer interchangeably to the state |ψ and the operator Ψ as the 'string field'.
Truncation and Yang-Mills complex Turning to the spectrum of off-shell states, one infers from (A.13) that it goes vastly beyond the spectrum of Yang-Mills. In order to remedy this, we shall decompose the full Hilbert space H by means of two number operators: N = θ µ θ * µ + γβ * − βγ * = N θ + N γ + N β , N t = t a t a * , (A.15) whose choice will be clarified in the following. The Hilbert space thus decomposes as a double direct sum as follows: 16) where in the above formula we denoted the eigenvalues with the same symbol as the operators (A.15). The basis elements (A.12), for instance, belong to the subspaces H m,n according to |i, j, k| µ 1 ···µp a 1 ···aq ∈ H p+i+j,q . The Hilbert subspace describing Yang-Mills is given by H 1,1 which, as we will now show, is isomorphic to the L ∞ complex X .