Homotopy Double Copy of Noncommutative Gauge Theories

Abstract

We discuss the double-copy formulation of Moyal–Weyl-type noncommutative gauge theories from the homotopy algebraic perspective of factorisations of $L_{\infty }$-algebras. We define new noncommutative scalar field theories with rigid colour symmetries taking the role of the zeroth copy, where the deformed colour algebra plays the role of a kinematic algebra; some of these theories have a trivial classical limit but exhibit colour–kinematics duality, from which we construct the double copy theory explicitly. We show that noncommutative gauge theories exhibit a twisted form of colour–kinematics duality, which we use to show that their double copies match with the commutative case. We illustrate this explicitly for Chern–Simons theory, and for Yang–Mills theory where we obtain a modified Kawai–Lewellen–Tye relationship whose momentum kernel is linked to a binoncommutative biadjoint scalar theory. We reinterpret rank-one noncommutative gauge theories as double copy theories and discuss how our findings tie in with recent discussions of Moyal–Weyl deformations of self–dual Yang–Mills theory and gravity.

1. Introduction

The double copy provides a novel perspective on observables in gravitational theories and related field theories by extracting information from gauge theory, offering an efficient tool for the calculation of perturbative gravity amplitudes while at the same time simplifying calculations in gauge theories. It has led to new insights into the geometric and algebraic structures underlying scattering amplitudes and suggests profound relations between a wide range of theories with various disparate properties. Very loosely put, it is a map between theories which follows the slogan

$$\begin{array}{c}\hfill \mathrm{Gravity}={\left(\mathrm{Gauge}\mathrm{Theory}\right)}^2\end{array}$$

(1)

though both sides have broad meanings and generalisations. There has been an explosion of intense activity on the subject in recent years from all sorts of directions as well as with different goals and applications in mind. The literature is by now vast and extensive; see e.g., refs. [1,2,3] for reviews and more complete lists of references.

The aim of this paper is to understand how standard noncommutative gauge theories, such as those which arise naturally from string theory, fit into the paradigm of colour–kinematics duality and the double copy of gauge theory to gravity. As an offspring of our investigations, we shall encounter some novel noncommutative scalar field theories with rigid colour symmetries that have no commutative counterparts. These have not appeared in the literature before and are worthy of further studies in their own right. They form the building blocks of the double copy relations considered in the present paper.

In this section, we give a short and informal introduction to the subject, providing some motivation and background to what we set out to do in the present paper. We start by briefly sketching the key ideas behind three double copy relations that will play a prominent role in our treatment. We then summarise the main ideas and results of this paper.

1.1. KLT Double Copy

The Kawai–Lewellen–Tye (KLT) relations [4] lie historically at the origins of the double copy. Symbolically, they prescribe the squaring operation (1) in the form

$$\begin{array}{c}\hfill {\mathcal{A}}_n^{\mathrm{L}\otimes \mathrm{R}}=\sum _{w,w^{\prime }}{𝓐}_n^{\mathrm{L}}\left(w\right){\mathit{S}}_n\left(w\right|w^{\prime }){𝓐}_n^{\mathrm{R}}\left(w^{\prime }\right).\end{array}$$

(2)

The left-hand side involves the n-point tree amplitudes in the double copy theory labelled by ${}^{\mathrm{L}\otimes \mathrm{R}}$, while the right-hand side involves colour-stripped n-point amplitudes of potentially different “left” and “right” theories called single copies, and respectively, labelled by ${}^{\mathrm{L}}$ and ${}^{\mathrm{R}}$. The sum runs over two choices from $(n-3)!$ of the $n!$ possible single-trace colour orderings of n external particles. The KLT momentum kernel ${\mathit{S}}_n\left(w\right|w^{\prime })$ is a polynomial of degree $n-3$ in n-point Mandelstam variables encoding the kinematic invariants of the scattering process.

A simple example is the four-graviton tree amplitude in Einstein gravity. In the evident notation, it can be expressed in terms of the product of colour-ordered four-gluon scattering amplitudes of Yang–Mills theory as

$$\begin{array}{c}\hfill {\mathcal{A}}_4^{\mathrm{GR}}={\mathit{S}}_4\left(1234\right|1243){𝓐}_4^{\mathrm{YM}}\left(1234\right){𝓐}_4^{\mathrm{YM}}\left(1243\right)\mathrm{with}{\mathit{S}}_4\left(1234\right|1243)=s,\end{array}$$

(3)

where $(s,t,u)=(s_{12},s_{23},s_{13})$ and $s_{ij}={(p_i+p_j)}^2$ are Mandelstam variables.

The KLT relations originate from string theory, where closed string states factorise into products of left-moving and right-moving open string states. In this case, ${𝓐}_n^{\mathrm{L}/\mathrm{R}}$ represent the colour-stripped left/right open string disk amplitudes, while ${\mathcal{A}}_n^{\mathrm{L}\otimes \mathrm{R}}$ represents the closed string sphere amplitude. At a topological level, it can be understood from the gluing of two disks, regarded as northern and southern hemispheres, into a sphere. Its low-energy limit gives the field theory relations (2).

1.2. BCJ Double Copy

The Bern–Carrasco–Johansson (BCJ) relations [5] appeared over 20 years after the KLT relations. The starting point is to reorganise the full (colour-dressed) left/right tree amplitudes in the form

$$\begin{array}{c}\hfill {\mathcal{A}}_n^{\mathrm{L}/\mathrm{R}}=\sum _{\Gamma }\frac{c_{\Gamma }{n}_{\Gamma }^{\mathrm{L}/\mathrm{R}}}{D_{\Gamma }},\end{array}$$

(4)

where the sum runs through all $(2n-5)!!$ trivalent graphs $\Gamma$ with n external legs. Here $c_{\Gamma }$ are colour factors, i.e., combinations of generators of the gauge algebra, while the numerator factors $n_{\Gamma }^{\mathrm{L}/\mathrm{R}}$ are kinematic weights made from Lorentz-invariant contractions of external momenta, polarisations, flavour, fermion wavefunctions, and so on. The denominators $D_{\Gamma }={\prod }_{e\in \Gamma }{s}_e$ come from propagators, where $s_e=p_e^2$ are Mandelstam invariants of the momenta $p_e$ flowing through internal lines $e\in \Gamma$.

The key requirement of the choice of decomposition (4) is colour–kinematics duality: On each subgraph, a generalised Jacobi identity among colour numerators $c_{{\Gamma }_s}+c_{{\Gamma }_t}+c_{{\Gamma }_u}=0$ inherited from the gauge algebra is obeyed identically among kinematic numerators $n_{{\Gamma }_u}^{\mathrm{L}/\mathrm{R}}+n_{{\Gamma }_t}^{\mathrm{L}/\mathrm{R}}+n_{{\Gamma }_u}^{\mathrm{L}/\mathrm{R}}=0$ under particle permutations, and individual factors are antisymmetric.

Consider, for example, the tree-level on-shell four-gluon amplitude in non-abelian gauge theory whose s-channel cubic graph is

(5)

The associated colour factor is $c_s=f^{a_1{a}_{2}b}{f}^{a_3{a}_{4}b}$ where $f^{abc}$ are structure constants for a gauge Lie algebra $\mathfrak{g}$. When summed over the three Mandelstam channels, obtained by degree 3 cyclic permutations of the colour labels, they satisfy $c_s+c_t+c_u=0$ as a consequence of the Jacobi identity for $\mathfrak{g}$. Then the same Jacobi identity must be satisfied by the kinematic numerator $n_s$ of the graph (5), obtained by contracting the Feynman rules with incoming on-shell polarisation vectors and using crossing symmetry.

When colour–kinematics duality holds for at least one of the left and right theories, then the replacement of colour factors $c_{\Gamma }$ with kinematic factors $n_{\Gamma }^{\mathrm{R}/\mathrm{L}}$ in ${\mathcal{A}}_n^{\mathrm{L}/\mathrm{R}}$ results in the tree amplitudes of another field theory, called the “double copy” theory. This prescribes the squaring operation (1) in the form

$$\begin{array}{c}\hfill {\mathcal{A}}_n^{\mathrm{L}\otimes \mathrm{R}}=\sum _{\Gamma }\frac{n_{\Gamma }^{\mathrm{L}}{n}_{\Gamma }^{\mathrm{R}}}{D_{\Gamma }}.\end{array}$$

(6)

This theory is invariant under diffeomorphisms. It is a gravitational theory with a dynamical spin-two field if there are dynamical spin one-gauge fields in each numerator. On the other hand, it may not be readily identifiable with a gravity theory when no on-shell spin-two states exist.

For the double copy of Yang–Mills theory with itself, Equation (6) expresses the four-graviton amplitude (3) as

$$\begin{array}{c}\hfill {\mathcal{A}}_4^{\mathrm{GR}}=\frac{{\left(n_s^{\mathrm{YM}}\right)}^2}{s}+\frac{{\left(n_t^{\mathrm{YM}}\right)}^2}{t}+\frac{{\left(n_u^{\mathrm{YM}}\right)}^2}{u}.\end{array}$$

(7)

For amplitudes of $𝓝=4$ supergravity, one may double copy amplitudes of $𝓝=4$ supersymmetric Yang–Mills theory with amplitudes of pure Yang–Mills theory, and so on.

The statement of colour–kinematics duality amounts to replacing the colour structure constants $f^{abc}$ by momentum-dependent factors that represent structure constants of an infinite-dimensional kinematic algebra [6,7,8]. From this perspective, gravity is a gauge theory for which colour is substituted by kinematics, thus reminiscent of old ideas of viewing gravity as a gauge theory whose gauge algebra is the infinite-dimensional Lie algebra of diffeomorphisms. One of the main problems in the double copy relationship between gauge theory and gravity is a detailed understanding of this kinematic algebra.

Colour-ordered amplitudes that satisfy the BCJ amplitude relations [5] along with the Kleiss–Kuijf relations [9] ensure that the double copy amplitude (6) does not depend on the choice of bases for the decomposition of Equation (4). These can be used to directly obtain the KLT formula (2). The KLT momentum kernel can be understood in this setting as the inverse of a matrix of bi-coloured scalar amplitudes [10], whose role is to ensure the correct propagator structure of the double copy amplitudes. These arise as the tree amplitudes of a cubic biadjoint scalar field theory, which result instead from the replacement of kinematic factors $n_{\Gamma }^{\mathrm{L}/\mathrm{R}}$ in ${\mathcal{A}}_n^{\mathrm{L}/\mathrm{R}}$ with a second set of colour factors ${\overline{c}}_{\Gamma }$ for another gauge algebra. They are labelled by ${}^{\mathrm{BAS}}$ and take the form

$$\begin{array}{c}\hfill {\mathcal{A}}_n^{\mathrm{BAS}}=\sum _{\Gamma }\frac{c_{\Gamma }{\overline{c}}_{\Gamma }}{D_{\Gamma }}.\end{array}$$

(8)

The amplitude (8) has manifest colour–kinematics duality, with either of the two colour factors $c_{\Gamma }$ or ${\overline{c}}_{\Gamma }$ regarded as kinematical numerators. The bi-adjoint scalar theory thus serves as the “identity model” in the double copy operations: its doubly colour-ordered amplitudes behave like identity matrices in KLT double copies with any other single-copy theory, while replacing $c_{\Gamma }$ in ${\mathcal{A}}_n^{\mathrm{L}/\mathrm{R}}$ with one of the two adjoint scalar colour factors returns the same amplitude. As such, it is called the “zeroth-copy” theory.

Altogether, the chain of relations among the theories involved in the double copy construction can be depicted symbolically as

(9)

The double copy procedure can also be applied to loop amplitudes at the level of momentum space integrands, provided that colour–kinematics duality holds on all unitarity cuts, therefore enabling the double copy construction of gravitational amplitudes [11].

1.3. Homotopy Double Copy

Moving forward another 13 years, the more recent remarkable perspective of Borsten, Jurčo, Kim, Macrelli, Sämann and Wolf [12] reformulates the double copy relations by exploiting the fact that the kinematical and dynamical data of any field theory are organised by homotopy algebras. By the strictification theorem for $L_{\infty }$-algebras, any theory that can be quantised in the Batalin–Vilkovisky (BV) formalism is perturbatively equivalent, via the addition of suitable auxiliary fields, to a theory with only cubic interaction vertices. The underlying $L_{\infty }$-algebras are quasi-isomorphic.

Let ${\mathfrak{L}}^{\mathrm{L}/\mathrm{R}}$ denote the strict $L_{\infty }$-algebras of left/right theories with only cubic interactions based on a colour Lie algebra $\mathfrak{g}$ of internal symmetries. Suppose that they can be factorised into tensor products

$$\begin{array}{c}\hfill {\mathfrak{L}}^{\mathrm{L}/\mathrm{R}}=\mathfrak{g}\otimes \left({\mathfrak{Kin}}^{\mathrm{L}/\mathrm{R}}{\otimes }_{{\tau }^{\mathrm{L}/\mathrm{R}}}\mathfrak{Scal}\right).\end{array}$$

(10)

Here ${\mathfrak{Kin}}^{\mathrm{L}/\mathrm{R}}$ are finite-dimensional ‘kinematical’ vector spaces encoding the kinematic degrees of freedom of the field theories, $\mathfrak{Scal}$ is the strict $L_{\infty }$-algebra of a cubic scalar field theory encoding the trivalent interactions, and ${\mathfrak{Kin}}^{\mathrm{L}/\mathrm{R}}{\otimes }_{{\tau }^{\mathrm{L}/\mathrm{R}}}\mathfrak{Scal}$ denote the twisted tensor products with twist data ${\tau }^{\mathrm{L}/\mathrm{R}}$ which define kinematical strict $C_{\infty }$-algebras.

If at least one of the factorisations (10) is compatible with colour–kinematics duality, then one can double copy the field theory by replacing the colour factor $\mathfrak{g}$ with the kinematic factor ${\mathfrak{Kin}}^{\mathrm{R}/\mathrm{L}}$ in ${\mathfrak{L}}^{\mathrm{L}/\mathrm{R}}$ and twist the tensor product by ${\tau }^{\mathrm{R}/\mathrm{L}}$. This results in a field theory organised by the strict $L_{\infty }$-algebra

$$\begin{array}{c}\hfill {\mathfrak{L}}^{\mathrm{L}\otimes \mathrm{R}}={\mathfrak{Kin}}^{\mathrm{L}}{\otimes }_{{\tau }^{\mathrm{L}}}\left({\mathfrak{Kin}}^{\mathrm{R}}{\otimes }_{{\tau }^{\mathrm{R}}}\mathfrak{Scal}\right).\end{array}$$

(11)

This realises the squaring operation (1) as the “homotopy double copy”. It provides a precise mathematical description of the bilinear multiplicative structure underlying the double copy operation on the space of certain classes of field theories.

In this framework, the KLT momentum kernels are constructed from the strict $L_{\infty }$-algebra of a biadjoint scalar theory, which is obtained by instead replacing the kinematic vector spaces ${\mathfrak{Kin}}^{\mathrm{L}/\mathrm{R}}$ by a second colour Lie algebra $\overline{\mathfrak{g}}$ in the factorisation (10) to obtain the strict $L_{\infty }$-algebra

$$\begin{array}{c}\hfill \mathfrak{BAS}=\mathfrak{g}\otimes \left(\overline{\mathfrak{g}}\otimes \mathfrak{Scal}\right).\end{array}$$

(12)

This is called the “homotopy zeroth copy”. Note that the same scalar $L_{\infty }$-algebra $\mathfrak{Scal}$ is common to all three factorisations.

In this setting, the chain of relations (9) is depicted by

(13)

Altogether this gives a complete and elegant algebraic formulation of the double copy operations at all levels, which moreover has the potential to elucidate the algebraic origins of colour–kinematics duality [13,14,15], and to give an off-shell non-perturbative definition of the double copy.

1.4. Noncommutative Gauge Theories and Gravity

In this paper, we are interested in how noncommutative gauge theories fit into the double copy paradigm. We focus on noncommutative field theories of the type which arise as low-energy limits of open string theory with stacks of D-branes in B-fields, such as noncommutative Yang–Mills theory with the Moyal–Weyl star-product; see e.g., [16,17] for early reviews of the subject. Although in this paper we deal only with the simplest Moyal–Weyl deformations to clearly illustrate the main ideas and constructions, many aspects of our constructions such as planar equivalence and colour-stripping should work also for more general (twisted) Poisson structures induced by non-constant B-fields, and in particular for more general twist deformations.

To avoid undue suspense, let us right away state the main message of this paper: the double copy operation (1) in the noncommutative world is replaced by

$$\begin{array}{c}\hfill \mathrm{Ordinary}\mathrm{Gravity}={\left(\mathrm{Noncommutative}\mathrm{Gauge}\mathrm{Theory}\right)}^2\end{array}$$

(14)

where we use the adjective ‘ordinary’ to distinguish commutative theories from their deformed counterparts. In other words, we find that the double copy of a noncommutative gauge theory is not a deformed gravitational theory, but rather coincides with the result of the ordinary double copy. Although this somewhat mundane conclusion may seem obvious to experts, it raises some interesting conceptual and theoretical questions whose answers lead to novel perspectives on the double copy. We discuss in detail how these expectations are borne out from the perspective of the homotopy double copy, drawing on the homotopy algebras organising the relevant noncommutative gauge theories that are discussed in [18,19]. We believe that the homotopy algebraic manipulations used to arrive at the slogan (14) are interesting and useful, and further point towards various generalisations of our work.

The standard examples of double copy relations mostly involve renormalizable field theories, but higher derivative operators are also expected to take part in some form of the double copy. Noncommutative gauge theories provide such an example: they are non-local because they involve an infinite tower of higher derivative operators, and they are generally non-renormalizable because they are plagued by the famous UV/IR mixing problem [20]. As such, in this paper we work only with noncommutative field theories at the tree level: when we speak of ‘double copy’ in this context we mean at the level of tree amplitudes or homotopy algebras at the classical (Lagrangian) level. UV/IR mixing becomes problematic only beyond one-loop, and the double copy operation (14) perhaps offers a different perspective on the non-renormalizability of certain gravitational theories.

Another question that can be addressed by our result (14) is the extent to which colour–kinematics duality depends on Poincaré invariance. If colour–kinematics duality and the double copy are truly intrinsic properties of certain quantum field theories, they should exist in some form in backgrounds that break spacetime symmetries. Our results show that this is the case for noncommutative deformations, which explicitly break Lorentz symmetry. This may explain why ordinary gravity is recovered: in certain Lorentz-violating scenarios, field theory scattering amplitudes are still severely constrained by unitarity and locality. The existence of massless spin-two particles forces three-particle amplitudes (including gravitons) to be Lorentz-invariant, and hence cubic graviton interactions in Minkowski space must be those of general relativity up to certain unique higher derivative corrections; this is conjectured to be true for all n-particle amplitudes in [21]. This fits in nicely with our perspective on ordinary Einstein gravity as the double copy of noncommutative Yang–Mills theory.

A somewhat mysterious issue is the precise meaning of colour–kinematics duality in noncommutative gauge theories, which are invariant under a deformed gauge symmetry that ensures consistency of the theory: a unitary colour algebra $\mathfrak{g}=\mathfrak{u}\left(N\right)$ combines intricately with kinematical degrees of freedom of the fields into an infinite-dimensional Lie algebra ${\mathfrak{u}}_{★}\left(N\right)$ parametrizing noncommutative gauge transformations. This leads to the well-known colour–kinematics mixing in noncommutative theories, which in turn spoils standard colour-stripping and colour–kinematics duality.

In noncommutative field theories, the colour–kinematics approach to the double copy is extended by considering generalised numerators $n_{\Gamma }$ that simultaneously depend on both colour and kinematics, while still satisfying Jacobi-like identities. This provides a novel and precise realisation of the kinematic Lie algebras underlying some theories. It is similar in spirit to the double copy of effective field theories which involve (finitely many) higher derivative operators, see e.g., [22,23,24], while providing an explicit realisation of the complementary construction of colour–kinematics duality for algebraic relations satisfied by the symmetric structure constants $d^{abc}$ of the unitary Lie algebra $\mathfrak{g}=\mathfrak{u}\left(N\right)$ [25]. We show that this modification is most naturally explained through the lens of twisted tensor products of homotopy algebras, leading to deformations of the standard constructions that we call ‘twisted homotopy factorisation’ and ‘twisted colour–kinematics duality’.

Our modifications of colour–kinematics duality, and ultimately the operation (14), are most naturally explained by the ultraviolet completion of noncommutative field theory in tree-level open bosonic string amplitudes with B-fields. In particular, we examine their connections with the BCJ and KLT amplitude relations. Higher derivative generalisations of the KLT relations are discussed in [26,27]; our KLT approach to the double copy is taken by generalising the momentum kernel by an infinite tower of higher derivative operators. The double copy amplitudes obtained with our modified KLT kernel can be equivalently achieved by the traditional kernel.

The B-field modified momentum kernel in the KLT relations thus also involves infinitely many higher derivative corrections. It satisfies the requisite minimal rank condition of [27] and is given by the inverse matrix of the colour-ordered amplitudes in a ‘binoncommutative biadjoint scalar theory’. This new scalar theory plays the role of the zeroth-copy theory for double copies of noncommutative gauge theories. It reduces to various interesting noncommutative scalar field theories by taking rank-one specialisations; in a certain sense, these theories are the most novel and interesting aspects of the story we tell in the following. One of these theories, which we call the ‘adjoint scalar theory’, is a simple example with rigid colour symmetry where all our considerations can be made very explicit. It is an unusual example of a noncommutative scalar field theory, in the sense that its commutative limit is a free theory.

The adjoint scalar theory has an interesting homotopy double copy which is a noncommutative scalar field theory that we construct and describe explicitly. Although the classical limit is trivial, its semi-classical limit reproduces some known topological theories such as the special Galileon theory in two dimensions and self–dual gravity in four dimensions. In these cases, the semi-classical colour–kinematics replacements involve replacing gauge algebras with algebras of area-preserving diffeomorphisms, which are naturally explained as mutual isomorphisms between the (noncommutative) theories. These relate the biadjoint scalar theory to the Zakharaov–Mikhailov and special Galileon theories in two dimensions, and to self–dual gauge and gravity theories in four dimensions. The non-perturbative double copy proposals in this case [28,29] are thus naturally explained via the elegant formulation of the double copy prescription using homotopy algebras, which enables an off-shell Lagrangian-level formulation and allows for replacement of colour algebras by kinematic algebras beyond amplitude level.

As is well-known, the rank-one specialisations of noncommutative theories with gauge symmetries are interesting interacting field theories of photons, unlike their commutative counterparts; like the adjoint scalar theory, in the classical limit, they become free theories. These are low-energy effective theories on a single D-brane in a background B-field, and they have long been believed to contain gravitation [30]. Superficial evidence for this comes from the interplay between noncommutative gauge transformations and spacetime diffeomorphisms [31,32]: spatial translations are equivalent to gauge transformations (up to global symmetry transformations), and thus these theories are at least toy models of general relativity, the only other theory that shares this property. More substantial arguments revolve around emergent gravity phenomena in noncommutative gauge theories; see e.g., refs. [33,34,35,36,37,38] and references therein for a partial list of early works in this direction.

The rank-one noncommutative gauge theories have no colour symmetries, but they are invariant under a kinematic Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$ of infinitesimal noncommutative $\mathsf{U}\left(1\right)$ gauge transformations. Here the adjoint scalar theory discussed above plays a decisive role which offers a new perspective on the emergence of gravitation in these theories: ${\mathfrak{u}}_{★}\left(1\right)$ gauge theories can be constructed via the homotopy double copy of the adjoint scalar theory with the corresponding commutative gauge theory for any colour Lie algebra $\mathfrak{g}$. This double copy operation is written symbolically as

$$\begin{array}{c}\hfill \begin{array}{c}\mathrm{Noncommutative}\\ \mathsf{U}\left(1\right)\mathrm{Gauge}\mathrm{Theory}\end{array}=\begin{array}{c}\mathrm{Adjoint}\\ \mathrm{Scalar}\mathrm{Theory}\end{array}\otimes \begin{array}{c}\mathrm{Ordinary}\\ \mathrm{Gauge}\mathrm{Theory}\end{array}\end{array}$$

(15)

In the case of noncommutative Chern–Simons theory, we corroborate the relationship (15) by showing explicitly that the left-hand side realises a subalgebra of diffeomorphisms of spacetime. However, the most compelling evidence comes from the self–dual sector of noncommutative Yang–Mills theory in four dimensions: rank-one noncommutative self–dual Yang–Mills theory is equivalent to a known noncommutative deformation of self–dual gravity, in which case the double copy operation (15) becomes

$$\begin{array}{c}\hfill \begin{array}{c}\mathrm{Noncommutative}\\ \mathrm{Self}–\mathrm{Dual}\mathrm{Gravity}\end{array}=\begin{array}{c}\mathrm{Adjoint}\\ \mathrm{Scalar}\mathrm{Theory}\end{array}\otimes \begin{array}{c}\mathrm{Ordinary}\mathrm{Self}–\mathrm{Dual}\\ \mathrm{Yang}–\mathrm{Mills}\mathrm{Theory}\end{array}\end{array}$$

(16)

Altogether, we arrive at an intricate web of relations among various commutative and noncommutative theories, which we display symbolically in Figure 1.

1.5. Outline

This paper works through the double copy operations for noncommutative field theories in a detailed and expository manner. Since the standard noncommutative field theories are organised by homotopy algebras largely in parallel to their commutative counterparts, some parts of this work can be read as a review upon taking classical limits. However, some of our homotopy double copy constructions, for example in the cases of the adjoint scalar theory, Chern–Simons theory, and the first-order formalism of Yang–Mills theory in four dimensions, have not appeared before in the literature as far as we are aware. On the other hand, we carefully analyse the modifications due to the noncommutativity of colour–kinematics duality and the precise forms of the kinematic Lie algebras underlying certain theories. The outline of the remainder of this paper is as follows.

In Section 2, we study two scalar field theories which form the “commutative skeletons” of all theories studied in this paper. We formulate them in the language of $L_{\infty }$-algebras and use this to illustrate a simple application of homotopy factorisation techniques.

In Section 3, we begin by reviewing the Moyal–Weyl deformation of scalar field theories with and without rigid colour symmetry. We apply the homotopy factorisation procedure to a noncommutative version of the biadjoint scalar theory (designated as ‘${\mathfrak{u}}_{★}\left(N\right)$ biadjoint scalar’ in Figure 1) and its rank-one limit, the adjoint scalar theory. Most importantly, we observe that colour-stripping in the deformation to noncommutative theories fits into the twisted homotopy factorisation introduced in [12]. We show explicitly that the adjoint scalar theory exhibits colour–kinematics duality at the amplitude level involving the kinematic Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$, from which we construct its homotopy double copy with kinematic factors based on the twisted tensor product of ${\mathfrak{u}}_{★}\left(1\right)$ with itself in Feynman rules. Guided by this result, we define the binoncommutative biadjoint scalar field theory (designated as ‘${\mathfrak{u}}_{★}\left(N\right){\otimes }_{★}{\mathfrak{u}}_{★}\left(\overline{N}\right)$ biadjoint scalar’ in Figure 1), seen as the double copy of the noncommutative biadjoint scalar theory. We discuss how our constructions encompass already known double copies in the literature, including the special Galileon theory in two dimensions as well as (noncommutative) self–dual gravity in four dimensions.

In Section 4 we start by reviewing the Moyal–Weyl deformation of theories involving differential forms. We then apply the twisted homotopy factorisation to noncommutative Chern–Simons gauge theory and explicitly identify a twisted form of colour–kinematics duality at the amplitude level involving the kinematic Lie algebra of volume-preserving diffeomorphisms (in the Lorenz gauge). This allows for the homotopy double copy of Chern–Simons theory with itself to be constructed, which matches with results from [39]. We furthermore realise the rank-one limit of noncommutative Chern–Simons theory as a double copy of the adjoint scalar theory with commutative Chern–Simons theory, together with an explicit diffeomorphism invariance identified at the amplitude level.

In Section 5 we first review the homotopy algebraic structure of noncommutative Yang–Mills theory. We then show that the twisted homotopy factorisation procedure is compatible with strictification in the second-order formalism, inspired by the construction of [12] in the commutative case. We also directly exhibit the twisted homotopy factorisation of the first-order formalism in four dimensions. We then proceed to consider the ultraviolet completion of noncommutative Yang–Mills theory to open string theory with background B-fields. We introduce a modification of the KLT relations to account for the B-field, making use of the factorisation of noncommutativity phase factors in open string amplitudes. In the low-energy limit, this validates our understanding that gravitational amplitudes can be built from two copies of noncommutative Yang–Mills amplitudes with a modified momentum kernel; this is also used to exhibit the corresponding modifications of the BCJ amplitude relations. We show that this modified kernel is exactly sourced by the binoncommutative biadjoint scalar theory. The rank-one limit is further shown to reproduce noncommutative self–dual gravity as the double copy of the adjoint scalar theory with commutative self–dual Yang–Mills theory; this points towards a more precise gravitational interpretation of rank-one noncommutative Yang–Mills theory beyond the self–dual sector.

In Section 6 we conclude with some final remarks summarising the main points of this paper and briefly discuss some future directions.

Supplementing this paper is an extensive Appendix A, in which we provide a pedagogical review of various aspects of homotopical algebra and their applications in quantum field theory. We discuss $L_{\infty }$-algebra methods for quantum field theory, twisted tensor products of homotopy algebras, as well as the construction of scattering amplitudes from Berends–Giele and minimal model recursion relations.

2. Building Blocks: Scalar Field Theories

2.1. Cubic Scalar Field Theory

An important theory in our discussions of the various factorisations of $L_{\infty }$-algebras that we will encounter is the massless scalar field theory with $g{\varphi }^3$ interaction on d-dimensional Minkowski spacetime ${\mathbb{R}}^{1,d-1}$; it forms the skeleton theory encoding trivalent interactions on which the propagating degrees of freedom of all field theories considered in this paper will be built. We use standard coordinates $x^0,x^1,\dots ,x^{d-1}$ on ${\mathbb{R}}^{1,d-1}$, where $x^0$ represents time, and use the shorthand notation ${\partial }_{\mu }$ for the partial derivative $\frac{\partial }{\partial x^{\mu }}$. The standard Minkowski metric of signature $(+-\cdots -)$ is denoted ${\eta }_{\mu \nu }$, and the inner product of two vectors $v=\left(v^{\mu }\right)$ and $v^{\prime }=\left(v^{\prime }{}^{\mu }\right)$ in ${\mathbb{R}}^{1,d-1}$ is $v·v^{\prime }={\eta }_{\mu \nu }{v}^{\mu }{v}^{\prime }{}^{\nu }=v_{\mu }{v}^{\prime }{}^{\mu }$. (Unless otherwise explicitly stated, implicit summation over repeated indices is always understood throughout this paper). The standard volume form on ${\mathbb{R}}^{1,d-1}$ is

$$\begin{array}{c}\hfill {\mathrm{d}}^{d}x=\mathrm{d}{x}^0\wedge \mathrm{d}{x}^1\wedge \cdots \wedge \mathrm{d}{x}^{d-1},\end{array}$$

(17)

and $\square ={\eta }^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }={\partial }_{\mu }{\partial }^{\mu }$ is the wave operator.

The action functional is given by

$$S_{\mathrm{Scal}}\left[\varphi \right]=\int {\mathrm{d}}^{d}x\frac{1}{2}\varphi \square \varphi -\frac{g}{3!}{\varphi }^3,$$

(18)

for a function (regarded as a zero-form) $\varphi \in {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$. The three-point Feynman vertex in momentum space is

(19)

This theory is organised into a cyclic strict $L_{\infty }$-algebra $\mathfrak{Scal}$, whose underlying cochain complex ${\mathsf{Ch}}_{\mathrm{Scal}}:=\mathsf{Ch}\left(\mathfrak{Scal}\right)$ is

$$\begin{array}{c}\hfill {\mathsf{Ch}}_{\mathrm{Scal}}=\left({\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)[-1]\stackrel{\square }{\to }{\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)[-2]\right)\end{array}$$

(20)

with differential ${\mu }_1^{\mathrm{Scal}}=\square$; there are no gauge symmetries so the degree 0 and 3 subspaces are trivial. The only non-trivial higher bracket is the 2-bracket ${\mu }_2^{\mathrm{Scal}}:{\Omega }^0\left({\mathbb{R}}^{1,d-1}\right){[-1]}^{\otimes 2}⟶{\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)[-2]$ that scales pointwise multiplication of functions as (This can be regarded as the pointwise multiplication in momentum space, or equivalently the convolution product in position space, and vice versa).

$$\begin{array}{c}\hfill {\mu }_2^{\mathrm{Scal}}({\varphi }_1,{\varphi }_2)=-g{\varphi }_1{\varphi }_2,\end{array}$$

(21)

which is symmetric, or graded antisymmetric with our degree conventions, as required. The Maurer–Cartan Equation (A17) yields the equation of motion

$$\begin{array}{c}\hfill {\mu }_1^{\mathrm{Scal}}\left(\varphi \right)+\frac{1}{2}{\mu }_2^{\mathrm{Scal}}(\varphi ,\varphi )=\square \varphi -\frac{g}{2}{\varphi }^2=0.\end{array}$$

(22)

The cyclic inner product of degree $-3$ is given by the single non-vanishing pairing between fields and antifields

$$\begin{array}{c}\hfill {\langle \varphi ,{\varphi }^{+}\rangle }_{\mathrm{Scal}}=\int {\mathrm{d}}^{d}x\varphi {\varphi }^{+}\end{array}$$

(23)

for $\varphi \in {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)[-1]$ and ${\varphi }^{+}\in {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)[-2]$. With this inner product, the Maurer–Cartan functional (A20) for the cubic scalar field theory is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_{\mathrm{Scal}}\left[\varphi \right]& =\frac{1}{2}{\langle \varphi ,{\mu }_1^{\mathrm{Scal}}\left(\varphi \right)\rangle }_{\mathrm{Scal}}+\frac{1}{3!}{\langle \varphi ,{\mu }_2^{\mathrm{Scal}}(\varphi ,\varphi )\rangle }_{\mathrm{Scal}},\hfill \end{array}\end{array}$$

(24)

which is just the action functional of Equation (18).

In summary, the cubic scalar field theory is organised into the cyclic strict $L_{\infty }$-algebra

$$\mathfrak{Scal}=\left({\mathsf{Ch}}_{\mathrm{Scal}},{\mu }_2^{\mathrm{Scal}},{\langle -,-\rangle }_{\mathrm{Scal}}\right).$$

(25)

In this paper we will build other field theories by adding extra data to the $L_{\infty }$-algebra $\mathfrak{Scal}$ that preserve the $L_{\infty }$-structure.

2.2. Biadjoint Scalar Theory

A theory that will become very useful in our discussion of the double copy is the biadjoint scalar theory on ${\mathbb{R}}^{1,d-1}$, which has a rigid $\mathsf{G}\times \overline{\mathsf{G}}$ symmetry under a pair of compact Lie groups $\mathsf{G}$ and $\overline{\mathsf{G}}$ equipped with bi-invariant metrics. Let $\mathfrak{g}$ and $\overline{\mathfrak{g}}$ be corresponding quadratic Lie algebras with brackets ${[-,-]}_{\mathfrak{g}}$ and ${[-,-]}_{\overline{\mathfrak{g}}}$, and invariant bilinear forms ${\mathrm{Tr}}_{\mathfrak{g}}$ and ${\mathrm{Tr}}_{\overline{\mathfrak{g}}}$. Choose a basis of generators $\left\{T^a\right\}$ for $\mathfrak{g}$ with structure constants $f^{abc}$, i.e., ${[T^a,T^b]}_{\mathfrak{g}}=f^{abc}{T}^c$, and normalisation ${\mathrm{Tr}}_{\mathfrak{g}}(T^a\otimes T^b)={\delta }^{ab}$. Similarly, choose a basis $\{{\overline{T}}^{\overline{a}}\}$ for $\overline{\mathfrak{g}}$ with ${[{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}]}_{\overline{\mathfrak{g}}}=f^{\overline{a}\overline{b}\overline{c}}{\overline{T}}^{\overline{c}}$, and normalisation ${\mathrm{Tr}}_{\overline{\mathfrak{g}}}({\overline{T}}^{\overline{a}}\otimes {\overline{T}}^{\overline{b}})={\delta }^{\overline{a}\overline{b}}$.

Let ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})$ be the space of zero-forms on ${\mathbb{R}}^{1,d-1}$ in the biadjoint representation of $\mathsf{G}\times \overline{\mathsf{G}}$ on $\mathfrak{g}\otimes \overline{\mathfrak{g}}$; we stress that $\mathfrak{g}\otimes \overline{\mathfrak{g}}$ is not a Lie algebra. An element $\varphi \in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})$ can be written as $\varphi ={\varphi }_{a\overline{a}}{T}^a\otimes {\overline{T}}^{\overline{a}}$ with ${\varphi }_{a\overline{a}}\in {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$. The biadjoint scalar theory on ${\mathbb{R}}^{1,d-1}$ is defined by the action functional

$$\begin{array}{c}\hfill S_{\mathrm{BAS}}\left[\varphi \right]=\int {\mathrm{d}}^{d}x\frac{1}{2}{\varphi }^{a\overline{a}}\square {\varphi }_{a\overline{a}}-\frac{g}{3!}{f}^{abc}{\overline{f}}^{\overline{a}\overline{b}\overline{c}}{\varphi }^{a\overline{a}}{\varphi }^{b\overline{b}}{\varphi }^{c\overline{c}}.\end{array}$$

(26)

The equation of motion derived from Equation (26) has the form

$$\begin{array}{c}\hfill \square {\varphi }^{a\overline{a}}-\frac{g}{2}{f}^{abc}{\overline{f}}^{\overline{a}\overline{b}\overline{c}}{\varphi }^{b\overline{b}}{\varphi }^{c\overline{c}}=0.\end{array}$$

(27)

This theory is organised by the cyclic strict $L_{\infty }$-algebra $\mathfrak{BAS}$ whose underlying two-term cochain complex ${\mathsf{Ch}}_{\mathrm{BAS}}:=\mathsf{Ch}\left(\mathfrak{BAS}\right)$ is simply formed with the differential ${\mu }_1^{\mathrm{BAS}}=\square$, acting as ${\mu }_1^{\mathrm{BAS}}\left(\varphi \right)=\square {\varphi }_{a\overline{a}}{T}^a\otimes {\overline{T}}^{\overline{a}}$:

$$\begin{array}{c}\hfill {\mathsf{Ch}}_{\mathrm{BAS}}=\left({\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})[-1]\stackrel{\square }{\to }{\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})[-2]\right).\end{array}$$

(28)

To write the remaining structure maps, we note that there is a symmetric bilinear operation on ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})$ given by

$$\begin{array}{c}\hfill 〚{\varphi }_1,{\varphi }_2{〛}_{\mathfrak{g}\otimes \overline{\mathfrak{g}}}=f^{abc}{\overline{f}}^{\overline{a}\overline{b}\overline{c}}{\varphi }_1^{a\overline{a}}{\varphi }_2^{b\overline{b}}{T}^c\otimes {\overline{T}}^{\overline{c}},\end{array}$$

(29)

and a local bilinear form ${\mathrm{Tr}}_{\mathfrak{g}\otimes \overline{\mathfrak{g}}}:{\Omega }^0{({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})}^{\otimes 2}⟶{\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$ defined as

$$\begin{array}{c}\hfill {\mathrm{Tr}}_{\mathfrak{g}\otimes \overline{\mathfrak{g}}}({\varphi }_1\otimes {\varphi }_2)={\delta }_{ab}{\delta }_{\overline{a}\overline{b}}{\varphi }_1^{a\overline{a}}{\varphi }_2^{b\overline{b}}.\end{array}$$

(30)

The single non-vanishing higher bracket ${\mu }_2^{\mathrm{BAS}}:{\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}}){[-1]}^{\otimes 2}⟶{\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})[-2]$ is then given by

$${\mu }_2^{\mathrm{BAS}}({\varphi }_1,{\varphi }_2)=-g〚{\varphi }_1,{\varphi }_2{〛}_{\mathfrak{g}\otimes \overline{\mathfrak{g}}}=-g{f}^{abc}{\overline{f}}^{\overline{a}\overline{b}\overline{c}}{\varphi }_1^{a\overline{a}}{\varphi }_2^{b\overline{b}}{T}^c\otimes {\overline{T}}^{\overline{c}}.$$

(31)

This is symmetric because the bracket (29) is symmetric. The cyclic inner product of degree $-3$ is given by the non-zero pairing

$$\begin{array}{c}\hfill {\langle \varphi ,{\varphi }^{+}\rangle }_{\mathrm{BAS}}=\int {\mathrm{d}}^{d}x{\mathrm{Tr}}_{\mathfrak{g}\otimes \overline{\mathfrak{g}}}(\varphi \otimes {\varphi }^{+})=\int {\mathrm{d}}^{d}x{\varphi }^{a\overline{a}}{\varphi }_{a\overline{a}}^{+},\end{array}$$

(32)

for $\varphi \in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})[-1]$ and ${\varphi }^{+}\in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})[-2]$.

Altogether this defines a cyclic strict $L_{\infty }$-algebra

$$\mathfrak{BAS}=\left({\mathsf{Ch}}_{\mathrm{BAS}},{\mu }_2^{\mathrm{BAS}},{\langle -,-\rangle }_{\mathrm{BAS}}\right).$$

(33)

This $L_{\infty }$-algebra is well-known and is very useful for computations in the double copy. From the Maurer–Cartan Equation (A17) we obtain the equation of motion

$$\begin{array}{c}\hfill {\mu }_1^{\mathrm{BAS}}\left(\varphi \right)+\frac{1}{2}{\mu }_2^{\mathrm{BAS}}(\varphi ,\varphi )=\square \varphi -\frac{g}{2}〚\varphi ,\varphi {〛}_{\mathfrak{g}\otimes \overline{\mathfrak{g}}}=0,\end{array}$$

(34)

which in components coincides with Equation (27). The Maurer–Cartan functional (A20) gives the action functional

$$\begin{array}{cc}\hfill S_{\mathrm{BAS}}\left[\varphi \right]& =\frac{1}{2}{\langle \varphi ,{\mu }_1^{\mathrm{BAS}}\left(\varphi \right)\rangle }_{\mathrm{BAS}}+\frac{1}{3!}{\langle \varphi ,{\mu }_2^{\mathrm{BAS}}(\varphi ,\varphi )\rangle }_{\mathrm{BAS}}\hfill \\ \hfill & =\int {\mathrm{d}}^{d}x{\mathrm{Tr}}_{\mathfrak{g}\otimes \overline{\mathfrak{g}}}\left(\frac{1}{2}\varphi \square \varphi -\frac{g}{3!}\varphi {\left[[\varphi ,\varphi ]\right]}_{\mathfrak{g}\otimes \overline{\mathfrak{g}}}\right),\hfill \end{array}$$

(35)

whose component form coincides with Equation (26).

2.2.1. Homotopy Factorisation

The three-point Feynman vertex of the biadjoint scalar theory is given by

(36)

We observe the factorisation of Equation (36) in terms of the colour structure carried by the pair of Lie algebras $(\mathfrak{g},\overline{\mathfrak{g}})$ and the cubic interaction vertex for the scalar field theory in Equation (19), which manifests the double copy structure between the colour group $\mathsf{G}$ and its ‘dual’ colour group $\overline{\mathsf{G}}$. (Since the degrees of freedom here are scalars rather than gluons, the notion of ‘colour’ should really be referred to as ‘flavour’. We prefer to uniformly use the term ‘colour’ for internal symmetries throughout this paper, always understood as a rigid symmetry for scalar fields). This motivates the idea of factorisation of $L_{\infty }$-structures in terms of a Lie structure and a $C_{\infty }$-structure.

It is easy to demonstrate that the cyclic strict biadjoint scalar $L_{\infty }$-algebra $\mathfrak{BAS}$ factorises in terms of the cyclic strict scalar $L_{\infty }$-algebra $\mathfrak{Scal}$ from Section 2.1 as

$$\begin{array}{c}\hfill \mathfrak{BAS}=\mathfrak{g}\otimes (\overline{\mathfrak{g}}\otimes \mathfrak{Scal}),\end{array}$$

(37)

with each Lie algebra regarded as a cyclic differential graded (dg) Lie algebra sitting in degree 0 with the zero differential. For this, we identify ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})$ with $\mathfrak{g}\otimes \overline{\mathfrak{g}}\otimes {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$ and write each element $\varphi$ of ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \overline{\mathfrak{g}})$ in the form $\varphi =T^a\otimes {\overline{T}}^{\overline{a}}\otimes {\varphi }_{a\overline{a}}$ with ${\varphi }_{a\overline{a}}\in {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$. With this identification it follows that (37) holds at the level of the underlying graded vector spaces.

Recalling the differential ${\mu }_1^{\mathrm{Scal}}=\square$ on $\mathfrak{Scal}$, we see that the differential on the biadjoint complex splits as

$$\begin{array}{c}\hfill {\mu }_1^{\mathrm{BAS}}(T^a\otimes {\overline{T}}^{\overline{a}}\otimes {\varphi }_{a\overline{a}})=T^a\otimes {\overline{T}}^{\overline{a}}\otimes \square {\varphi }_{a\overline{a}}=T^a\otimes {\overline{T}}^{\overline{a}}\otimes {\mu }_1^{\mathrm{Scal}}\left({\varphi }_{a\overline{a}}\right).\end{array}$$

(38)

On the other hand, the bracket (31) factorises in terms of the bracket (21) on $\mathfrak{Scal}$ as

$$\begin{array}{cc}\hfill {\mu }_2^{\mathrm{BAS}}(T^a\otimes {\overline{T}}^{\overline{a}}\otimes {\varphi }_1^{a\overline{a}},T^b\otimes {\overline{T}}^{\overline{b}}\otimes {\varphi }_2^{b\overline{b}})& =T^c\otimes {\overline{T}}^{\overline{c}}\otimes \left(-g{f}^{abc}{\overline{f}}^{\overline{a}\overline{b}\overline{c}}{\varphi }_1^{a\overline{a}}{\varphi }_2^{b\overline{b}}\right)\hfill \\ \hfill & ={[T^a,T^b]}_{\mathfrak{g}}\otimes {[{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}]}_{\overline{\mathfrak{g}}}\otimes \left(-g{\varphi }_1^{a\overline{a}}{\varphi }_2^{b\overline{b}}\right)\hfill \\ \hfill & ={[T^a,T^b]}_{\mathfrak{g}}\otimes {[{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}]}_{\overline{\mathfrak{g}}}\otimes {\mu }_2^{\mathrm{Scal}}({\varphi }_1^{a\overline{a}},{\varphi }_2^{b\overline{b}}).\hfill \end{array}$$

(39)

This establishes (37) at the level of dg-Lie algebras.

Finally, we factorise the cyclic inner product (32) with respect to the inner product (23) on $\mathfrak{Scal}$ as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\langle T^a\otimes {\overline{T}}^{\overline{a}}\otimes {\varphi }_{a\overline{a}},T^b\otimes {\overline{T}}^{\overline{b}}\otimes {\varphi }_{b\overline{b}}^{+}\rangle }_{\mathrm{BAS}}& ={\delta }^{ab}{\delta }^{\overline{a}\overline{b}}\int {\mathrm{d}}^{d}x{\varphi }_{a\overline{a}}{\varphi }_{b\overline{b}}^{+}\hfill \\ \hfill & ={\mathrm{Tr}}_{\mathfrak{g}}(T^a\otimes T^b){\mathrm{Tr}}_{\overline{\mathfrak{g}}}({\overline{T}}^{\overline{a}}\otimes {\overline{T}}^{\overline{b}}){\langle {\varphi }_{a\overline{a}},{\varphi }_{b\overline{b}}^{+}\rangle }_{\mathrm{Scal}},\hfill \end{array}\end{array}$$

(40)

which shows (37) at the full level of cyclic strict $L_{\infty }$-algebras.

2.2.2. Amplitudes

Scattering amplitudes for the biadjoint scalar theory can be computed efficiently using the minimal model of the $L_{\infty }$-algebra $\mathfrak{BAS}$, which also encodes its perturbiner expansion, as we review in Appendix A.3. For example, the tree-level four-point off-shell amplitude is found in this way to be

(41)

where we introduced the Mandelstam variables

$$\begin{array}{c}\hfill s={(p_1+p_2)}^2,t={(p_2+p_3)}^2\mathrm{and}u={(p_1+p_3)}^2,\end{array}$$

(42)

along with the colour factors

$$\begin{array}{c}\hfill c_s=g{f}^{a_1{a}_{2}b}{f}^{a_3{a}_{4}b},c_t=g{f}^{a_3{a}_{1}b}{f}^{a_2{a}_{4}b}\mathrm{and}{c}_u=g{f}^{a_2{a}_{3}b}{f}^{a_1{a}_{4}b},\end{array}$$

(43)

and similarly for the barred colour factors. See [40] for further details. Note that $c_s+c_u+c_t=0$ as a consequence of the Jacobi identity for $\mathfrak{g}$, and similarly for $\overline{\mathfrak{g}}$.

2.2.3. Colour Ordering and Decomposition

Tree-level amplitudes of the biadjoint scalar theory, such as the four-point amplitude ${\mathcal{A}}_4(p,a,\overline{a})$ of Equation (41), are organised in terms of bi-invariant partial amplitudes as [10]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{A}}_n^{\mathrm{BAS}}(p,a,\overline{a})& =\sum _{\sigma ,{\sigma }^{\prime }\in S_n/{\mathbb{Z}}_n}{\mathrm{Tr}}_{\mathfrak{g}}(T^{a_{\sigma \left(1\right)}}\cdots T^{a_{\sigma \left(n\right)}}){\mathrm{Tr}}_{\overline{\mathfrak{g}}}({\overline{T}}^{{\overline{a}}_{{\sigma }^{\prime }\left(1\right)}}\cdots {\overline{T}}^{{\overline{a}}_{{\sigma }^{\prime }\left(n\right)}})\hfill \\ \hfill & \times {𝓐}_n^{\mathrm{BAS}}\left(\sigma \left(1\right),\dots ,\sigma \left(n\right)|{\sigma }^{\prime }\left(1\right),\dots ,{\sigma }^{\prime }\left(n\right)\right),\hfill \end{array}\end{array}$$

(44)

where the sums run over non-cyclic orderings. This defines the colour decomposition of biadjoint scalar amplitudes. The partial amplitudes ${𝓐}_n^{\mathrm{BAS}}$ each appear four times.

For a permutation $\sigma \in S_n$ on n letters, a colour ordering is an equivalence class of n-tuples $\left[\sigma \right(1),\dots ,\sigma (n\left)\right]$ with respect to the equivalence relationship

$$\begin{array}{c}\hfill \left(\sigma \left(1\right),\dots ,\sigma (n-1),\sigma \left(n\right)\right)\sim \left(\sigma \left(n\right),\sigma \left(1\right),\dots ,\sigma (n-1)\right)\sim \left(\sigma \left(n\right),\sigma (n-1),\dots ,\sigma \left(1\right)\right)\end{array}$$

(45)

on $S_n$ generated by the action of the subgroup ${\mathbb{Z}}_n⋊{\mathbb{Z}}_2\subset S_n$; the first identification takes care of cyclic ordering symmetry while the second implements the Kleiss–Kuijf relations. A canonical representative of a colour ordering has $\sigma \left(1\right)=1$ and $\sigma \left(2\right)<\sigma \left(n\right)$. Given a colour ordering $\left[\sigma \right(1),\dots ,\sigma (n\left)\right]$ in canonical form, the associated colour factor is

$$\begin{array}{c}\hfill C\left(\sigma \right):={\mathrm{Tr}}_{\mathfrak{g}}(T^{a_{\sigma \left(1\right)}}\cdots T^{a_{\sigma \left(n\right)}})+{(-1)}^n{\mathrm{Tr}}_{\mathfrak{g}}(T^{a_{\sigma \left(n\right)}}\cdots T^{a_{\sigma \left(1\right)}}),\end{array}$$

(46)

and similarly, we define the colour factor $\overline{C}\left(\sigma \right)$ for the dual Lie algebra $\overline{\mathfrak{g}}$. There are $\frac{1}{2}(n-1)!$ colour factors $C\left(\sigma \right)$ and $\overline{C}\left(\sigma \right)$, and $\sigma$ is called their planar orderings.

In terms of these, the n-point amplitudes become

$$\begin{array}{c}\hfill {\mathcal{A}}_n^{\mathrm{BAS}}(p,a,\overline{a})=\sum _{\sigma ,{\sigma }^{\prime }\in S_n/{\mathbb{Z}}_n⋊{\mathbb{Z}}_2}C\left(\sigma \right)\overline{C}\left({\sigma }^{\prime }\right){𝓐}_n^{\mathrm{BAS}}\left(\sigma \left(1\right),\dots ,\sigma \left(n\right)|{\sigma }^{\prime }\left(1\right),\dots ,{\sigma }^{\prime }\left(n\right)\right).\end{array}$$

(47)

The partial amplitudes ${𝓐}_n^{\mathrm{BAS}}$ have a simple expression as a sum over tree-level Feynman diagrams of the cubic scalar field theory from Section 2.1 which are planar with respect to both orderings [10,41]. This is of course the amplitude incarnation of the homotopy factorisation (37). For example, the bi-colour-ordered four-point amplitude is given by

$$\begin{array}{c}\hfill {𝓐}_4^{\mathrm{BAS}}(1,2,3,4|1,2,3,4)=\frac{1}{t}-\frac{1}{s}.\end{array}$$

(48)

Conversely, the cubic scalar tree-level amplitudes can be computed by summing over all colour orderings of ${𝓐}_n^{\mathrm{BAS}}$ [42].

3. Noncommutative Scalar Theories with Rigid Colour Symmetries

3.1. Moyal–Weyl Deformation of Scalar Fields

In the following we will describe natural noncommutative deformations of the biadjoint scalar theory from Section 2.2. Although the formalism can be quite generally applied using any Drinfel’d twist deformation in the universal enveloping algebra of the Lie algebra of vector fields on ${\mathbb{R}}^{1,d-1}$, for definiteness we work with the simplest and best-known example of the Moyal–Weyl twist. This is a case that is best understood both algebraically and geometrically, and it is the natural one which will arise in our later discussions of the relationship between scattering amplitudes of noncommutative Yang–Mills theory and open string theory. It is also the case which has appeared in recent discussions of the double copy.

The Moyal–Weyl deformation of the algebra of functions ${\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$ is parametrized by a constant Poisson bivector $\theta =\frac{1}{2}{\theta }^{\mu \nu }{\partial }_{\mu }\wedge {\partial }_{\nu }$ on ${\mathbb{R}}^{1,d-1}$, and is defined by replacing the commutative pointwise multiplication of functions ${\varphi }_1{\varphi }_2$ with the associative but noncommutative star-product

$$\begin{array}{c}\hfill {\varphi }_1★{\varphi }_2={\varphi }_{1}exp\left(-\frac{\mathrm{i}t}{2}{\overleftarrow{\partial }}_{\mu }{\theta }^{\mu \nu }{\overrightarrow{\partial }}_{\nu }\right){\varphi }_2,\end{array}$$

(49)

which is understood by expanding the exponential of the bidifferential operator as a formal power series in the deformation parameter $t\in \mathbb{R}$. (There is also a convergent integral convolution formula for the Moyal–Weyl product [17], whose asymptotic expansion in t coincides with (49)). The product (49) is associative and the commutative limit $t=0$ returns the usual pointwise product of ${\varphi }_1$ and ${\varphi }_2$, i.e., ${\varphi }_1★{\varphi }_2={\varphi }_1{\varphi }_2+O\left(t\right)$. It is not real, but it is Hermitian:

$$\begin{array}{c}\hfill \overline{{\varphi }_1★{\varphi }_2}={\varphi }_2★{\varphi }_1,\end{array}$$

(50)

for ${\varphi }_1,{\varphi }_2\in {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$.

If we restrict to Schwartz functions, then the star-product obeys the well-known integration by parts identity

$$\begin{array}{c}\hfill \int {\mathrm{d}}^{d}x{\varphi }_1★{\varphi }_2=\int {\mathrm{d}}^{d}x{\varphi }_2★{\varphi }_1=\int {\mathrm{d}}^{d}x{\varphi }_1{\varphi }_2,\end{array}$$

(51)

which will translate to the statement that free field theories are unaffected by the Moyal–Weyl deformation. This identity implies that integration defines a trace on the deformed algebra of functions, i.e., it is cyclic with respect to the star-product (49).

3.1.1. The Kinematical Lie Algebra ${\mathfrak{u}}_{★}\left(1\right)$

Using the star-product we define the star-commutator of two functions ${\varphi }_1,{\varphi }_2\in {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$ by

$$\begin{array}{c}\hfill {[{\varphi }_1\stackrel{★}{,}{\varphi }_2]}_{\mathfrak{u}\left(1\right)}:={\varphi }_1★{\varphi }_2-{\varphi }_2★{\varphi }_1=-2\mathrm{i}{\varphi }_{1}sin\left(\frac{t}{2}{\overleftarrow{\partial }}_{\mu }{\theta }^{\mu \nu }{\overrightarrow{\partial }}_{\nu }\right){\varphi }_2.\end{array}$$

(52)

By construction the star-commutator defines a Lie bracket, and it is a deformation of the Poisson bracket ${\{{\varphi }_1,{\varphi }_2\}}_{\theta }={\theta }^{\mu \nu }{\partial }_{\mu }{\varphi }_1{\partial }_{\nu }{\varphi }_2$ in the sense that its semi-classical limit is given by

$$\begin{array}{c}\hfill {[{\varphi }_1\stackrel{★}{,}{\varphi }_2]}_{\mathfrak{u}\left(1\right)}=-\mathrm{i}t{\{{\varphi }_1,{\varphi }_2\}}_{\theta }+O\left(t^2\right).\end{array}$$

(53)

We write ${\mathfrak{u}}_{★}\left(1\right)$ for the infinite-dimensional Lie algebra $\left({\Omega }^0({\mathbb{R}}^{1,d-1}),{[-\stackrel{★}{,}-]}_{\mathfrak{u}\left(1\right)}\right)$. In the following, this will appear naturally as a kinematical Lie algebra in our considerations of the double copy duality.

A basis for the kinematical Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$ is given via Fourier transformation by the plane waves

$$\begin{array}{c}\hfill e_k\left(x\right):={\mathrm{e}}^{\mathrm{i}k·x}\end{array}$$

(54)

for $x\in {\mathbb{R}}^{1,d-1}$ and $k\in {\left({\mathbb{R}}^{1,d-1}\right)}^{\ast }$. They obey the star-commutation relations

$$\begin{array}{c}\hfill {\left[e_k\stackrel{★}{,}{e}_p\right]}_{\mathfrak{u}\left(1\right)}=\int {\mathrm{d}}^{d}q\overline{F}(k,p,q)e_q\mathrm{with}\overline{F}(k,p,q)=2\mathrm{i}sin\left(\frac{t}{2}k·\theta p\right)\delta (k+p-q).\end{array}$$

(55)

A detailed description of the infinite-dimensional Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$ is found in [32], including its realisation as a deformation of the Poisson–Lie algebra $\mathfrak{sdiff}\left({\mathbb{R}}^{1,d-1}\right)$ of symplectic diffeomorphisms of ${\mathbb{R}}^{1,d-1}$. In [32,43] its relationship to the infinite unitary Lie algebra $\mathfrak{u}(\infty )$ is also described. Using an alternative monomial basis of generators for ${\mathfrak{u}}_{★}\left(1\right)$ in $d=2$ dimensions, it is shown in [44] that the deformation of the Poisson bracket to the Moyal–Weyl bracket corresponds to a deformation of the $w_{1+\infty }$ algebra of Poisson diffeomorphisms of the plane to a certain type of $W_{1+\infty }$ algebra called the symplecton algebra; see also [32] for other features of ${\mathfrak{u}}_{★}\left(1\right)$ in monomial and other bases as well as in arbitrary dimensionalities.

3.1.2. The Colour Lie Algebra ${\mathfrak{u}}_{★}\left(\overline{N}\right)$

The construction above can be generalised to include colour degrees of freedom in the following way. Let $\overline{\mathfrak{g}}$ be a matrix Lie algebra that is closed under both commutators and anticommutators; for definiteness, we can take $\overline{\mathfrak{g}}=\mathfrak{u}\left(\overline{N}\right)$. Denote the Lie bracket of $\mathfrak{u}\left(\overline{N}\right)$ by ${[-,-]}_{\mathfrak{u}\left(\overline{N}\right)}$, and choose a basis $\left\{{\overline{T}}^{\overline{a}}\right\}$ of anti-Hermitian matrices for $\mathfrak{u}\left(\overline{N}\right)$ with structure constants ${\overline{f}}^{\overline{a}\overline{b}\overline{c}}$, i.e., ${[{\overline{T}}^{\overline{a}},T^{\overline{b}}]}_{\mathfrak{u}\left(\overline{N}\right)}={\overline{f}}^{\overline{a}\overline{b}\overline{c}}{\overline{T}}^{\overline{c}}$.

Let ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right))$ be the space of functions with values in $\mathfrak{u}\left(\overline{N}\right)$. This is an associative algebra under the composition of matrix multiplication with the pointwise product of functions, which as usual we make into a Lie algebra under the commutator bracket:

$$\begin{array}{c}\hfill {[{\varphi }_1,{\varphi }_2]}_{\mathfrak{u}\left(\overline{N}\right)}={\varphi }_1{\varphi }_2-{\varphi }_2{\varphi }_1.\end{array}$$

(56)

There is a vector space isomorphism over $\mathbb{R}$

$$\begin{array}{c}\hfill {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right))\simeq \mathfrak{u}\left(\overline{N}\right)\otimes {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right),\end{array}$$

(57)

and this is also true at the level of Lie algebras: the Lie algebra ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right))$ factorises into the tensor product of the Lie algebra $\mathfrak{u}\left(\overline{N}\right)$ with the commutative algebra ${\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {[{\varphi }_1,{\varphi }_2]}_{\mathfrak{u}\left(\overline{N}\right)}={[{\overline{T}}^{\overline{a}}\otimes {\varphi }_1^{\overline{a}},{\overline{T}}^{\overline{b}}\otimes {\varphi }_2^{\overline{b}}]}_{\mathfrak{u}\left(\overline{N}\right)}={[{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}]}_{\mathfrak{u}\left(\overline{N}\right)}\otimes ({\varphi }_1^{\overline{a}}{\varphi }_2^{\overline{b}}),\end{array}\end{array}$$

(58)

where ${\varphi }_1^{\overline{a}},{\varphi }_2^{\overline{b}}\in {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$ and we used commutativity of the pointwise multiplication of functions.

Let us now pass to the Moyal–Weyl deformation and define the star-commutator of functions in ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right))$ by

$$\begin{array}{c}\hfill {[{\varphi }_1\stackrel{★}{,}{\varphi }_2]}_{\mathfrak{u}\left(\overline{N}\right)}={\varphi }_1★{\varphi }_2-{\varphi }_2★{\varphi }_1,\end{array}$$

(59)

where here the associative operation ★ means the composition of matrix multiplication with the star-product (49). This bracket again makes ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right))$ into an infinite-dimensional Lie algebra, which we denote by ${\mathfrak{u}}_{★}\left(\overline{N}\right)$. However, while the vector space isomorphism (57) still holds, it is no longer an isomorphism of Lie algebras, as now noncommutativity implies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {[{\varphi }_1\stackrel{★}{,}{\varphi }_2]}_{\mathfrak{u}\left(\overline{N}\right)}={[{\overline{T}}^{\overline{a}}\otimes {\varphi }_1^{\overline{a}}\stackrel{★}{,}{\overline{T}}^{\overline{b}}\otimes {\varphi }_2^{\overline{b}}]}_{\mathfrak{u}\left(\overline{N}\right)}& ={[{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}]}_{\mathfrak{u}\left(\overline{N}\right)}\otimes \frac{1}{2}\left({\varphi }_1^{\overline{a}}★{\varphi }_2^{\overline{b}}+{\varphi }_2^{\overline{b}}★{\varphi }_1^{\overline{a}}\right)\hfill \\ \hfill & +{\{{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}\}}_{\mathfrak{u}\left(\overline{N}\right)}\otimes \frac{1}{2}\left({\varphi }_1^{\overline{a}}★{\varphi }_2^{\overline{b}}-{\varphi }_2^{\overline{b}}★{\varphi }_1^{\overline{a}}\right).\hfill \end{array}\end{array}$$

(60)

This lack of factorisation, which is the well-known intertwining between colour and kinematical degrees of freedom in noncommutative field theories, will make colour–kinematics duality somewhat subtle in these instances. Indeed, as noted by e.g., [12], colour-stripping is only possible in theories whose interactions are constructed exclusively from Lie algebra commutators ${[{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}]}_{\mathfrak{u}\left(\overline{N}\right)}$, even if all fields are valued in the adjoint representation of $\mathfrak{u}\left(\overline{N}\right)$. This is not the case in noncommutative field theories, whose interactions also involve the anticommutators ${\{{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}\}}_{\mathfrak{u}\left(\overline{N}\right)}$. A related theory that violates the criterion is the non-abelian Dirac–Born–Infeld theory, whose fields are all $\mathrm{ad}\left(\mathfrak{u}\left(\overline{N}\right)\right)$-valued but whose interactions also involve ${\{{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}\}}_{\mathfrak{u}\left(\overline{N}\right)}$. We shall discuss in detail below how to handle this difficulty in our theories: it is precisely the mixing of kinematics with the various colour weights which will permit a double copy construction. This is analogous to the extensions of colour–kinematics duality discussed in [22,23,24,25] for theories involving higher derivative operators and gauge algebra anticommutators.

Let us rewrite the star-commutator in terms of the symmetric d-coefficients ${\overline{d}}^{\overline{a}\overline{b}\overline{c}}$ defined by ${\{{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}\}}_{\mathfrak{u}\left(\overline{N}\right)}=\mathrm{i}{\overline{d}}^{\overline{a}\overline{b}\overline{c}}{\overline{T}}^{\overline{c}}$ to obtain

$$\begin{array}{c}\hfill {\left[{\varphi }_1\stackrel{★}{,}{\varphi }_2\right]}_{\mathfrak{u}\left(\overline{N}\right)}=\frac{1}{2}\left({\overline{f}}^{\overline{a}\overline{b}\overline{c}}{\left\{{\varphi }_1^{\overline{a}}\stackrel{★}{,}{\varphi }_2^{\overline{b}}\right\}}_{\mathfrak{u}\left(1\right)}+\mathrm{i}{\overline{d}}^{\overline{a}\overline{b}\overline{c}}{\left[{\varphi }_1^{\overline{a}}\stackrel{★}{,}{\varphi }_2^{\overline{b}}\right]}_{\mathfrak{u}\left(1\right)}\right){\overline{T}}^{\overline{c}},\end{array}$$

(61)

where we introduced the star-anticommutator of functions

$$\begin{array}{c}\hfill {\left\{{\varphi }_1^{\overline{a}}\stackrel{★}{,}{\varphi }_2^{\overline{b}}\right\}}_{\mathfrak{u}\left(1\right)}:={\varphi }_1^{\overline{a}}★{\varphi }_2^{\overline{b}}+{\varphi }_2^{\overline{b}}★{\varphi }_1^{\overline{a}}=2{\varphi }_1^{\overline{a}}cos\left(\frac{t}{2}{\overleftarrow{\partial }}_{\mu }{\theta }^{\mu \nu }{\overrightarrow{\partial }}_{\nu }\right){\varphi }_2^{\overline{b}}.\end{array}$$

(62)

A momentum space basis for ${\mathfrak{u}}_{★}\left(\overline{N}\right)$ is given by the $\mathfrak{u}\left(\overline{N}\right)$-valued plane waves

$$e_k^{\overline{a}}\left(x\right):={\mathrm{e}}^{\mathrm{i}k·x}{\overline{T}}^{\overline{a}}\in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right)),$$

(63)

whose star-commutation relations

$${\left[e_k^{\overline{a}}\stackrel{★}{,}{e}_p^{\overline{b}}\right]}_{\mathfrak{u}\left(\overline{N}\right)}=\int {\mathrm{d}}^{d}q{\overline{F}}^{\overline{a}\overline{b}\overline{c}}(k,p,q)e_q^{\overline{c}}$$

(64)

can be expressed in terms of structure constants

$$\begin{array}{cc}\hfill {\overline{F}}^{\overline{a}\overline{b}\overline{c}}(k,p,q)& =\left({\overline{f}}^{\overline{a}\overline{b}\overline{c}}cos\left(\frac{t}{2}k·\theta p\right)+\mathrm{i}{\overline{d}}^{\overline{a}\overline{b}\overline{c}}sin\left(\frac{t}{2}k·\theta p\right)\right)\delta (k+p-q).\hfill \end{array}$$

(65)

These relations again highlight the mixing of colour and kinematics in the noncommutative algebra of $\mathfrak{u}\left(\overline{N}\right)$-valued functions, which allows the incorporation of noncommutative field theories into the standard $L_{\infty }$-algebra formalism [18,19]. See [45] for a description of the infinite-dimensional Lie algebra ${\mathfrak{u}}_{★}\left(\overline{N}\right)$ as a deformation of a Kac–Moody algebra in $d=2$ dimensions, where the colour–kinematics mixing in the context of the double copy was also noted.

3.2. Noncommutative Biadjoint Scalar Theory

A natural noncommutative deformation of the biadjoint scalar theory of Section 2.2 is defined by replacing the second Lie algebra $\overline{\mathfrak{g}}$ by the infinite-dimensional Lie algebra ${\mathfrak{u}}_{★}\left(\overline{N}\right)$, using the Lie bracket (61). It is organised by a cyclic strict $L_{\infty }$-algebra ${\mathfrak{BAS}}_{★}$ whose cochain complex $\mathsf{Ch}\left({\mathfrak{BAS}}_{★}\right)={\mathsf{Ch}}_{\mathrm{BAS}}$ coincides with that of the commutative case (28) with $\overline{\mathfrak{g}}=\mathfrak{u}\left(\overline{N}\right)$, which again identifies the differential ${\mu }_1^{★\mathrm{BAS}}={\mu }_1^{\mathrm{BAS}}=\square$.

The higher bracket ${\mu }_2^{★\mathrm{BAS}}:{\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \mathfrak{u}\left(\overline{N}\right)){[-1]}^{\otimes 2}⟶{\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \mathfrak{u}\left(\overline{N}\right))[-2]$ is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_2^{★\mathrm{BAS}}({\varphi }_1,{\varphi }_2)=& -g〚{\varphi }_1,{\varphi }_2{〛}_{\mathfrak{g}\otimes {\mathfrak{u}}_{★}\left(\overline{N}\right)}\hfill \\ \hfill :=& -\frac{g}{2}{f}^{abc}\left({\overline{f}}^{\overline{a}\overline{b}\overline{c}}{\left\{{\varphi }_1^{a\overline{a}}\stackrel{★}{,}{\varphi }_2^{b\overline{b}}\right\}}_{\mathfrak{u}\left(1\right)}+\mathrm{i}{\overline{d}}^{\overline{a}\overline{b}\overline{c}}{\left[{\varphi }_1^{a\overline{a}}\stackrel{★}{,}{\varphi }_2^{b\overline{b}}\right]}_{\mathfrak{u}\left(1\right)}\right)T^c\otimes {\overline{T}}^{\overline{c}}.\hfill \end{array}\end{array}$$

(66)

Note that

$$\begin{array}{c}\hfill {[[T^a\otimes {\overline{T}}^{\overline{a}}\otimes {\varphi }_1^{a\overline{a}},T^b\otimes {\overline{T}}^{\overline{b}}\otimes {\varphi }_2^{b\overline{b}}]]}_{\mathfrak{g}\otimes {\mathfrak{u}}_{★}\left(\overline{N}\right)}={[T^a,T^b]}_{\mathfrak{g}}\otimes {[{\overline{T}}^{\overline{a}}\otimes {\varphi }_1^{a\overline{a}}\stackrel{★}{,}{\overline{T}}^{\overline{b}}\otimes {\varphi }_2^{b\overline{b}}]}_{\mathfrak{u}\left(\overline{N}\right)},\end{array}$$

(67)

and hence the 2-bracket is symmetric.

The cyclic inner product of degree $-3$ is given by the non-zero pairing

$$\begin{array}{c}\hfill {\langle \varphi ,{\varphi }^{+}\rangle }_{\mathrm{BAS}}^{★}=\int {\mathrm{d}}^{d}x{\mathrm{Tr}}_{\mathfrak{g}\otimes \mathfrak{u}\left(\overline{N}\right)}(\varphi ★{\varphi }^{+})=\int {\mathrm{d}}^{d}x{\varphi }^{a\overline{a}}★{\varphi }_{a\overline{a}}^{+}={\langle \varphi ,{\varphi }^{+}\rangle }_{\mathrm{BAS}},\end{array}$$

(68)

for $\varphi \in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \mathfrak{u}\left(\overline{N}\right))[-1]$ and ${\varphi }^{+}\in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \mathfrak{u}\left(\overline{N}\right))[-2]$, where we used Equation (51). Note that

$$\begin{array}{c}\hfill {\langle T^a\otimes {\overline{T}}^{\overline{a}}\otimes {\varphi }_{a\overline{a}},T^b\otimes {\overline{T}}^{\overline{b}}\otimes {\varphi }_{b\overline{b}}^{+}\rangle }_{\mathrm{BAS}}^{★}={\mathrm{Tr}}_{\mathfrak{g}}(T^a\otimes T^b){\mathrm{Tr}}_{{\mathfrak{u}}_{★}\left(\overline{N}\right)}({\overline{T}}^{\overline{a}}{\varphi }_{a\overline{a}}★{\overline{T}}^{\overline{b}}{\varphi }_{b\overline{b}}^{+}),\end{array}$$

(69)

where ${\mathrm{Tr}}_{{\mathfrak{u}}_{★}\left(\overline{N}\right)}=\int \circ {\mathrm{Tr}}_{\mathfrak{u}\left(\overline{N}\right)}$ defines a cyclic trace on the noncommutative algebra of functions on ${\mathbb{R}}^{1,d-1}$ with values in the matrix Lie algebra $\mathfrak{u}\left(\overline{N}\right)$.

Altogether, we obtain a cyclic strict $L_{\infty }$-algebra

$$\begin{array}{c}\hfill {\mathfrak{BAS}}_{★}=\left({\mathsf{Ch}}_{\mathrm{BAS}},{\mu }_2^{★\mathrm{BAS}},{\langle -,-\rangle }_{\mathrm{BAS}}\right).\end{array}$$

(70)

The underlying free field theory from Section 2.2 is unchanged and only the interactions undergo noncommutative deformation. Nevertheless, for presentation, we continue to delineate all operations with the symbol ‘★’ to emphasise that we are working with a noncommutative field theory.

The Maurer–Cartan Equation (A17) gives the equation of motion for $\varphi \in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \mathfrak{u}\left(\overline{N}\right))$ as

$$\begin{array}{c}\hfill {\mu }_1^{★\mathrm{BAS}}\left(\varphi \right)+\frac{1}{2}{\mu }_2^{★\mathrm{BAS}}(\varphi ,\varphi )=\square \varphi -\frac{g}{2}〚\varphi ,\varphi {〛}_{\mathfrak{g}\otimes {\mathfrak{u}}_{★}\left(\overline{N}\right)}=0,\end{array}$$

(71)

which on component functions in ${\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)$ reads as

$$\begin{array}{c}\hfill \square {\varphi }^{a\overline{a}}-\frac{g}{2}{f}^{abc}{\overline{\lambda }}^{\overline{a}\overline{b}\overline{c}}{\varphi }^{b\overline{b}}★{\varphi }^{c\overline{c}}=0,\end{array}$$

(72)

where

$$\begin{array}{c}\hfill {\overline{\lambda }}^{\overline{a}\overline{b}\overline{c}}={\overline{f}}^{\overline{a}\overline{b}\overline{c}}+\mathrm{i}{\overline{d}}^{\overline{a}\overline{b}\overline{c}}.\end{array}$$

(73)

The Maurer–Cartan functional (A20) is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_{\mathrm{BAS}}^{★}\left[\varphi \right]& =\frac{1}{2}{\langle \varphi ,{\mu }_1^{★\mathrm{BAS}}\left(\varphi \right)\rangle }_{\mathrm{BAS}}^{★}+\frac{1}{3!}{\langle \varphi ,{\mu }_2^{★\mathrm{BAS}}(\varphi ,\varphi )\rangle }_{\mathrm{BAS}}^{★}\hfill \\ \hfill & =\int {\mathrm{d}}^{d}x{\mathrm{Tr}}_{\mathfrak{g}\otimes \mathfrak{u}\left(\overline{N}\right)}\left(\frac{1}{2}\varphi ★\square \varphi -\frac{g}{3!}\varphi ★{\left[[\varphi ,\varphi ]\right]}_{\mathfrak{g}\otimes {\mathfrak{u}}_{★}\left(\overline{N}\right)}\right)\hfill \\ \hfill & =\int {\mathrm{d}}^{d}x\frac{1}{2}{\varphi }^{a\overline{a}}★\square {\varphi }_{a\overline{a}}-\frac{g}{3!}{f}^{abc}{\overline{\lambda }}^{\overline{a}\overline{b}\overline{c}}{\varphi }^{a\overline{a}}★{\varphi }^{b\overline{b}}★{\varphi }^{c\overline{c}}.\hfill \end{array}\end{array}$$

(74)

Remark 1.

We stress that the noncommutative biadjoint scalar theory possesses a rigid symmetry under $\mathfrak{g}\oplus \mathfrak{u}\left(\overline{N}\right)$, because the $\mathrm{ad}\left(\mathfrak{u}\left(\overline{N}\right)\right)$-action preserves independently the $\mathfrak{u}\left(\overline{N}\right)$ commutator and anticommutator in Equation (60). In particular, we may define different noncommutative field theories with the same rigid symmetry by replacing ${\overline{\lambda }}^{\overline{a}\overline{b}\overline{c}}$ by ${\overline{f}}^{\overline{a}\overline{b}\overline{c}}$, $\mathrm{i}{\overline{d}}^{\overline{a}\overline{b}\overline{c}}$ or any linear combination of the two sets of structure constants, i.e., there exists a two-parameter family of noncommutative deformations of the biadjoint scalar theory defined by the more general structure constants

$$\begin{array}{c}\hfill {\overline{\lambda }}_{\overline{u},\overline{v}}^{\overline{a}\overline{b}\overline{c}}:=\overline{u}{\overline{f}}^{\overline{a}\overline{b}\overline{c}}+\mathrm{i}\overline{v}{\overline{d}}^{\overline{a}\overline{b}\overline{c}},\overline{u},\overline{v}\in \mathbb{R}.\end{array}$$

(75)

In the following we will mostly concentrate on the natural member of this family induced by the structure constants of the Lie algebra ${\mathfrak{u}}_{★}\left(\overline{N}\right)$, with $\overline{u}=\overline{v}=1$, as above.

3.2.1. Twisted Homotopy Factorisation

Let us now consider the problem of factorising the cyclic strict $L_{\infty }$-algebra ${\mathfrak{BAS}}_{★}$. The three-point Feynman vertex of the noncommutative biadjoint scalar theory is given by

(76)

The lack of factorisation of the star-commutator ${[-\stackrel{★}{,}-]}_{\mathfrak{u}\left(\overline{N}\right)}$, discussed in Section 3.1, means that we cannot simply strip off the rigid $\mathfrak{u}\left(\overline{N}\right)$ symmetry, like we did with $\overline{\mathfrak{g}}$ in the biadjoint theory from Section 2.2. Instead, we show that the infinite-dimensional Lie algebra ${\mathfrak{u}}_{★}\left(\overline{N}\right)$ admits a ‘twisted factorisation’, with the twist implementing the intertwining of colour and kinematical degrees of freedom of the scalar fields.

For this, we factorise ${\mathfrak{BAS}}_{★}$ into three parts: a colour part, a twisted colour part, and a cyclic strict $L_{\infty }$-algebra which fully describes the trivalent interactions. There are two steps: In the first step, we show that ${\mathfrak{BAS}}_{★}$ admits a factorisation into the colour Lie algebra $\mathfrak{g}$ and a kinematical cyclic strict $C_{\infty }$-algebra ${\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(\overline{N}\right)}$, which corresponds to colour-stripping the cyclic strict $L_{\infty }$-algebra ${\mathfrak{BAS}}_{★}$.

The cochain complex underlying ${\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(\overline{N}\right)}$ is

$$\begin{array}{c}\hfill \mathsf{Ch}({\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(\overline{N}\right)})=\left({\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right))[-1]\stackrel{\square }{\to }{\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right))[-2]\right),\end{array}$$

(77)

which again identifies the differential $m_1^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}=\square$ acting as $m_1^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}\left(\varphi \right)=\square {\varphi }_{\overline{a}}{\overline{T}}^{\overline{a}}$. The product $m_2^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}:{\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right)){[-1]}^{\otimes 2}⟶{\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right))[-2]$ is defined by

$$\begin{array}{c}\hfill m_2^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}({\varphi }_1,{\varphi }_2)=-g{\left[{\varphi }_1\stackrel{★}{,}{\varphi }_2\right]}_{\mathfrak{u}\left(\overline{N}\right)}.\end{array}$$

(78)

This operation is graded commutative, and for degree reasons, it satisfies the Leibniz rule as well as the graded associativity relationship. The cyclic structure is defined by ${\mathrm{Tr}}_{{\mathfrak{u}}_{★}\left(\overline{N}\right)}=\int \circ {\mathrm{Tr}}_{\mathfrak{u}\left(\overline{N}\right)}$.

This cyclic strict $C_{\infty }$-algebra allows for a factorisation

$$\begin{array}{c}\hfill {\mathfrak{BAS}}_{★}=\mathfrak{g}\otimes {\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(\overline{N}\right)}.\end{array}$$

(79)

For this, we identify the vector space ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g}\otimes \mathfrak{u}\left(\overline{N}\right))$ in the usual way with the tensor product $\mathfrak{g}\otimes {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(\overline{N}\right))$, recall the definition of the differential ${\mu }_1^{★\mathrm{BAS}}$ and the 2-bracket ${\mu }_2^{★\mathrm{BAS}}$, and use Equation (67) as well as Equation (69). It then easily follows that the structure maps of ${\mathfrak{BAS}}_{★}$ factor through ${\mu }_1^{★\mathrm{BAS}}=\mathbb{1}\otimes m_1^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}$, ${\mu }_2^{★\mathrm{BAS}}={[-,-]}_{\mathfrak{g}}\otimes m_2^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}$ and ${\langle -,-\rangle }_{\mathrm{BAS}}^{★}={\mathrm{Tr}}_{\mathfrak{g}}\otimes {\mathrm{Tr}}_{{\mathfrak{u}}_{★}\left(\overline{N}\right)}$.

The second step consists of stripping off the kinematical factor encoding the trivalent interactions. This uses the notion of twisted tensor product of homotopy algebras from [12], (The general notion of ‘twist’ in this context should not be confused with Drinfel’d twist deformation in the context of noncommutative geometry, though in our setting they both play a similar role in defining a noncommutative deformation). which we review in Appendix A.2. Regarding $\mathfrak{u}\left(\overline{N}\right)$ as a graded vector space sitting in degree 0, we wish to compute the twisted tensor product between $\mathfrak{u}\left(\overline{N}\right)$ and the cyclic strict $L_{\infty }$-algebra $\mathfrak{Scal}$ of the commutative $g{\varphi }^3$-theory on ${\mathbb{R}}^{1,d-1}$ from Section 2.1. The twist datum ${\tau }^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}=\left({\tau }_1^{{\mathfrak{u}}_{★}\left(\overline{N}\right)},{\tau }_2^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}\right)$ consists of a pair of maps ${\tau }_1^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}:\mathfrak{u}\left(\overline{N}\right)⟶\mathfrak{u}\left(\overline{N}\right)\otimes \mathsf{End}\left(L\right)$ and ${\tau }_2^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}:\mathfrak{u}\left(\overline{N}\right)\otimes \mathfrak{u}\left(\overline{N}\right)⟶\mathfrak{u}\left(\overline{N}\right)\otimes \mathsf{End}\left(L\right)\otimes \mathsf{End}\left(L\right)$, where

$$\begin{array}{c}\hfill L={\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)[-1]\oplus {\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)[-2]\end{array}$$

(80)

is the graded vector space underlying the cochain complex ${\mathsf{Ch}}_{\mathrm{Scal}}$ from Equation (20).

We set

$$\begin{array}{c}\hfill {\tau }_1^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}\left({\overline{T}}^{\overline{a}}\right)={\overline{T}}^{\overline{a}}\otimes \mathbb{1}\end{array}$$

(81)

and

$$\begin{array}{c}{\tau }_2^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}({\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}})={[{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}]}_{\mathfrak{u}\left(\overline{N}\right)}\otimes cos\left(\frac{t}{2}{\theta }^{\mu \nu }{\partial }_{\mu }\otimes {\partial }_{\nu }\right)+\mathrm{i}{\{{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}\}}_{\mathfrak{u}\left(\overline{N}\right)}\otimes sin\left(\frac{t}{2}{\theta }^{\mu \nu }{\partial }_{\mu }\otimes {\partial }_{\nu }\right).\end{array}$$

(82)

Following the prescription of [12], the tensor product $\mathfrak{u}\left(\overline{N}\right)\otimes L$ now carries the structure of a cyclic strict $C_{\infty }$-algebra with differential

$$\begin{array}{c}\hfill m_1^{{\tau }^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}}({\overline{T}}^{\overline{a}}\otimes {\varphi }_{\overline{a}})={\overline{T}}^{\overline{a}}\otimes {\mu }_1^{\mathrm{Scal}}\left({\varphi }_{\overline{a}}\right)={\overline{T}}^{\overline{a}}\otimes \square {\varphi }_{\overline{a}},\end{array}$$

(83)

and product

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill m_2^{{\tau }^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}}({\overline{T}}^{\overline{a}}\otimes {\varphi }_1^{\overline{a}},{\overline{T}}^{\overline{b}}\otimes {\varphi }_2^{\overline{b}})& ={[{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}]}_{\mathfrak{u}\left(\overline{N}\right)}\otimes {\mu }_2^{\mathrm{Scal}}\circ cos\left(\frac{t}{2}{\theta }^{\mu \nu }{\partial }_{\mu }\otimes {\partial }_{\nu }\right)({\varphi }_1^{\overline{a}}\otimes {\varphi }_2^{\overline{b}})\hfill \\ \hfill & +\mathrm{i}{\{{\overline{T}}^{\overline{a}},{\overline{T}}^{\overline{b}}\}}_{\mathfrak{u}\left(\overline{N}\right)}\otimes {\mu }_2^{\mathrm{Scal}}\circ sin\left(\frac{t}{2}{\theta }^{\mu \nu }{\partial }_{\mu }\otimes {\partial }_{\nu }\right)({\varphi }_1^{\overline{a}}\otimes {\varphi }_2^{\overline{b}})\hfill \\ \hfill & =-g{[{\overline{T}}^{\overline{a}}\otimes {\varphi }_1^{\overline{a}}\stackrel{★}{,}{\overline{T}}^{\overline{b}}\otimes {\varphi }_2^{\overline{b}}]}_{\mathfrak{u}\left(\overline{N}\right)},\hfill \end{array}\end{array}$$

(84)

where we used Equations (21) and (A44). The cyclic structure does not twist. This cyclic strict $C_{\infty }$-algebra is the twisted tensor product of $\mathfrak{u}\left(\overline{N}\right)$ and $\mathfrak{Scal}$, denoted $\mathfrak{u}\left(\overline{N}\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}}\mathfrak{Scal}$. From the definitions it immediately follows that $m_1^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}=m_1^{{\tau }^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}}$ and $m_2^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}=m_2^{{\tau }^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}}$, and hence the cyclic strict $C_{\infty }$-algebra ${\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(\overline{N}\right)}$ factorises as

$$\begin{array}{c}\hfill {\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(\overline{N}\right)}=\mathfrak{u}\left(\overline{N}\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}}\mathfrak{Scal}.\end{array}$$

(85)

Altogether, we have shown that the cyclic strict $L_{\infty }$-algebra of the noncommutative biadjoint scalar theory on ${\mathbb{R}}^{1,d-1}$ admits the factorisation

$$\begin{array}{c}\hfill {\mathfrak{BAS}}_{★}=\mathfrak{g}\otimes \left(\mathfrak{u}\left(\overline{N}\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}}\mathfrak{Scal}\right).\end{array}$$

(86)

This is one of the main messages of this paper: colour-stripping is achievable in noncommutative field theories through a suitable notion of twisted factorisation, which enables one to disentangle colour and kinematical degrees of freedom. From this perspective noncommutativity is completely absorbed into the twist maps ${\tau }^{{\mathfrak{u}}_{★}\left(\overline{N}\right)}$. This will also apply to the noncommutative theories with local gauge symmetry that we consider later.

3.3. Adjoint Scalar Theory

The limiting rank-one case $\overline{N}=1$ of the noncommutative biadjoint scalar theory is an interesting noncommutative scalar field theory that does not have a non-trivial commutative counterpart, in the sense that it becomes non-interacting in the commutative limit. We, therefore, refrain from using the adjective ‘noncommutative’ and simply refer to it as the adjoint scalar theory on ${\mathbb{R}}^{1,d-1}$. We begin by briefly summarising its main features, which follow by setting $\overline{N}=1$ everywhere in Section 3.2.

The cyclic strict $L_{\infty }$-algebra $\mathfrak{AS}$ of this theory has cochain complex

$$\begin{array}{c}\hfill \mathsf{Ch}\left(\mathfrak{AS}\right)=\left({\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g})[-1]\stackrel{\square }{\to }{\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g})[-2]\right),\end{array}$$

(87)

and the single non-vanishing higher bracket

$$\begin{array}{c}\hfill {\mu }_2^{\mathrm{AS}}({\varphi }_1,{\varphi }_2)=-g{\left[[T^a\otimes {\varphi }_1^a,T^b\otimes {\varphi }_2^b]\right]}_{\mathrm{AS}}=-\mathrm{i}g{[T^a,T^b]}_{\mathfrak{g}}\otimes {\left[{\varphi }_1^a\stackrel{★}{,}{\varphi }_2^b\right]}_{\mathfrak{u}\left(1\right)},\end{array}$$

(88)

where we abbreviate $〚-,-{〛}_{\mathrm{AS}}:=〚-,-{〛}_{\mathfrak{g}\otimes {\mathfrak{u}}_{★}\left(1\right)}$. Note that this binary operation is not the antisymmetric Lie bracket ${\left[{\varphi }_1\stackrel{★}{,}{\varphi }_2\right]}_{\mathfrak{g}}$; in particular, it vanishes in the commutative limit $t=0$. The cyclic structure is given by

$$\begin{array}{c}\hfill {\langle \varphi ,{\varphi }^{+}\rangle }_{\mathrm{AS}}=\int {\mathrm{d}}^{d}x{\mathrm{Tr}}_{\mathfrak{g}}(\varphi ★{\varphi }^{+})=\int {\mathrm{d}}^{d}x{\varphi }^a★{\varphi }_a^{+}.\end{array}$$

(89)

The equations of motion

$$\begin{array}{c}\hfill \square {\varphi }^a-\mathrm{i}g{f}^{abc}{\varphi }^b★{\varphi }^c=0\end{array}$$

(90)

of the adjoint scalar theory are the variational equations of the action functional

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_{\mathrm{AS}}\left[\varphi \right]& =\int {\mathrm{d}}^{d}x\frac{1}{2}{\varphi }^a★\square {\varphi }_a-\frac{\mathrm{i}g}{3}{f}^{abc}{\varphi }^a★{\varphi }^b★{\varphi }^c.\hfill \end{array}\end{array}$$

(91)

The three-point Feynman vertex defined by Equation (91) is

(92)

This is completely antisymmetric due to symmetry of the structure constants of ${\mathfrak{u}}_{★}\left(1\right)$ from Equation (55) under cyclic permutations of the momenta, which follows from momentum conservation.

Remark 2.

The adjoint scalar theory is formally obtained from the standard biadjoint scalar theory of Section 2.2 by replacing the second Lie algebra $\overline{\mathfrak{g}}$ with the kinematical Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$, via the prescription

$$\begin{array}{c}\hfill {\varphi }^{a\overline{a}}⟶{\varphi }^a,{\overline{f}}^{\overline{a}\overline{b}\overline{c}}⟶\overline{F}(k,p,q)and{\mathrm{Tr}}_{\mathfrak{g}\otimes \overline{\mathfrak{g}}}⟶\int \circ {\mathrm{Tr}}_{\mathfrak{g}}\end{array}$$

(93)

in the Lagrangian of the commutative biadjoint scalar theory. This line of reasoning was used in [28] to discuss aspects of a non-perturbative Lagrangian-level double copy between these field theories.

Twisted Factorisation

The factorisation of the adjoint scalar $L_{\infty }$-algebra $\mathfrak{AS}$ follows from the $\overline{N}=1$ limit of the twisted factorisation (86), but with a crucial simplification. We treat the Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$ as a kinematical factor by identifying $\mathfrak{u}\left(1\right)\simeq \mathbb{R}$ and regarding it as the kinematical vector space of the adjoint scalar theory:

$$\begin{array}{c}\hfill {\mathfrak{Kin}}_{{\wedge }^0}:=\mathbb{R}.\end{array}$$

(94)

The tensor products (over $\mathbb{R}$) with the one-dimensional vector space $\mathbb{R}$ are trivial, so that we may identify $\mathbb{R}\otimes W\simeq W$ for any real vector space W. The definition of the twist datum ${\tau }^{{\mathfrak{u}}_{★}\left(1\right)}$ correspondingly simplifies to

$$\begin{array}{c}\hfill {\tau }_1^{{\mathfrak{u}}_{★}\left(1\right)}\left(1\right)=\mathbb{1}\mathrm{and}{\tau }_2^{{\mathfrak{u}}_{★}\left(1\right)}\left(1\right)=2\mathrm{i}sin\left(\frac{t}{2}{\theta }^{\mu \nu }{\partial }_{\mu }\otimes {\partial }_{\nu }\right),\end{array}$$

(95)

from which one recovers the star-commutator bracket ${[-\stackrel{★}{,}-]}_{\mathfrak{u}\left(1\right)}$.

Therefore, the cyclic strict $L_{\infty }$-algebra of the adjoint scalar theory on ${\mathbb{R}}^{1,d-1}$ factorises as

$$\begin{array}{c}\hfill \mathfrak{AS}=\mathfrak{g}\otimes ({\mathfrak{Kin}}_{{\wedge }^0}{\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(1\right)}}\mathfrak{Scal}).\end{array}$$

(96)

This agrees with the prescription in (93), and it identifies the commutative biadjoint scalar theory as the zeroth copy of the adjoint scalar theory: replacing the kinematic factor ${\mathfrak{Kin}}_{{\wedge }^0}=\mathbb{R}$ in the factorisation (96) with the second colour factor $\overline{\mathfrak{g}}$ yields the cyclic strict $L_{\infty }$-algebra (37), which is the correct homotopy algebraic structure in the zeroth-copy prescription.

3.4. Colour–Kinematics Duality

In terms of the factorisation (96), the double copy prescription for the adjoint scalar theory is immediately apparent. However, we must first verify that the factorisation is compatible with colour–kinematics duality. To do this, we will demonstrate how colour–kinematics duality is manifest in the Berends–Giele currents of the adjoint scalar theory. This framework realises the double copy as a generalisation of the colour–kinematics dual formulation (relying on kinematic Jacobi identities), rather than the KLT formulation (relying on relations among colour-ordered amplitudes) that we will encounter in Section 5.

3.4.1. Perturbative Calculations

We first give the perturbiner expansion, reviewed in Appendix A.3, for the adjoint scalar theory. The contracting homotopy satisfying the Hodge–Kodaira decomposition for the projection to the minimal model of the $L_{\infty }$-algebra $\mathfrak{AS}$ is simply the massless Feynman propagator

$$\begin{array}{c}\hfill ({\mu }_1^{\mathrm{AS}}\circ G_{\mathrm{F}})(x,y)=\mathrm{i}\delta (x-y)\mathrm{with}{G}_{\mathrm{F}}\left({\mathrm{e}}^{\mathrm{i}p·x}\right)=-\frac{\mathrm{i}}{p^2}{\mathrm{e}}^{\mathrm{i}p·x},\end{array}$$

(97)

where ${\mu }_1^{\mathrm{AS}}=\square$. From a perturbiner element ${\varphi }^{\circ }\in {\mathsf{H}}^1\left(\mathfrak{AS}\right)=ker(\square )$, we look for a quasi-isomorphism ${\psi }_n:{\mathsf{H}}^1{\left(\mathfrak{AS}\right)}^{\otimes n}⟶L^1$ which is given by the recursion relations (A38). Given ${\varphi }^{\circ }$, a perturbiner expansion for the interacting theory is nothing but the Maurer–Cartan field under this quasi-isomorphism, namely

$$\varphi =\sum _{n=1}^{\infty }\frac{1}{n!}{\psi }_n\left({\varphi }^{\circ \otimes n}\right)\in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g})[-1].$$

(98)

We now consider the decomposition in terms of on-shell multiparticle solutions. This differs from the Formula (98) that is used to reconstruct a Maurer–Cartan element in the full interacting theory from identical free fields. Denote by $\varphi \left(i\right)\in {\mathsf{H}}^1\left(\mathfrak{AS}\right)$ a field in the minimal model corresponding to the ‘i-th’ adjoint scalar. Start by decomposing the quasi-isomorphism maps in terms of plane waves with coefficients $J(1\cdots n)$, called the Berends–Giele currents:

$${\psi }_n\left(\varphi \left(1\right),\dots ,\varphi \left(n\right)\right)=J(1\cdots n){\mathrm{e}}^{\mathrm{i}{p}_{1\cdots n}·x}\in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g})[-1],$$

(99)

where $p_{1\cdots n}:=p_1+\cdots +p_n$. These currents are sometimes written as ${\varphi }_{1\cdots n}$; they are simply the Fourier coefficients of the quasi-isomorphism applied to n different external states.

The recursion relations for the currents are extracted from the quasi-isomorphisms ${\psi }_n$, and one finds for $n>1$

$$\begin{array}{cc}\hfill J(1\cdots n)& =\frac{g}{2}\sum _{i=1}^{n-1}\sum _{\sigma \in \mathrm{Sh}(i;n)}{G}_{\mathrm{F}}\circ 〚J(\sigma \left(1\right),\dots ,\sigma \left(i\right)),J(\sigma (i+1),\dots ,\sigma \left(n\right)){〛}_{\mathrm{AS}},\hfill \end{array}$$

(100)

where we understand $G_{\mathrm{F}}$ as acting on the plane wave basis representing the double bracket. Writing $\varphi \left(i\right)={\varphi }_i{\mathrm{e}}^{\mathrm{i}{p}_i·x}$ for ${\varphi }_i\in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{g})$, the first few currents are given by

$$\begin{array}{cc}\hfill J\left(1\right)& ={\varphi }_1,\hfill \\ \hfill J\left(12\right)& =-\mathrm{i}g\frac{〚{\varphi }_1,{\varphi }_2{〛}_{\mathrm{AS}}}{s_{12}},\hfill \\ \hfill J\left(123\right)& =-{\left(\mathrm{i}g\right)}^2\left(\frac{〚{\varphi }_1,〚{\varphi }_2,{\varphi }_3{〛}_{\mathrm{AS}}{〛}_{\mathrm{AS}}}{s_{23}{s}_{123}}+\frac{〚{\varphi }_2,〚{\varphi }_3,{\varphi }_1{〛}_{\mathrm{AS}}{〛}_{\mathrm{AS}}}{s_{31}{s}_{123}}+\frac{〚{\varphi }_3,〚{\varphi }_1,{\varphi }_2{〛}_{\mathrm{AS}}{〛}_{\mathrm{AS}}}{s_{12}{s}_{123}}\right),\hfill \end{array}$$

(101)

and so on, where $s_{ij\cdots }={(p_i+p_j+\cdots )}^2$ are Mandelstam invariants. This verifies that the currents are cyclic symmetric.

The brackets on the minimal model, for the projection $\mathsf{p}:L^1⟶{\mathsf{H}}^1\left(\mathfrak{AS}\right)$, are given by Equation (A40):

$$\begin{array}{c}\hfill \begin{array}{c}{\mu }_n^{\circ \mathrm{AS}}\left(\varphi \left(1\right),\dots ,\varphi \left(n\right)\right)\hfill \\ =-\frac{g}{2}\sum _{i=1}^{n-1}\sum _{\sigma \in \mathrm{Sh}(i;n)}\mathsf{p}\circ 〚J(\sigma \left(1\right),\dots ,\sigma \left(i\right)),J(\sigma (i+1),\dots ,\sigma \left(n\right)){〛}_{\mathrm{AS}}{\mathrm{e}}^{\mathrm{i}{p}_{1\cdots n}·x}.\hfill \end{array}\end{array}$$

(102)

For the first few brackets, this gives

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_2^{\circ \mathrm{AS}}({\varphi }_1,{\varphi }_2)& =-g〚{\varphi }_1,{\varphi }_2{〛}_{\mathrm{AS}}{\mathrm{e}}^{\mathrm{i}{p}_{12}·x},\hfill \\ \hfill {\mu }_3^{\circ \mathrm{AS}}({\varphi }_1,{\varphi }_2,{\varphi }_3)& =\mathrm{i}{g}^2(\frac{〚{\varphi }_1,〚{\varphi }_2,{\varphi }_3{〛}_{\mathrm{AS}}{〛}_{\mathrm{AS}}}{s_{23}}+\frac{〚{\varphi }_2,〚{\varphi }_3,{\varphi }_1{〛}_{\mathrm{AS}}{〛}_{\mathrm{AS}}}{s_{31}}\hfill \\ \hfill & +\frac{〚{\varphi }_3,〚{\varphi }_1,{\varphi }_2{〛}_{\mathrm{AS}}{〛}_{\mathrm{AS}}}{s_{12}}){\mathrm{e}}^{\mathrm{i}{p}_{123}·x},\hfill \end{array}\end{array}$$

(103)

and so on.

Ordered tree-level amplitudes are given by the cyclic structure of the minimal model, taken on distinct perturbiner elements as in Equation (A48):

$$\begin{array}{c}\hfill {𝓜}_n^{\mathrm{AS}}(1,\dots ,n)={\langle {\varphi }_1,{\mu }_{n-1}^{\circ \mathrm{AS}}({\varphi }_2,\dots ,{\varphi }_n)\rangle }_{\circ \mathrm{AS}}.\end{array}$$

(104)

The full amplitude ${\mathcal{A}}_n^{\mathrm{AS}}(p,a)$ is given by the sum over all planar ordered permutations (cf. Section 2.2).

For example, the full tree-level off-shell four-point amplitude is given by a sum of three terms

$$\begin{array}{c}\hfill {\mathcal{A}}_4^{\mathrm{AS}}(p,a)=\sum _{\sigma \in \mathrm{Sh}(1;3)}{𝓜}_4^{\mathrm{AS}}\left(1,\sigma \left(2\right),\sigma \left(3\right),\sigma \left(4\right)\right)=\frac{c_s{n}_s}{s}+\frac{c_t{n}_t}{t}+\frac{c_u{n}_u}{u},\end{array}$$

(105)

where the kinematical numerators are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill n_s& =g\int {\mathrm{d}}^{d}q\overline{F}(p_1,p_2,q)\overline{F}(q,p_3,p_4)=-4gsin\left(\frac{t}{2}{p}_1·\theta p_2\right)sin\left(\frac{t}{2}{p}_3·\theta p_4\right),\hfill \\ \hfill n_t& =g\int {\mathrm{d}}^{d}q\overline{F}(p_3,p_1,q)\overline{F}(q,p_2,p_4)=-4gsin\left(\frac{t}{2}{p}_3·\theta p_1\right)sin\left(\frac{t}{2}{p}_2·\theta p_4\right),\hfill \\ \hfill n_u& =g\int {\mathrm{d}}^{d}q\overline{F}(p_2,p_3,q)\overline{F}(q,p_1,p_4)=-4gsin\left(\frac{t}{2}{p}_2·\theta p_3\right)sin\left(\frac{t}{2}{p}_1·\theta p_4\right),\hfill \end{array}\end{array}$$

(106)

and we suppressed the overall delta-functions enforcing momentum conservation. These satisfy the off-shell kinematic Jacobi identity $n_s+n_t+n_u=0$ for any deformation parameter t, as a consequence of the Jacobi identity for the infinite-dimensional Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$. This mirrors the Jacobi identity $c_s+c_u+c_t=0$ satisfied by the colour numerators from Equation (43).

3.4.2. Colour–Kinematics Duality

Using these perturbative computations we show colour–kinematics duality directly from the $L_{\infty }$-recursion relations. We use the evaluation map $\mathrm{ev}:\mathrm{Sh}(i,n-1)⟶{\mathcal{W}}_i\otimes {\mathcal{W}}_{n-i}$ from Equation (A59), into the tensor product of ordered words, to translate the recursion relations into the language of binary trees [46].

For the adjoint scalar theory, the Berends–Giele currents (100) and minimal model brackets (102) can be written as a sum over deconcatenations of the ordered word $w=1\cdots n$ into non-empty ordered words $w_1$ and $w_2$:

$$\begin{array}{cc}\hfill J\left(w\right)& =\frac{g}{2}\sum _{w=w_1\bigsqcup w_2}{G}_{\mathrm{F}}\circ 〚J\left(w_1\right),J\left(w_2\right){〛}_{\mathrm{AS}},\hfill \\ \hfill {\mu }_n^{\circ \mathrm{AS}}\left(\varphi \left(1\right),\dots ,\varphi \left(n\right)\right)& =-\frac{g}{2}\sum _{w=w_1\bigsqcup w_2}\mathsf{p}\circ 〚J\left(w_1\right),J\left(w_2\right){〛}_{\mathrm{AS}}{\mathrm{e}}^{\mathrm{i}{p}_{1\cdots n}·x}.\hfill \end{array}$$

(107)

We have observed in Equation (88) that the bracket $〚-,-{〛}_{\mathrm{AS}}$ of the adjoint scalar theory has a tensor product factorisation into the Lie bracket on the colour Lie algebra $\mathfrak{g}$ and the Lie bracket on the kinematic Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$, as illustrated by the coefficient of the three-point vertex (92).

Given a word $w=k_1\cdots k_n\in {\mathcal{W}}_n$ labelling n external particles, we denote its length by $\left|w\right|=n$ and factorise the symmetric bracket into the tensor product of maps,

$$\begin{array}{c}\hfill C:{\mathcal{W}}_n\stackrel{{\ell }_{\mathrm{c}}}{\to }({\mathcal{L}}_n,{[-,-]}_{\mathfrak{g}})\stackrel{\mathrm{col}}{\to }\mathfrak{g},K:{\mathcal{W}}_n\stackrel{{\ell }_{\mathrm{k}}}{\to }({\mathcal{L}}_n,{[-\stackrel{★}{,}-]}_{\mathfrak{u}\left(1\right)})\stackrel{\mathrm{kin}}{\to }{\mathfrak{u}}_{★}\left(1\right),\end{array}$$

(108)

as $〚-,-{〛}_{\mathrm{AS}}\left(w\right)=\left(\mathrm{col}\circ {\ell }_{\mathrm{c}}\left(w\right)\right)\otimes \left(\mathrm{kin}\circ {\ell }_{\mathrm{k}}\left(w\right)\right)$. We have made explicit the left bracketing maps ${\ell }_{\mathrm{c}}$ and ${\ell }_{\mathrm{k}}$ from Equation (A52) for colour and kinematical numerators into the multilinear Lie polynomials with colour and kinematical Lie brackets, respectively. In components we find

$$\begin{array}{cc}\hfill C\left(w\right)& ={[\cdots {[{[T^{a_{k_1}},T^{a_{k_2}}]}_{\mathfrak{g}},T^{a_{k_3}}]}_{\mathfrak{g}},\dots ,T^{a_{k_n}}]}_{\mathfrak{g}},\hfill \\ \hfill K\left(w\right)& ={[\cdots {\left[{\left[{\varphi }_{k_1}^{a_{k_1}}\stackrel{★}{,}{\varphi }_{k_2}^{a_{k_2}}\right]}_{\mathfrak{u}\left(1\right)}\stackrel{★}{,}{\varphi }_{k_3}^{a_{k_3}}\right]}_{\mathfrak{u}\left(1\right)}\stackrel{★}{,}\dots \stackrel{★}{,}{\varphi }_{k_n}^{a_{k_n}}]}_{\mathfrak{u}\left(1\right)}.\hfill \end{array}$$

(109)

Finally, one arrives at

$$J\left(w\right)=-\frac{1}{2}\sum _{w=w_1\bigsqcup w_2}\frac{{\left(\mathrm{i}g\right)}^{\left|w\right|-1}}{s_w}C\left(w\right)\otimes K\left(w\right),$$

(110)

with recursion relations for the colour and kinematic numerators given by

$$\begin{array}{cc}\hfill C\left(w\right)& =\mathrm{col}\circ \sum _{w=w_1\bigsqcup w_2}{[{\ell }_{\mathrm{c}}\left(w_1\right),{\ell }_{\mathrm{c}}\left(w_2\right)]}_{\mathfrak{g}},\hfill \\ \hfill K\left(w\right)& =\mathrm{kin}\circ \sum _{w=w_1\bigsqcup w_2}{\left[{\ell }_{\mathrm{k}}\left(w_1\right)\stackrel{★}{,}{\ell }_{\mathrm{k}}\left(w_2\right)\right]}_{\mathfrak{u}\left(1\right)}.\hfill \end{array}$$

(111)

One can replace the recursion (110) in terms of the binary tree map from Equation (A58) by defining $[C\otimes K]\circ w:=C\left(w\right)\otimes K\left(w\right)$ for any word w, which yields

$$\begin{array}{c}\hfill J\left(w\right)=-\frac{1}{2}{\left(\mathrm{i}g\right)}^{\left|w\right|-1}[C\otimes K]\circ b\left(w\right).\end{array}$$

(112)

One obtains the tree-level amplitude with partial ordering $1w\in {\mathcal{W}}_n$ by simply cancelling the overall pole in the current $J\left(w\right)$. Indeed, from Equation (104), cyclicity of the inner product on the minimal model implies that there is only one such component, namely

$${𝓜}_n^{\mathrm{AS}}(1,w)=-\mathrm{i}{s}_{1w}J\left(w\right){\left(2\pi \right)}^d\delta \left(p_{1w}\right),$$

(113)

where the overall momentum conservation comes from integrating over spacetime. Notice that the Moyal–Weyl product is trivialised by momentum conservation.

The full n-point amplitude, obtained by summing over all ordered words, can then be written as

$$\begin{array}{c}\hfill {\mathcal{A}}_n^{\mathrm{AS}}(p,a)=\sum _{\Gamma \in {\mathcal{T}}_{3,n}}\frac{c_{\Gamma }{n}_{\Gamma }}{D_{\Gamma }},\end{array}$$

(114)

generalising Equation (105). Here ${\mathcal{T}}_{3,n}$ denotes the set of all trivalent trees with n external edges; it has cardinality $(2n-5)!!$. Associated with any tree $\Gamma \in {\mathcal{T}}_{3,n}$ there is a denominator $D_{\Gamma }={\prod }_{e\in \Gamma }{s}_e$ with propagators $s_e$ assigned to each internal edge e of $\Gamma$. The colour factors $c_{\Gamma }$ are contractions of the structure constants of the colour Lie algebra $\mathfrak{g}$ associated with $\Gamma$, while $n_{\Gamma }$ are the kinematic parts of the numerators involving analogous contractions of the structure constants of the kinematic Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$.

We conclude that, since the colour algebra based on $\mathfrak{g}$ naturally obeys the generalised Jacobi identities, or equivalently the STU-relations on any Jacobi subgraph, the kinematic numerators also obey them. This is an instance of colour–kinematics duality: the infinite-dimensional kinematic Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$ which determines the kinematic numerators is dual to the colour Lie algebra $\mathfrak{g}$. Moreover, from the perspective of Equation (112), the effect of the zeroth-copy construction is to replace the kinematic numerators $K\left(w\right)$ with a second set of colour numerators $\overline{C}\left(w\right)$.

The colour–kinematics duality here is implied by the factorisation of the three-point vertex (92) into $f^{abc}\overline{F}(k,p,q)$ and holds off-shell, as is evident from the homotopy algebraic perspective. At loop level, all integrands are computed using Equation (92), and hence all loop-level kinematic Jacobi identities are automatically satisfied, even off-shell. Below we shall construct the action functional of the double copied theory using Maurer–Cartan theory.

3.5. Homotopy Double Copy

Our discussion of colour–kinematics duality in Section 3.4 justifies the use of the double copy prescription. The factorisation of the Berends–Giele currents in Equation (112), with respect to the (untwisted) tensor product $\mathfrak{g}\otimes {\mathfrak{u}}_{★}\left(1\right)$ of Lie algebras, is a manifestation of the factorisation (96) of the strict $L_{\infty }$-algebra $\mathfrak{AS}$. This permits us to exploit the powerful techniques of $L_{\infty }$-algebras: the homotopy double copy construction [12] replaces the colour factor $\mathfrak{g}$ in the factorisation (96) of $\mathfrak{AS}$ with another copy of the “twisted” kinematic factor ${\mathfrak{Kin}}_{{\wedge }^0}=\mathbb{R}$, producing the cyclic strict $L_{\infty }$-algebra

$$\begin{array}{c}\hfill \widehat{\mathfrak{AS}}={\mathfrak{Kin}}_{{\wedge }^0}{\otimes }_{{\tau }^{{\mathfrak{u}}_{\overline{★}}\left(1\right)}}({\mathfrak{Kin}}_{{\wedge }^0}{\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(1\right)}}\mathfrak{Scal})\end{array}$$

(115)

as a twisted tensor product between the graded vector space ${\mathfrak{Kin}}_{{\wedge }^0}$ (concentrated in degree 0) and the $C_{\infty }$-algebra ${\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(1\right)}={\mathfrak{Kin}}_{{\wedge }^0}{\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(1\right)}}\mathfrak{Scal}$.

The underlying graded vector space is identified as $\mathbb{R}\otimes (\mathbb{R}\otimes L)\simeq L$, and the twisted differential is ${\widehat{\mu }}_1^{\mathrm{AS}}=m_1^{{\tau }^{{\mathfrak{u}}_{★}\left(1\right)}}=\square$, so that again the cochain complex $\mathsf{Ch}(\widehat{\mathfrak{AS}}$) is the cochain complex ${\mathsf{Ch}}_{\mathrm{Scal}}$ from Equation (20). The bracket of the double copy is given by the doubly twisted bracket

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{\mu }}_2^{\mathrm{AS}}({\varphi }_1,{\varphi }_2)& ={\mu }_2^{\mathrm{Scal}}\circ 2\mathrm{i}sin\left(\frac{\overline{t}}{2}{\overline{\theta }}^{\mu \nu }{\partial }_{\mu }\otimes {\partial }_{\nu }\right)\circ 2\mathrm{i}sin\left(\frac{t}{2}{\theta }^{\lambda \rho }{\partial }_{\lambda }\otimes {\partial }_{\rho }\right)({\varphi }_1\otimes {\varphi }_2),\hfill \end{array}\end{array}$$

(116)

where we use two independent expansion parameters $t,\overline{t}\in \mathbb{R}$ to keep track of the double Moyal–Weyl deformation, determined by star-products $★,\overline{★}$ which quantise two generally independent constant Poisson bivectors $\theta ,\overline{\theta }$ on ${\mathbb{R}}^{1,d-1}$. Note that this is symmetric in ${\varphi }_1$ and ${\varphi }_2$, as required for a strict $L_{\infty }$-algebra bracket. It may be written explicitly as a formal power series

$$\begin{array}{c}\hfill \begin{array}{c}{\widehat{\mu }}_2^{\mathrm{AS}}({\varphi }_1,{\varphi }_2)\hfill \\ =\mathrm{i}\kappa \sum _{n=0}^{\infty }\frac{{(-1)}^n{\overline{t}}^{2n+1}}{(2n+1)!}{\overline{\theta }}^{{\mu }_1{\nu }_1}\cdots {\overline{\theta }}^{{\mu }_{2n+1}{\nu }_{2n+1}}{[{\partial }_{{\mu }_1}\cdots {\partial }_{{\mu }_{2n+1}}{\varphi }_1\stackrel{★}{,}{\partial }_{{\nu }_1}\cdots {\partial }_{{\nu }_{2n+1}}{\varphi }_2]}_{\mathfrak{u}\left(1\right)},\hfill \end{array}\end{array}$$

(117)

where we denote the coupling as $g=\frac{\kappa }{2}$ in the double copied theory, as it acquires a different engineering dimension.

The Maurer–Cartan equation associated with $\widehat{\mathfrak{AS}}$ yields the field equation which governs the dynamics of the double copy as

$$\begin{array}{c}\square \varphi +\frac{\mathrm{i}\kappa }{2}\sum _{n=0}^{\infty }\frac{{(-1)}^n{\overline{t}}^{2n+1}}{(2n+1)!}{\overline{\theta }}^{{\mu }_1{\nu }_1}\cdots {\overline{\theta }}^{{\mu }_{2n+1}{\nu }_{2n+1}}{[{\partial }_{{\mu }_1}\cdots {\partial }_{{\mu }_{2n+1}}\varphi \stackrel{★}{,}{\partial }_{{\nu }_1}\cdots {\partial }_{{\nu }_{2n+1}}\varphi ]}_{\mathfrak{u}\left(1\right)}=0.\end{array}$$

(118)

It can be derived as the stationary locus of the Maurer–Cartan functional, which using the cyclic inner product of $\mathfrak{Scal}$ from Equation (23) reads

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{S}}_{\mathrm{AS}}\left[\varphi \right]=\int {\mathrm{d}}^{d}x& \frac{1}{2}\varphi ★\square \varphi -\frac{\mathrm{i}\kappa }{6}\sum _{n=0}^{\infty }\frac{{(-1)}^n{\overline{t}}^{2n+1}}{(2n+1)!}{\overline{\theta }}^{{\mu }_1{\nu }_1}\cdots {\overline{\theta }}^{{\mu }_{2n+1}{\nu }_{2n+1}}\hfill \\ \hfill & \times \varphi {[{\partial }_{{\mu }_1}\cdots {\partial }_{{\mu }_{2n+1}}\varphi \stackrel{★}{,}{\partial }_{{\nu }_1}\cdots {\partial }_{{\nu }_{2n+1}}\varphi ]}_{\mathfrak{u}\left(1\right)}.\hfill \end{array}\end{array}$$

(119)

The three-point Feynman vertex following from (119) is

(120)

where the kinematic structure constants are given by Equation (55).

Remark 3.

This scalar theory is a double noncommutative deformation of the cubic scalar field theory of Section 2.1. It can be obtained from the adjoint scalar theory by performing substitutions analogous to Equation (93) for the remaining colour factor, or equivalently by the substitutions

$$\begin{array}{c}\hfill {\varphi }^{a\overline{a}}⟶\varphi ,f^{abc}{\overline{f}}^{\overline{a}\overline{b}\overline{c}}⟶F(k,p,q)\overline{F}(k,p,q)and{\mathrm{Tr}}_{\mathfrak{g}\otimes \overline{\mathfrak{g}}}⟶\int \end{array}$$

(121)

directly in the Lagrangian of the commutative biadjoint scalar theory of Section 2.2. Hence the double copy of the adjoint scalar theory can in this sense be regarded as a biadjoint scalar theory based on the twisted tensor product $\mathfrak{u}\left(1\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(1\right)}}{\mathfrak{u}}_{\overline{★}}\left(1\right)$ of Lie algebras.

Perturbative Calculations

To obtain the Berends–Giele currents and n-point amplitudes of the double copy, one simply needs to perform calculations analogous to those presented for the adjoint scalar in Section 3.4. The double copied homotopy algebra is a strict $L_{\infty }$-algebra with bracket ${\widehat{\mu }}_2^{\mathrm{AS}}$ and contracting homotopy $\widehat{\mathsf{h}}={\widehat{G}}_{\mathrm{F}}$ satisfying

$$({\widehat{\mu }}_1^{\mathrm{AS}}\circ {\widehat{G}}_{\mathrm{F}})(x,y)=\mathrm{i}\delta (x-y)\mathrm{with}{\widehat{G}}_{\mathrm{F}}\left({\mathrm{e}}^{\mathrm{i}p·x}\right)=-\frac{\mathrm{i}}{p^2}{\mathrm{e}}^{\mathrm{i}p·x}.$$

(122)

This is just the scalar contracting homotopy acting on the double copied fields, which are scalars.

For example, the tree-level off-shell four-point amplitude is found in this way to be

$$\begin{array}{c}\hfill {\widehat{\mathcal{A}}}_4^{\mathrm{AS}}\left(p\right)=\sum _{\sigma \in \mathrm{Sh}(1;3)}{\widehat{𝓜}}_4^{\mathrm{AS}}\left(1,\sigma \left(2\right),\sigma \left(3\right),\sigma \left(4\right)\right)=\frac{n_s{\overline{n}}_s}{s}+\frac{n_t{\overline{n}}_t}{t}+\frac{n_u{\overline{n}}_u}{u},\end{array}$$

(123)

as expected from the replacement rule (121). More generally, the n-point amplitudes follow by replacing the colour factors $C\left(w\right)$ with a second set of kinematic numerators $\overline{K}\left(w\right)$ in Equation (112) and can be written in the form

$$\begin{array}{c}\hfill {\widehat{\mathcal{A}}}_n^{\mathrm{AS}}\left(p\right)=\sum _{\Gamma \in {\mathcal{T}}_{3,n}}\frac{n_{\Gamma }{\overline{n}}_{\Gamma }}{D_{\Gamma }}.\end{array}$$

(124)

3.6. Binoncommutative Biadjoint Scalar Theory

The homotopy double copy construction of Section 3.5 inspires a more general class of noncommutative deformations of the biadjoint scalar theory from Section 2.2. They are obtained from the noncommutative biadjoint scalar theory of Section 3.2 by replacing the tensor product $\mathfrak{g}\otimes {\mathfrak{u}}_{\overline{★}}\left(\overline{N}\right)$ with the twisted tensor product $\mathfrak{u}\left(N\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(N\right)}}{\mathfrak{u}}_{\overline{★}}\left(\overline{N}\right)$. The double copy theory of Section 3.5 is then recovered in the rank-one limits $N=\overline{N}=1$. This ‘binoncommutative’ biadjoint scalar theory will turn out to be the zeroth-copy theory for noncommutative gauge theories. In particular, it will play an important role when we study the KLT relations in association with noncommutative Yang–Mills theory in Section 5.

The binoncommutative biadjoint scalar theory is organised by a cyclic strict $L_{\infty }$-algebra ${\mathfrak{BAS}}_{★\overline{★}}$ which is given by the twisted tensor product construction of Section 3.2.1, now applied to both Lie algebra factors:

$${\mathfrak{BAS}}_{★\overline{★}}:=\mathfrak{u}\left(N\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(N\right)}}\left(\mathfrak{u}\left(\overline{N}\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{\overline{★}}\left(\overline{N}\right)}}\mathfrak{Scal}\right).$$

(125)

The underlying cochain complex $\mathsf{Ch}\left({\mathfrak{BAS}}_{★\overline{★}}\right)$ is given by Equation (28) with $\mathfrak{g}=\mathfrak{u}\left(N\right)$ and $\overline{\mathfrak{g}}=\mathfrak{u}\left(\overline{N}\right)$, which as usual identifies the differential as ${\mu }_1^{★\overline{★}\mathrm{BAS}}={\mu }_1^{\mathrm{BAS}}=\square$. The higher bracket ${\mu }_2^{★\overline{★}\mathrm{BAS}}$ is the binoncommutative product

$$\begin{array}{cc}\hfill {\mu }_2^{★\overline{★}\mathrm{BAS}}({\varphi }_1,{\varphi }_2)& =T^c\otimes {\overline{T}}^{\overline{c}}\otimes {\mu }_2^{\mathrm{Scal}}\circ \left(f^{abc}cos\left(\frac{t}{2}{\theta }^{\mu \nu }{\partial }_{\mu }\otimes {\partial }_{\nu }\right)-d^{abc}sin\left(\frac{t}{2}{\theta }^{\mu \nu }{\partial }_{\mu }\otimes {\partial }_{\nu }\right)\right)\hfill \\ \hfill & \circ \left({\overline{f}}^{\overline{a}\overline{b}\overline{c}}cos\left(\frac{\overline{t}}{2}{\overline{\theta }}^{\lambda \rho }{\partial }_{\lambda }\otimes {\partial }_{\rho }\right)-{\overline{d}}^{\overline{a}\overline{b}\overline{c}}sin\left(\frac{\overline{t}}{2}{\overline{\theta }}^{\lambda \rho }{\partial }_{\lambda }\otimes {\partial }_{\rho }\right)\right)\left({\varphi }_1^{a\overline{a}}\otimes {\varphi }_2^{b\overline{b}}\right)\hfill \end{array}$$

(126)

of elements ${\varphi }_1=T^a\otimes {\overline{T}}^{\overline{a}}\otimes {\varphi }_1^{a\overline{a}}$ and ${\varphi }_2=T^b\otimes {\overline{T}}^{\overline{b}}\otimes {\varphi }_2^{b\overline{b}}$ in ${\Omega }^0\left({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right)\otimes \mathfrak{u}\left(\overline{N}\right)\right)[-1]$. The cyclic inner product is again identified as the pairing (32) of fields $\varphi \in {\Omega }^0\left({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right)\otimes \mathfrak{u}\left(\overline{N}\right)\right)[-1]$ with antifields ${\varphi }^{+}\in {\Omega }^0\left({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right)\otimes \mathfrak{u}\left(\overline{N}\right)\right)[-2]$.

One can expand the bracket (126) analogously to what we did in Section 3.5, and formally develop the Maurer–Cartan theory underlying the cyclic $L_{\infty }$-algebra ${\mathfrak{BAS}}_{★\overline{★}}$. As with the noncommutative biadjoint scalar theory of Section 3.2, this theory has a rigid symmetry under the adjoint action of $\mathfrak{u}\left(N\right)\oplus \mathfrak{u}\left(\overline{N}\right)$ and can be extended to a four-parameter family of binoncommutative biadjoint scalar theories with structure constants ${\lambda }_{u,v}^{abc}$ and ${\overline{\lambda }}_{\overline{u},\overline{v}}^{\overline{a}\overline{b}\overline{c}}$ (cf. Remark 1). The resulting formulas are even more complicated than those of Section 3.5, and they will not be needed explicitly in this paper.

In momentum space, this theory has a remarkably much simpler representation, giving the three-point vertex in its Feynman diagram expansion in terms of two copies of the structure constants from Equation (65) as

(127)

Note that the momentum-dependent structure constants $F^{abc}$ and ${\overline{F}}^{\overline{a}\overline{b}\overline{c}}$ obey the Jacobi identities of the infinite-dimensional Lie algebras ${\mathfrak{u}}_{★}\left(N\right)$ and ${\mathfrak{u}}_{\overline{★}}\left(\overline{N}\right)$, respectively, which from this perspective can therefore also be regarded as kinematic Lie algebras. Looking at the twisted homotopy factorisation (86), this means that the binoncommutative biadjoint scalar theory can be regarded as a double copy between the noncommutative biadjoint scalar theory and itself, with the replacement of colour by kinematics involving twisted tensor products.

Colour Ordering and Decomposition

The tree-level amplitudes of the binoncommutative biadjoint scalar theory admit a colour decomposition on both copies of the colour algebra. By writing the structure constants of $\mathfrak{u}\left(N\right)$ as $f^{abc}=2{\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left({[T^a,T^b]}_{\mathfrak{u}\left(N\right)}{T}^c\right)$ and $\mathrm{i}{d}^{abc}=2{\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left({\{T^a,T^b\}}_{\mathfrak{u}\left(N\right)}{T}^c\right)$, the structure constants of ${\mathfrak{u}}_{★}\left(N\right)$ given in Equation (65) can be written as

$$\begin{array}{cc}\hfill F^{abc}(p_1,p_2,p_3)& =2\left({\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(T^a{T}^b{T}^c\right){\mathrm{e}}^{-\frac{\mathrm{i}t}{2}{p}_1·\theta p_2}-{\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(T^b{T}^a{T}^c\right){\mathrm{e}}^{-\frac{\mathrm{i}t}{2}{p}_2·\theta p_1}\right)\hfill \\ \hfill & \times \delta (p_1+p_2+p_3).\hfill \end{array}$$

(128)

This is an explicit factorisation for a given colour ordering, a common feature of noncommutative field theories with Moyal–Weyl deformation. In the commutative limit $t\to 0$, this returns $f^{abc}$ as expected.

In the binoncommutative biadjoint scalar theory, the same factorisation appears as a decomposition in terms of two sets of orderings of colour indices as

$$\begin{array}{c}{F}^{abc}(p_1,p_2,p_3){\overline{F}}^{\overline{a}\overline{b}\overline{c}}(p_1,p_2,p_3)\hfill \\ =4\left({\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(T^a{T}^b{T}^c\right){\mathrm{e}}^{-\frac{\mathrm{i}t}{2}{p}_1·\theta p_2}-{\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(T^b{T}^a{T}^c\right){\mathrm{e}}^{-\frac{\mathrm{i}t}{2}{p}_2·\theta p_1}\right)\hfill \\ \times \left({\mathrm{Tr}}_{\mathfrak{u}\left(\overline{N}\right)}\left({\overline{T}}^{\overline{a}}{\overline{T}}^{\overline{b}}{\overline{T}}^{\overline{c}}\right){\mathrm{e}}^{-\frac{\mathrm{i}\overline{t}}{2}{p}_1·\overline{\theta }{p}_2}-{\mathrm{Tr}}_{\mathfrak{u}\left(\overline{N}\right)}\left({\overline{T}}^{\overline{b}}{\overline{T}}^{\overline{a}}{\overline{T}}^{\overline{c}}\right){\mathrm{e}}^{-\frac{\mathrm{i}\overline{t}}{2}{p}_2·\overline{\theta }{p}_1}\right)\hfill \\ \times \delta (p_1+p_2+p_3).\hfill \end{array}$$

(129)

The commutative limit $t,\overline{t}\to 0$ similarly reduces to $f^{abc}{\overline{f}}^{\overline{a}\overline{b}\overline{c}}$. The rank-one theory studied in Section 3.5 also follows from this formula, which for $N=\overline{N}=1$ recovers only the sinus part of Equation (129).

The binoncommutative biadjoint scalar theory studied here differs from the commutative biadjoint scalar theory of Section 2.2 only in the three-point vertex. Crucially, this vertex admits a colour factorisation with only phase factors that depend solely on the orderings of momenta. We conclude that the n-point partial amplitudes of the binoncommutative biadjoint scalar theory, defined as the summands in the expansion of the full amplitudes

$${\mathcal{A}}_n^{★\overline{★}\mathrm{BAS}}(p,a,\overline{a})=\sum _{\sigma ,{\sigma }^{\prime }\in S_n/{\mathbb{Z}}_n⋊{\mathbb{Z}}_2}C\left(\sigma \right)\overline{C}\left({\sigma }^{\prime }\right){𝓐}_n^{★\overline{★}\mathrm{BAS}}\left(\sigma \left(1\right),\dots ,\sigma \left(n\right)|{\sigma }^{\prime }\left(1\right),\dots ,{\sigma }^{\prime }\left(n\right)\right),$$

(130)

factor through these phases.

For example, the first term in the three-point partial amplitude given by the diagram (127) and the decomposition (129) is

$$\begin{array}{cc}\hfill {𝓐}_3^{★\overline{★}\mathrm{BAS}}(1,2,3|1,2,3)& =-4\mathrm{i}g{\mathrm{e}}^{-\frac{\mathrm{i}t}{2}{p}_1·\theta p_2-\frac{\mathrm{i}\overline{t}}{2}{p}_1·\overline{\theta }{p}_2}\delta (p_1+p_2+p_3)\hfill \\ \hfill & ={\mathrm{e}}^{-\frac{\mathrm{i}t}{2}{p}_1·\theta p_2-\frac{\mathrm{i}\overline{t}}{2}{p}_1·\overline{\theta }{p}_2}{𝓐}_3^{\mathrm{BAS}}(1,2,3|1,2,3),\hfill \end{array}$$

(131)

where ${𝓐}_n^{\mathrm{BAS}}$ is the commutative n-point biadjoint scalar partial amplitude, defined in Section 2.2.

We interpret this factorisation as a slight generalisation of the well-known fact that noncommutative scalar field theory differs from its commutative counterpart only by a phase in planar graphs [17,47]. The upshot is that one can dress commutative colour-stripped biadjoint scalar amplitudes by the corresponding phase factors to obtain tree-level binoncommutative biadjoint scalar amplitudes. This fact will be of use in Section 5.

3.7. Applications: Special Galileons and Self–Dual Gravity

As some concrete applications of the formalism we have developed thus far, as well as a glimpse towards some of our later double copy constructions, let us now look at two special “topological” realisations of the generic double copy map from the adjoint scalar theory of Section 3.3 to the theory of Section 3.5. Here we work mostly at the level of equations of motion for brevity, but analogous statements also hold off-shell at the level of Maurer–Cartan functionals.

3.7.1. Zakharov–Mikhailov Theory and Special Galileons

In $d=2$ dimensions with the Moyal–Weyl star-product (49), the adjoint scalar theory was considered in [28] as a ‘stringy deformation’ of Zakharov–Mikhailov theory [48], which is classically equivalent to the principal chiral model. We set $t=\overline{t}={\alpha }^{\prime }$, regarded as a string tension parametrizing an infinite tower of higher dimension operators, and take ${\theta }^{\mu \nu }={\overline{\theta }}^{\mu \nu }={ϵ}^{\mu \nu }$ to be the Levi–Civita symbol in two dimensions with ${ϵ}^{01}=1$. Let $g=\frac{\overline{g}}{{\alpha }^{\prime }}$, and take the semi-classical limit ${\alpha }^{\prime }\to 0$ with $\overline{g}$ finite.

Recall from Section 3.1 that in the semi-classical limit the star-commutator $\mathrm{i}g{\left[{\varphi }_1\stackrel{★}{,}{\varphi }_2\right]}_{\mathfrak{u}\left(1\right)}$ is replaced by the Poisson bracket $\overline{g}\{{\varphi }_1,{\varphi }_2\}$, where

$$\begin{array}{c}\hfill \{{\varphi }_1,{\varphi }_2\}={ϵ}^{\mu \nu }{\partial }_{\mu }{\varphi }_1{\partial }_{\nu }{\varphi }_2.\end{array}$$

(132)

The kinematic algebra ${\mathfrak{u}}_{★}\left(1\right)$ reduces to the Poisson–Lie algebra $\mathfrak{sdiff}\left({\mathbb{R}}^{1,1}\right)$ of area-preserving diffeomorphisms of ${\mathbb{R}}^{1,1}$, and the kinematical numerators $n_s$ become $-4g(p_1\times p_2)(p_3\times p_4)$, and so on, where $p_i\times p_j:={ϵ}^{\mu \nu }{p}_i{}_{\mu }{p}_j{}_{\nu }$.

The equations of motion (90) of the adjoint scalar theory reduce to

$$\begin{array}{c}\hfill \square \varphi -\frac{\overline{g}}{2}{ϵ}^{\mu \nu }{[{\partial }_{\mu }\varphi ,{\partial }_{\nu }\varphi ]}_{\mathfrak{g}}=0.\end{array}$$

(133)

These are just the equations of motion for the non-linear sigma-model in two dimensions, with fields $h\in {\Omega }^0({\mathbb{R}}^{1,1},\mathsf{G})$, which read as ${\partial }^{\mu }{j}_{\mu }=0$ for the left-invariant currents $j=h^{-1}\mathrm{d}h\in {\Omega }^1({\mathbb{R}}^{1,1},\mathfrak{g})$. These imply that there exists an adjoint scalar $\varphi \in {\Omega }^0({\mathbb{R}}^{1,1},\mathfrak{g})$ such that $j_{\mu }=\overline{g}{ϵ}_{\mu \nu }{\partial }^{\nu }\varphi$, and the Maurer–Cartan equation $\mathrm{d}j+\frac{1}{2}{[j,j]}_{\mathfrak{g}}=0$ coincides with the equations of motion (133). Therefore, in this limit, the adjoint scalar field theory for $d=2$ reduces to the usual Zakharov–Mikhailov theory [48].

On the other hand, the equation of motion (118) of the homotopy double copy theory in the limit ${\alpha }^{\prime }\to 0$ becomes

$$\begin{array}{c}\hfill \square \varphi -\frac{\overline{\kappa }}{4}{ϵ}^{\mu \nu }\{{\partial }_{\mu }\varphi ,{\partial }_{\nu }\varphi \}=0,\end{array}$$

(134)

which is the equation of motion for the special Galileon theory in two dimensions [49,50,51]. The special Galileon theory is invariant under the Galilean-type transformations $\varphi \left(x\right)⟼\varphi \left(x\right)+b·x+c$ of the scalar field $\varphi \in {\Omega }^0\left({\mathbb{R}}^{1,1}\right)$ for $b\in {\mathbb{R}}^{1,1}$ and $c\in \mathbb{R}$, and it is quasi-isomorphic to a two-dimensional free theory. Hence the double copy of the adjoint scalar theory can be regarded as a ‘stringy deformation’ of the special Galileon theory in two dimensions.

One application of the semi-classical double copy construction is to integrability. Since the Zakharov–Mikhailov theory is classically equivalent to the principal chiral model, it is likewise integrable. Applying the colour–kinematics duality map, the special Galileon theory is also integrable; see [28] for the explicit map of the Lax connection which furnishes an infinite tower of conserved currents. However, integrability does not seem to persist generally beyond the semi-classical limits in the full noncommutative theories, as we discuss below.

3.7.2. Self–Dual Yang–Mills Theory and Gravity

Self–dual Yang–Mills theory and gravity provide a four-dimensional realisation of our homotopy double copy construction: for $d=2$ the adjoint scalar theory was considered in [52] as a noncommutative deformation of self–dual Yang–Mills theory, while Equation (118) is the equation of motion for doubly deformed self–dual gravity considered in [52]. Below we discuss this realisation in a bit more detail. In the semi-classical limit, the explicit relationship of these $d=4$ theories and their double copy duality to the $d=2$ theories discussed above is elucidated by [29].

Let $A\in {\Omega }^1({\mathbb{R}}^{1,3},\mathfrak{g})$ be a gauge field with curvature $F=\mathrm{d}A+\frac{\overline{g}}{2}{[A,A]}_{\mathfrak{g}}$. We decompose four-dimensional Minkowski space as ${\mathbb{R}}^{1,3}\simeq {\mathbb{R}}^{1,1}\times {\mathbb{R}}^2$, with light-cone coordinates $(x^{+},x^{-})$ on ${\mathbb{R}}^{1,1}$ and complex coordinates $(z,\overline{z})$ on ${\mathbb{R}}^2\simeq \mathbb{C}$. Let ${\ast }_{\mathrm{H}}$ be the Hodge duality operator on ${\mathbb{R}}^{1,3}$, which acts on two-forms with ${\ast }_{\mathrm{H}}^2=-\mathbb{1}$. Then the component form of the self-duality equation $F=\mathrm{i}{\ast }_{\mathrm{H}}F$ can be written as

$$\begin{array}{c}\hfill F_{-z}=F_{+\overline{z}}=0\mathrm{and}{F}_{-+}=F_{z\overline{z}}.\end{array}$$

(135)

From $F_{-z}=0$ we may choose the light-cone gauge $A_{-}=A_z=0$. From $F_{-+}=F_{z\overline{z}}$ it follows that there exists an adjoint scalar $\varphi \in {\Omega }^0({\mathbb{R}}^{1,3},\mathfrak{g})$ such that $A_{+}=-{\partial }_z\varphi$ and $A_{\overline{z}}=-{\partial }_{-}\varphi$; the scalar field $\varphi$ represents the single polarisation state remaining in a gluon after projection to the self–dual sector. The remaining equation $F_{+\overline{z}}=0$ then yields

$$\begin{array}{c}\hfill \square \varphi -\overline{g}{[{\partial }_z\varphi ,{\partial }_{-}\varphi ]}_{\mathfrak{g}}=0,\end{array}$$

(136)

where $\square =-{\partial }_{+}{\partial }_{-}+{\partial }_z{\partial }_{\overline{z}}$, which is just the equation of motion (133) with $\mu ,\nu \in \{z,-\}$. In this case, the semi-classical kinematic algebra generates area-preserving diffeomorphisms of the null $(z,x^{-})$-plane, which is the known kinematic Lie algebra of self–dual Yang–Mills theory [6]. The corresponding semi-classical limit of the Maurer–Cartan functional (91) for the adjoint scalar theory is the cubic action functional for self–dual Yang–Mills theory in the Leznov gauge [53,54,55].

The analogue in asymptotically flat gravity is the self-duality equation for the Riemann curvature tensor, which encodes both the equations of motion and the algebraic Bianchi identity. With the light-cone gauge choice for the metric, the semi-classical double copy equation of motion (134) coincides with Plebański’s second heavenly equation for self–dual gravity [56]:

$$\begin{array}{c}\hfill \square \varphi -\frac{\overline{\kappa }}{2}\{{\partial }_z\varphi ,{\partial }_{-}\varphi \}=0,\end{array}$$

(137)

where here the scalar field $\varphi \in {\Omega }^0\left({\mathbb{R}}^{1,3}\right)$ represents the positive helicity state of the graviton, and the constant $\overline{\kappa }$ controls the deformation away from flat space. This involves two copies of the area-preserving diffeomorphism algebra. The corresponding semi-classical limit of the Maurer–Cartan functional (119) is the action functional for the string field theory of $𝓝=2$ strings [57].

Like the Zakharov–Mikhailov and special Galileon theories, self–dual Yang–Mills theory and gravity are integrable theories, each admitting an infinite tower of conserved charges: On the gauge theory side there is a Lax pair $(𝓛,𝓜)$, which with our gauge choice is given by

$$\begin{array}{c}\hfill 𝓛={\partial }_{+}-\lambda ({\partial }_{\overline{z}}-{\partial }_{+}\varphi )\mathrm{and}𝓜={\partial }_z-\lambda ({\partial }_{-}-{\partial }_z\varphi )\end{array}$$

(138)

for a spectral parameter $\lambda \in \mathbb{C}{\mathbb{P}}^1$, and analogously on the gravity side where integrability is linked to its infinite-dimensional $w_{1+\infty }$ symmetry. This is consistent with Ward’s conjecture: all integrable theories are related to self–dual Yang–Mills theory by replacing its structure constants $f^{abc}$ with other structure constants, which can be thought of as a “symmetry reduction”. Here we observe the natural sequence of symmetry reductions similarly to Equation (9), starting from the biadjoint scalar theory of Section 2.2, to self–dual Yang–Mills theory, and finally to self–dual gravity.

Let us now look at the generic case of finite deformation parameters $t\ne \overline{t}$, with the understanding as above that spacetime indices are always restricted to directions along the null $(z,x^{-})$-plane. Then the Maurer–Cartan Equation (118) for the homotopy double copy theory coincides with the “doubly deformed” Plebański equation for self–dual gravity considered in [52]. This theory of noncommutative gravity is not integrable, due to the breakdown of the Jacobi identities. In the semi-classical limit $\overline{t}\to 0$ it reduces to the deformed Plebański equation of noncommutative gravity [58,59,60]

$$\begin{array}{c}\hfill \square \varphi -\frac{\mathrm{i}\overline{\kappa }}{2}{\left[{\partial }_z\varphi \stackrel{★}{,}{\partial }_{-}\varphi \right]}_{\mathfrak{u}\left(1\right)}=0.\end{array}$$

(139)

This is now integrable with an infinite tower of conserved currents; integrability here relies crucially on the semi-classical form of the kinematic structure constants ${\overline{F}}_0(p_1,p_2,p_3)$ of the symplecton $W_{1+\infty }$ algebra [52]. The Moyal–Weyl deformation of self–dual gravity is one-loop exact and has also appeared in recent parallel discussions of celestial holography [45,61,62,63,64]; in [65] it is interpreted as the $T\overline{T}$-deformation of self–dual gravity.

Consider now what happens beyond the semi-classical limit of the adjoint scalar theory. The self–dual Yang–Mills equations are then deformed to

$$\begin{array}{c}\hfill \square \varphi +\frac{g}{2}〚\varphi ,\varphi {〛}_{\mathrm{AS}}=0,\end{array}$$

(140)

where the bracket $〚-,-{〛}_{\mathrm{AS}}$ is defined in Section 3.3. This is obtained as one of two single copies of deformed self–dual gravity in the semi-classical limit, as $\overline{t}\to 0$ alone, of the $L_{\infty }$-algebra (115): replacing the inner kinematical vector space ${\mathfrak{Kin}}_{{\wedge }^0}$ with the colour algebra $\mathfrak{g}$ gives the standard self–dual Yang–Mills theory above, which is integrable, whereas replacing the outer factor of ${\mathfrak{Kin}}_{{\wedge }^0}$ with $\mathfrak{g}$ leads to the theory with deformed equations of motion (140), which is not integrable. The main technical issue is that the bracket operation $〚-,-{〛}_{\mathrm{AS}}$ is not a Lie bracket, as it violates the (ungraded) Jacobi identities.

As we have shown in Section 3.4, Equation (140) are the natural gauge field equations that arise in the homotopy double copy prescription, and which are consistent with colour–kinematics duality. However, they differ from the field equations of the usual noncommutative self–dual Yang–Mills theory, which has a non-trivial commutative limit to ordinary self–dual Yang–Mills theory and is defined using the star-commutator bracket:

$$\begin{array}{c}\hfill \square \varphi -\mathrm{i}g{\left[{\partial }_z\varphi \stackrel{★}{,}{\partial }_{-}\varphi \right]}_{\mathfrak{u}\left(N\right)}=0.\end{array}$$

(141)

These are the Euler–Lagrange equations which follow from varying the noncommutative cubic action functional [66]

$$\begin{array}{c}\hfill S_{\mathrm{YM}+}^{★}\left[\varphi \right]=\int {\mathrm{d}}^{4}x{\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(\frac{1}{2}\varphi ★\square \varphi -\frac{\mathrm{i}g}{3!}{ϵ}^{\mu \nu }\varphi ★{\left[{\partial }_{\mu }\varphi \stackrel{★}{,}{\partial }_{\nu }\varphi \right]}_{\mathfrak{u}\left(N\right)}\right)\end{array}$$

(142)

for the Leznov prepotential $\varphi \in {\Omega }^0({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))$.

Noncommutative $\mathsf{U}\left(N\right)$ self–dual Yang–Mills theory with action functional (142) is the semi-classical limit $\overline{t}\to 0$ of the rank $\overline{N}=1$ binoncommutative biadjoint scalar theory of Section 3.6 in $d=4$ dimensions. As we discussed in Section 3.1, the star-commutator bracket is not compatible with standard colour–kinematics duality, due to the appearance of anticommutators of Lie algebra generators in Equation (60) which obstructs an immediate (untwisted) homotopy factorisation of the $L_{\infty }$-structure. On the other hand, this theory is integrable [67,68].

The noncommutative instanton Equation (141) in the rank-one limit $N=1$ coincides with the noncommutative Plebański Equation (139), while the action functional (142) for $N=1$ is the semi-classical limit of the Maurer–Cartan functional (119) as $\overline{t}\to 0$. It follows that self–dual ${\mathfrak{u}}_{★}\left(1\right)$ Yang–Mills theory is the double copy of the adjoint scalar theory with ordinary self–dual Yang–Mills theory, for any gauge algebra $\mathfrak{g}$. This double copy theory is precisely noncommutative self–dual gravity. In particular, the integrability of gravity is “inherited” from the corresponding gauge theories via the double copy and is guaranteed by integrability of at least one of the two gauge-theory single copies [52,61]. In what follows we shall assert that such double copy interpretations of rank-one noncommutative gauge theories hold in general.

4. Noncommutative Chern–Simons Theory

4.1. Moyal–Weyl Deformation of Differential Forms

To study general noncommutative gauge theories, we first discuss the extension of the Moyal–Weyl deformation of scalar fields from Section 3.1 to include differential forms of arbitrary degree. Let ${\Omega }^{•}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$ be the exterior algebra of differential forms on ${\mathbb{R}}^{1,d-1}$ valued in the matrix Lie algebra $\mathfrak{g}=\mathfrak{u}\left(N\right)$. This is a strict $A_{\infty }$-algebra with the de Rham differential $\mathrm{d}$ and the composition of matrix multiplication with the exterior product of forms, which as usual we make into a strict $L_{\infty }$-algebra under the commutator bracket:

$$\begin{array}{c}\hfill {[\alpha ,\beta ]}_{\mathfrak{u}\left(N\right)}=\alpha \wedge \beta -{(-1)}^{\left|\alpha \right|\left|\beta \right|}\beta \wedge \alpha ,\end{array}$$

(143)

where $\left|\alpha \right|$ denotes the degree of a homogeneous form $\alpha \in {\Omega }^{•}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$.

As a vector space

$$\begin{array}{c}\hfill {\Omega }^{•}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))\simeq \mathfrak{u}\left(N\right)\otimes {\Omega }^{•}\left({\mathbb{R}}^{1,d-1}\right),\end{array}$$

(144)

and this is also true at the level of homotopy algebras: the strict $L_{\infty }$-algebra ${\Omega }^{•}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$ factorises into the tensor product of the Lie algebra $\mathfrak{u}\left(N\right)$ with the strict $C_{\infty }$-algebra ${\Omega }^{•}\left({\mathbb{R}}^{1,d-1}\right)$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{d}\alpha & =\mathrm{d}(T^a\otimes {\alpha }_a)=T^a\otimes \mathrm{d}{\alpha }_a,\hfill \\ \hfill {[\alpha ,\beta ]}_{\mathfrak{u}\left(N\right)}& ={[T^a\otimes {\alpha }_a,T^b\otimes {\beta }_b]}_{\mathfrak{u}\left(N\right)}={[T^a,T^b]}_{\mathfrak{u}\left(N\right)}\otimes ({\alpha }_a\wedge {\beta }_b),\hfill \end{array}\end{array}$$

(145)

where ${\alpha }_a,{\beta }_b\in {\Omega }^{•}\left({\mathbb{R}}^{1,d-1}\right)$ and we used graded commutativity of the exterior product of forms: ${\alpha }_a\wedge {\beta }_b={(-1)}^{|{\alpha }_a\left|\right|{\beta }_b|}{\beta }_b\wedge {\alpha }_a$. This is a special instance of a more general statement that is relevant for us: The tensor product of any $L_{\infty }$-algebra $\mathfrak{L}$ with ${\Omega }^{•}\left({\mathbb{R}}^{1,d-1}\right)$ is an $L_{\infty }$-algebra.

The Moyal–Weyl deformation of ${\Omega }^{•}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$ is defined by deforming the exterior product to the star-product

$$\begin{array}{c}\hfill \alpha {\wedge }_{★}\beta :=\alpha \wedge exp\left(-\frac{\mathrm{i}t}{2}{\overleftarrow{\pounds }}_{\mu }{\theta }^{\mu \nu }{\overrightarrow{\pounds }}_{\nu }\right)\beta ,\end{array}$$

(146)

where ${\pounds }_{\mu }:={\pounds }_{{\partial }_{\mu }}$ are Lie derivatives along the holonomic frame of vector fields on ${\mathbb{R}}^{1,d-1}$. The conventional noncommutative gauge theories are realised by deforming the commutator (143) to the star-commutator on ${\Omega }^{•}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$:

$$\begin{array}{c}\hfill {\left[\alpha \stackrel{★}{,}\beta \right]}_{\mathfrak{u}\left(N\right)}=\alpha {\wedge }_{★}\beta -{(-1)}^{\left|\alpha \right|\left|\beta \right|}\beta {\wedge }_{★}\alpha .\end{array}$$

(147)

This bracket again makes ${\Omega }^{•}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$ into a strict $L_{\infty }$-algebra.

As before, while the vector space isomorphism (144) still holds, it is no longer an isomorphism of strict $L_{\infty }$-algebras, as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left[\alpha \stackrel{★}{,}\beta \right]}_{\mathfrak{u}\left(N\right)}={[T^a\otimes {\alpha }_a\stackrel{★}{,}{T}^b\otimes {\beta }_b]}_{\mathfrak{u}\left(N\right)}& =\frac{1}{2}{[T^a,T^b]}_{\mathfrak{u}\left(N\right)}\otimes {\left\{{\alpha }_a\stackrel{★}{,}{\beta }_b\right\}}_{\mathfrak{u}\left(1\right)}\hfill \\ \hfill & +\frac{1}{2}{\{T^a,T^b\}}_{\mathfrak{u}\left(N\right)}\otimes {\left[{\alpha }_a\stackrel{★}{,}{\beta }_b\right]}_{\mathfrak{u}\left(1\right)},\hfill \end{array}\end{array}$$

(148)

where ${\left\{{\alpha }_a\stackrel{★}{,}{\beta }_b\right\}}_{\mathfrak{u}\left(1\right)}:={\alpha }_a{\wedge }_{★}{\beta }_b+{(-1)}^{\left|\alpha \right|\left|\beta \right|}{\beta }_b{\wedge }_{★}{\alpha }_a$. As previously, in what follows we shall understand colour-stripping in noncommutative gauge theories in a twisted sense, which is compatible with colour–kinematics duality in this setting.

4.2. The $L_{\infty }$-Structure of Noncommutative Chern–Simons Theory

Consider standard noncommutative $\mathsf{U}\left(N\right)$ Chern–Simons gauge theory on ${\mathbb{R}}^{1,2}$, which is defined by the action functional

$$\begin{array}{c}\hfill S_{\mathrm{CS}}^{★}\left[A\right]=\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(\frac{1}{2}A{\wedge }_{★}\mathrm{d}A+\frac{g}{3!}A{\wedge }_{★}{\left[A\stackrel{★}{,}A\right]}_{\mathfrak{u}\left(N\right)}\right),\end{array}$$

(149)

where $A\in {\Omega }^1({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ is the gauge field and $g=\sqrt{2\pi /k}$ is the gauge coupling constant with k the Chern–Simons level. Solutions of the corresponding field equation are flat noncommutative connections, $F_A^{★}=0$, where

$$\begin{array}{c}\hfill F_A^{★}=\mathrm{d}A+\frac{g}{2}{\left[A\stackrel{★}{,}A\right]}_{\mathfrak{u}\left(N\right)}\end{array}$$

(150)

is the field strength in ${\Omega }^2({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$. The noncommutative Chern–Simons functional is invariant under infinitesimal star-gauge transformations ${\delta }_c^{★}A=\mathrm{d}c+g{\left[c\stackrel{★}{,}A\right]}_{\mathfrak{u}\left(N\right)}$ with $c\in {\Omega }^0({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$. The Bianchi identity ${\nabla }_A^{★}{F}_A^{★}=0$ in ${\Omega }^3({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ is the Noether identity corresponding to this gauge symmetry, where ${\nabla }_A^{★}:{\Omega }^p({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))⟶{\Omega }^{p+1}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ is the covariant derivative

$$\begin{array}{c}\hfill {\nabla }_A^{★}=\mathrm{d}+g{[A\stackrel{★}{,}-]}_{\mathfrak{u}\left(N\right)}.\end{array}$$

(151)

This noncommutative gauge theory is organised by the cyclic strict $L_{\infty }$-algebra

$$\begin{array}{c}\hfill {\mathfrak{CS}}_{★}=\left(\left({\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right)),{\mu }_1^{★\mathrm{CS}}=\mathrm{d}\right),{\mu }_2^{★\mathrm{CS}}=g{[-\stackrel{★}{,}-]}_{\mathfrak{u}\left(N\right)},{\langle -,-\rangle }_{\mathrm{CS}}^{★}\right),\end{array}$$

(152)

whose underlying cochain complex $\mathsf{Ch}\left({\mathfrak{CS}}_{★}\right)=\left({\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right)),\mathrm{d}\right)$ is the de Rham complex of differential forms on ${\mathbb{R}}^{1,2}$ valued in the matrix Lie algebra $\mathfrak{u}\left(N\right)$. The cyclic structure of degree $-3$ is given by the pairing of $\mathfrak{u}\left(N\right)$-valued differential forms in complementary degrees:

$$\begin{array}{c}\hfill {\langle \alpha ,{\alpha }^{+}\rangle }_{\mathrm{CS}}^{★}=\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(\alpha {\wedge }_{★}{\alpha }^{+}\right)=\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}(\alpha \wedge {\alpha }^{+}),\end{array}$$

(153)

for $\alpha \in \{c,A\}$, with cyclicity ensured by the $\mathrm{ad}\left(\mathfrak{u}\right(N\left)\right)$-invariance of the trace pairing ${\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}:\mathfrak{u}\left(N\right)\otimes \mathfrak{u}\left(N\right)⟶\mathbb{R}$.

For this strict $L_{\infty }$-algebra, the Maurer–Cartan theory from Appendix A.2 reproduces the standard curvature and Bianchi identity given in degree 2 and 3, respectively, by

$$\begin{array}{cc}\hfill f_A^{★\mathrm{CS}}& ={\mu }_1^{★\mathrm{CS}}\left(A\right)+\frac{1}{2}{\mu }_2^{★\mathrm{CS}}(A,A)=F_A^{★},\hfill \\ \hfill {\mathrm{d}}_A^{★\mathrm{CS}}{f}_A^{★\mathrm{CS}}& ={\mu }_1^{★\mathrm{CS}}\left(f_A^{★\mathrm{CS}}\right)+{\mu }_2^{★\mathrm{CS}}(A,f_A^{★\mathrm{CS}})={\nabla }_A^{★}{F}_A^{★}=0,\hfill \end{array}$$

(154)

and the noncommutative Chern–Simons functional

$$\begin{array}{c}\hfill S_{\mathrm{CS}}^{★}\left[A\right]=\frac{1}{2}{\langle A,{\mu }_1^{★\mathrm{CS}}\left(A\right)\rangle }_{\mathrm{CS}}^{★}+\frac{1}{3!}{\langle A,{\mu }_2^{★\mathrm{CS}}(A,A)\rangle }_{\mathrm{CS}}^{★},\end{array}$$

(155)

invariant under the star-gauge transformations ${\delta }_c^{★}A={\mu }_1^{★\mathrm{CS}}\left(c\right)+{\mu }_2^{★\mathrm{CS}}(c,A)$.

Batalin–Vilkovisky Formalism

The Batalin–Vilkovisky (BV) complex corresponding to the $L_{\infty }$-algebra ${\mathfrak{CS}}_{★}$ consists of BRST ghosts $c\in {\Omega }^0({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$, gauge fields $A\in {\Omega }^1({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ along with their antifields $c^{+}\in {\Omega }^3({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ and $A^{+}\in {\Omega }^2({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$. The non-zero differentials ${\mu }_1^{★\mathrm{CS}}:{\Omega }^p({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))⟶{\Omega }^{p+1}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ are

$$\begin{array}{c}\hfill {\mu }_1^{★\mathrm{CS}}\left(c\right)=\mathrm{d}c,{\mu }_1^{★\mathrm{CS}}\left(A\right)=\mathrm{d}A,{\mu }_1^{★\mathrm{CS}}\left(A^{+}\right)=\mathrm{d}{A}^{+},\end{array}$$

(156)

while the non-zero 2-brackets ${\mu }_2^{★\mathrm{CS}}:{\Omega }^p({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))\otimes {\Omega }^q({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))⟶{\Omega }^{p+q}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ are given by

$$\begin{array}{cc}\hfill {\mu }_2^{★\mathrm{CS}}(c_1,c_2)=g{\left[c_1\stackrel{★}{,}{c}_2\right]}_{\mathfrak{u}\left(N\right)},& {\mu }_2^{★\mathrm{CS}}(c,A)=g{\left[c\stackrel{★}{,}A\right]}_{\mathfrak{u}\left(N\right)},\hfill \\ \hfill {\mu }_2^{★\mathrm{CS}}(c,A^{+})=g{\left[c\stackrel{★}{,}{A}^{+}\right]}_{\mathfrak{u}\left(N\right)},& {\mu }_2^{★\mathrm{CS}}(A_1,A_2)=g{\left[A_1\stackrel{★}{,}{A}_2\right]}_{\mathfrak{u}\left(N\right)},\hfill \\ \hfill {\mu }_2^{★\mathrm{CS}}(A,A^{+})=g{\left[A\stackrel{★}{,}{A}^{+}\right]}_{\mathfrak{u}\left(N\right)},& {\mu }_2^{★\mathrm{CS}}(c,c^{+})=g{\left[c\stackrel{★}{,}{c}^{+}\right]}_{\mathfrak{u}\left(N\right)}.\hfill \end{array}$$

(157)

Consider now a superfield element $\mathit{A}\in \mathsf{Fun}\left({\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))\left[1\right]\right)\otimes {\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$. With the local coordinate functions ${\vartheta }^{{\mu }_1\cdots {\mu }_p}$ on ${\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))\left[1\right]$ of degree $|{\vartheta }^{{\mu }_1\cdots {\mu }_p}|=1-p$, we can express the superfield as

$$\mathit{A}=\vartheta \otimes c+{\vartheta }^{\mu }\otimes A_{\mu }+{\vartheta }^{\mu \nu }\otimes A_{\mu \nu }^{+}+{\vartheta }^{\mu \nu \rho }\otimes c_{\mu \nu \rho }^{+}.$$

(158)

Since $\mathit{A}$ is a degree 1 element by construction, we retrieve the full BV action functional from the Maurer–Cartan functional of $\mathit{A}$ as (see Appendix A.2)

$$\begin{array}{cc}\hfill S_{\mathrm{BV}}^{★}\left[\mathit{A}\right]& =\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(\frac{1}{2}A{\wedge }_{★}\mathrm{d}A+\frac{g}{3!}A{\wedge }_{★}{\left[A\stackrel{★}{,}A\right]}_{\mathfrak{u}\left(N\right)}+A^{+}{\wedge }_{★}{\nabla }_A^{★}c+\frac{g}{2}{c}^{+}{\wedge }_{★}{\left[c\stackrel{★}{,}c\right]}_{\mathfrak{u}\left(N\right)}\right).\hfill \end{array}$$

(159)

In the commutative limit $t=0$, this reproduces the superfield Chern–Simons action functional of [39] entirely from the underlying $L_{\infty }$-structure.

The Seiberg–Witten map [69] gives rise to a quasi-isomorphism between a noncommutative gauge theory and a commutative gauge theory. (See e.g., refs. [70,71] for recent explicit elucidations in the language of homotopy algebras). The local BRST cohomology of standard Chern–Simons theory implies that any consistent deformation of commutative $\mathsf{U}\left(N\right)$ Chern–Simons theory is trivial; in particular, the noncommutative deformation is trivial [72,73]. It follows that, under the Seiberg–Witten map, the $L_{\infty }$-algebra ${\mathfrak{CS}}_{★}$ of noncommutative $\mathsf{U}\left(N\right)$ Chern–Simons theory is quasi-isomorphic to the $L_{\infty }$-algebra $\mathfrak{CS}$ of ordinary $\mathsf{U}\left(N\right)$ Chern–Simons theory. This will account for some classical features of the homotopy double copy construction that we explain below. However, it does not imply that the corresponding quantum field theories are equivalent, and indeed we shall see that the amplitudes in the two theories are not the same.

4.3. Twisted Homotopy Factorisation

Colour-stripping of noncommutative Chern–Simons theory can be done in two equivalent ways, each of which will prove useful for our subsequent analysis. Following what we did in Section 3.2.1, we can first strip off the infinite-dimensional colour Lie algebra ${\mathfrak{u}}_{★}\left(N\right)$ by extending the twist datum ${\tau }^{{\mathfrak{u}}_{★}\left(N\right)}=\left({\tau }_1^{{\mathfrak{u}}_{★}\left(N\right)},{\tau }_2^{{\mathfrak{u}}_{★}\left(N\right)}\right)$ to arbitrary degree differential forms, which, by abuse of notation, we continue to denote with the same symbol. Therefore, the twist datum now consists of the map ${\tau }_1^{{\mathfrak{u}}_{★}\left(N\right)}:\mathfrak{u}\left(N\right)⟶\mathfrak{u}\left(N\right)\otimes \mathsf{End}\left({\Omega }^{•}\left({\mathbb{R}}^{1,2}\right)\right)$ defined by

$$\begin{array}{c}\hfill {\tau }_1^{{\mathfrak{u}}_{★}\left(N\right)}\left(T^a\right)=T^a\otimes \mathbb{1},\end{array}$$

(160)

and the map ${\tau }_2^{{\mathfrak{u}}_{★}\left(N\right)}:\mathfrak{u}\left(N\right)\otimes \mathfrak{u}\left(N\right)⟶\mathfrak{u}\left(N\right)\otimes \mathsf{End}\left({\Omega }^{•}\left({\mathbb{R}}^{1,2}\right)\right)\otimes \mathsf{End}\left({\Omega }^{•}\left({\mathbb{R}}^{1,2}\right)\right)$ defined by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\tau }_2^{{\mathfrak{u}}_{★}\left(N\right)}(T^a,T^b)& ={[T^a,T^b]}_{\mathfrak{u}\left(N\right)}\otimes cos\left(\frac{t}{2}{\theta }^{\mu \nu }{\pounds }_{\mu }\otimes {\pounds }_{\nu }\right)\hfill \\ \hfill & +\mathrm{i}{\{T^a,T^b\}}_{\mathfrak{u}\left(N\right)}\otimes sin\left(\frac{t}{2}{\theta }^{\mu \nu }{\pounds }_{\mu }\otimes {\pounds }_{\nu }\right).\hfill \end{array}\end{array}$$

(161)

By completely analogous calculations to those of Section 3.2.1, it is now straightforward to show that the cyclic strict $L_{\infty }$-algebra ${\mathfrak{CS}}_{★}$ admits a homotopy factorisation

$$\begin{array}{c}\hfill {\mathfrak{CS}}_{★}=\mathfrak{u}\left(N\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(N\right)}}{\mathfrak{C}}_{{\Omega }^{•}}\end{array}$$

(162)

into the twisted tensor product of the quadratic colour Lie algebra $\mathfrak{u}\left(N\right)$ with the kinematical strict $C_{\infty }$-algebra

$$\begin{array}{c}\hfill {\mathfrak{C}}_{{\Omega }^{•}}=\left(\left({\Omega }^{•}\left({\mathbb{R}}^{1,2}\right),m_1^{{\Omega }^{•}}=\mathrm{d}\right),m_2^{{\Omega }^{•}}=g\wedge ,{\langle -,-\rangle }_{{\Omega }^{•}}\right),\end{array}$$

(163)

with cyclic structure ${\langle -,-\rangle }_{{\Omega }^{•}}$ defined by integration of exterior products of forms in complementary degree. This is just the exterior algebra of differential forms on ${\mathbb{R}}^{1,2}$, which represents the infinitesimal diffeomorphism invariance of classical Chern–Simons theory.

Next, we factorise ${\mathfrak{C}}_{{\Omega }^{•}}$ into a finite-dimensional kinematic vector space ${\mathfrak{Kin}}_{{\wedge }^{•}}$ and a cyclic strict $L_{\infty }$-algebra encoding the trivalent interactions. As a graded vector space, we identify ${\Omega }^{•}\left({\mathbb{R}}^{1,2}\right)$ with the tensor product ${\wedge }^{•}{\left({\mathbb{R}}^{1,2}\right)}^{\ast }\otimes {\Omega }^0\left({\mathbb{R}}^{1,2}\right)$ by writing any one-form $\alpha ={\alpha }_{\mu }\mathrm{d}{x}^{\mu }\in {\Omega }^1\left({\mathbb{R}}^{1,2}\right)$ as $\alpha =e^{\mu }\otimes {\alpha }_{\mu }$, where ${\alpha }_{\mu }\in {\Omega }^0\left({\mathbb{R}}^{1,2}\right)$ and $e^{\mu }$ is the natural basis of covectors in ${\mathbb{R}}^{1,2}$ relative to the rectangular coordinates $\left(x^{\mu }\right)$.

At the level of homotopy algebras, the cyclic strict $C_{\infty }$-algebra ${\mathfrak{C}}_{{\Omega }^{•}}$ factorises as a twisted tensor product

$$\begin{array}{c}\hfill {\mathfrak{C}}_{{\Omega }^{•}}={\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}\mathfrak{Scal}\end{array}$$

(164)

of the 1-shifted exterior algebra

$$\begin{array}{c}\hfill {\mathfrak{Kin}}_{{\wedge }^{•}}:={\wedge }^{•}{\left({\mathbb{R}}^{1,2}\right)}^{\ast }\left[1\right]\end{array}$$

(165)

with the cyclic strict $L_{\infty }$-algebra $\mathfrak{Scal}$ from Section 2.1 for scalar field theory with underlying graded vector space (80) on ${\mathbb{R}}^{1,2}$. The twist datum ${\tau }^{{\Omega }^{•}}=\left({\tau }_1^{{\Omega }^{•}},{\tau }_2^{{\Omega }^{•}}\right)$ is defined as follows. The map ${\tau }_1^{{\Omega }^{•}}:{\mathfrak{Kin}}_{{\wedge }^{•}}⟶{\mathfrak{Kin}}_{{\wedge }^{•}}\otimes \mathsf{End}\left(L\right)$ is defined by

$$\begin{array}{c}\hfill {\tau }_1^{{\Omega }^{•}}\left(v\right)=(e^{\mu }\wedge v)\otimes \frac{{\partial }_{\mu }}{\square },\end{array}$$

(166)

while ${\tau }_2^{{\Omega }^{•}}:{\mathfrak{Kin}}_{{\wedge }^{•}}\otimes {\mathfrak{Kin}}_{{\wedge }^{•}}⟶{\mathfrak{Kin}}_{{\wedge }^{•}}\otimes \mathsf{End}\left(L\right)\otimes \mathsf{End}\left(L\right)$ is given as

$$\begin{array}{c}\hfill {\tau }_2^{{\Omega }^{•}}(v,w)=(v\wedge w)\otimes \mathbb{1}\otimes \mathbb{1},\end{array}$$

(167)

for all $v,w\in {\mathfrak{Kin}}_{{\wedge }^{•}}={\wedge }^{•}{\left({\mathbb{R}}^{1,2}\right)}^{\ast }\left[1\right]$.

It is then straightforward to check that this twisting reproduces the differential: $m_1^{{\tau }^{{\Omega }^{•}}}=m_1^{{\Omega }^{•}}$; e.g., on degree 1 fields we find

$$\begin{array}{c}\hfill m_1^{{\tau }^{{\Omega }^{•}}}(e^{\mu }\otimes {\alpha }_{\mu })=(e^{\nu }\wedge e^{\mu })\otimes {\mu }_1^{\mathrm{Scal}}\left(\frac{{\partial }_{\nu }}{\square }{\alpha }_{\mu }\right)=(e^{\nu }\wedge e^{\mu })\otimes {\partial }_{\nu }{\alpha }_{\mu }=\mathrm{d}\alpha .\end{array}$$

(168)

Similarly we reproduce the product: $m_2^{{\tau }^{{\Omega }^{•}}}=m_2^{{\Omega }^{•}}$; e.g., on degree 1 fields we find

$$\begin{array}{c}\hfill m_2^{{\tau }^{{\Omega }^{•}}}(e^{\mu }\otimes {\alpha }_{\mu },e^{\nu }\otimes {\beta }_{\nu })=(e^{\mu }\wedge e^{\nu })\otimes {\mu }_2^{\mathrm{Scal}}({\alpha }_{\mu },{\beta }_{\nu })=(e^{\mu }\wedge e^{\nu })\otimes g{\alpha }_{\mu }{\beta }_{\nu }=g\alpha \wedge \beta .\end{array}$$

(169)

Note, however, that due to the tensor product of the underlying graded vector spaces ${\mathfrak{Kin}}_{{\wedge }^{•}}={\wedge }^{•}{\left({\mathbb{R}}^{1,2}\right)}^{\ast }\left[1\right]$ and $L={\Omega }^0\left({\mathbb{R}}^{1,2}\right)[-1]\oplus {\Omega }^0\left({\mathbb{R}}^{1,2}\right)[-2]$, the factorisation (164) introduces some redundancy. For example, the cochain complex underlying the twisted tensor product is

(170)

Finally, the cyclic structure is recovered by wedging forms in ${\wedge }^{•}{\left({\mathbb{R}}^{1,2}\right)}^{\ast }$ of complementary degrees and applying the Hodge duality operator ${\mathrm{Tr}}_{{\wedge }^{•}}:{\wedge }^3{\left({\mathbb{R}}^{1,2}\right)}^{\ast }⟶\mathbb{R}$ in three dimensions. Then ${\mathrm{Tr}}_{{\wedge }^{•}}\otimes {\langle -,-\rangle }_{\mathrm{Scal}}$ reproduces the inner product ${\langle -,-\rangle }_{{\Omega }^{•}}$ on ${\mathfrak{C}}_{{\Omega }^{•}}$.

Altogether, we have shown that the cyclic strict $L_{\infty }$-algebra of noncommutative Chern–Simons gauge theory admits the factorisation

$$\begin{array}{c}\hfill {\mathfrak{CS}}_{★}=\mathfrak{u}\left(N\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(N\right)}}\left({\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}\mathfrak{Scal}\right).\end{array}$$

(171)

In particular, replacing ${\mathfrak{Kin}}_{{\wedge }^{•}}$ with a copy of another colour Lie algebra $\mathfrak{u}\left(\overline{N}\right)$ and corresponding twisted tensor product yields the cyclic strict $L_{\infty }$-algebra (125). This identifies the $d=3$ binoncommutative biadjoint scalar theory of Section 3.6 as the zeroth copy of noncommutative Chern–Simons theory.

This last statement is perhaps more transparent if one notes that the factorisation (171) can be equivalently written as the twisted tensor product

$$\begin{array}{c}\hfill {\mathfrak{CS}}_{★}={\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}{\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(N\right)},\end{array}$$

(172)

where the cyclic strict $C_{\infty }$-algebra ${\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(N\right)}$ itself factorises as in Equation (85). In Equation (172) the twist datum is defined in a completely analogous way to the twist datum above in terms of maps ${\tau }_1^{{\Omega }^{•}}:{\mathfrak{Kin}}_{{\wedge }^{•}}⟶{\mathfrak{Kin}}_{{\wedge }^{•}}\otimes \mathsf{End}\left(\mathfrak{u}\left(N\right)\otimes L\right)$ and ${\tau }_2^{{\Omega }^{•}}:{\left({\mathfrak{Kin}}_{{\wedge }^{•}}\right)}^{\otimes 2}⟶{\mathfrak{Kin}}_{{\wedge }^{•}}\otimes \mathsf{End}(\mathfrak{u}\left(N\right)$$\otimes L{)}^{\otimes 2}$. For example, on degree 1 fields we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_1^{{\tau }^{{\Omega }^{•}}}(e^{\mu }\otimes {\alpha }_{\mu })& =(e^{\nu }\wedge e^{\mu })\otimes m_1^{{\mathfrak{u}}_{★}\left(N\right)}\left(\frac{{\partial }_{\nu }}{\square }{\alpha }_{\mu }\right)=(e^{\nu }\wedge e^{\mu })\otimes {\partial }_{\nu }{\alpha }_{\mu }=\mathrm{d}\alpha ,\hfill \\ \hfill {\mu }_2^{{\tau }^{{\Omega }^{•}}}(e^{\mu }\otimes {\alpha }_{\mu },e^{\nu }\otimes {\beta }_{\nu })& =(e^{\mu }\wedge e^{\nu })\otimes \left(-m_2^{{\mathfrak{u}}_{★}\left(N\right)}({\alpha }_{\mu },{\beta }_{\nu })\right)\hfill \\ \hfill & =(e^{\mu }\wedge e^{\nu })\otimes g{\left[{\alpha }_{\mu }\stackrel{★}{,}{\beta }_{\nu }\right]}_{\mathfrak{u}\left(N\right)}=g{\left[\alpha \stackrel{★}{,}\beta \right]}_{\mathfrak{u}\left(N\right)}.\hfill \end{array}\end{array}$$

(173)

Compared to Equation (171), which is a factorisation into a colour Lie algebra and a kinematic $C_{\infty }$-algebra, in Equation (172) the factorisation is into a kinematic vector space and a colour $C_{\infty }$-algebra.

4.4. Twisted Colour–Kinematics Duality

Perturbative computations in noncommutative Chern–Simons theory are facilitated using the factorisation (172) to decompose degree 1 fields as $A=e^{\mu }\otimes A_{\mu }$ in terms of coordinate functions $A_{\mu }:{\mathbb{R}}^{1,2}⟶{\mathfrak{u}}_{(}N)$, with bracket

$$\begin{array}{c}\hfill {\left[A_{\mu }\stackrel{★}{,}{B}_{\nu }\right]}_{\mathfrak{u}\left(N\right)}=A_{\mu }★B_{\nu }-B_{\nu }★A_{\mu }\end{array}$$

(174)

for $A_{\mu },B_{\nu }\in {\Omega }^0({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$. In this form, the three-gluon Feynman vertex can be computed directly by expanding the action functional (149) as

$$\begin{array}{c}\hfill S_{\mathrm{CS}}^{★}\left[A\right]=\int {\mathrm{d}}^{3}x{ϵ}^{\mu \nu \rho }{\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(A_{\mu }★{\partial }_{\nu }{A}_{\rho }+\frac{g}{3!}{A}_{\mu }★{\left[A_{\nu }\stackrel{★}{,}{A}_{\rho }\right]}_{\mathfrak{u}\left(N\right)}\right),\end{array}$$

(175)

where ${ϵ}^{\mu \nu \rho }$ is the Levi–Civita symbol in three dimensions with ${ϵ}^{012}=1$. It is given in terms of the structure constants of the infinite-dimensional Lie algebra ${\mathfrak{u}}_{★}\left(N\right)=({\Omega }^0({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right)),$${[-\stackrel{★}{,}-]}_{\mathfrak{u}\left(N\right)})$ as

(176)

In the following we shall set up the perturbiner expansion for noncommutative Chern–Simons theory in the $L_{\infty }$-algebra formalism and following the discussion from Section 3.4 we establish a twisted form of colour–kinematics duality compatible with the twisted colour-stripping.

4.4.1. Perturbiner Solutions

As Chern–Simons fields have no propagating degrees of freedom, all currents vanish on-shell and there are no interactions; in particular, the pure gauge theory has no non-trivial S-matrix elements. From the homotopy algebraic perspective, this is an easy consequence of the fact that the de Rham cohomology of ${\mathbb{R}}^{1,2}$ is trivial:

$$\begin{array}{c}{\mathsf{H}}^0\left({\mathfrak{CS}}_{★}\right)=ker\left(\mathrm{d}\right)\otimes \mathfrak{u}\left(N\right),{\mathsf{H}}^3\left({\mathfrak{CS}}_{★}\right)=ker\left(\delta \right)\otimes \mathfrak{u}\left(N\right),{\mathsf{H}}^1\left({\mathfrak{CS}}_{★}\right)={\mathsf{H}}^2\left({\mathfrak{CS}}_{★}\right)=0,\end{array}$$

(177)

where $\delta ={\ast }_{\mathrm{H}}\mathrm{d}{\ast }_{\mathrm{H}}$ is the codifferential corresponding to the exterior differential $\mathrm{d}$ and the Hodge duality operator ${\ast }_{\mathrm{H}}$ induced by the Minkowski metric. Correlation functions with external states lying in ${\mathsf{H}}^1\left({\mathfrak{CS}}_{★}\right)$ are thus always trivially zero.

As we did in Section 3.5, below we will construct the off-shell action functional of the double copied theory using Maurer–Cartan theory, so we should first establish that colour–kinematics duality holds off-shell from the homotopy algebraic perspective. Off-shell correlation functions are obtained by projecting onto “harmonic” states ${𝓗}^{•}\left({\mathfrak{CS}}_{★}\right)=ker(\square )$ rather than the minimal model [13,74,75], where $\square =\mathrm{d}\delta +\delta \mathrm{d}:{\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))⟶{\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ is the Hodge–d’Alembertian operator acting on forms; in Minkowski signature this is a bigger space than ${\mathsf{H}}^{•}\left({\mathfrak{CS}}_{★}\right)$. Correlation functions on this space will be non-zero as, unlike the projection to the cohomology, the space of harmonic forms contains configurations that are not pure gauge and hence can propagate.

Chern–Simons perturbation theory requires a “Hodge inverse” of the differential ${\mu }_1^{★\mathrm{CS}}=\mathrm{d}$, which is constructed using □ together with the Hodge–Kodaira decomposition (A35). We look for a contracting homotopy $\mathsf{h}$ satisfying a Hodge–Kodaira decomposition for projection ${\mathsf{p}}_{ker(\square )}:{\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))⟶{𝓗}^{•}\left({\mathfrak{CS}}_{★}\right)$ onto harmonic states:

$$\mathbb{1}={\mathsf{p}}_{ker(\square )}+{\mu }_1^{★\mathrm{CS}}\circ \mathsf{h}+\mathsf{h}\circ {\mu }_1^{★\mathrm{CS}}.$$

(178)

The contracting homotopy is the partial inverse of the differential ${\mu }_1^{★\mathrm{CS}}$ on the space of pure gauge configurations $\mathrm{im}\left(\mathrm{d}\right)\oplus \mathrm{im}\left(\delta \right)$, extended trivially to all of ${\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ by

$$\begin{array}{c}\hfill \mathsf{h}:\mathrm{im}\left(\mathrm{d}\right)\oplus \mathrm{im}\left(\delta \right)⟶{\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))\mathrm{and}ker\left(\mathsf{h}\right)=ker(\square ).\end{array}$$

(179)

It can be constructed from the partial inverse $\mathsf{D}$ of the d’Alembertian □ on the subspace of pure gauge configurations by extending it trivially to $ker(\square )$:

$$\mathsf{D}{|}_{\mathrm{im}\left(\mathrm{d}\right)\oplus \mathrm{im}\left(\delta \right)}=\frac{1}{\square }\mathrm{and}ker\left(\mathsf{D}\right)=ker(\square ).$$

(180)

A choice of contracting homotopy $\mathsf{h}:{\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))⟶{\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ of degree $-1$ completing the Hodge–Kodaira decomposition (178) is then

$$\mathsf{h}=\delta \circ \mathsf{D}.$$

(181)

Acting on gauge fields and ghost antifields, the maps ${\mathsf{h}}^{\left(1\right)}:{\Omega }^1({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))⟶{\Omega }^0({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ and ${\mathsf{h}}^{\left(3\right)}:{\Omega }^3({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))⟶{\Omega }^2({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ are, respectively, given by

$$\begin{array}{c}\hfill {\mathsf{h}}^{\left(1\right)}\left(A\right)=\frac{{\partial }_{\mu }}{\square }{A}^{\mu }\mathrm{and}{\mathsf{h}}^{\left(3\right)}{\left(c^{+}\right)}_{\mu \nu }=\frac{{\partial }^{\rho }}{\square }{c}_{\mu \nu \rho }^{+}.\end{array}$$

(182)

We set ${\mathsf{h}}^{\left(1\right)}\left(A\right)=0$, which imposes the Lorenz gauge $\delta A={\partial }_{\mu }{A}^{\mu }=0$. Acting on antifields in degree 2, the contracting homotopy is then given by the Feynman propagator for Chern–Simons theory: ${\mathsf{h}}^{\left(2\right)}=C_{\mathrm{F}}:{\Omega }^2({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))⟶{\Omega }^1({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ where

$$C_{\mathrm{F}}{\left(A^{+}\right)}_{\rho }={ϵ}_{\rho \sigma \alpha }{ϵ}^{\mu \nu \sigma }\frac{{\partial }^{\alpha }}{\square }{A}_{\mu \nu }^{+}.$$

(183)

Indeed, using these formulas, and the fact that the distribution $\frac{1}{\square }$ commutes with the operators $\mathrm{d}$, $\delta$ and ${\ast }_{\mathrm{H}}$, we verify the projector onto harmonic forms fits into Equation (178).

Given a perturbiner element $A\left(i\right)\in {𝓗}^1\left({\mathfrak{CS}}_{★}\right)$ corresponding to the ‘i-th’ gluon, we look for a quasi-isomorphism ${\psi }_n:{𝓗}^1{\left({\mathfrak{CS}}_{★}\right)}^{\otimes n}⟶{\Omega }^1({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))$ which is again given by the $L_{\infty }$-recursion relations. We work in terms of word combinatorics, where an ordered word $w=k_1\cdots k_n\in {\mathcal{W}}_n$ represents n external gluons. The pullback to ordered words on n letters has the physical interpretation of performing a colour ordering expansion, as is often done in Yang–Mills theory. The Berends–Giele currents $J_{\mu }:{\mathcal{W}}_n⟶\mathfrak{u}\left(N\right)$ are the coefficients of the plane wave expansions

$${\psi }_n\left(A\left(1\right),\dots ,A\left(n\right)\right)=e^{\mu }\otimes J_{\mu }\left(w\right){\mathrm{e}}^{\mathrm{i}{p}_{1\cdots n}·x}\in {\Omega }^1({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right)),$$

(184)

where $p_{1\cdots n}:=p_1+\cdots +p_n$. The currents $J\left(w\right):=e^{\mu }\otimes J_{\mu }\left(w\right)$ are coclosed in $ker\left({\mathsf{h}}^{\left(1\right)}\right)$, $p_w·J\left(w\right)=0$, from how we have built the contracting homotopy $\mathsf{h}$.

Since □ is a second-order differential operator, the star-commutator of two currents is not generally in $ker(\square )$. Therefore, upon using the recursion relations from Equations (A38) and (A40), successive application of the 2-bracket ${\mu }_2^{★\mathrm{CS}}$ of the full theory moves us out of the harmonic states. Similarly, the codifferential can be expressed as $\delta =\frac{\partial }{\partial e^{\mu }}\otimes {\partial }^{\mu }$ and so is a second-order differential operator of degree $-1$. Hence the 2-bracket ${\mu }_2^{★\mathrm{CS}}$ of two currents is also not generally coclosed, but this is restored upon following with the Chern–Simons propagator ${\mathsf{h}}^{\left(2\right)}=C_{\mathrm{F}}$.

The recursion relations for the quasi-isomorphism give

$$\begin{array}{cc}\hfill J\left(w\right)& =-\frac{g}{2}\sum _{w=w_1\bigsqcup w_2}{e}^{\rho }\otimes \left({\left(C_{\mathrm{F}}\right)}_{\rho }^{\mu \nu }\circ {\left[J_{\mu }\left(w_1\right)\stackrel{★}{,}{J}_{\nu }\left(w_2\right)\right]}_{\mathfrak{u}\left(N\right)}\right),\hfill \\ \hfill {\mu }_n^{\circ ★\mathrm{CS}}\left(A\left(1\right),\dots ,A\left(n\right)\right)& =\frac{g}{2}\sum _{w=w_1\bigsqcup w_2}\left(e^{\mu }\wedge e^{\nu }\right)\otimes \left({\mathsf{p}}_{ker(\square )}\circ {\left[J_{\mu }\left(w_1\right)\stackrel{★}{,}{J}_{\nu }\left(w_2\right)\right]}_{\mathfrak{u}\left(N\right)}{\mathrm{e}}^{\mathrm{i}{p}_{1\cdots n}·x}\right).\hfill \end{array}$$

(185)

The images of the n-brackets ${\mu }_n^{\circ ★\mathrm{CS}}$ are coclosed in $ker\left({\mathsf{h}}^{\left(2\right)}\right)$, by our construction of the contracting homotopy $\mathsf{h}$. The final projection to the harmonic states ${\mathsf{p}}_{ker(\square )}:{\Omega }^{•}({\mathbb{R}}^{1,2},\mathfrak{u}\left(N\right))⟶{𝓗}^{•}\left({\mathfrak{CS}}_{★}\right)$ also enforces momentum conservation $s_{1\cdots n}=0$, where $s_w$ are the Mandelstam variables.

The colour-ordered partial amplitudes are given by substitution into the cyclic structure on the harmonic states. This gives

$${𝓜}_n^{★\mathrm{CS}}(1,\dots ,n)={\langle A\left(1\right),{\mu }_{n-1}^{\circ ★\mathrm{CS}}(A\left(2\right),\dots ,A\left(n\right))\rangle }_{\circ \mathrm{CS}}^{★}$$

(186)

for the n-gluon amplitudes.

4.4.2. Colour–Kinematics Duality

With the Feynman propagator from Equation (183), and understanding star-commutators as living in ${\mathfrak{u}}_{★}\left(N\right)$, the recursion relations can be written as

$$\begin{array}{cc}\hfill J_{\rho }\left(w\right)& =-\frac{g}{2}\sum _{w=w_1\bigsqcup w_2}{\delta }_{\left[\alpha \rho \right]}^{\mu \nu }\frac{p_w^{\alpha }}{s_w}{\left[J_{\mu }\left(w_1\right)\stackrel{★}{,}{J}_{\nu }\left(w_2\right)\right]}_{\mathfrak{u}\left(N\right)}\hfill \\ \hfill & =-\frac{g}{2s_w}\sum _{w=w_1\bigsqcup w_2}\left(\left(p_{w_2}·J^a{\left(p_{w_1}\right)}^{♯}\right)J_{\rho }^b\left(p_{w_2}\right)-J_{\rho }^a\left(p_{w_1}\right)\left(p_{w_1}·J^b{\left(p_{w_2}\right)}^{♯}\right)\right)\hfill \\ \hfill & \times F^{abc}(p_{w_1},p_{w_2},p_w)T^c,\hfill \end{array}$$

(187)

where ${\delta }_{\left[\alpha \rho \right]}^{\mu \nu }:={\delta }_{\alpha }^{\mu }{\delta }_{\rho }^{\nu }-{\delta }_{\rho }^{\mu }{\delta }_{\alpha }^{\nu }$, and we have decomposed the Berends–Giele currents as tensor products $J_{\mu }\left(w\right)=J_{\mu }^a\left(p_w\right)\otimes e_{p_w}^a$ using the momentum space basis of the infinite-dimensional Lie algebra ${\mathfrak{u}}_{★}\left(N\right)$ introduced in Equation (63). The superscript ${}^{♯}$ indicates the operation which sends a covector in ${\left({\mathbb{R}}^{1,2}\right)}^{\ast }$ to its dual vector in ${\mathbb{R}}^{1,2}$.

The coefficients $J^a\left(p_w\right)$ can be thought of in terms of a basis of covectors ${\epsilon }_{p_w}\in {\left({\mathbb{R}}^{1,2}\right)}^{\ast }$, which are dual to gluon polarisation vectors ${\epsilon }_{p_w}^{♯}\in {\mathbb{R}}^{1,2}$. In Lorenz gauge, a given momentum $p_w\in {\mathbb{R}}^{1,2}$ is orthogonal to the corresponding polarisation vector: $p_w·{\epsilon }_{p_w}^{♯}=0$, corresponding to projection to states lying in $ker\left(\delta \right)$. These covectors on ${\mathbb{R}}^{1,2}$ naturally form an infinite-dimensional Lie algebra under the bracket

$${[{\epsilon }_{p_{w_1}},{\epsilon }_{p_{w_2}}]}_{\mathfrak{diff}\left({\mathbb{R}}^{1,2}\right)}:=\left({\epsilon }_{p_{w_1}}^{♯}·p_{w_2}\right){\epsilon }_{p_{w_2}}-\left({\epsilon }_{p_{w_2}}^{♯}·p_{w_1}\right){\epsilon }_{p_{w_1}}.$$

(188)

In particular, the bracket of two covectors is transverse to the total momentum they carry, i.e., $p_{12}·{[{\epsilon }_{p_1},{\epsilon }_{p_2}]}_{\mathfrak{diff}\left({\mathbb{R}}^{1,2}\right)}^{♯}=0$. This is recognised as the bracket of the Lie algebra $\mathfrak{diff}\left({\mathbb{R}}^{1,2}\right)$ of infinitesimal diffeomorphisms of ${\mathbb{R}}^{1,2}$. In Lorenz gauge, this restricts to the Lie subalgebra ${\mathfrak{diff}}_{\mathrm{vol}}\left({\mathbb{R}}^{1,2}\right)$ of volume-preserving diffeomorphisms, i.e., divergence-free vector fields on ${\mathbb{R}}^{1,2}$.

We can therefore express the recursion relations for the Berends–Giele currents in terms of a twisted tensor product $\mathfrak{u}\left(N\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(N\right)}}{\mathfrak{diff}}_{\mathrm{vol}}\left({\mathbb{R}}^{1,2}\right)$ of two Lie algebras, which defines a symmetric bracket operation

$$\begin{array}{cc}\hfill 〚J\left(w_1\right),J\left(w_2\right){〛}_{\mathrm{CS}}^{★}:=& e^{\rho }\otimes {\delta }_{\left[\alpha \rho \right]}^{\mu \nu }{p}_{w_1\bigsqcup w_2}^{\alpha }{\left[J_{\mu }\left(w_1\right)\stackrel{★}{,}{J}_{\nu }\left(w_2\right)\right]}_{\mathfrak{u}\left(N\right)}\hfill \\ \hfill =& {[J^a\left(p_{w_1}\right),J^b\left(p_{w_2}\right)]}_{\mathfrak{diff}\left({\mathbb{R}}^{1,2}\right)}\otimes {\left[e_{p_{w_1}}^a\stackrel{★}{,}{e}_{p_{w_2}}^b\right]}_{\mathfrak{u}\left(N\right)}.\hfill \end{array}$$

(189)

The intertwining of momenta in Equation (189) simply reflects the twisted factorisation of colour degrees of freedom in Equation (171), and the related absence of full diffeomorphism invariance in the Moyal–Weyl deformation of Chern–Simons gauge theory.

The recursion relations thus become

$$\begin{array}{cc}\hfill J\left(w\right)& =-\frac{g}{2}\sum _{w=w_1\bigsqcup w_2}\frac{〚J\left(w_1\right),J\left(w_2\right){〛}_{\mathrm{CS}}^{★}}{s_w},\hfill \end{array}$$

(190)

with currents $J\left(w\right)\in ker\left({\mathsf{h}}^{\left(1\right)}\right)$. At the level of amplitudes, we scatter harmonic gluon states $A\left(1\right)=e^{\mu }\otimes e_{p_1}^a\otimes A_{\mu }^a\left(1\right)$ with $A_{\mu }^a\left(1\right)\in {\Omega }^0\left({\mathbb{R}}^{1,2}\right)$ satisfying $\square A_{\mu }^a\left(1\right)=0$. Then the n-point colour-ordered partial amplitude is written as

$$\begin{array}{cc}\hfill {𝓜}_n^{★\mathrm{CS}}(1,w)& ={〈A\left(1\right),\frac{g}{2}\sum _{w=w_1\bigsqcup w_2}{\mathsf{p}}_{ker(\square )}\left({\left[J_{}\left(w_1\right)\stackrel{★}{,}J\left(w_2\right)\right]}_{\mathfrak{u}\left(N\right)}{\mathrm{e}}^{\mathrm{i}{p}_w·x}\right)〉}_{\circ \mathrm{CS}}^{★}.\hfill \end{array}$$

(191)

The projection operator enforces $p_w^2=0$, while $p_1^2=0$ already for the first state. Altogether one finds

$$\begin{array}{cc}\hfill {𝓜}_n^{★\mathrm{CS}}(1,w)& =\frac{g}{2}\sum _{w=w_1\bigsqcup w_2}{ϵ}^{\mu \nu \rho }{F}^{abc}(p_{w_1},p_{w_2},p_1)J_{\mu }^a\left(w_1\right)J_{\nu }^b\left(w_2\right)A_{\rho }^c\left(1\right){\left(2\pi \right)}^3\delta \left(p_{1w}\right).\hfill \end{array}$$

(192)

The full amplitude is obtained by summing over planar ordered permutations of $n-1$ gluons and performing the recursion (190) for the currents $J\left(w\right)$.

By running through the same argument from Section 3.4 for the adjoint scalar theory, this demonstrates a twisted form of colour–kinematics duality for currents of noncommutative Chern–Simons theory, between the Lie algebras underlying volume-preserving diffeomorphisms of ${\mathbb{R}}^{1,2}$ and noncommutative gauge transformations. In the commutative limit $t=0$, this identifies ${\mathfrak{diff}}_{\mathrm{vol}}\left({\mathbb{R}}^{1,2}\right)$ as the kinematic Lie algebra underlying true (off-shell) colour–kinematics duality in ordinary Chern–Simons theory [39]. Furthermore, from the perspective of Equation (189), the zeroth-copy construction corresponds to replacing the kinematic factors $J\left(w\right)$ with a set of twisted colour factors $C_{\overline{★}}\left(w\right)$ for the second rigid Lie algebra $\mathfrak{u}\left(\overline{N}\right)$ of the $d=3$ binoncommutative biadjoint scalar theory from Section 3.6.

4.5. Homotopy Double Copy

The twisted colour–kinematics duality discussed in Section 4.4 makes it clear that, in noncommutative gauge theories of the type considered in this paper, the double copy involves a replacement of the full twisted colour-stripping by kinematical factors. From the factorisation (171), the double copy prescription is clear: the homotopy double copy construction replaces the twisted tensor product with the colour factor $\mathfrak{u}\left(N\right)$ by the twisted tensor product with another copy of the kinematical factor ${\mathfrak{Kin}}_{{\wedge }^{•}}={\wedge }^{•}{\left({\mathbb{R}}^{1,2}\right)}^{\ast }\left[1\right]$, which gives the cyclic strict $L_{\infty }$-algebra

$$\begin{array}{c}\hfill \widehat{\mathfrak{CS}}={\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}\left({\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}\mathfrak{Scal}\right).\end{array}$$

(193)

The second kinematical twist datum is defined by maps ${\tau }_1^{{\Omega }^{•}}:{\mathfrak{Kin}}_{{\wedge }^{•}}⟶{\mathfrak{Kin}}_{{\wedge }^{•}}\otimes \mathsf{End}\left({\mathfrak{Kin}}_{{\wedge }^{•}}\otimes L\right)$ and ${\tau }_2^{{\Omega }^{•}}:{\mathfrak{Kin}}_{{\wedge }^{•}}\otimes {\mathfrak{Kin}}_{{\wedge }^{•}}⟶{\mathfrak{Kin}}_{{\wedge }^{•}}\otimes \mathsf{End}\left({\mathfrak{Kin}}_{{\wedge }^{•}}\otimes L\right)\otimes \mathsf{End}\left({\mathfrak{Kin}}_{{\wedge }^{•}}\otimes L\right)$.

Most notably, this construction asserts the general statement that double copies of noncommutative gauge theories are not deformed and coincide with their commutative counterparts. In particular, the double copy theory organised by the $L_{\infty }$-algebra (193) coincides with that obtained in [39] from a more physical perspective. In what follows we will unravel some further details about this non-local higher-spin theory, which involves two copies of the volume-preserving diffeomorphism algebra ${\mathfrak{diff}}_{\mathrm{vol}}\left({\mathbb{R}}^{1,2}\right)$.

In the following we regard the vector space ${\wedge }^{•}{\left({\mathbb{R}}^{1,2}\right)}^{\ast }\otimes {\wedge }^{•}{\left({\mathbb{R}}^{1,2}\right)}^{\ast }\otimes {\Omega }^0\left({\mathbb{R}}^{1,2}\right)$ as the tensor product ${\Omega }^{•,•}\left({\mathbb{R}}^{1,2}\right):={\Omega }^{•}\left({\mathbb{R}}^{1,2}\right){\otimes }_{{\Omega }^0\left({\mathbb{R}}^{1,2}\right)}{\Omega }^{•}\left({\mathbb{R}}^{1,2}\right)$ of ${\Omega }^{•}\left({\mathbb{R}}^{1,2}\right)$ with itself, considered to be a module over ${\Omega }^0\left({\mathbb{R}}^{1,2}\right)$; we can think of its elements as differential forms on ${\mathbb{R}}^{1,2}$ valued in the exterior algebra ${\Omega }^{•}\left({\mathbb{R}}^{1,2}\right)$. For example, an element $H\in {\Omega }^{1,1}\left({\mathbb{R}}^{1,2}\right)$ can be decomposed as $H=e^{\overline{\mu }}\otimes H_{\overline{\mu }}$ relative to a basis $e^{\overline{\mu }}$ corresponding to a rectangular coordinate system $\left(x^{\overline{\mu }}\right)$, with $H_{\overline{\mu }}\in {\Omega }^1\left({\mathbb{R}}^{1,2}\right)$. We may then further expand $H_{\overline{\mu }}=e^{\mu }\otimes H_{\overline{\mu }\mu }$ relative to the given basis $e^{\mu }$, where $H_{\overline{\mu }\mu }\in {\Omega }^0\left({\mathbb{R}}^{1,2}\right)$; we regard $x^{\mu }$ and $x^{\overline{\mu }}$ as independent variables so that in particular the corresponding partial derivatives ${\partial }_{\mu }$ and ${\partial }_{\overline{\mu }}$ commute. This enables the definition of a tensor product differential $\mathrm{d}\otimes \mathrm{d}$, sending ${\Omega }^{1,1}\left({\mathbb{R}}^{1,2}\right)$ to ${\Omega }^{2,2}\left({\mathbb{R}}^{1,2}\right)$

The graded vector space ${\mathfrak{Kin}}_{{\wedge }^{•}}\otimes {\mathfrak{Kin}}_{{\wedge }^{•}}\otimes L$ underlying the $L_{\infty }$-algebra $\widehat{\mathfrak{CS}}$ has 16 homogeneous components. Any zero differential appearing in the redundant part of Equation (170) yields a zero differential in the double copy, making the cochain complex of $\widehat{\mathfrak{CS}}$ “diagonal” in the sense that its non-trivial part is

(194)

For example, on fields of degree 1, the differential ${\widehat{\mu }}_1^{\mathrm{CS}}$ of $\widehat{\mathfrak{CS}}$ is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{\mu }}_1^{\mathrm{CS}}(e^{\overline{\mu }}\otimes e^{\mu }\otimes H_{\overline{\mu }\mu })& =(e^{\overline{\nu }}\wedge e^{\overline{\mu }})\otimes m_1^{{\tau }^{{\Omega }^{•}}}\left(e^{\mu }\otimes \frac{{\partial }_{\overline{\nu }}}{\square }{H}_{\overline{\mu }\mu }\right)\hfill \\ \hfill & =(e^{\overline{\nu }}\wedge e^{\overline{\mu }})\otimes (e^{\nu }\wedge e^{\mu })\otimes \frac{1}{\square }{\partial }_{\nu }{\partial }_{\overline{\nu }}{H}_{\overline{\mu }\mu }\hfill \end{array}\end{array}$$

(195)

for $H_{\overline{\mu }\mu }\in {\Omega }^0\left({\mathbb{R}}^{1,2}\right)$, because we treat the left and right factors of the kinematical vector space ${\mathfrak{Kin}}_{{\wedge }^{•}}$ as independent.

The 2-bracket ${\widehat{\mu }}_2^{\mathrm{CS}}:{\Omega }^{p,q}\left({\mathbb{R}}^{1,2}\right)[-i]\otimes {\Omega }^{r,s}\left({\mathbb{R}}^{1,2}\right)[-j]⟶{\Omega }^{p+r,q+s}\left({\mathbb{R}}^{1,2}\right)[-i-j]$ of $\widehat{\mathfrak{CS}}$ in addition acts non-trivially on “off-diagonal” fields. It is easily identified as the map which scales the tensor product $\frac{\kappa }{2}\wedge \otimes \wedge$ of independent exterior products on the left and right kinematical vector spaces. For example, the bracket of two fields of degree 1 is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{\mu }}_2^{\mathrm{CS}}(e^{\overline{\mu }}\otimes e^{\mu }\otimes H_{\overline{\mu }\mu },e^{\overline{\nu }}\otimes e^{\nu }\otimes H_{\overline{\nu }\nu }^{\prime })& =(e^{\overline{\mu }}\wedge e^{\overline{\nu }})\otimes m_2^{{\tau }^{{\Omega }^{•}}}(e^{\mu }\otimes H_{\overline{\mu }\mu },e^{\nu }\otimes H_{\overline{\nu }\nu }^{\prime })\hfill \\ \hfill & =(e^{\overline{\mu }}\wedge e^{\overline{\nu }})\otimes (e^{\mu }\wedge e^{\nu })\otimes \frac{\kappa }{2}{H}_{\overline{\mu }\mu }{H}_{\overline{\nu }\nu }^{\prime }.\hfill \end{array}\end{array}$$

(196)

The cyclic structure is obtained by composing the tensor product of Hodge duality operators with wedge products in complementary degrees in each kinematic factor:

$$\begin{array}{c}\hfill {\langle -,-\rangle }_{\widehat{\mathrm{CS}}}:={\mathrm{Tr}}_{{\wedge }^{•}}\otimes {\mathrm{Tr}}_{{\wedge }^{•}}\otimes {\langle -,-\rangle }_{\mathrm{Scal}}.\end{array}$$

(197)

Altogether, this makes the twisted tensor product (193) into a cyclic strict $L_{\infty }$-algebra.

In Maurer–Cartan theory for the $L_{\infty }$-algebra $\widehat{\mathfrak{CS}}$, the underlying homogeneous spaces of the “diagonal” cochain complex (194) contain all relevant data of the double copy field theory, because only that part of the complex is non-trivial. The Maurer–Cartan equation

$$\begin{array}{c}\hfill {\widehat{f}}_H^{\mathrm{CS}}:={\widehat{\mu }}_1^{\mathrm{CS}}\left(H\right)+\frac{1}{2}{\widehat{\mu }}_2^{\mathrm{CS}}(H,H)=0\end{array}$$

(198)

for $H\in {\Omega }^{1,1}\left({\mathbb{R}}^{1,2}\right)$ gives the equation of motion

$$\begin{array}{c}\hfill {ϵ}^{\overline{\mu }\overline{\nu }\overline{\rho }}{ϵ}^{\mu \nu \rho }\left(\frac{1}{\square }{\partial }_{\overline{\nu }}{\partial }_{\nu }{H}_{\overline{\rho }\rho }+\frac{\kappa }{4}{H}_{\overline{\nu }\nu }{H}_{\overline{\rho }\rho }\right)=0.\end{array}$$

(199)

These are the variational equations for the non-local classical action functional which is extracted from the Maurer–Cartan functional

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{S}}_{\mathrm{CS}}\left[H\right]& =\frac{1}{2}{\langle H,{\widehat{\mu }}_1^{\mathrm{CS}}\left(H\right)\rangle }_{\widehat{\mathrm{CS}}}+\frac{1}{3!}{\langle H,{\widehat{\mu }}_2^{\mathrm{CS}}(H,H)\rangle }_{\widehat{\mathrm{CS}}}\hfill \\ \hfill & =\int {\mathrm{d}}^{3}x{ϵ}^{\overline{\mu }\overline{\nu }\overline{\rho }}{ϵ}^{\mu \nu \rho }\left(H_{\overline{\mu }\mu }\frac{1}{\square }{\partial }_{\overline{\nu }}{\partial }_{\nu }{H}_{\overline{\rho }\rho }+\frac{\kappa }{6}{H}_{\overline{\mu }\mu }{H}_{\overline{\nu }\nu }{H}_{\overline{\rho }\rho }\right).\hfill \end{array}\end{array}$$

(200)

This theory is invariant under the variations ${\widehat{\delta }}_{C}H={\widehat{\mu }}_1^{\mathrm{CS}}\left(C\right)+{\widehat{\mu }}_2^{\mathrm{CS}}(H,H)$ for a gauge parameter $C\in {\Omega }^{0,0}\left({\mathbb{R}}^{1,2}\right)\simeq {\Omega }^0\left({\mathbb{R}}^{1,2}\right)$, which reads explicitly as

$$\begin{array}{c}\hfill {\widehat{\delta }}_C{H}_{\overline{\mu }\mu }=\frac{1}{\square }{\partial }_{\overline{\mu }}{\partial }_{\mu }C+\frac{\kappa }{2}C{H}_{\overline{\mu }\mu }.\end{array}$$

(201)

The Noether identity corresponding to this local gauge symmetry is the off-shell Bianchi identity for the curvature in Equation (198): ${\widehat{\mu }}_1^{\mathrm{CS}}\left({\widehat{f}}_H^{\mathrm{CS}}\right)+{\widehat{\mu }}_2^{\mathrm{CS}}(H,{\widehat{f}}_H^{\mathrm{CS}})=0$.

The three-particle Feynman vertex of this double copy field theory is

(202)

The Maurer–Cartan functional (200) matches precisely with the non-local gauge-fixed action functional of [39], which was obtained by identifying $H_{\overline{\mu }\mu }=A_{\overline{\mu }}\otimes A_{\mu }$ as a tensor product of fields in momentum space, and writing down its propagator and interaction vertex by double copying the Chern–Simons kinematic numerators in Lorenz gauge. In our approach, the result of the double copy procedure applied to Chern–Simons theory is immediate, with its local gauge symmetry automatically identified, and with the gauge-fixing prescription needed for the colour–kinematics duality of Section 4.4 manifest in the homotopy double copy construction without additional input.

Our perspective also allows for a simple construction of the superspace formulation of the double copy in [39] that includes all bosonic fields and ghosts: Applying the Maurer–Cartan functional to a superfield $\mathit{H}\in \mathsf{Fun}\left({\Omega }^{•,•}\left({\mathbb{R}}^{1,2}\right)\left[1\right]\right)\otimes {\Omega }^{•,•}\left({\mathbb{R}}^{1,2}\right)$ yields the BV action functional

$${\widehat{S}}_{\mathrm{BV}}\left[\mathit{H}\right]=\frac{1}{2}{\langle \mathit{H},{\widehat{\mu }}_1^{\mathrm{CS}\mathrm{ext}}\left(\mathit{H}\right)\rangle }_{\widehat{\mathrm{CS}}\mathrm{ext}}+\frac{1}{3!}{\langle \mathit{H},{\widehat{\mu }}_2^{\mathrm{CS}\mathrm{ext}}(\mathit{H},\mathit{H})\rangle }_{\widehat{\mathrm{CS}}\mathrm{ext}},$$

(203)

where $\mathit{H}=\overline{\mathit{A}}\otimes \mathit{A}$ is a double copy of two Chern–Simons superfields (158) and the notation is defined in Appendix A.2. Therefore, the homotopy double copy framework offers a more systematic and rigorous method of tackling the problem.

4.6. ${\mathfrak{u}}_{★}\left(1\right)$ Chern–Simons Theory as a Double Copy

As with the adjoint scalar theory from Section 3.3, the rank-one limit $N=1$ of noncommutative Chern–Simons theory is an interacting theory of photons with no non-trivial commutative counterpart. The noncommutative $\mathsf{U}\left(1\right)$ Chern–Simons gauge theory on ${\mathbb{R}}^{1,2}$ is defined by the action functional

$$\begin{array}{c}\hfill {\widehat{S}}_{{\mathrm{CS}}_{★}\left(1\right)}\left[A\right]=\int \frac{1}{2}A{\wedge }_{★}\mathrm{d}A+\frac{\kappa }{6}A{\wedge }_{★}A{\wedge }_{★}A,\end{array}$$

(204)

for $A\in {\Omega }^1\left({\mathbb{R}}^{1,2}\right)$. This theory is organised by the cyclic strict $L_{\infty }$-algebra

$$\begin{array}{c}\hfill \widehat{{\mathfrak{CS}}_{★}\left(1\right)}=\left(\left({\Omega }^{•}\left({\mathbb{R}}^{1,2}\right),{\widehat{\mu }}_1^{{\mathrm{CS}}_{★}\left(1\right)}=\mathrm{d}\right),{\widehat{\mu }}_2^{{\mathrm{CS}}_{★}\left(1\right)}=\frac{\kappa }{2}{[-\stackrel{★}{,}-]}_{\mathfrak{u}\left(1\right)},{\langle -,-\rangle }_{{\Omega }^{•}}\right),\end{array}$$

(205)

whose underlying cochain complex is just the de Rham complex of differential forms on ${\mathbb{R}}^{1,2}$.

As the notation suggests, we regard this noncommutative gauge theory without colour degrees of freedom itself as a double copy of a field theory with colour symmetry. This is analogous to the rank-one limits $N=\overline{N}=1$ of the binoncommutative biadjoint scalar theory of Section 3.6, which results in the homotopy double copy theory of Section 3.5. The two equivalent twisted factorisations (171) and (172) offer two (equivalent) perspectives on the origins of this double copied theory, which corroborates the old suggestions, discussed in Section 1, that noncommutative $\mathsf{U}\left(1\right)$ gauge theories realise models of gravity in certain senses, despite the absence of propagating spin-two states.

The twisted factorisation (172) for $N=1$ is given by

$$\begin{array}{c}\hfill \widehat{{\mathfrak{CS}}_{★}\left(1\right)}={\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}{\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(1\right)}.\end{array}$$

(206)

We use the factorisation (85), together with the interpretation of the Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$ as a kinematical factor with vector space ${\mathfrak{Kin}}_{{\wedge }^0}=\mathbb{R}$ from Section 3.3, to write the factorisation of the cyclic strict $L_{\infty }$-algebra of $\mathsf{U}\left(1\right)$ noncommutative Chern–Simons theory as

$$\begin{array}{c}\hfill \widehat{{\mathfrak{CS}}_{★}\left(1\right)}={\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}\left({\mathfrak{Kin}}_{{\wedge }^0}{\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(1\right)}}\mathfrak{Scal}\right).\end{array}$$

(207)

The underlying cochain complex in this factorisation is again given by Equation (170). Comparing Equations (96) and (207), and recalling the colour–kinematics duality of Section 3.4, we conclude that the noncommutative $\mathsf{U}\left(1\right)$ Chern–Simons theory is a double copy of the $d=3$ adjoint scalar theory of Section 3.3 for any colour algebra $\mathfrak{g}$.

To better understand this statement, we note that the construction of Section 4.3 shows that ordinary Chern–Simons gauge theory based on a quadratic Lie algebra $\mathfrak{g}$ is organised by a cyclic strict $L_{\infty }$-algebra $\mathfrak{CS}$ which admits the factorisation

$$\begin{array}{c}\hfill \mathfrak{CS}=\mathfrak{g}\otimes \left({\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}\mathfrak{Scal}\right),\end{array}$$

(208)

because the colour-stripping is not twisted in the commutative case. Instead of the double copy (115) of the adjoint scalar theory with itself that we considered in Section 3.5, we can take its double copy with commutative Chern–Simons theory for any gauge algebra $\mathfrak{g}$, which involves the kinematical vector space ${\mathfrak{Kin}}_{{\wedge }^{•}}$ and leads to the double copy prescription of Equation (207).

We denote this double copy operation symbolically by

$$\begin{array}{c}\hfill {\mathfrak{u}}_{★}\left(1\right)\mathrm{Chern}–\mathrm{Simons}=\mathrm{Adjoint}\mathrm{Scalar}\otimes \mathrm{Ordinary}\mathrm{Chern}–\mathrm{Simons}\end{array}$$

(209)

with reducible fields $A_{\mu }^{{\mathfrak{u}}_{★}\left(1\right)}=\varphi \otimes A_{\mu }^{\mathrm{CS}}$. The twisted tensor product construction of Equation (207) combines the kinematic numerators of the adjoint scalar theory, based on the Lie algebra ${\mathfrak{u}}_{★}\left(1\right)$, with those of commutative Chern–Simons theory, based on the Lie algebra ${\mathfrak{diff}}_{\mathrm{vol}}\left({\mathbb{R}}^{1,2}\right)$.

Alternatively, again with the understanding of ${\mathfrak{u}}_{★}\left(1\right)$ as a kinematical Lie algebra analogously to Section 3.3, the twisted factorisation (171) for $N=1$ is given by

$$\begin{array}{c}\hfill \widehat{{\mathfrak{CS}}_{★}\left(1\right)}={\mathfrak{Kin}}_{{\wedge }^0}{\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(1\right)}}\left({\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}\mathfrak{Scal}\right).\end{array}$$

(210)

Comparing Equations (208) and (210), and recalling the colour–kinematics duality of Section 4.4, we conclude now that the noncommutative $\mathsf{U}\left(1\right)$ Chern–Simons theory is a double copy of the commutative Chern–Simons theory for any gauge algebra $\mathfrak{g}$ with the adjoint scalar theory. This perspective is very natural, as the propagators in the commutative and noncommutative gauge theories are the same while the three-point interaction vertex is copied as

$$\begin{array}{c}\hfill -\mathrm{i}g{ϵ}_{\mu \nu \rho }{f}^{abc}⟶-\frac{\mathrm{i}\kappa }{2}{ϵ}_{\mu \nu \rho }F(k,p,q)\end{array}$$

(211)

in terms of the structure constants of the Lie algebras $\mathfrak{g}$ and ${\mathfrak{u}}_{★}\left(1\right)$. As the twisted tensor product operations commute, i.e., formally ${\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}{\mathfrak{Kin}}_{{\wedge }^0}{\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(1\right)}}={\mathfrak{Kin}}_{{\wedge }^0}{\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(1\right)}}{\mathfrak{Kin}}_{{\wedge }^{•}}{\otimes }_{{\tau }^{{\Omega }^{•}}}$, the two double copy operations lead to the same theory (209), as expected on heuristic grounds.

Diffeomorphism Invariance

The rank-one limit of the brackets (189) is given by

$$\begin{array}{c}〚J\left(w_1\right),J\left(w_2\right){〛}_{\mathrm{CS}}^{★}={[{\epsilon }_{p_{w_1}},{\epsilon }_{p_{w_2}}]}_{\mathfrak{diff}\left({\mathbb{R}}^{1,2}\right)}\otimes 2\mathrm{i}sin\left(\frac{t}{2}{p}_{w_1}·\theta p_{w_2}\right)e_{p_{w_3}}\delta (p_{w_1}+p_{w_2}-p_{w_3}).\end{array}$$

(212)

This infinite-dimensional Lie algebra is generated by the divergence-free vector fields

$${\epsilon }_{\mu }\left(p\right)={\mathrm{e}}^{\mathrm{i}p·x}{\Pi }_{\mu \nu }\left(p\right){\partial }^{\nu }$$

(213)

for a transverse projection tensor ${\Pi }_{\mu \nu }\left(p\right)$ with ${\Pi }_{\mu \nu }\left(p\right)p^{\mu }=0$. Their deformed Lie bracket is

$$\begin{array}{c}\hfill {\left[{\epsilon }_{\mu }\left(p_1\right)\stackrel{★}{,}{\epsilon }_{\nu }\left(p_2\right)\right]}_{\mathfrak{u}\left(1\right)}=\int {\mathrm{d}}^3{p}_3{\mathtt{F}}_{\mu \nu }{}^{\rho }(p_1,p_2,p_3){\epsilon }_{\rho }\left(p_3\right),\end{array}$$

(214)

where

$$\begin{array}{cc}\hfill {\mathtt{F}}_{\mu \nu }{}^{\rho }(p_1,p_2,p_3)& =2\mathrm{i}sin\left(\frac{t}{2}{p}_1·\theta p_2\right){\Pi }_{\mu {\mu }^{\prime }}\left(p_1\right){ϵ}^{{\mu }^{\prime }{\nu }^{\prime }\rho }{\Pi }_{{\nu }^{\prime }\nu }\left(p_2\right)\delta (p_1+p_2-p_3).\hfill \end{array}$$

(215)

This is the twisted tensor product of the Lie algebra of infinitesimal volume-preserving diffeomorphisms of ${\mathbb{R}}^{1,2}$ with the abelian Lie algebra $\mathfrak{u}\left(1\right)$. In the semi-classical limit, it is the Poisson–Lie algebra of symplectic diffeomorphisms of ${\mathbb{R}}^{1,2}$.

In the commutative case, a double copy theory which is mapped from a gauge theory should be invariant under diffeomorphisms. Here we find that ${\mathfrak{u}}_{★}\left(1\right)$ Chern–Simons theory, interpreted as a double copied theory, realises a subalgebra of ${\mathfrak{diff}}_{\mathrm{vol}}\left({\mathbb{R}}^{1,2}\right)$ consisting of deformed symplectic diffeomorphisms of spacetime, in the sense of [32]. This vindicates our interpretation of the ${\mathfrak{u}}_{★}\left(1\right)$ Chern–Simons theory as a genuine double copy, but with reduced diffeomorphism symmetry at the amplitude level.

5. Noncommutative Yang–Mills Theory

5.1. The $L_{\infty }$-Structure of Noncommutative Yang–Mills Theory

Consider the quadratic Lie algebra $\left(\mathfrak{u}\left(N\right),{[-,-]}_{\mathfrak{u}\left(N\right)},{\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\right)$ with the usual normalisation of the trace of generators ${\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(T^a{T}^b\right)={\delta }^{ab}$. We work on d-dimensional Minkowski spacetime ${\mathbb{R}}^{1,d-1}$, with Hodge duality operator denoted ${\ast }_{\mathrm{H}}:{\Omega }^p\left({\mathbb{R}}^{1,d-1}\right)⟶{\Omega }^{d-p}\left({\mathbb{R}}^{1,d-1}\right)$. The corresponding codifferential $\delta :{\Omega }^p\left({\mathbb{R}}^{1,d-1}\right)⟶{\Omega }^{p-1}\left({\mathbb{R}}^{1,d-1}\right)$ is given by $\delta ={(-1)}^{dp+1}{\ast }_{\mathrm{H}}\mathrm{d}{\ast }_{\mathrm{H}}$.

The classical action functional for standard $\mathsf{U}\left(N\right)$ noncommutative Yang–Mills theory on ${\mathbb{R}}^{1,d-1}$ is given by

$$S_{\mathrm{YM}}^{★}\left[A\right]=\frac{1}{2}\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(F_A^{★}{\wedge }_{★}{\ast }_{\mathrm{H}}{F}_A^{★}\right),$$

(216)

with gauge field $A\in {\Omega }^1({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$ and noncommutative field strength $F_A^{★}\in {\Omega }^2({\mathbb{R}}^{1,d-1},$$\mathfrak{u}\left(N\right))$ given by Equation (150), where g is the Yang–Mills coupling constant. The noncommutative Yang–Mills functional is invariant under infinitesimal star-gauge transformations ${\delta }_c^{★}A=\mathrm{d}c+g{\left[c\stackrel{★}{,}A\right]}_{\mathfrak{u}\left(N\right)}$ with $c\in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$.

This noncommutative gauge theory is organised into a cyclic $L_{\infty }$-algebra

$${\mathfrak{YM}}_{★}=\left(\mathsf{Ch}\left({\mathfrak{YM}}_{★}\right),{\mu }_2^{★\mathrm{YM}},{\mu }_3^{★\mathrm{YM}},{\langle -,-\rangle }_{\mathrm{YM}}^{★}\right).$$

(217)

The underlying cochain complex $\mathsf{Ch}\left({\mathfrak{YM}}_{★}\right)$ is given by

$$\begin{array}{cc}\hfill {\Omega }^0({\mathbb{R}}^{1,d-1},& \mathfrak{u}\left(N\right))\stackrel{\mathrm{d}}{\to }{\Omega }^1({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))[-1]\hfill \\ & \stackrel{\delta \mathrm{d}}{\to }{\Omega }^1({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))[-2]\stackrel{\delta }{\to }{\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))[-3],\hfill \end{array}$$

(218)

which identifies the differential as ${\mu }_1^{★\mathrm{YM}}\left(c\right)=\mathrm{d}c$ on ghosts c in ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$, as ${\mu }_1^{★\mathrm{YM}}\left(A\right)=\delta \mathrm{d}A$ on gauge fields A in ${\Omega }^1({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))[-1]$, and as ${\mu }_1^{★\mathrm{YM}}\left(A^{+}\right)=\delta A^{+}$ on antifields $A^{+}$ in ${\Omega }^1({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))[-2]$.

Together with the ghost antifields $c^{+}\in {\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))[-3]$, the non-zero 2-brackets are given by

$$\begin{array}{cc}\hfill {\mu }_2^{★\mathrm{YM}}(c_1,c_2)=g{\left[c_1\stackrel{★}{,}{c}_2\right]}_{\mathfrak{u}\left(N\right)},& {\mu }_2^{★\mathrm{YM}}(c,A)=g{\left[c\stackrel{★}{,}A\right]}_{\mathfrak{u}\left(N\right)},{\mu }_2^{★\mathrm{YM}}(c,A^{+})=g{\left[c\stackrel{★}{,}{A}^{+}\right]}_{\mathfrak{u}\left(N\right)},\hfill \\ \hfill {\mu }_2^{★\mathrm{YM}}(c,c^{+})=g{\left[c\stackrel{★}{,}{c}^{+}\right]}_{\mathfrak{u}\left(N\right)},& {\mu }_2^{★\mathrm{YM}}(A,A^{+})=g{\left[A\stackrel{★}{,}{A}^{+}\right]}_{\mathfrak{u}\left(N\right)},\hfill \\ \hfill {\mu }_2^{★\mathrm{YM}}(A_1,A_2)=g\left(\delta \right[A_1& \stackrel{★}{,}{A}_2{]}_{\mathfrak{u}\left(N\right)}+{\ast }_{\mathrm{H}}{\left[A_1\stackrel{★}{,}{\ast }_{\mathrm{H}}\mathrm{d}{A}_2\right]}_{\mathfrak{u}\left(N\right)}+{\ast }_{\mathrm{H}}{\left[{\ast }_{\mathrm{H}}\mathrm{d}{A}_1\stackrel{★}{,}{A}_2\right]}_{\mathfrak{u}\left(N\right)}).\hfill \end{array}$$

(219)

The single non-zero higher bracket ${\mu }_3^{★\mathrm{YM}}:{\Omega }^1({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right)){)[-1]}^{\otimes 3}⟶{\Omega }^1({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$$[-3]$ acts as

$$\begin{array}{cc}\hfill {\mu }_3^{★\mathrm{YM}}(A_1,A_2,A_3)& =g^2({\ast }_{\mathrm{H}}{\left[A_1\stackrel{★}{,}{\ast }_{\mathrm{H}}{\left[A_2\stackrel{★}{,}{A}_3\right]}_{\mathfrak{u}\left(N\right)}\right]}_{\mathfrak{u}\left(N\right)}+{\ast }_{\mathrm{H}}{\left[A_2\stackrel{★}{,}{\ast }_{\mathrm{H}}{\left[A_3\stackrel{★}{,}{A}_1\right]}_{\mathfrak{u}\left(N\right)}\right]}_{\mathfrak{u}\left(N\right)}\hfill \\ \hfill & +{\ast }_{\mathrm{H}}{\left[A_3\stackrel{★}{,}{\ast }_{\mathrm{H}}{\left[A_1\stackrel{★}{,}{A}_2\right]}_{\mathfrak{u}\left(N\right)}\right]}_{\mathfrak{u}\left(N\right)}).\hfill \end{array}$$

(220)

The cyclic structure of $L_{\infty }$-degree $-3$ is given by the Hodge inner product of differential forms in the same exterior degree:

$${\langle \alpha ,{\alpha }^{+}\rangle }_{\mathrm{YM}}^{★}=\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(\alpha {\wedge }_{★}{\ast }_{\mathrm{H}}{\alpha }^{+}\right)=\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}(\alpha \wedge {\ast }_{\mathrm{H}}{\alpha }^{+}),$$

(221)

for $\alpha \in \{c,A\}$.

Applying the Maurer–Cartan theory from Appendix A.2 for the cyclic $L_{\infty }$-algebra (217), the Maurer–Cartan curvature $f_A^{★\mathrm{YM}}\in {\Omega }^1({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))[-2]$ from Equation (A17) reads

$$f_A^{★\mathrm{YM}}={\mu }_1^{★\mathrm{YM}}\left(A\right)+\frac{1}{2!}{\mu }_2^{★\mathrm{YM}}(A,A)+\frac{1}{3!}{\mu }_3^{★\mathrm{YM}}(A,A,A)=-{\ast }_{\mathrm{H}}{\nabla }_A^{★}{\ast }_{\mathrm{H}}{F}_A^{★},$$

(222)

where ${\nabla }_A^{★}:{\Omega }^p({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))⟶{\Omega }^{p+1}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$ is the covariant derivative given by Equation (151). The Maurer–Cartan equation $f_A^{★\mathrm{YM}}=0$ is therefore equivalent to the noncommutative Yang–Mills equations ${\nabla }_A^{★}{\ast }_{\mathrm{H}}{F}_A^{★}=0$.

The Maurer–Cartan–Bianchi identity in ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))[-3]$ is the Noether identity for the star-gauge symmetry ${\delta }_c^{★}A={\mu }_1^{★\mathrm{YM}}\left(c\right)+{\mu }_2^{★\mathrm{YM}}(c,A)$ of noncommutative Yang–Mills theory, which is satisfied off-shell. It expresses the fact that the Maurer–Cartan curvature $f_A^{★\mathrm{YM}}$ is covariantly constant for the Maurer–Cartan covariant derivative

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{d}}_A^{★\mathrm{YM}}{f}_A^{★\mathrm{YM}}& ={\mu }_1^{★\mathrm{YM}}\left(f_A^{★\mathrm{YM}}\right)+{\mu }_2^{★\mathrm{YM}}(A,f_A^{★\mathrm{YM}})\hfill \\ \hfill & =-{\ast }_{\mathrm{H}}{\nabla }_A^{★}{\ast }_{\mathrm{H}}{f}_A^{★\mathrm{YM}}={\ast }_{\mathrm{H}}{\left({\nabla }_A^{★}\right)}^2{\ast }_{\mathrm{H}}{F}_A^{★}={\ast }_{\mathrm{H}}{\left[F_A^{★}\stackrel{★}{,}{\ast }_{\mathrm{H}}{F}_A^{★}\right]}_{\mathfrak{u}\left(N\right)}=0,\hfill \end{array}\end{array}$$

(223)

which follows directly from the symmetry properties of the four-form ${\left[F_A^{★}\stackrel{★}{,}{\ast }_{\mathrm{H}}{F}_A^{★}\right]}_{\mathfrak{u}\left(N\right)}$ in the Lie algebra ${\mathfrak{u}}_{★}\left(N\right)$.

Finally, the Maurer–Cartan functional

$$\begin{array}{cc}\hfill S_{\mathrm{YM}}^{★}\left[A\right]& =\frac{1}{2}{〈A,{\mu }_1^{★\mathrm{YM}}\left(A\right)〉}_{\mathrm{YM}}^{★}+\frac{1}{3!}{〈A,{\mu }_2^{★\mathrm{YM}}(A,A)〉}_{\mathrm{YM}}^{★}+\frac{1}{4!}{〈A,{\mu }_3^{★\mathrm{YM}}(A,A,A)〉}_{\mathrm{YM}}^{★}\hfill \end{array}$$

(224)

recovers the noncommutative Yang–Mills functional from Equation (216).

5.1.1. Batalin–Vilkovisky Formalism

Following the same steps as in Section 4.2, the superspace extension of the noncommutative Yang–Mills functional (216) yields the BV action functional (See e.g., [76] for the BRST extension including the antighost of the ghost field c and the Nakanishi–Lautrup auxiliary field).

$$\begin{array}{c}\hfill S_{\mathrm{BV}}^{★}\left[\mathit{A}\right]=\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(\frac{1}{2}{F}_A^{★}{\wedge }_{★}{\ast }_{\mathrm{H}}{F}_A^{★}+A^{+}{\wedge }_{★}{\ast }_{\mathrm{H}}{\nabla }_A^{★}c+\frac{g}{2}{c}^{+}{\wedge }_{★}{\ast }_{\mathrm{H}}{\left[c\stackrel{★}{,}c\right]}_{\mathfrak{u}\left(N\right)}\right).\end{array}$$

(225)

In contrast to the noncommutative Chern–Simons theory of Section 4, the noncommutative deformation here corresponds to a non-trivial local BRST cohomology class and so it is non-trivial [72,73]: the $L_{\infty }$-algebra ${\mathfrak{YM}}_{★}$ is not quasi-isomorphic to the $L_{\infty }$-algebra $\mathfrak{YM}$ of ordinary Yang–Mills theory. This explains why the double copy construction of Section 4 was so straightforward; for noncommutative Yang–Mills theory we must work harder.

5.1.2. Colour Ordering and Decomposition

A common way to organise n-point amplitude calculations in gauge theories is by summing over inequivalent orderings of colour factors, which yields reduction formulas from $n!$ down to $(n-1)!$ inequivalent amplitudes. As with the biadjoint scalar theory from Section 2.2, tree-level scattering amplitudes of gluons are organised in terms of gauge-invariant partial amplitudes and colour structures as [77]

$$\begin{array}{c}\hfill {\mathcal{A}}_n^{\mathrm{YM}}(p,\zeta ,a)=\sum _{\sigma \in S_n/{\mathbb{Z}}_n}{\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}(T^{a_{\sigma \left(1\right)}}\dots T^{a_{\sigma \left(n\right)}}){𝓐}_n^{\mathrm{YM}}\left(\sigma \left(1\right),\dots ,\sigma \left(n\right)\right).\end{array}$$

(226)

Each Yang–Mills partial amplitude appears twice due to the ordering properties

$$\begin{array}{c}\hfill {𝓐}_n^{\mathrm{YM}}(1,\dots ,n-1,n)={𝓐}_n^{\mathrm{YM}}(n,1,\dots ,n-1)={(-1)}^n{𝓐}_n^{\mathrm{YM}}(n,n-1,\cdots ,1),\end{array}$$

(227)

which enables one to restrict the sum to planar orderings as

$$\begin{array}{c}\hfill {\mathcal{A}}_n^{\mathrm{YM}}(p,\zeta ,a)=\sum _{\sigma \in S_n/{\mathbb{Z}}_n⋊{\mathbb{Z}}_2}C\left(\sigma \right){𝓐}_n^{\mathrm{YM}}\left(\sigma \left(1\right),\cdots ,\sigma \left(n\right)\right).\end{array}$$

(228)

In [78,79] it is shown that noncommutative gauge theories still respect the colour decomposition. As we have seen already in Equation (128), the structure constants $F^{abc}(p_1,p_2,p_3)$ of ${\mathfrak{u}}_{★}\left(N\right)$ can be decomposed to give the three-point vertex for noncommutative Yang–Mills theory in the form

(229)

where for simplicity we suppress momentum conserving delta-functions. Note that the two summands are considered to be parts of different colour-ordered amplitudes. Similarly, the four-point vertex can be decomposed into a sum over inequivalent orderings with phase factors dressing the summands as

(230)

Since tree-level amplitudes involve only planar diagrams, colour-ordered amplitudes in noncommutative Yang–Mills theory factorise with the appropriate phase factor [78,79]. Given an ordered word $w=k_1\cdots k_{n-1}{k}_n\in {\mathcal{W}}_n$ labelling colour, polarisations and momenta, for rank $N>1$ one finds that the noncommutative n-point Yang–Mills partial amplitudes ${𝓐}_n^{★\mathrm{YM}}\left(w\right)$ are related to the commutativen-point partial amplitudes ${𝓐}_n^{\mathrm{YM}}\left(w\right)$ through the simple relationship

$$\begin{array}{cc}\hfill {𝓐}_n^{★\mathrm{YM}}\left(w\right)& ={\Theta }_n\left(w\right){𝓐}_n^{\mathrm{YM}}\left(w\right),\hfill \end{array}$$

(231)

with the overall momentum-dependent phase factor

$$\begin{array}{cc}\hfill {\Theta }_n\left(w\right)& =exp\left(-\frac{\mathrm{i}t}{2}\sum _{i<j}{p}_{k_i}·\theta p_{k_j}\right).\hfill \end{array}$$

(232)

Due to antisymmetry of the bivector $\theta$, the phase is cyclically invariant: ${\Theta }_n\left(w\right)={\Theta }_n\left(w^{⥀}\right)$ where $w^{⥀}:=k_n{k}_1\cdots k_{n-1}\in {\mathcal{W}}_n$, and symmetric under Kleiss–Kuijf reflection: ${\Theta }_n\left(w\right)={\Theta }_n{\left(\overline{w}\right)}^{-1}$ where $\overline{w}=k_n{k}_{n-1}\cdots k_1\in {\mathcal{W}}_n$. The ordering properties (227) then imply

$$\begin{array}{c}\hfill {𝓐}_n^{★\mathrm{YM}}\left(w\right)={𝓐}_n^{★\mathrm{YM}}\left(w^{⥀}\right)={(-1)}^n{\Theta }_n{\left(\overline{w}\right)}^{-2}{𝓐}_n^{★\mathrm{YM}}\left(\overline{w}\right).\end{array}$$

(233)

This argument will be corroborated later when we embed the gauge theory as a low-energy limit of open string theory with a constant background B-field. It suggests that a twisted homotopy factorisation is possible for noncommutative Yang–Mills theory.

5.2. Strictification and Twisted Homotopy Factorisation

5.2.1. Strictification in the Second-Order Formalism

Just like ordinary Yang–Mills theory, because of the non-trivial 3-bracket in Equation (220), the $L_{\infty }$-structure of noncommutative Yang–Mills theory is not strict. The twisted homotopy factorisation, and hence the homotopy double copy construction, only makes sense for strict $L_{\infty }$-algebras. This means finding a perturbatively equivalent theory with no vertices of order higher than three.

If one further wishes to double copy this theory with itself, a cubic Lagrangian whose Feynman diagrams produce tree-level gluon amplitudes in twisted colour–kinematics dual form must be sought; in the commutative case this is always possible [80]. This is conducted order by order in the multiplicity, by finding quasi-isomorphisms between the original theory and a strictified theory (organised by a strict $L_{\infty }$-algebra) that has manifest colour–kinematics duality. It extends the underlying cochain complex $\mathsf{Ch}\left({\mathfrak{YM}}_{★}\right)$ to include auxiliary fields that enforce both the strict $L_{\infty }$-structure and the colour–kinematics duality, such that putting the auxiliary fields on-shell yields the higher order vertex.

We generalise the arguments of [12] to construct a strictification of noncommutative Yang–Mills theory that produces twisted colour–kinematics dual gluon amplitudes up to multiplicity four. For this, we introduce an auxiliary field $G\in {\Omega }^{1,2}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$, where

$$\begin{array}{c}\hfill {\Omega }^{1,2}\left({\mathbb{R}}^{1,d-1}\right):={\Omega }^1\left({\mathbb{R}}^{1,d-1}\right){\otimes }_{{\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)}{\Omega }^2\left({\mathbb{R}}^{1,d-1}\right).\end{array}$$

(234)

Written in component form, the cubic noncommutative Yang–Mills functional we consider is

$$\begin{array}{cc}\hfill S_{{\mathrm{YM}}_2}^{★}[A,G]& =\frac{1}{2}\int {\mathrm{d}}^{d}x{\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}(A_{\mu }★\square A^{\mu }-2g{A}_{\nu }★{\partial }_{\mu }{\left[A^{\mu }\stackrel{★}{,}{A}^{\nu }\right]}_{\mathfrak{u}\left(N\right)}\hfill \\ \hfill & +\sqrt{2}g{\partial }_{\overline{\alpha }}\left({\left[A_{\mu }\stackrel{★}{,}{A}_{\nu }\right]}_{\mathfrak{u}\left(N\right)}\right)★G^{\overline{\alpha }\mu \nu }-G_{\overline{\alpha }\mu \nu }★\square G^{\overline{\alpha }\mu \nu }).\hfill \end{array}$$

(235)

The resulting noncommutative gauge theory is perturbatively equivalent to the original theory after integration over the auxiliary field G, whose equation of motion is

$$\begin{array}{c}\hfill \square G_{\overline{\alpha }\mu \nu }=\frac{1}{2\sqrt{2}}g{\partial }_{\overline{\alpha }}{\left[A_{\mu }\stackrel{★}{,}{A}_{\nu }\right]}_{\mathfrak{u}\left(N\right)}.\end{array}$$

(236)

Substituting Equation (236) into Equation (235) gives back the noncommutative Yang–Mills functional (224), because of the $\mathrm{ad}\left({\mathfrak{u}}_{★}\left(N\right)\right)$-invariance of the cyclic structure. This defines an $L_{\infty }$-quasi-isomorphism ${\psi }^{★{\mathrm{YM}}_2}:{\mathfrak{YM}}_{★}⟶{\mathfrak{YM}}_{★2}$ whose non-vanishing components on gauge fields is given by

$$\begin{array}{c}\hfill {\psi }_1^{★{\mathrm{YM}}_2}\left(A\right)=A\mathrm{and}{\psi }_2^{★{\mathrm{YM}}_2}(A_1,A_2)=\frac{g}{2\sqrt{2}\square }\mathrm{d}\otimes {\left[A_1\stackrel{★}{,}{A}_2\right]}_{\mathfrak{u}\left(N\right)},\end{array}$$

(237)

so that $G={\psi }_2^{★{\mathrm{YM}}_2}(A,A)$.

The new cyclic strict $L_{\infty }$-algebra is

$$\begin{array}{c}\hfill {\mathfrak{YM}}_{★2}=\left(\mathsf{Ch}\left({\mathfrak{YM}}_{★2}\right),{\mu }_2^{★{\mathrm{YM}}_2},{\langle -,-\rangle }_{{\mathrm{YM}}_2}^{★}\right).\end{array}$$

(238)

In the following we focus only on its subspaces in degrees 1 and 2 for illustration and drop the antighost as well as various auxiliary fields including the Nakanishi–Lautrup field. Its cochain complex $\mathsf{Ch}\left({\mathfrak{YM}}_{★2}\right)$ extends the four-term Yang–Mills cochain complex (218) as

(239)

with 2-bracket modified to

$$\begin{array}{c}{\mu }_2^{★{\mathrm{YM}}_2}\left(\left(\begin{array}{c}{A}_1\\ G_1\end{array}\right),\left(\begin{array}{c}{A}_2\\ G_2\end{array}\right)\right)\hfill \\ =\left(\begin{array}{c}{\mu }_2^{★\mathrm{YM}}(A_1,A_2)+\sqrt{2}g{\ast }_{\mathrm{H}}\left({\left[A_1\stackrel{★}{,}{\ast }_{\mathrm{H}}(\delta \otimes \mathbb{1})G_2\right]}_{\mathfrak{u}\left(N\right)}+{\left[A_2\stackrel{★}{,}{\ast }_{\mathrm{H}}(\delta \otimes \mathbb{1})G_1\right]}_{\mathfrak{u}\left(N\right)}\right)\\ \sqrt{2}g\mathrm{d}\otimes {\left[A_1\stackrel{★}{,}{A}_2\right]}_{\mathfrak{u}\left(N\right)}\end{array}\right).\hfill \end{array}$$

(240)

The cyclic structure is extended by

$$\begin{array}{c}\hfill {\langle G,G^{+}\rangle }_{{\mathrm{YM}}_2}^{★}=\int {\mathrm{d}}^{d}x{\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(G_{\overline{\alpha }\mu \nu }★G^{+\overline{\alpha }\mu \nu }\right),\end{array}$$

(241)

for $G\in {\Omega }^{1,2}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))[-1]$ and $G^{+}\in {\Omega }^{1,2}({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))[-2]$.

We will first factorise the kinematic dependence from the differential form part, much like we did in Equation (172). For this, we introduce a graded vector space organising the differential forms for this theory through

$${\mathfrak{Kin}}_{{\mathrm{YM}}_2}:=\cdots \oplus \begin{array}{c}{\wedge }^1{\left({\mathbb{R}}^{1,d-1}\right)}^{\ast }\\ \oplus \\ {\wedge }^1{\left({\mathbb{R}}^{1,d-1}\right)}^{\ast }\otimes {\wedge }^2{\left({\mathbb{R}}^{1,d-1}\right)}^{\ast }\end{array}\oplus \begin{array}{c}{\wedge }^1{\left({\mathbb{R}}^{1,d-1}\right)}^{\ast }[-1]\\ \oplus \\ {\wedge }^1{\left({\mathbb{R}}^{1,d-1}\right)}^{\ast }\otimes {\wedge }^2{\left({\mathbb{R}}^{1,d-1}\right)}^{\ast }[-1]\end{array}\oplus \cdots .$$

(242)

We write $e^{\mu }$ for the basis of covectors on ${\mathbb{R}}^{1,d-1}$ in degree 0 and $e^{+\mu }$ in degree 1. We will also abbreviate $e^{\overline{\alpha }\mu \nu }:=e^{\overline{\alpha }}\otimes (e^{\mu }\wedge e^{\nu })$ and $e^{+\overline{\alpha }\mu \nu }:=e^{+\overline{\alpha }}\otimes (e^{+\mu }\wedge e^{+\nu })$, similarly to our conventions from Section 4.5. This space carries the non-vanishing inner products

$$\begin{array}{c}\hfill \langle \langle e^{\mu },e^{+\nu }\rangle \rangle :={\eta }^{\mu \nu }\mathrm{and}\langle \langle e^{\overline{\alpha }\mu \nu },e^{+\overline{\beta }\rho \lambda }\rangle \rangle :=\frac{1}{2}{\eta }^{\overline{\alpha }\overline{\beta }}({\eta }^{\mu \rho }{\eta }^{\nu \lambda }-{\eta }^{\mu \lambda }{\eta }^{\nu \rho }).\end{array}$$

(243)

Following [12], we introduce a twist datum ${\tau }^{{\mathrm{YM}}_2}=\left({\tau }_1^{{\mathrm{YM}}_2},{\tau }_2^{{\mathrm{YM}}_2}\right)$ for $\mathfrak{u}\left(N\right)\otimes L$, where L is the graded vector space (80). Its non-trivial actions on basis vectors in ${\mathfrak{Kin}}_{{\mathrm{YM}}_2}$ are given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\tau }_1^{{\mathrm{YM}}_2}\left(e^{\mu }\right)& =e^{+\mu }\otimes \mathbb{1},{\tau }_1^{{\mathrm{YM}}_2}\left(e^{\overline{\alpha }\mu \nu }\right)=-e^{+\overline{\alpha }\mu \nu }\otimes \mathbb{1},\hfill \\ \hfill {\tau }_2^{{\mathrm{YM}}_2}(e^{\mu },e^{\nu })& =3\left(e^{+\mu }\otimes ({\partial }^{\nu }\otimes \mathbb{1}+\mathbb{1}\otimes {\partial }^{\nu })-e^{+\nu }\otimes ({\partial }^{\mu }\otimes \mathbb{1}+\mathbb{1}\otimes {\partial }^{\mu })\right)\hfill \\ \hfill & +\sqrt{2}{e}^{+\overline{\alpha }\mu \nu }\otimes ({\partial }_{\overline{\alpha }}\otimes \mathbb{1}+\mathbb{1}\otimes {\partial }_{\overline{\alpha }}),\hfill \\ \hfill {\tau }_2^{{\mathrm{YM}}_2}(e^{\mu },e^{\overline{\alpha }\nu \rho })& =-\frac{1}{\sqrt{2}}\left({\eta }^{\mu \nu }{e}^{+\rho }\otimes \mathbb{1}\otimes {\partial }^{\overline{\alpha }}-{\eta }^{\mu \rho }{e}^{+\nu }\otimes \mathbb{1}\otimes {\partial }^{\overline{\alpha }}\right),\hfill \\ \hfill {\tau }_2^{{\mathrm{YM}}_2}(e^{\overline{\alpha }\nu \rho },e^{\mu })& =\frac{1}{\sqrt{2}}\left({\eta }^{\mu \nu }{e}^{+\rho }\otimes {\partial }^{\overline{\alpha }}\otimes \mathbb{1}-{\eta }^{\mu \rho }{e}^{+\nu }\otimes {\partial }^{\overline{\alpha }}\otimes \mathbb{1}\right).\hfill \end{array}\end{array}$$

(244)

Let ${\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(N\right)}$ be the cyclic strict $C_{\infty }$-algebra which was introduced in Section 3.2.1 for the twisted homotopy factorisation of the star-commutator. It can be used to factorise the star-commutators of coordinate functions $A_{\mu },G_{\alpha \mu \nu }:{\mathbb{R}}^{1,d-1}⟶\mathfrak{u}\left(N\right)$ of the gauge fields and the auxiliary fields. In this way, we obtain the twisted homotopy factorisation

$${\mathfrak{YM}}_{★2}={\mathfrak{Kin}}_{{\mathrm{YM}}_2}{\otimes }_{{\tau }^{{\mathrm{YM}}_2}}{\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(N\right)}.$$

(245)

To see this, note that, after twisting against the $C_{\infty }$-algebra ${\mathfrak{C}}_{{\mathfrak{u}}_{★}\left(N\right)}$, the first line of Equation (244) easily reproduces the differential ${\mu }_1^{★{\mathrm{YM}}_2}(A,G)=(\delta \mathrm{d}A,-(\delta \mathrm{d}\otimes \mathbb{1})G)$ of Equation (239) on gauge fields and auxiliary fields. On gauge fields $A_1,A_2\in {\Omega }^1({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$, the third equality reproduces the non-strict noncommutative Yang–Mills 2-bracket ${\mu }_2^{★\mathrm{YM}}(A_1,A_2)$ from Equation (219), as well as the last term proportional to $\mathrm{d}\otimes {\left[A_1\stackrel{★}{,}{A}_2\right]}_{\mathfrak{u}\left(N\right)}$ of the strict 2-bracket on degree 1 fields in Equation (240) which lives in ${\Omega }^1\left({\mathbb{R}}^{1,d-1}\right){\otimes }_{{\Omega }^0\left({\mathbb{R}}^{1,d-1}\right)}{\Omega }^2\left({\mathbb{R}}^{1,d-1}\right)$$\otimes \mathfrak{u}\left(N\right)$:

$$\begin{array}{cc}\hfill {\mu }_2^{{\tau }^{{\mathrm{YM}}_2}}(e^{\mu }\otimes A_{1\mu },e^{\nu }\otimes A_{2\nu })& =3g{e}^{+\mu }\otimes \left({\left[{\partial }^{\nu }{A}_{1\mu }\stackrel{★}{,}{A}_{2\nu }\right]}_{\mathfrak{u}\left(N\right)}+{\left[A_{1\mu }\stackrel{★}{,}{\partial }^{\nu }{A}_{2\nu }\right]}_{\mathfrak{u}\left(N\right)}\right)\hfill \\ \hfill & -3g{e}^{+\nu }\otimes \left({\left[{\partial }^{\mu }{A}_{1\mu }\stackrel{★}{,}{A}_{2\nu }\right]}_{\mathfrak{u}\left(N\right)}+{\left[A_{1\mu }\stackrel{★}{,}{\partial }^{\mu }{A}_{2\nu }\right]}_{\mathfrak{u}\left(N\right)}\right)\hfill \\ \hfill & +\sqrt{2}g{e}^{+\overline{\alpha }\mu \nu }\otimes \left({\left[{\partial }_{\overline{\alpha }}{A}_{1\mu }\stackrel{★}{,}{A}_{2\nu }\right]}_{\mathfrak{u}\left(N\right)}+{\left[A_{1\mu }\stackrel{★}{,}{\partial }_{\overline{\alpha }}{A}_{2\nu }\right]}_{\mathfrak{u}\left(N\right)}\right).\hfill \end{array}$$

(246)

Similarly, the last two equalities of the twist data from Equation (244) reproduce the mixed terms among $A_i$ and $G_i$ in Equation (240). The cyclic structure ${\langle -,-\rangle }_{{\mathrm{YM}}_2}$ is easily seen to be reproduced by $\langle \langle -,-\rangle \rangle \otimes {\mathrm{Tr}}_{{\mathfrak{u}}_{★}\left(N\right)}$.

Finally, the $C_{\infty }$-algebra part of the theory has already been factorised in Equation (85) using the twist (160) applied to coordinate functions in ${\Omega }^0({\mathbb{R}}^{1,d-1},\mathfrak{u}\left(N\right))$. Therefore, we have constructed a twisted colour–kinematics dual factorisation

$${\mathfrak{YM}}_{★2}={\mathfrak{Kin}}_{{\mathrm{YM}}_2}{\otimes }_{{\tau }^{{\mathrm{YM}}_2}}\left(\mathfrak{u}\left(N\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(N\right)}}\mathfrak{Scal}\right).$$

(247)

This factorisation allows one to compute double copies up to multiplicity four scattering [12] with any theory, regardless of whether its strictification has manifest colour–kinematics duality.

Alternatively, by simply extracting the colour structure first analogously to Equation (171), we find that the factorisation (247) commutes with the twist data defined by Equation (244). Therefore, we can write the equivalent twisted homotopy factorisation

$${\mathfrak{YM}}_{★2}=\mathfrak{u}\left(N\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(N\right)}}\left({\mathfrak{Kin}}_{{\mathrm{YM}}_2}{\otimes }_{{\tau }^{{\mathrm{YM}}_2}}\mathfrak{Scal}\right).$$

(248)

5.2.2. First-Order Formalism

In the foregoing discussion we set $d=4$, because a much more tractable way to strictify noncommutative Yang–Mills theory in four dimensions is to rewrite it in the first-order formalism. Following [81], this reformulates the theory as a non-topological deformation of noncommutative $BF$-theory with only cubic vertices; we will organise it below into a cyclic strict $L_{\infty }$-algebra akin to that of the noncommutative Chern–Simons theory of Section 4.2. This strictification is not manifestly twisted colour–kinematics dual, and so it cannot be used to double copy itself. However, in this form the theory also admits a twisted homotopy factorisation, which can be used to double copy noncommutative Yang–Mills theory with any other theory that has manifest colour–kinematics duality and compute arbitrary n-point scattering amplitudes.

In four spacetime dimensions, the Hodge duality operator defines a complex structure on the vector space ${\Omega }^2({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))$, i.e., an endomorphism ${\ast }_{\mathrm{H}}:{\Omega }^2({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))⟶{\Omega }^2({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))$, acting solely on the differential form part, such that ${\ast }_{\mathrm{H}}^2=-\mathbb{1}$. The cubic noncommutative Yang–Mills functional for an auxiliary $\mathfrak{u}\left(N\right)$-valued two-form field $B\in {\Omega }^2({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))$ and a gauge field $A\in {\Omega }^1({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))$ is given by

$$S_{{\mathrm{YM}}_1}^{★}[A,B]=\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(\frac{1}{2}B{\wedge }_{★}{\ast }_{\mathrm{H}}B-B{\wedge }_{★}{F}_A^{★}\right),$$

(249)

where as usual the corresponding noncommutative field strength is $F_A^{★}=\mathrm{d}A+\frac{g}{2}{\left[A\stackrel{★}{,}A\right]}_{\mathfrak{u}\left(N\right)}$ in ${\Omega }^2({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))$. Dropping the metric-dependent term defines noncommutative $BF$-theory, whose restriction to self–dual fields is the action functional of noncommutative self–dual Yang–Mills theory.

The equations of motion for B and A are found using $\mathrm{ad}\left(\mathfrak{u}\right(N\left)\right)$-invariance of the trace. They, respectively, read as

$$F_A^{★}={\ast }_{\mathrm{H}}B\mathrm{and}{\nabla }_A^{★}B=0,$$

(250)

where ${\nabla }_A^{★}:{\Omega }^p({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))⟶{\Omega }^{p+1}({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))$ is the star-gauge covariant derivative (151). Integrating out B by imposing the on-shell condition in Equation (249), we recover the noncommutative Yang–Mills functional (216). For a gauge parameter $c\in {\Omega }^0({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))$, the action functional (249) is invariant under the star-gauge transformations ${\delta }_{c}A=\mathrm{d}c+g{\left[c\stackrel{★}{,}A\right]}_{\mathfrak{u}\left(N\right)}$ and ${\delta }_{c}B=g{\left[c\stackrel{★}{,}B\right]}_{\mathfrak{u}\left(N\right)}$.

Noncommutative Yang–Mills theory in the first-order formalism is organised by the cyclic strict $L_{\infty }$-algebra

$${\mathfrak{YM}}_{★1}=\left(\mathsf{Ch}\left({\mathfrak{YM}}_{★1}\right),{\mu }_2^{★{\mathrm{YM}}_1},{\langle -,-\rangle }_{{\mathrm{YM}}_1}^{★}\right),$$

(251)

which is quasi-isomorphic to the cyclic $L_{\infty }$-algebra (217) through an injective $L_{\infty }$-morphism ${\psi }^{★{\mathrm{YM}}_2}:{\mathfrak{YM}}_{★}\hookrightarrow {\mathfrak{YM}}_{★1}$. The BV cochain complex $\mathsf{Ch}\left({\mathfrak{YM}}_{★1}\right)$ associated with this $L_{\infty }$-algebra is

(252)

which identifies the differential ${\mu }_1^{★{\mathrm{YM}}_1}$ of degree 1 acting on homogeneous elements as

$$\begin{array}{c}{\mu }_1^{★{\mathrm{YM}}_1}\left(c\right)=(\mathrm{d}c,0),{\mu }_1^{★{\mathrm{YM}}_1}(A,B)=(\mathrm{d}A-{\ast }_{\mathrm{H}}B,\mathrm{d}B),{\mu }_1^{★{\mathrm{YM}}_1}(B^{+},A^{+})=\mathrm{d}{A}^{+}\end{array}.$$

(253)

The non-zero 2-brackets ${\mu }_2^{★{\mathrm{YM}}_1}$ are given by

$$\begin{array}{cc}\hfill {\mu }_2^{★{\mathrm{YM}}_1}(c_1,c_2)=g{\left[c_1\stackrel{★}{,}{c}_2\right]}_{\mathfrak{u}\left(N\right)}& ,{\mu }_2^{★{\mathrm{YM}}_1}\left(c,\left(\begin{array}{c}A\\ B\end{array}\right)\right)=g\left(\begin{array}{c}{\left[c\stackrel{★}{,}A\right]}_{\mathfrak{u}\left(N\right)}\\ {\left[c\stackrel{★}{,}B\right]}_{\mathfrak{u}\left(N\right)}\end{array}\right),\hfill \\ \hfill {\mu }_2^{★{\mathrm{YM}}_1}\left(c,\left(\begin{array}{c}{B}^{+}\\ A^{+}\end{array}\right)\right)=g\left(\begin{array}{c}{\left[c\stackrel{★}{,}{B}^{+}\right]}_{\mathfrak{u}\left(N\right)}\\ {\left[c\stackrel{★}{,}{A}^{+}\right]}_{\mathfrak{u}\left(N\right)}\end{array}\right)& ,{\mu }_2^{★{\mathrm{YM}}_1}(c,c^{+})=g{\left[c\stackrel{★}{,}{c}^{+}\right]}_{\mathfrak{u}\left(N\right)},\hfill \\ \hfill {\mu }_2^{★{\mathrm{YM}}_1}\left(\left(\begin{array}{c}{A}_1\\ B_1\end{array}\right),\left(\begin{array}{c}{A}_2\\ B_2\end{array}\right)\right)& =g\left(\begin{array}{c}{\left[A_1\stackrel{★}{,}{A}_2\right]}_{\mathfrak{u}\left(N\right)}\\ {\left[A_1\stackrel{★}{,}{B}_2\right]}_{\mathfrak{u}\left(N\right)}+{\left[A_2\stackrel{★}{,}{B}_1\right]}_{\mathfrak{u}\left(N\right)}\end{array}\right),\hfill \\ \hfill {\mu }_2^{★{\mathrm{YM}}_1}\left(\left(\begin{array}{c}A\\ B\end{array}\right),\left(\begin{array}{c}{B}^{+}\\ A^{+}\end{array}\right)\right)& =g\left({\left[A\stackrel{★}{,}{A}^{+}\right]}_{\mathfrak{u}\left(N\right)}+{\left[B\stackrel{★}{,}{B}^{+}\right]}_{\mathfrak{u}\left(N\right)}\right).\hfill \end{array}$$

(254)

Finally, as in Section 4.2, the cyclic structure ${\langle -,-\rangle }_{{\mathrm{YM}}_1}^{★}$ pairs forms of complementary degrees:

$$\begin{array}{c}\hfill {\langle \alpha ,{\alpha }^{+}\rangle }_{{\mathrm{YM}}_1}^{★}:=\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}\left(\alpha {\wedge }_{★}{\alpha }^{+}\right)=\int {\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}(\alpha \wedge {\alpha }^{+}),\end{array}$$

(255)

for $\alpha \in \{c,A,B\}$.

5.2.3. Factorisation in the First-Order Formalism

We shall now demonstrate the homotopy factorisation of first-order noncommutative Yang–Mills theory, which is similar to that of noncommutative Chern–Simons theory. We first factorise the colour algebra using the twisting map from Section 4.3, now applied to differential forms on ${\mathbb{R}}^{1,3}$. It reproduces the star-commutator on ${\Omega }^{•}({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))$ as a twisted tensor product with ${\tau }_2^{{\mathfrak{u}}_{★}\left(N\right)}$, and similarly the differential on ${\Omega }^{•}({\mathbb{R}}^{1,3},\mathfrak{u}\left(N\right))$ as a twisted tensor product with ${\tau }_1^{{\mathfrak{u}}_{★}\left(N\right)}$.

This yields the factorisation

$${\mathfrak{YM}}_{★1}=\mathfrak{u}\left(N\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(N\right)}}{\mathfrak{C}}_{{\mathrm{YM}}_1},$$

(256)

where ${\mathfrak{C}}_{{\mathrm{YM}}_1}$ is the colour-stripped $C_{\infty }$-algebra with underlying cochain complex

(257)

Its brackets are given by colour-stripping the $L_{\infty }$-structure of Equation (253), so the differentials $m_1^{{\mathrm{YM}}_1}$ are of the same form, while the 2-brackets $m_2^{{\mathrm{YM}}_1}$ are simply given by exterior products as

$$\begin{array}{cc}\hfill m_2^{{\mathrm{YM}}_1}(c_1,c_2)=g{c}_1{c}_2,& m_2^{{\mathrm{YM}}_1}\left(c,\left(\begin{array}{c}A\\ B\end{array}\right)\right)=g\left(\begin{array}{c}cA\\ cB\end{array}\right),m_2^{{\mathrm{YM}}_1}\left(c,\left(\begin{array}{c}{B}^{+}\\ A^{+}\end{array}\right)\right)=g\left(\begin{array}{c}c{B}^{+}\\ c{A}^{+}\end{array}\right)\hfill \\ \hfill m_2^{{\mathrm{YM}}_1}(c,c^{+})=gc{c}^{+},& m_2^{{\mathrm{YM}}_1}\left(\left(\begin{array}{c}{A}_1\\ B_1\end{array}\right),\left(\begin{array}{c}{A}_2\\ B_2\end{array}\right)\right)=g\left(\begin{array}{c}{A}_1\wedge A_2\\ A_1\wedge B_2+A_2\wedge B_1\end{array}\right),\hfill \\ \hfill m_2^{{\mathrm{YM}}_1}\left(\left(\begin{array}{c}A\\ B\end{array}\right),\left(\begin{array}{c}{B}^{+}\\ A^{+}\end{array}\right)\right)& =g\left(A\wedge A^{+}+B\wedge B^{+}\right).\hfill \end{array}$$

(258)

We further factorise this $C_{\infty }$-algebra as a twisted tensor product

$$\begin{array}{c}\hfill {\mathfrak{C}}_{{\mathrm{YM}}_1}={\mathfrak{Kin}}_{{\mathrm{YM}}_1}{\otimes }_{{\tau }^{{\mathrm{YM}}_1}}\mathfrak{Scal},\end{array}$$

(259)

where the kinematical vector space is identified as

$${\mathfrak{Kin}}_{{\mathrm{YM}}_1}:={\wedge }^0{\left({\mathbb{R}}^{1,3}\right)}^{\ast }\left[1\right]\oplus \begin{array}{c}{\wedge }^1{\left({\mathbb{R}}^{1,3}\right)}^{\ast }\\ \oplus \\ {\wedge }^2{\left({\mathbb{R}}^{1,3}\right)}^{\ast }\end{array}\oplus \begin{array}{c}{\wedge }^2{\left({\mathbb{R}}^{1,3}\right)}^{\ast }[-1]\\ \oplus \\ {\wedge }^3{\left({\mathbb{R}}^{1,3}\right)}^{\ast }[-1]\end{array}\oplus {\wedge }^4{\left({\mathbb{R}}^{1,3}\right)}^{\ast }[-2].$$

(260)

For a basis of covectors $e^{\mu }$ on ${\mathbb{R}}^{1,3}$, we abbreviate $e^{\mu \nu \cdots }:=e^{\mu }\wedge e^{\nu }\wedge \cdots$; we distinguish the bases of ${\wedge }^2{\left({\mathbb{R}}^{1,3}\right)}^{\ast }$ in degrees 0 and 1 by denoting them, respectively, as $e^{\mu \nu }$ and $e^{+\mu \nu }$. As in Section 4.3, this vector space is equipped with a pairing given by wedging forms in complementary degrees and applying the Hodge duality operator ${\mathrm{Tr}}_{{\wedge }^{•}}:{\wedge }^4{\left({\mathbb{R}}^{1,3}\right)}^{\ast }⟶\mathbb{R}$ to the resulting top form on ${\mathbb{R}}^{1,3}$, which on basis elements is given by

$$\begin{array}{c}\hfill {\mathrm{Tr}}_{{\wedge }^{•}}\left(e^{\mu \nu \rho \sigma }\right)={ϵ}^{\mu \nu \rho \sigma },\end{array}$$

(261)

where ${ϵ}^{\mu \nu \rho \sigma }$ is the Levi–Civita symbol in four dimensions with ${ϵ}^{0123}=1$.

The twist datum ${\tau }^{{\mathrm{YM}}_1}=\left({\tau }_1^{{\mathrm{YM}}_1},{\tau }_2^{{\mathrm{YM}}_1}\right)$ is defined as follows. The non-zero values of the twisting map ${\tau }_1^{{\mathrm{YM}}_1}:{\mathfrak{Kin}}_{{\mathrm{YM}}_1}⟶{\mathfrak{Kin}}_{{\mathrm{YM}}_1}\otimes \mathsf{End}\left(L\right)$ for the graded vector space (80) are given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\tau }_1^{{\mathrm{YM}}_1}\left(1\right)=e^{\mu }\otimes \frac{{\partial }_{\mu }}{\square }& ,{\tau }_1^{{\mathrm{YM}}_1}\left(e^{\mu }\right)=e^{+\mu \nu }\otimes \frac{{\partial }_{\nu }}{\square },{\tau }_1^{{\mathrm{YM}}_1}\left(e^{\mu \nu \rho }\right)=e^{\mu \nu \rho \sigma }\otimes \frac{{\partial }_{\sigma }}{\square },\hfill \\ \hfill & {\tau }_1^{{\mathrm{YM}}_1}\left(e^{\mu \nu }\right)=-e^{+\rho \sigma }\otimes {ϵ}_{\rho \sigma }{}^{\mu \nu }\frac{1}{\square }+e^{\mu \nu \rho }\otimes \frac{{\partial }_{\rho }}{\square }.\hfill \end{array}\end{array}$$

(262)

This recovers the colour-stripped differential: $m_1^{{\tau }^{{\mathrm{YM}}_1}}=m_1^{{\mathrm{YM}}_1}$. For example, on degree 1 fields $(A,B)$ the differential decomposes into

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill m_1^{{\tau }^{{\mathrm{YM}}_1}}(e^{\mu }\otimes A_{\mu }+e^{\nu \rho }\otimes B_{\nu \rho })& =e^{+\mu \nu }\otimes {\mu }_1^{\mathrm{Scal}}\left(\frac{{\partial }_{\nu }}{\square }{A}_{\mu }\right)\hfill \\ \hfill & -e^{+\rho \sigma }\otimes {\mu }_1^{\mathrm{Scal}}\left(\frac{1}{\square }{ϵ}_{\rho \sigma }{}^{\mu \nu }{B}_{\mu \nu }\right)+e^{\nu \rho \sigma }\otimes {\mu }_1^{\mathrm{Scal}}\left(\frac{{\partial }_{\sigma }}{\square }{B}_{\nu \rho }\right)\hfill \\ \hfill & =e^{+\sigma \mu }\otimes ({\partial }_{\sigma }{A}_{\mu }-{ϵ}_{\sigma \mu }{}^{\nu \rho }{B}_{\nu \rho })+e^{\mu \nu \rho }\otimes {\partial }_{\mu }{B}_{\nu \rho }\hfill \\ \hfill & =(\mathrm{d}A-{\ast }_{\mathrm{H}}B,\mathrm{d}B).\hfill \end{array}\end{array}$$

(263)

The colour-stripped 2-bracket $m_2^{{\tau }^{{\mathrm{YM}}_1}}=m_2^{{\mathrm{YM}}_1}$ is recovered using the following twisting map ${\tau }_2^{{\mathrm{YM}}_1}:{\mathfrak{Kin}}_{{\mathrm{YM}}_1}\otimes {\mathfrak{Kin}}_{{\mathrm{YM}}_1}⟶{\mathfrak{Kin}}_{{\mathrm{YM}}_1}\otimes \mathsf{End}\left(L\right)\otimes \mathsf{End}\left(L\right)$. Any bracket involving a degree 0 element is twisted in the same way, recovering the first four brackets of Equation (258) from

$$\begin{array}{c}\hfill {\tau }_2^{{\mathrm{YM}}_1}(1,v)=v\otimes \mathbb{1}\otimes \mathbb{1},\end{array}$$

(264)

for all $v\in {\mathfrak{Kin}}_{{\mathrm{YM}}_1}$. Brackets between two degree 1 fields as well as brackets between a degree 1 field and a degree 2 field are recovered, respectively, by the non-vanishing values

$$\begin{array}{cc}\hfill {\tau }_2^{{\mathrm{YM}}_1}(e^{\mu },e^{\nu })=e^{+\mu \nu }\otimes \mathbb{1}\otimes \mathbb{1},& {\tau }_2^{{\mathrm{YM}}_1}(e^{\mu },e^{\nu \rho })=e^{\mu \nu \rho }\otimes \mathbb{1}\otimes \mathbb{1},\hfill \\ \hfill {\tau }_2^{{\mathrm{YM}}_1}(e^{\mu \nu },e^{\rho })=-e^{\mu \nu \rho }\otimes \mathbb{1}\otimes \mathbb{1},& {\tau }_2^{{\mathrm{YM}}_1}(e^{\mu },e^{\nu \rho \sigma })=e^{\mu \nu \rho \sigma }\otimes \mathbb{1}\otimes \mathbb{1},\hfill \\ \hfill {\tau }_2^{{\mathrm{YM}}_1}(e^{\mu \nu },e^{+\rho \sigma })=e^{\mu \nu \rho \sigma }\otimes \mathbb{1}\otimes \mathbb{1},& {\tau }_2^{{\mathrm{YM}}_1}(e^{+\mu \nu },e^{\rho \sigma })=-e^{\mu \nu \rho \sigma }\otimes \mathbb{1}\otimes \mathbb{1}.\hfill \end{array}$$

(265)

For example, 2-brackets between fields in degrees 1 and 2 are generated by the twisted bracket using the non-zero twist data from Equation (265) to reduce to top forms in ${\wedge }^4{\left({\mathbb{R}}^{1,3}\right)}^{\ast }$:

$$\begin{array}{c}{m}_2^{{\tau }^{{\mathrm{YM}}_1}}\left(e^{\mu }\otimes A_{\mu }+e^{\nu \rho }\otimes B_{\nu \rho },e^{+\alpha \sigma }\otimes B_{\alpha \sigma }^{+}+e^{\beta \lambda \kappa }\otimes A_{\beta \lambda \kappa }^{+}\right)\hfill \\ =e^{\mu \beta \lambda \kappa }\otimes {\mu }_2^{\mathrm{Scal}}(A_{\mu },A_{\beta \lambda \kappa }^{+})+e^{\nu \rho \alpha \sigma }\otimes {\mu }_2^{\mathrm{Scal}}(B_{\nu \rho },B_{\alpha \sigma }^{+})\hfill \\ =g{e}^{\mu \nu \rho \sigma }\otimes \left(A_{\mu }{A}_{\nu \rho \sigma }^{+}+B_{\mu \nu }{B}_{\rho \sigma }^{+}\right)\hfill \\ =g\left(A\wedge A^{+}+B\wedge B^{+}\right).\hfill \end{array}$$

(266)

Finally, as in Section 4.3 the cyclic structure (255) is reproduced by ${\mathrm{Tr}}_{{\wedge }^{•}}\otimes {\langle -,-\rangle }_{\mathrm{Scal}}$, and altogether we have shown that the twisted homotopy factorisation of the cyclic strict $L_{\infty }$-algebra underlying noncommutative Yang–Mills theory in the first-order formalism is given by

$${\mathfrak{YM}}_{★1}=\mathfrak{u}\left(N\right){\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(N\right)}}\left({\mathfrak{Kin}}_{{\mathrm{YM}}_1}{\otimes }_{{\tau }^{{\mathrm{YM}}_1}}\mathfrak{Scal}\right).$$

(267)

As in Section 4.3, both factorisations (247) and (267) identify the binoncommutative biadjoint scalar theory of Section 3.6 as the zeroth copy of noncommutative Yang–Mills theory.

5.3. Noncommutative Gauge Theories on D-Branes

We will now aim to understand the structural features of noncommutative Yang–Mills amplitudes that we have discussed as a natural consequence of the embedding of the noncommutative gauge theory into string theory as a low-energy limit of open strings in constant B-field backgrounds.

5.3.1. Open Strings in Kalb–Ramond Fields

Consider the open string sigma-model with fields $X=\left(X^{\mu }\right)$ mapping from a Euclidean worldsheet $\Sigma$ with boundary conditions corresponding to a stack of Dp-branes filling flat space ${\mathbb{R}}^{1,p}$ in a closed string background with metric g, Kalb–Ramond two-form b and dilaton $\varphi$, all of which are assumed to be constant. The D-brane worldvolume supports an abelian gauge field A of constant curvature F. This combines with the Kalb–Ramond field to the gauge-invariant Born–Infeld field strength $B:=b+F$, which can be regarded as an electromagnetic field on the D-branes to which the open string endpoints are charged. We assume that A is a sum of gauge fields restricted to each connected component of the boundary $\partial \Sigma$, so that different branes can support independent $\mathsf{U}\left(1\right)$ gauge fluxes; this allows for open strings to end on D-branes with different overall background B-fields. For simplicity we suppose that the gauge flux F has the same rank $r\le p+1$ as b, and that $g_{\mu \nu }=0$ whenever $\mu \in \{0,1,\cdots ,r-1\}$ and $\nu \notin \{0,1,\cdots ,r-1\}$.

At tree level in open string perturbation theory, the worldsheet $\Sigma$ is a disk, or the conformally equivalent complex upper-half plane ${\mathbb{H}}^{+}\subset \mathbb{C}$. In the boundary conformal field theory on ${\mathbb{H}}^{+}$, the presence of the B-field modifies the Neumann boundary conditions in the longitudinal directions to the Dp-branes to the mixed boundary conditions

$$\begin{array}{c}\hfill ({\partial }_z-{\partial }_{\overline{z}})gX+2\pi {\alpha }^{\prime }({\partial }_z+{\partial }_{\overline{z}})BX{|}_{\partial {\mathbb{H}}^{+}}=0,\end{array}$$

(268)

where $\mathrm{Im}\left(z\right)\ge 0$ and ${\alpha }^{\prime }$ is the string Regge slope. The bulk propagator on the worldsheet ${\mathbb{H}}^{+}$ is then [82]

$$\begin{array}{c}\hfill {\langle X^{\mu }(z,\overline{z})X^{\nu }(z^{\prime },{\overline{z}}^{\prime })\rangle }_{{\mathbb{H}}^{+}}=-2\pi {\alpha }^{\prime }\left(g^{\mu \nu }log|z-z^{\prime }|+D^{\mu \nu }log|z-{\overline{z}}^{\prime }|+c^{\mu \nu }\right),\end{array}$$

(269)

with the projector

$$\begin{array}{c}\hfill D=-\frac{1}{g}+\frac{2}{g+2\pi {\alpha }^{\prime }B}\end{array}$$

(270)

and $c^{\mu \nu }$ are arbitrary integration constants.

This Green’s function is defined on the double cover $\mathbb{C}$ of the upper-half plane ${\mathbb{H}}^{+}$ by worldsheet parity $\Omega :z⟼\overline{z}$ and is obtained using the method of images: starting from the free Green’s function $-2\pi {\alpha }^{\prime }{g}^{\mu \nu }log{|z-z^{\prime }|}^2$ for the two-dimensional Laplace equation, one enforces the mixed boundary conditions by adding to it the contribution of an image charge symmetric with respect to reflection through the real line $z=\overline{z}$, which corresponds to worldsheet parity. The propagator (269) is single-valued if the logarithmic branch cut is placed in the complex lower-half plane ${\mathbb{H}}^{-}\subset \mathbb{C}$.

The effective target space geometry seen by the open strings ending on the Dp-branes consists of the open string metric G and the Poisson bivector $\theta$. They are related to the parameters $(g,b)$ of the closed string background through the open-closed relations [69]

$$\begin{array}{c}\hfill G=g-{\left(2\pi {\alpha }^{\prime }\right)}^{2}B\frac{1}{g}B\mathrm{and}\theta =-{\left(2\pi {\alpha }^{\prime }\right)}^2\frac{1}{g+2\pi {\alpha }^{\prime }B}B\frac{1}{g-2\pi {\alpha }^{\prime }B}.\end{array}$$

(271)

This parametrization allows us to write the string propagator as

$$\begin{array}{cc}\hfill {\langle X^{\mu }(z,\overline{z})X^{\nu }(z^{\prime },{\overline{z}}^{\prime })\rangle }_{{\mathbb{H}}^{+}}& =-2\pi {\alpha }^{\prime }(g^{\mu \nu }log|z-z^{\prime }|-g^{\mu \nu }log|z-{\overline{z}}^{\prime }|+G^{\mu \nu }log{|z-{\overline{z}}^{\prime }|}^2\hfill \\ \hfill & -\frac{\mathrm{i}{\theta }^{\mu \nu }}{2\pi {\alpha }^{\prime }}log\left(\frac{z-{\overline{z}}^{\prime }}{\overline{z}-z^{\prime }}\right)+c^{\mu \nu }).\hfill \end{array}$$

(272)

On the boundary $\partial {\mathbb{H}}^{+}=\mathbb{R}$, the choice of constant counterterm $c^{\mu \nu }=-\frac{\mathrm{i}}{2}{\theta }^{\mu \nu }$ reduces the propagator to the correlation function [69]

$${\langle X^{\mu }\left(\tau \right)X^{\nu }\left({\tau }^{\prime }\right)\rangle }_{\partial {\mathbb{H}}^{+}}=-{\alpha }^{\prime }{G}^{\mu \nu }log{(\tau -{\tau }^{\prime })}^2+\frac{\mathrm{i}}{2}{\theta }^{\mu \nu }\mathrm{sgn}(\tau -{\tau }^{\prime }).$$

(273)

This implies that the operator product expansion of tachyon vertex operators in the limit $\tau \to {\tau }^{\prime }$ with $\tau >{\tau }^{\prime }$ is given by

$$\begin{array}{c}\hfill {\mathrm{e}}^{\mathrm{i}p·X}\left(\tau \right){\mathrm{e}}^{\mathrm{i}q·X}\left({\tau }^{\prime }\right)={|\tau -{\tau }^{\prime }|}^{{\alpha }^{\prime }p·Gq}exp\left(\frac{\mathrm{i}}{2}p·\theta q\right){\mathrm{e}}^{\mathrm{i}(p+q)·X}\left({\tau }^{\prime }\right)+\cdots ,\end{array}$$

(274)

whereas usual the ellipses denote less singular terms as $\tau \to {\tau }^{\prime }$. The term in the boundary propagator (273) involving $\theta$ is a piecewise-constant function of $\tau$ and ${\tau }^{\prime }$, so it does not contribute to correlation functions of $\tau$-derivatives of X.

Consider open strings with $\mathsf{U}\left(N\right)$ Chan–Paton factors, and the tree-level scattering of n gluons of momenta $p_i$, polarisations ${\zeta }_i$, and Chan–Paton wavefunctions ${\lambda }_i=T^{a_i}$, with $i=1,\dots ,n$. The scattering amplitude for an ordering of colour factors $(a_1,\dots ,a_n)$ and insertion points $({\tau }_1,\dots ,{\tau }_n)$ on $\partial {\mathbb{H}}^{+}$ is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {𝓜}_n^{\mathrm{open}}{(p,\zeta ,a)}_{G,\theta }& ={\mathrm{Tr}}_{\mathfrak{u}\left(N\right)}(T^{a_1}\cdots T^{a_n})\hfill \\ \hfill & \times {\int }_{{(\partial {\mathbb{H}}^{+})}^{\times n}}\mathrm{d}\mu ({\tau }_1,\dots ,{\tau }_n){〈\prod _{i=1}^n{\zeta }_i·\frac{\mathrm{d}X}{\mathrm{d}\tau }{\mathrm{e}}^{\mathrm{i}{p}_i·X}\left({\tau }_i\right)〉}_{G,\theta },\hfill \end{array}\end{array}$$

(275)

where the subscripts ${}_{G,\theta }$ indicate that we evaluate correlation functions and amplitudes as functions of the open string parameters $(G,\theta )$. The term multiplying the trace is the colour-stripped open string amplitude ${𝓐}_n^{\mathrm{open}}{(1,\dots ,n)}_{G,\theta }$. The vertex operators are inserted on $\partial {\mathbb{H}}^{+}$ in a definite cyclic order, and the measure $\mathrm{d}\mu$ refers to the integral over the positions ${\tau }_i$ modulo the action of $\mathsf{SL}(2,\mathbb{R})$ by boundary conformal transformations.

The only $\theta$-dependence of the amplitude (275) is in the phase factor $exp\left(\frac{\mathrm{i}}{2}{\sum }_{i<j}{p}_i·\theta p_j\right)$ that comes from the expectation value of products of the tachyon vertex operators ${\mathrm{e}}^{\mathrm{i}{p}_i·X\left({\tau }_i\right)}$ in Equation (274). This factor arises from the correlation function inside the $\mathrm{d}\mu$ integral, and it is a piecewise-constant function of ${\tau }_1,\dots ,{\tau }_n$ that depends only on their cyclic ordering. Since the cyclic ordering is kept fixed in evaluating the integral that gives the scattering amplitude, this factor multiplies an otherwise $\theta$-independent amplitude. Rescaling $\theta \to t\theta$ for a parameter $t\in \mathbb{R}$, the colour-stripped amplitude ${𝓐}_n^{\mathrm{open}}{(1,\dots ,n)}_{G,t\theta }$ thus factors as

$$\begin{array}{cc}\hfill {𝓐}_n^{\mathrm{open}}{(1,\dots ,n)}_{G,t\theta }& =exp\left(-\frac{\mathrm{i}t}{2}\sum _{i<j}{p}_i·\theta p_j\right){𝓐}_n^{\mathrm{open}}{(1,\dots ,n)}_{G,t=0}\hfill \end{array}.$$

(276)

In the limit $B\to 0$, the flat space string amplitudes are recovered.

We may interpret these equations in terms of an ordering of momenta $(p_1,\dots ,p_n)$ viewed as the word $w=1\cdots n\in {\mathcal{W}}_n$. The phase factor appearing in Equation (276) may then be denoted as ${\Theta }_n\left(w\right)$. This is the same as the phase (232) factoring tree-level amplitudes in noncommutative Yang–Mills theory. The full amplitude is given by summing over the $(n-3)!$ inequivalent orderings. Since all the data of partial amplitudes respects an ordering $w\in {\mathcal{W}}_n$, all terms in Equation (276) can be written in terms of w.

5.3.2. Seiberg–Witten Limit

To make the link with noncommutative gauge theories, we take the Seiberg–Witten limit [69]. This scales the closed string metric to zero while keeping both the open string metric G and Poisson bivector $\theta$ fixed. It can be achieved by taking the usual low-energy limit ${\alpha }^{\prime }\to 0$, but now correlated with the limit $g_{\mu \nu }\sim {\left({\alpha }^{\prime }\right)}^2\to 0$, or equivalently with the limit $B_{\mu \nu }\sim {\left({\alpha }^{\prime }\right)}^{-2}\to \infty$. In either of these scaling limits, the boundary propagator (273) becomes

$${\langle X^{\mu }\left(\tau \right)X^{\nu }\left({\tau }^{\prime }\right)\rangle }_{\partial {\mathbb{H}}^{+}}^{\mathrm{SW}}=\frac{\mathrm{i}}{2}{\theta }^{\mu \nu }\mathrm{sgn}(\tau -{\tau }^{\prime }).$$

(277)

In the Seiberg–Witten limit, only the phase in the operator product expansion (274) remains as an overall pre-factor. This has the effect of replacing the ordinary multiplication of wavefunctions by the Moyal–Weyl star-product which quantises the Poisson structure $\theta$. The closed string propagator (272) diverges in this limit, as can also be seen at the level of the sigma-model action functional where the bulk kinetic term disappears and the worldsheet theory becomes a topological field theory on the boundary with a degenerate phase space. BV quantisation of this topological string theory in the first-order formalism reproduces star-products of fields in correlation functions of boundary observables [83,84]. The tree-level S-matrix is generated by the spacetime effective action functional of noncommutative Yang–Mills theory on ${\mathbb{R}}^{1,p}$ [69], with the constant metric G replacing the $d=p+1$ Minkowski metric $\eta$ everywhere in Section 5.1.

Application of the Seiberg–Witten limit to Equation (276) relates colour-ordered tree-level scattering amplitudes in $\mathsf{U}\left(N\right)$ noncommutative Yang–Mills theory with $N>1$ to those of the corresponding commutative theory through the same phase factor (232):

$${𝓐}_n^{★\mathrm{YM}}{\left(w\right)}_G={\Theta }_n\left(w\right){𝓐}_n^{\mathrm{YM}}{\left(w\right)}_G,$$

(278)

for $w\in {\mathcal{W}}_n$. Here the subscript ${}_G$ emphasises that all kinematic invariants used in the computation of amplitudes through this relationship are evaluated with respect to the constant open string metric.

5.4. Bern–Carrasco–Johansson Relations

The Bern–Carrasco–Johansson (BCJ) relations [5] are linear relations, whose coefficients are rational functions of Mandelstam variables, between tree-level partial amplitudes with different cyclic orderings that differ by the insertion position of a single gluon. They further reduce the number of independent subamplitudes at multiplicity n to $(n-3)!$. After their advent it was realised that a natural explanation for them appears as the low-energy limit of monodromy relations in open string theory, see e.g., [85].

BCJ relations for noncommutative Yang–Mills amplitudes are obtained in [79], building on the BCFW recursion formulas discussed in [78]. This requires some care, as the noncommutative amplitudes contain essential singularities in the complex plane. As we now explicitly demonstrate, these relations can also be derived from the same set of monodromy equations for open string scattering amplitudes in a constant Kalb–Ramond field that were found in [86].

For an ordering of punctures $w=k_1\cdots k_n\in {\mathcal{W}}_n$, the tachyonic part of the colour-stripped open string amplitude in Equation (275) is given by an iterated integral of the corresponding Koba–Nielsen factor

$$\begin{array}{c}\hfill {𝓐}_n^{\mathrm{open}}{\left(w\right)}_{G,t\theta }={\Theta }_n\left(w\right){\int }_{\mathbb{D}\left(w\right)}\prod _{i=1}^n\mathrm{d}{\tau }_{k_i}\prod _{i<j}{\left({\tau }_{k_i}-{\tau }_{k_j}\right)}^{{\alpha }^{\prime }{p}_{k_i}·G{p}_{k_j}}\end{array}$$

(279)

over the domain $\mathbb{D}\left(w\right)=\{({\tau }_{k_1},\dots ,{\tau }_{k_n})\in {\mathbb{R}}^n|{\tau }_{k_1}<\cdots <{\tau }_{k_n}\}$. For illustration, we choose the trivial ordering $w=1\cdots n$. We may then choose to single out ${\tau }_1$ and integrate it along the boundary of the worldsheet $\mathbb{R}$. The integrand is singular at ${\tau }_1={\tau }_i$ for $i>1$, where it has poles, and analytic everywhere else. Using this information, we analytically continue the integral over ${\tau }_1$ to an integral over a suitable closed contour in the complex plane.

For each pole ${\tau }_i$, the integrand picks up a monodromy factor ${\mathrm{e}}^{-2\pi \mathrm{i}{\alpha }^{\prime }{p}_1·G{p}_i}$ while the noncommutativity phase picks up a factor ${\mathrm{e}}^{\mathrm{i}t{p}_1·\theta p_i}$. In addition, integration around each chamber ${\tau }_i<{\tau }_1<{\tau }_{i+1}$ gives the partial amplitude ${𝓐}_n^{\mathrm{open}}{(2,\dots .i,1,i+1,\dots ,n)}_{G,t\theta }$. Introducing the massless Mandelstam invariants $s_{k_i{k}_j}:=2p_{k_i}·G{p}_{k_j}$, and deforming the contour to infinity, it follows that the result of the contour integration yields

$$\begin{array}{c}{𝓐}_n^{\mathrm{open}}{(1,\dots ,n)}_{G,t\theta }\hfill \\ +{\Theta }_n(1\cdots n)\sum _{i=2}^{n-1}\frac{{\mathrm{e}}^{-\pi \mathrm{i}{\alpha }^{\prime }(s_{12}+\cdots +s_{1i})}}{{\Theta }_n(2\cdots i1i+1\cdots n)}{𝓐}_n^{\mathrm{open}}{(2,\dots ,i,1,i+1,\dots ,n)}_{G,t\theta }=0.\hfill \end{array}$$

(280)

Now we take the Seiberg–Witten limit, in which the noncommutativity factor survives the low-energy limit ${\alpha }^{\prime }\to 0$. In this way, we obtain the Ward identities

$$\sum _{i=1}^{n-1}{\Theta }_n(n\cdots i+11i\cdots 21){𝓐}_n^{★\mathrm{YM}}{(1,2,\dots ,i,1,i+1,\cdots n)}_G=0$$

(281)

as well as the BCJ relations

$$\sum _{i=2}^{n-1}\frac{s_{12}+\cdots +s_{1i}}{{\Theta }_n(12\cdots i1i+1\cdots n)}{𝓐}_n^{★\mathrm{YM}}{(1,2,\dots ,i,1,i+1,\cdots n)}_G=0$$

(282)

for noncommutative Yang–Mills theory.

5.5. Kawai–Lewellen–Tye Relations

The factorisations discussed in Section 5.2 can be used to construct a variety of double copy theories from noncommutative Yang–Mills theory. As we saw already in Section 4.5, these dual theories do not undergo any noncommutative deformation and coincide with their classical double copies. In particular, the homotopy double copy of noncommutative Yang–Mills theory with itself in the second-order formalism yields ordinary perturbative gravity in d dimensions, or more exactly $𝓝=0$ supergravity which involves a two-form and a dilaton in addition to the graviton [1,12]. Again, we demonstrate that this is a natural consequence of properties of open string amplitudes in constant B-field backgrounds.

5.5.1. Kawai–Lewellen–Tye Relations with B-Fields

The Kawai–Lewellen–Tye (KLT) relations [4] are an explicit realisation of open-closed worldsheet duality for the tachyonic parts of string amplitudes. A first topological argument comes from the observation that one can glue two oppositely oriented copies of a disk along their boundaries to form a Riemann sphere. For open string amplitudes, punctures are inserted along the boundaries of the disks, so the gluing between amplitudes is a function only of kinematic invariants and the ordering of vertices along the two boundaries. The KLT relations have been understood more recently from several perspectives: through a more computationally efficient paradigm [26,87], through an ultraviolet completion in ${\alpha }^{\prime }$ of biadjoint scalar theory called Z-theory [88], and through intersection theory on the moduli space of n-punctured Riemann spheres [26,89]. The one-loop extension of the tree-level KLT relations, involving torus and annulus amplitudes, is found in [90].

Here we consider the tachyonic part ${𝓐}_n^{\mathrm{open}}{\left(w\right)}_G$ of the open string amplitude for a given ordering of punctures (and associated momenta) $w\in {\mathcal{W}}_n$, and kinematical invariants calculated with respect to the open string metric G. A closed form of the KLT relationship is given in terms of a momentum kernel ${\mathit{S}}_n{\left(w\right|w^{\prime })}_G$ linking the colour-stripped partial amplitudes to tree-level closed string amplitudes through

$${\mathcal{A}}_{n,G}^{\mathrm{closed}}\left(p\right)={\left(-\frac{\mathrm{i}}{4}\right)}^{n-3}\sum _{w,w^{\prime }\in {\mathcal{W}}_{n-3}}{𝓐}_n^{\mathrm{open}}{\left(w\right)}_G{\mathit{S}}_n{\left(w\right|w^{\prime })}_G{𝓐}_n^{\mathrm{open}}{\left(w^{\prime }\right)}_G.$$

(283)

The reduced number of word letters here is due to the $\mathsf{SL}(2,\mathbb{C})$ conformal invariance of the worldsheet theory, which enables one to fix three insertion points; following the standard choice we fix $z_1=0,$ $z_{n-1}=1$ and $z_n=\infty$. The n-point momentum kernel ${\mathit{S}}_n{\left(w\right|w^{\prime })}_G$ is not unique: it is derived from an iterated integral over $n-3$ complex variables, so there are $n-3$ equivalent contour integrals to choose from.

One possible choice is found in [91]. For $w=k_1\cdots k_n$ and fixed ordering of the right insertion $w^{\prime }=1\cdots n$, define the operation

$$\mathtt{H}{(k_i,k_j)}_{w^{\prime }}=\left\{\begin{array}{c}1\mathrm{if}\mathrm{the}\mathrm{ordering}\mathrm{of}{k}_i,k_j\mathrm{is}\mathrm{opposite}\mathrm{in}{w}^{\prime },\hfill \\ 0\mathrm{otherwise}.\hfill \end{array}\right.$$

(284)

By conformal invariance, three points are fixed with labels $\{1,n-1,n\}$, so any summation over words in equations below is understood as leaving these letters fixed; for example, when writing $w\in {\mathcal{W}}_n$ we mean $w=1w^{\circ }n-1n$ for some $w^{\circ }\in {\mathcal{W}}_{n-3}$. We then set

$${\mathit{S}}_n{\left(w\right|1\cdots n)}_G={\left(\frac{2}{\pi {\alpha }^{\prime }}\right)}^{n-3}{\displaystyle \prod _{i=2}^{n-2}}sin\pi {\alpha }^{\prime }\left(p_1·G{p}_{k_i}+{\displaystyle \sum _{1\le j<l\le i}}\mathtt{H}{(k_j,k_l)}_{1\cdots n}{p}_{k_j}·G{p}_{k_l}\right),$$

(285)

and the full formula for the momentum kernel is simply obtained by permuting the right letters of $w^{\prime }\in {\mathcal{W}}_{n-3}$ in the algebra of ordered words on the letters $\{2,\dots ,n-2\}$. For example, unpacking Equation (285) we find the momentum kernel

$$\begin{array}{c}\hfill {\mathit{S}}_5{\left(12345\right|32145)}_G={\left(\frac{2}{\pi {\alpha }^{\prime }}\right)}^{2}sin\left(\pi {\alpha }^{\prime }{p}_1·G{p}_2\right)sin\left(\pi {\alpha }^{\prime }{p}_3·G(p_1+p_2)\right)\end{array}$$

(286)

involved in the gluing of multiplicity five string amplitudes.

The colour-stripped open string amplitudes appearing in the KLT formula (283) combine in an analytic way to compute closed string amplitudes. Using Equation (276) they can be written in terms of open string amplitudes with a Kalb–Ramond background. This scales separately the left and right partial amplitudes with respective phase factors, and leaves the KLT formula invariant under redefinition of the momentum kernel according to (Recall from Section 5.3 that a situation with $\theta \ne \overline{\theta }$ can arise if we allow for distinct worldvolume gauge fluxes on different stacks of D-branes, which give rise to distinct Born–Infeld field strengths $B\ne \overline{B}$ in a fixed closed string background $(g,b)$. Even when $\theta =\overline{\theta }$, the notation aids in distinguishing the left-moving and right-moving open string sectors involved in the gluing of amplitudes).

$${\mathit{S}}_n{\left(w\right|w^{\prime })}_{G,t\theta ,\overline{t}\overline{\theta }}:={\Theta }_n\left(\overline{w}\right){\mathit{S}}_n{\left(w\right|w^{\prime })}_G{\overline{\Theta }}_n\left({\overline{w}}^{\prime }\right).$$

(287)

We interpret this trivial modification as saying that, for a constant B-field, it is possible to glue open string amplitudes in a manner consistent with the fact that the Kalb–Ramond field is invisible to closed strings on a Riemann sphere. We thus propose the B-field modified KLT relations

$$\begin{array}{cc}\hfill {\mathcal{A}}_{n,G}^{\mathrm{closed}}\left(p\right)& ={\left(-\frac{\mathrm{i}}{4}\right)}^{n-3}\sum _{w,w^{\prime }\in {\mathcal{W}}_{n-3}}{𝓐}_n^{\mathrm{open}}{\left(w\right)}_{G,t\theta }{\mathit{S}}_n{\left(w\right|w^{\prime })}_{G,t\theta ,\overline{t}\overline{\theta }}{𝓐}_n^{\mathrm{open}}{\left(w^{\prime }\right)}_{G,\overline{t}\overline{\theta }},\hfill \end{array}$$

(288)

leaving unchanged the closed string scattering amplitudes.

5.5.2. Double Copy Relations

In the ${\alpha }^{\prime }\to 0$ limit, the closed string amplitudes taken with respect to the open string metric G reduce to pure gravity amplitudes ${\mathcal{A}}_{n,G}^{\mathrm{GR}}\left(p\right)$ on the background ${\mathbb{R}}^{1,p}$ with metric G. Therefore, the Seiberg–Witten limit of the KLT relations (288) gives the double copy relations

$$\begin{array}{cc}\hfill {\mathcal{A}}_{n,G}^{\mathrm{GR}}\left(p\right)& ={\left(-\frac{\mathrm{i}}{4}\right)}^{n-3}\sum _{w,w^{\prime }\in {\mathcal{W}}_{n-3}}{𝓐}_n^{★\mathrm{YM}}{\left(w\right)}_G{\mathit{S}}_n^{★\overline{★}\mathrm{YM}}{\left(w\right|w^{\prime })}_G{𝓐}_n^{\overline{★}\mathrm{YM}}{\left(w^{\prime }\right)}_G.\hfill \end{array}$$

(289)

In terms of the massless Mandelstam invariants $s_{ij}=2p_i·G{p}_j$ computed with respect to the spacetime metric G, the new field theory KLT kernel is

$${\mathit{S}}_n^{★\overline{★}\mathrm{YM}}{\left(w\right|w^{\prime })}_G={\Theta }_n\left(\overline{w}\right){\overline{\Theta }}_n\left({\overline{w}}^{\prime }\right)\prod _{i=2}^{n-2}\left(s_{1k_i}+\sum _{1\le j<l\le i}\mathtt{H}{(k_j,k_l)}_{w^{\prime }}{s}_{k_j{k}_l}\right),$$

(290)

giving ${\mathit{S}}_n^{★\overline{★}\mathrm{YM}}{\left(w\right|w^{\prime })}_G={\Theta }_n\left(\overline{w}\right){\mathit{S}}_n^{\mathrm{YM}}{\left(w\right|w^{\prime })}_G{\overline{\Theta }}_n\left({\overline{w}}^{\prime }\right)$, the momentum kernel for gluing two copies of noncommutative gauge theories with respect to the original momentum kernel. This provides the explicit double copy construction of noncommutative Yang–Mills theory with itself to ordinary perturbative gravity: an ordinary graviton can also be regarded as composed of two noncommutative gluons.

A well-known feature of the field theory KLT kernel, originally derived in [10], is that it can be expressed as the inverse matrix of double colour-ordered biadjoint amplitudes ${𝓐}_n^{\mathrm{BAS}}{\left(w\right|w^{\prime })}_G$ that were discussed in Section 2.2, here evaluated in the flat background metric G. This partial amplitude is the amplitude of a ${\varphi }^3$-theory restricted to an ordering of external momenta, and it is related to the commutative momentum kernel as ${𝓐}_n^{\mathrm{BAS}}{\left(w\right|w^{\prime })}_G={\mathit{S}}_n^{\mathrm{YM}}{\left(w\right|w^{\prime })}_G^{-1}$. In other words, tree amplitudes of the zeroth copy uniquely determine the double copy kernel. This connection has been more recently understood through different perspectives: the inverse of the string theory KLT kernel can be related to the doubly ordered tree amplitudes of the Z-theory prescription ([87], Section 2.2), while a simpler argument in the field theory limit is found in [92].

In the noncommutative field theory, this relationship applied to the modified momentum kernel (290) involves the subamplitudes

$$\begin{array}{c}\hfill {𝓐}_n^{★\overline{★}\mathrm{BAS}}{\left(w\right|w^{\prime })}_G={\Theta }_n\left(w\right){𝓐}_n^{\mathrm{BAS}}{\left(w\right|w^{\prime })}_G{\overline{\Theta }}_n\left(w^{\prime }\right)\end{array}$$

(291)

of the binoncommutative biadjoint scalar theory which we defined in Section 3.6: the interaction vertex (127) keeps track of the two copies of noncommutative phase factors that are required by the noncommutative KLT kernel. Therefore, the momentum kernel associated with the double copy of noncommutative gauge theory is sourced by the binoncommutative biadjoint scalar theory through

$${\mathit{S}}_n^{★\overline{★}\mathrm{YM}}{\left(w\right|w^{\prime })}_G={𝓐}_n^{★\overline{★}\mathrm{BAS}}{\left(w\right|w^{\prime })}_G^{-1}.$$

(292)

Crucially, the noncommutative corrections preserve the rank of the matrix of double colour-ordered amplitudes, which is $(n-3)!$, hence they satisfy the minimal rank condition and result in an admissible KLT kernel [27]. This further vindicates our understanding that double copies of noncommutative gauge theories are the same as the double copies of the corresponding commutative gauge theories.

Remark 4.

In the commutative case, the (non-strict) Yang–Mills $L_{\infty }$ algebra $\mathfrak{YM}$ can be viewed as a tensor product $\mathfrak{u}\left(N\right)\otimes {\mathfrak{C}}_{\mathfrak{YM}}$ of the gauge algebra with a kinematical $C_{\infty }$-algebra [93]. In a quasi-isomorphic description of $\mathfrak{YM}$ inspired by open string field theory, this was used by [94] to show that a subspace of the $C_{\infty }$-algebra ${\mathfrak{C}}_{\mathfrak{YM}}^{\mathrm{L}}\otimes {\mathfrak{C}}_{\mathfrak{YM}}^{\mathrm{R}}$, corresponding to states satisfying the level-matching constraints of closed string theory, induces the cubic truncation of the $L_{\infty }$-algebra of double field theory. Following the prescription of the present paper, the same construction using our notion of twisted colour-stripping also gives ordinary double field theory as a doubling of noncommutative Yang–Mills theory, which results in the same $C_{\infty }$-algebra ${\mathfrak{C}}_{\mathfrak{YM}}$.

5.6. ${\mathfrak{u}}_{★}\left(1\right)$ Yang–Mills Theory as a Double Copy

Most of what we have said so far in this section only applies to rank $N>1$, and it is natural to ask what the fate of noncommutative $\mathsf{U}\left(1\right)$ Yang–Mills theory is from the double copy perspective. This is an interacting theory with non-trivial amplitudes that cannot be simply related to its commutative counterpart, which is the non-interacting Maxwell theory; scattering amplitudes in this theory are studied in e.g., [95,96]. Instead, we will interpret it analogously to what we did in Section 4.6, by viewing $\mathfrak{u}\left(1\right)\simeq \mathbb{R}=:{\mathfrak{Kin}}_{{\wedge }^0}$ as the kinematic vector space underlying the adjoint scalar theory introduced in Section 3.3.

Consider the rank-one limit of the factorisation of noncommutative Yang–Mills theory in the first-order formalism from Equation (267), whose $L_{\infty }$-algebra is now elusively denoted as

$$\widehat{{\mathfrak{YM}}_{★1}\left(1\right)}={\mathfrak{Kin}}_{{\wedge }^0}{\otimes }_{{\tau }^{{\mathfrak{u}}_{★}\left(1\right)}}\left({\mathfrak{Kin}}_{{\mathrm{YM}}_1}{\otimes }_{{\tau }^{{\mathrm{YM}}_1}}\mathfrak{Scal}\right).$$

(293)

The underlying cochain complex in this factorisation is again given by Equation (257). Comparing Equations (96) and (293), and recalling the colour–kinematics duality of Section 3.4, we conclude that noncommutative $\mathsf{U}\left(1\right)$ Yang–Mills theory in the first-order formalism is a double copy of the adjoint scalar theory of Section 3.3 with commutative Yang–Mills theory in the first-order formalism, for any colour algebra $\mathfrak{g}$. Symbolically, this double copy relationship reads as

$$\begin{array}{c}\hfill {\mathfrak{u}}_{★}\left(1\right)\mathrm{Yang}–\mathrm{Mills}=\mathrm{Adjoint}\mathrm{Scalar}\otimes \mathrm{Ordinary}\mathrm{Yang}–\mathrm{Mills}.\end{array}$$

(294)

This renders an interpretation of rank-one noncommutative Yang–Mills theory as a gravitational theory, despite the absence of dynamical spin-two fields, thus harvesting old anticipations discussed in Section 1.

5.7. Noncommutative Self–Dual Yang–Mills Theory

We can make the double copy relations of this section somewhat more precise in the self–dual sector of Yang–Mills theory in $d=4$ dimensions, which we studied already in Section 3.7. Noncommutative self–dual Yang–Mills theory was originally studied in [97]. In this sector the first-order formalism of Section 5.2 is a $BF$-theory involving self–dual two-forms that describes (noncommutative) self–dual Yang–Mills theory. As shown by [66], this theory (in the Leznov gauge) can be obtained from quantising open $𝓝=2$ strings in a constant background B-field in the Seiberg–Witten zero-slope scaling limit. For rank $N>1$, our twisted homotopy factorisation is compatible with the twisted form of colour–kinematics duality that provides a double copy map of noncommutative $\mathsf{U}\left(N\right)$ self–dual Yang–Mills theory with itself to ordinary self–dual gravity (perturbed around the open string metric G), along similar lines as discussed in Section 3.7.

Restricting both sides of Equation (294) to the self–dual sector in $d=4$ dimensions provides further corroboration of this double copy relationship as a map to a gravitational theory. As we discussed in Section 3.7, self–dual ${\mathfrak{u}}_{★}\left(1\right)$ Yang–Mills theory is the same theory as noncommutative self–dual gravity whose equation of motion is the deformed Plebański Equation (139). There we also exhibited the explicit double copy construction of noncommutative self–dual gravity from the adjoint scalar theory with ordinary self–dual Yang–Mills theory. The connections of noncommutative self–dual gravity and Yang–Mills theory to Lorentz-invariant chiral higher-spin theories, discussed in [45], hints at possible interpretations of the double copy relationship (294) beyond the self–dual sector.

6. Final Remarks

In this paper, we have applied the homotopy algebraic formalism of [12] to the study of noncommutative gauge theories descending from the low-energy limit of open string theories with a stack of D-branes in constant B-fields. We resolve the issue of colour and kinematic degrees of freedom mixing, which naively obstruct factorisation and colour–kinematics duality, by introducing a twisted form of factorisation and colour–kine-matics duality. We then construct the double copy of noncommutative gauge theories in this framework and show that it coincides with the commutative limit. This apparently trivial result is substantiated by the well-known fact that noncommutativity is an open string effect which leads us to introduce a modified KLT relationship; nevertheless, the homotopy algebraic techniques used in this work are themselves interesting and may lead to further insights into the structure of noncommutative theories. We also introduced a corresponding zeroth copy for noncommutative field theories and checked that it plays the role of the inverse momentum kernel in our modified field theory KLT relationship.

We illustrated our arguments by studying noncommutative deformations of Chern–Simons theory in three dimensions as well as of Yang–Mills theory in both first- and second-order formalisms. We interpreted the special case of the rank-one limits of noncommutative gauge theories as double copies themselves with the purely noncommutative adjoint scalar theory. We also have reviewed applications of this formalism in the study of the self–dual sector of Yang–Mills theory and gravity by studying the semi-classical limits of a new binoncommutative scalar theory. The diverse relations between theories are succinctly summarised in Figure 1.

Let us address here one final question: is there a way to modify our picture in such a way that noncommutative Yang–Mills theory can be double copied to a noncommutative theory of gravity? We assert that the answer is affirmative. As is well-known, noncommutative gravity involves a twisted form of diffeomorphism invariance, see e.g., [36] for a review. This twisted symmetry does not fit nicely into the standard $L_{\infty }$-algebra formalism; see [98] for an explicit exposition. This in itself formally hints as to why the standard noncommutative gauge theories, which are organised systematically by $L_{\infty }$-algebras [18,19], do not homotopy double copy to noncommutative gravity. However, we believe that this obstacle can be overcome by passing from $L_{\infty }$-algebras to braided $L_{\infty }$-algebras, which organise noncommutative field theories with braided gauge symmetries [19,99]; this formalism was initially developed with a homotopy algebraic approach to noncommutative gravity in mind.

By working with homotopy algebras in a symmetric monoidal category with non-trivial braiding, one can avoid the twisted factorisations and colour–kinematics duality that we had to introduce to stay in the usual categories of homotopy algebras, and instead work with the standard concepts, albeit in a braided setting. By a suitable extension of the homotopy double copy prescription to braided homotopy algebras, the double copy theory in this case would be encoded by a braided $L_{\infty }$-algebra and so would have braided diffeomorphism symmetry. Hence it should describe noncommutative gravity. We plan to address this interesting perspective and its consequences in future work.