QMUL-PH-20-24

New heavenly double copies

Erick<EMAIL_ADDRESS>Hugo García-
<EMAIL_ADDRESS>Andrés<EMAIL_ADDRESS>
Ricardo<EMAIL_ADDRESS>and Chris D.
<EMAIL_ADDRESS>
a Centre for Research in String Theory, School of Physics and Astronomy,

Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK

b Departamento de Física,

Centro de Investigación y de Estudios Avanzados del Instituto Politécnico
Nacional

P.O. Box 14-740, CP. 07000, México D.F., México.

c Mani L. Bhaumik Institute for Theoretical Physics,

UCLA Department of Physics and Astronomy, Los Angeles, CA 90095, USA

The double copy relates scattering amplitudes and classical solutions in Yang-
Mills theory, gravity, and related field theories. Previous work has shown
that this has an explicit realisation in self-dual YM theory, where the
equation of motion can be written in a form that maps directly to Plebański’s
heavenly equation for self-dual gravity. The self-dual YM equation involves an
area-preserving diffeomorphism algebra, two copies of which appear in the
heavenly equation. In this paper, we show that this construction is a special
case of a wider family of heavenly-type examples, by (i) performing Moyal
deformations, and (ii) replacing the area-preserving diffeomorphisms with a
less restricted algebra. As a result, we obtain a double-copy interpretation
for hyper-Hermitian manifolds, extending the previously known hyper-Kähler
case. We also introduce a double-Moyal deformation of the heavenly equation.
The examples where the construction of Lax pairs is possible are manifestly
consistent with Ward’s conjecture, and suggest that the classical
integrability of the gravity-type theory may be guaranteed in general by the
integrability of at least one of two gauge-theory-type single copies.

##  1 Introduction

A major focus of modern theoretical physics is the remarkable web of
connections between different field theories of interest. In this paper, we
explore one such connection, the so-called double copy between gauge theory
and gravity, which has been the basis of much work over the past decade due to
its applications. The double copy originated in the study of perturbative
scattering amplitudes, first in string theory [1] and later in quantum field
theory and gravity [2, 3, 4], where it has been studied at various loop
orders, both with and without supersymmetry. More recently, the notion of
double-copy relations between theories has been extended to the context of the
classical limit of the theories using a variety of approaches. These include
relating exact, algebraically special solutions to the equations of motion of
the theories [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], as well as
applications to perturbative methods. Some examples of the latter are
constructing perturbative metrics directly [36], solving the equations of
motion in a world-line formalism [37, 38, 39, 40, 41, 42, 43, 44, 45, 46],
relating linearised solutions using a convolution approach [47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57], as well as applying double-copy ideas to the
computation of quantities of interest to gravitational wave astronomy, mainly
in the post-Minkowskian regime [58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68,
69]. Closer to the idea of this paper, there is a large body of literature on
webs of theories related by the double copy, e.g., [70, 71, 72, 73, 74, 75,
76, 77, 78]. A comprehensive review of these developments may be found in
refs. [79, 80].

A crucial idea, which fostered the development of the double copy, is the
colour-kinematics duality of Bern, Carrasco and Johansson (BCJ) [2]. This
‘duality’ remains a conjecture at loop level, but is well established for
tree-level (i.e., classical) gauge theory. It states that the scattering
amplitudes can be written in such a way that (apart from scalar-type
propagators) the kinematic dependence has the same algebraic properties as the
colour dependence, thus hinting that there is a kinematic analogue of the
colour Lie algebra in gauge theory. In particular, there is a kinematic
analogue for the Jacobi identity of the colour Lie algebra, which gives rise
to a notion of kinematic algebra. Upon taking the double copy, the colour
information is replaced by another copy of its kinematic analogue, so that two
kinematic algebras appear in gravity scattering amplitudes. In fact, starting
from two distinct gauge theories (e.g., degree of supersymmetry or self-
duality constraint), the two kinematic algebras in the gravity theory, each
extracted from one of the ‘single copies’, will be distinct. Despite some
progress, the mathematical interpretation of a kinematic algebra remains
elusive, partly because there is no understanding of the colour-kinematics
duality at the level of the action or the equations of motion. This would rely
on a formulation of gauge theory with a single cubic interaction vertex,
whereas the usual Yang-Mills action has also a four-point vertex. This cubic
vertex would have as colour factor the usual Lie algebra structure constant,
$f^{abc}$, and as kinematic factor the putative structure constant of a
kinematic algebra.

There is an exception, however, where the kinematic algebra has been fully
understood. Reference [81] circumvented the difficulties of the general
problem by restricting to the self-dual sector of pure Yang-Mills theory. In a
long-known formulation [82], the action of self-dual YM involves a single
adjoint-valued degree of freedom, and is manifestly cubic. It is then possible
to completely elucidate the kinematic algebra alluded to above, which turns
out to correspond to certain area-preserving diffeomorphisms (n.b. in the
mathematical literature, this group is often referred to as SDiff or SDiff2,
and the algebra as sdiff or sdiff2). Furthermore, the equations of motion can
be written in terms of an interaction involving the product of two sets of
structure constants, for the colour and kinematic Lie algebras respectively.
Thus, the colour-kinematics duality becomes explicit, and the double copy to
self-dual gravity straightforward. The equation of motion for self-dual
gravity arises naturally in the form of Plebański’s (second) heavenly equation
[83].

There have been many interesting results regarding kinematic algebras. They
have been explored via algorithms for obtaining colour-kinematics-dual
expressions for scattering amplitudes, e.g., in [84, 85, 86, 87, 88, 89, 90,
91, 92, 93, 94], and via a variety of algebraic and geometric approaches,
e.g., in [95, 96, 97, 98, 99]. And yet, to date, it has not been possible to
generalise the very explicit construction for the self-dual sectors to the
full Yang-Mills theory and gravity case. This motivates looking for examples
of kinematic algebras that go beyond those of ref. [81]. That is the main aim
of our paper, and we will seek to generalise the results of ref. [81] in two
key ways. Firstly, we may form qualitatively different types of gauge and
gravity theories by replacing the Lie and Poisson brackets appearing in the
former and latter with their Moyal deformations [100]. These arise in
alternative formulations of quantum mechanics [101, 102] as well as in the
study of field theories in non-commutative spacetimes; see e.g. refs. [103,
104] for pedagogical reviews. The Moyal bracket constitutes the most general
bracket one can write down for a Lie algebra of functions of two variables
[105], here corresponding to the two-dimensional space on which the
diffeomorphisms act. Secondly, we will consider replacing the sdiff algebra
discussed above with the set of arbitrary – rather than area-preserving – two-
dimensional diffeomorphisms (diff). Both of the above generalisations lead to
new instances of the double copy, in which the colour-kinematics duality is
manifest. These results are summarised in the table of theories of section 5.

A second motivation for our paper is to explore the classical integrability
(or otherwise) of the gauge and gravity theories related by the double copy,
an aspect that was not considered in ref. [81]. As is well-known, both self-
dual Yang-Mills theory and self-dual gravity are integrable, and thus admit an
infinite number of conserved charges. This integrability can be expressed in
terms of a Lax pair, and we will find a hitherto unexplored double copy
interpretation of Lax pairs in different theories. This suggests that
integrability of a gravity theory is “inherited” from its corresponding gauge
theories via the double copy. Also, in considering the more general theories
outlined above, we will find that integrability in the gravity theory can be
obtained even if only one of the gauge theories in the double copy is
integrable.

The structure of our paper is as follows. In section 2, we review the self-
dual double copy, and relate this to ideas regarding integrability. We
consider Moyal deformations in section 3. In section 4, we look at
generalising the algebra of diffeomorphisms in the gauge theory, and construct
a corresponding gravity theory. We summarise the web of theories studied in
this paper in section 5, and the general form of the different Lax pairs and
the infinite tower of linearised symmetries is examined as well. Finally, we
discuss our results and conclude in section 6.

##  2 The self-dual double copy

###  2.1 Review

In this section, we review salient details of the double copy in the self-dual
sector described in ref. [81]. This will set up useful notation for what
follows, whilst also allowing us to draw attention to aspects of
integrability, which were not examined in [81]. Our starting point is to
consider pure Yang-Mills theory. The (vacuum) equation of motion is given by

$D^{\mu}F_{\mu\nu}=0,$ (1)

where

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ig[A_{\mu},A_{\nu}]$
(2)

is the field strength tensor in terms of the gauge field $A_{\mu}\equiv
A^{a}_{\mu}\,{T}^{a}$, and ${T}^{a}$ is a generator of the Lie algebra
${\mathbf{g}}$ of the gauge group $G$. Also,
$D_{\mu}=\partial_{\mu}-igA_{\mu}$ is the covariant derivative, where the
gauge field in the second term acts in the adjoint representation. The
equations of motion take a particularly simple form if one considers self-dual
solutions, for which the field strength tensor satisfies

$F_{\mu\nu}=\frac{i}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}.$ (3)

One may introduce coordinates

$u=t-z,\quad v=t+z,\quad w=x+iy,\quad\bar{w}=x-iy,$ (4)

such that the Minkowski line element becomes

$ds^{2}=dudv-dwd\bar{w},$ (5)

and the self-duality conditions can be written as

$\displaystyle F_{uw}=0,$ (6) $\displaystyle F_{v\bar{w}}=0,$ (7)
$\displaystyle F_{uv}-F_{w\bar{w}}=0.$ (8)

From eq. (6) and a light-cone gauge choice, we may take

$A_{u}=A_{w}=0.$ (9)

It follows from eq. (8) that there must exist an adjoint-valued scalar
function $\Phi\equiv\Phi^{a}{T}^{a}$ such that

$A_{v}=-\partial_{w}\Phi,\quad A_{\bar{w}}=-\partial_{u}\Phi.$ (10)

Physically, $\Phi$ represents the single polarisation state that remains in
the gauge field upon projecting to the self-dual sector, and eq. (7) implies
that it satisfies the following equation of motion:

$\partial^{2}\Phi+ig[\partial_{w}\Phi,\partial_{u}\Phi]=0,$ (11)

where
$\partial^{2}\equiv\partial_{u}\partial_{v}-\partial_{w}\partial_{\bar{w}}$.
We follow Ref. [81], to rewrite this as an integral equation in momentum
space. Fourier transforming eq. (11) and rearranging yields

$\displaystyle\Phi(k)$ $\displaystyle=-ig\int
d^{D}xe^{-ix\cdot(p_{1}+p_{2}-k)}\int d\\!\\!{}^{-}\\!p_{1}\int
d\\!\\!{}^{-}\\!p_{2}\frac{1}{k^{2}}\left(p_{1w}p_{2u}-p_{1u}p_{2w}\right)\Phi(p_{1})\Phi(p_{2})$
$\displaystyle=-ig\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}X(p_{1},p_{2})\Phi(p_{1})\Phi(p_{2}),$
(12)

where we have employed the shorthand notation

$d\\!\\!{}^{-}\\!p\equiv\frac{d^{D}p}{(2\pi)^{D}},\qquad\delta\\!\\!\\!{}^{-}\\!(p)\equiv(2\pi)^{D}\delta^{(D)}(p),$
(13)

and, in the second line, have defined the kinematic structure constant

$X(p_{1},p_{2})=p_{1w}p_{2u}-p_{1u}p_{2w}.$ (14)

Antisymmetry of the latter under $p_{1}\leftrightarrow p_{2}$ means that one
may replace the product of scalar fields with a commutator, to finally write

$\Phi^{a}(k)=\frac{g}{2}\int
d\\!\\!{}^{-}\\!{p}_{1}\,d\\!\\!{}^{-}\\!{p}_{2}\,\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}\,X(p_{1},p_{2})\,f^{abc}\,\Phi^{b}(p_{1})\Phi^{c}(p_{2}).$
(15)

The $f^{abc}$ are the structure constants of the Lie algebra ${\mathbf{g}}$,
$[T^{a},T^{b}]=if^{abc}T^{c}$. The kinematic objects in eq. (14) are also the
structure constants of a Lie algebra, the Jacobi identity being

$X(p_{1},p_{2})\,X(p_{1}+p_{2},p_{3})+X(p_{2},p_{3})\,X(p_{2}+p_{3},p_{1})+X(p_{3},p_{1})\,X(p_{3}+p_{1},p_{2})=0.$
(16)

The kinematic Lie algebra is infinitely dimensional, with generators of the
form

$L_{k}=e^{-ik\cdot x}(-k_{w}\partial_{u}+k_{u}\partial_{w}),$ (17)

satisfying

$[L_{p_{1}},L_{p_{2}}]=iX(p_{1},p_{2})L_{p_{1}+p_{2}}.$ (18)

It is the Lie algebra sdiff of area-preserving diffeomorphisms in the $(w,u)$
plane. We can also interpret it as a Poisson algebra, with

$\\{e^{-ip_{1}x},e^{-ip_{2}x}\\}=-X(p_{1},p_{2})\,e^{-i(p_{1}+p_{2})x},$ (19)

where the Poisson bracket is

$\\{A,B\\}=\partial_{w}A\,\partial_{u}B-\partial_{u}A\,\partial_{w}B.$ (20)

We see that the self-dual YM equation (15) is precisely that expected from a
Lagrangian involving a cubic interaction only, whose Feynman rule involves the
product of structure constants for two Lie algebras, corresponding
respectively to colour, $f^{abc}$, and kinematics, $X(p,q)$. This makes
manifest the BCJ duality between colour and kinematics of ref. [2], in a
sector where the kinematic algebra is a straightforward Lie algebra.

Given eq. (15) and the interpretation given above, one may replace the colour
structure constants with a second set of kinematic structure constants, i.e.

$\displaystyle f^{abc}\rightarrow X(p_{1},p_{2}),$ (21)

obtaining the momentum-space equation of motion

$\phi(k)=\frac{\kappa}{2}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\,\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}\,X(p_{1},p_{2})^{2}\,\phi(p_{1})\phi(p_{2}),$
(22)

for a scalar field $\phi$. We now identify the coupling constant as the
gravitational coupling $\kappa=\sqrt{32\pi G_{N}}$, in terms of Newton’s
constant $G_{N}$. In coordinate space, eq. (22) is simply the Plebański
equation  for self-dual gravity, also known as the (second) heavenly equation:

$\partial^{2}\phi+\kappa\\{\partial_{w}\phi,\partial_{u}\phi\\}=0.$ (23)

Thus, the double copy is particularly clear, and amounts to simply replacing
the colour algebra with its kinematic counterpart. One can also go the other
way in eq. (21), replacing the area-preserving diffeomorphisms Lie algebra
with a second ‘colour’ Lie algebra $\tilde{\mathbf{g}}$ associated to a Lie
group $\tilde{G}$. In this case, one obtains solutions of the so-called
biadjoint scalar field theory, whose equation of motion is

$\partial^{2}\Phi^{aa^{\prime}}+yf^{abc}\tilde{f}^{a^{\prime}b^{\prime}c^{\prime}}\Phi^{bb^{\prime}}\Phi^{cc^{\prime}}=0,$
(24)

or, in momentum space,

$\Phi^{aa^{\prime}}(k)=y\,\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\,\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}\,f^{abc}\,\tilde{f}^{a^{\prime}b^{\prime}c^{\prime}}\,\Phi^{bb^{\prime}}(p_{1})\Phi^{cc^{\prime}}(p_{2}).$
(25)

Here, $y$ is a coupling constant, and $f^{abc}$,
$\tilde{f}^{a^{\prime}b^{\prime}c^{\prime}}$ are structure constants for the
two Lie groups. There is thus a hierarchy of theories as shown in figure 1,
which in the self-dual sector involves scalar field theories with a cubic
interaction. In moving to the right in the figure, one removes an adjoint
index from the field, and replaces a set of colour structure constants in the
field equation with a set of kinematic structure constants. The biadjoint
scalar theory plays a crucial role in the formulas expressing the double copy
of scattering amplitudes.

Figure 1: Various field theories and the relationships between them. Figure
taken from ref. [21].

From now on, we set the different coupling constants $(y,g,\kappa)$ to one.

This ‘heavenly’ example of the double copy is a different viewpoint to the
well-known story in the integrability literature where self-dual gravity is
recovered as a symmetry reduction of self-dual Yang-Mills theory, considering
the gauge group as the area-preserving diffeomorphism group on the two-surface
$\Sigma$, SDiff($\Sigma$) [106, 107, 108]. This fact has been used to build
hyper-Kähler metrics from solutions of two-dimensional reduced SDYM equations
[109, 110, 111]. The double copy motivates the consideration of the biadjoint
scalar theory mentioned above. Notice that self-dual Yang-Mills theory is a
‘symmetry reduction’ of the biadjoint scalar theory in the same way that self-
dual gravity is a symmetry reduction of self-dual Yang-Mills theory.

Let us also comment on an alternative description of the self-dual theories.
Whereas the second heavenly equation is eq. (23), the first heavenly equation
reads

$\\{\partial_{v}\Omega,\partial_{\bar{w}}\Omega\\}=1,$ (26)

where $\Omega$ is the Kähler potential, and we use the $(w,u)$ Poisson bracket
previously defined. The two heavenly equations are equivalent and arise due to
distinct gauge choices [83]. A somewhat analogous formulation is possible in
self-dual Yang-Mills, where instead of (10), the relevant equation is

$[\partial_{w}\Xi,\partial_{u}\Xi]=0,$ (27)

where $\Xi$ is Lie algebra-valued. This equation arises from a two-dimensional
sigma model with pure SU(N) Wess-Zumino term, and is related to eq. (26) in
the large-N limit [112], for reasons similar to those discussed in section
3.1.1. While equations (26) and (27) are closely related, the double copy
structure is not at all clear, as in the case of (23) and (11), which are
based on a cubic interaction vertex, the preferred setting for making the
double copy manifest. Still, it would be interesting to explore these and
other alternative formulations, in the search for further insights into the
double copy.

###  2.2 Lax pairs

There is an additional structure in the equations of motion (11) and (23) for
the self-dual theories, that did not receive attention in ref. [81], but which
will be useful to review for what follows. Namely, both self-dual Yang-Mills
and gravity are known to be integrable theories, admitting an infinite number
of conserved charges. The existence of a Lax pair indicates that a given
theory may be integrable, and we will be interested in tracing how
integrability is inherited by different theories in our double copy examples.
For self-dual YM, this Lax pair consists of two operators obtained from the
covariant derivative,

$L=D_{u}-\lambda D_{\bar{w}},\quad M=D_{w}-\lambda D_{v},$ (28)

where $\lambda\in{\mathbb{C}P}^{1}$ is an arbitrary spectral parameter. It is
then straightforward to show that the self-dual Yang-Mills equations of eqs.
(6–8) follow from the compatibility condition

$[L,M]=0.$ (29)

Note that for the explicit gauge choice of eqs. (9, 10), the above Lax pair
can also be written as

$L=\partial_{u}-\lambda(\partial_{\bar{w}}+i\partial_{u}\Phi),\qquad\qquad
M=\partial_{w}-\lambda(\partial_{v}+i\partial_{w}\Phi).$ (30)

The gravitational case can be expressed in a similar way. We can use known
results to classify the types of manifolds that arise, based on the relevant
Lax pair, as in [113, 114]. More specifically, let us take
${\mathbf{V}}=\\{U,V,W,\bar{W}\\}$ to be four independent holomorphic vector
fields on a four-dimensional complex manifold ${\cal M}$, such that

$L=U-\lambda\bar{W},\quad M=W-\lambda V,\quad\lambda\in{\mathbb{C}P}^{1},$
(31)

are operators satisfying $\mathcal{L}_{L}\upsilon=\mathcal{L}_{M}\upsilon=0$
and eq. (29), for all $\lambda$, where $\upsilon$ is a non-zero holomorphic
four-form and $\mathcal{L}$ denotes the Lie derivative. Then ${\bf V}$ forms a
null tetrad for a so-called hyper-Kähler metric on ${\cal M}$. To make contact
with the Plebański equation, we can choose coordinates such that

$\displaystyle U$ $\displaystyle=\partial_{u},$ $\displaystyle V$
$\displaystyle=\partial_{v}+{\phi_{ww}}\partial_{u}-{\phi_{uw}}\partial_{w},$
$\displaystyle W$ $\displaystyle=\partial_{w},$ $\displaystyle\bar{W}$
$\displaystyle=\partial_{\bar{w}}+{\phi_{uw}}\partial_{u}-{\phi_{uu}}\partial_{w},$
(32)

which corresponds to

$L=\partial_{u}-\lambda\left(\partial_{\bar{w}}+{\phi_{uw}}\partial_{u}-{\phi_{uu}}\partial_{w}\right),\quad
M=\partial_{w}-\lambda\left(\partial_{v}+{\phi_{ww}}\partial_{u}-{\phi_{uw}}\partial_{w}\right),$
(33)

where we use the notation
$\phi_{\mu\nu}\equiv\partial_{\mu}\partial_{\nu}\phi$. This Lax pair satisfies
$[L,M]=0$ if $\phi$ satisfies the Plebański equation (23). Furthermore, by
considering the holomorphic four-form

$\displaystyle\upsilon=du\wedge dv\wedge dw\wedge d\bar{w},$ (34)

the Lax pair satisfies also the conditions
$\mathcal{L}_{L}\upsilon=\mathcal{L}_{M}\upsilon=0$. Then, the vector fields
in eq. (32) form a null tetrad for a hyper-Kähler metric.

The Lax pairs (30) and (33) are naturally related, as expected by the double
copy. Let us introduce the Hamiltonian vector field in the $(u,w)$ plane,
$\Phi_{f}$, defined by its action on functions666This Hamiltonian vector field
is defined in the $(u,w)$ plane. However, the Hamiltonian functions depend on
the coordinates $(u,v,w,\bar{w})$.,

$\Phi_{f}(g)\equiv\\{f,g\\},$ (35)

so that

$[\Phi_{f},\Phi_{g}]=\Phi_{\\{f,g\\}}.$ (36)

Then, the Lax pair in self-dual gravity, in eq. (33), takes the form

$L=\partial_{u}-\lambda(\partial_{\bar{w}}+\partial_{u}\Phi_{\phi}),\qquad
M=\partial_{w}-\lambda(\partial_{v}+\partial_{w}\Phi_{\phi}),$ (37)

where the derivatives act only on the components of the Hamiltonian vector
fields, that is, we define
$\partial_{u}\Phi_{\theta_{n}}\equiv[\partial_{u},\Phi_{\theta_{n}}]$ and
$\partial_{w}\Phi_{\phi}\equiv[\partial_{w},\Phi_{\phi}]$. This Lax pair
mirrors the one in self-dual YM, eq. (30). In the self-dual gravity case, we
can also write the Lax pair as

$L=\partial_{u}-\lambda(\partial_{\bar{w}}+\Phi_{\partial_{u}\phi}),\qquad
M=\partial_{w}-\lambda(\partial_{v}+\Phi_{\partial_{w}\phi}).$ (38)

###  2.3 Infinite tower of conserved currents

Let us return to the question of integrability. With the Lax pairs in hand, it
is possible to construct recursively (formally at least) an infinite number of
symmetries, associated to the infinite-dimensional vector space of
deformations of solutions to the equations of motion. This brief review is
roughly based on the discussions in [115, 116]. Let us take the self-dual YM
example, with Lax pair given in (30). Notice that

$\displaystyle[\partial_{v}+i\partial_{w}\Phi,L\Psi]-[\partial_{\bar{w}}+i\partial_{u}\Phi,M\Psi]=\partial^{2}\Psi+i[\partial_{w}\Phi,\partial_{u}\Psi]+i[\partial_{w}\Psi,\partial_{u}\Phi]\,,$
(39)

where we have used the Jacobi identity and the equation of motion (11). Hence,
we obtain a solution $\Psi$ to the linearised equation of motion,
$\Phi\mapsto\Phi+\Psi$, by solving the system

$\displaystyle L\Psi=0,\qquad M\Psi=0\,.$ (40)

The compatibility condition of this overdetermined system is $[L,M]=0$. Now,
let us consider a solution to eq. (40) of the form

$\displaystyle\Psi(x,\lambda)=\sum_{n=0}^{\infty}\Psi_{n}(x)\lambda^{n}\,,$
(41)

where $\lambda$ is the spectral parameter. Since

$\displaystyle
L\Psi=\sum_{n=0}^{\infty}\left(\partial_{u}\Psi_{n}-\left(\partial_{\bar{w}}\Psi_{n-1}+i[\partial_{u}\Phi,\Psi_{n-1}]\right)\right)\lambda^{n},$
$\displaystyle
M\Psi=\sum_{n=0}^{\infty}\left(\partial_{w}\Psi_{n}-\left(\partial_{v}\Psi_{n-1}+i[\partial_{w}\Phi,\Psi_{n-1}]\right)\right)\lambda^{n},$
(42)

with $\Psi_{-1}\equiv 0$, the system (40) is solved by an infinite tower of
linearised solutions $\\{\Psi_{n}\\}$ constructed recursively via

$\displaystyle\partial_{u}\Psi_{n}=\partial_{\bar{w}}\Psi_{n-1}+i[\partial_{u}\Phi,\Psi_{n-1}],\qquad\partial_{w}\Psi_{n}=\partial_{v}\Psi_{n-1}+i[\partial_{w}\Phi,\Psi_{n-1}].$
(43)

For instance, we can take $\Psi_{0}=\Psi_{0}^{a}T^{a}$, where $\Psi_{0}^{a}$
are constants, so that $\Psi_{1}=i[\Phi,\Psi_{0}^{a}T^{a}]$, etc. We can also
express this hierarchy of infinitesimal symmetries in terms of a hierarchy of
conserved currents, by considering

$\displaystyle
J(x,\lambda)=(\partial_{w}\Psi)\partial_{u}-(\partial_{u}\Psi)\partial_{w},\qquad\text{such
that}\quad[\partial_{\mu},J^{\mu}]=0.$ (44)

The infinite tower of currents is given by the $\lambda$-expansion of
$J^{\mu}$. Equivalently, we can state that, if (43) represents commuting flows
of $\Psi_{n}$ along $u$ and $w$ (this is the compatibility condition
$\partial_{wu}\Psi_{n}=\partial_{uw}\Psi_{n}$), then $\Psi_{n}$ is a
linearised symmetry due to

$\displaystyle\partial^{2}\Psi_{n}+i[\partial_{w}\Phi,\partial_{u}\Psi_{n}]+i[\partial_{w}\Psi_{n},\partial_{u}\Phi]=i[\partial^{2}\Phi+i[\partial_{w}\Phi,\partial_{u}\Phi],\Psi_{n-1}]=0\,,$
(45)

where the last step follows from the equation of motion for $\Phi$. The first
equality above relies on the Jacobi identity.

The discussion for self-dual gravity is analogous. For the Plebański equation,
we can express the hierarchy of infinitesimal symmetries in terms of
Hamiltonian vector fields $\\{\Phi_{\theta_{n}}\\}$ recursively via

$\displaystyle\partial_{u}\Phi_{\theta_{n}}=\partial_{\bar{w}}\Phi_{\theta_{n-1}}+[\partial_{u}\Phi_{\phi},\Phi_{\theta_{n-1}}],\qquad\partial_{w}\Phi_{\theta_{n}}=\partial_{v}\Phi_{\theta_{n-1}}+[\partial_{w}\Phi_{\phi},\Phi_{\theta_{n-1}}].$
(46)

We can also write

$\displaystyle\Phi_{\partial_{u}\theta_{n}}=\Phi_{\partial_{\bar{w}}\theta_{n-1}}+\Phi_{\\{\partial_{u}\phi,\Phi_{\theta_{n-1}}\\}},\qquad\Phi_{\partial_{w}\theta_{n}}=\Phi_{\partial_{v}\theta_{n-1}}+\Phi_{\\{\partial_{w}\phi,\Phi_{\theta_{n-1}}\\}}\,.$
(47)

In analogy with the self-dual Yang-Mills case, we can take $\Phi_{\theta_{0}}$
as a constant Hamiltonian vector field, e.g.,
$\Phi_{\theta_{0}}=\partial_{u}-\partial_{w}$, so that
$\Phi_{\theta_{1}}=[\Phi_{\phi},\Phi_{\theta_{0}}]=\Phi_{\\{\phi,\theta_{0}\\}}$,
etc. For these two examples, we have $\theta_{0}=u+w$ and
$\theta_{1}=\phi_{w}-\phi_{u}$, respectively. Then the conserved currents are
obtained from

$\displaystyle J(x,\lambda)=\Phi_{\theta},\qquad\text{such
that}\quad[\partial_{\mu},J^{\mu}]=0\,.$ (48)

The analogous statement to (45) is

$\displaystyle\partial^{2}\theta_{n}+\\{\partial_{w}\phi,\partial_{u}\theta_{n}\\}+\\{\partial_{w}\theta_{n},\partial_{u}\phi\\}=\\{\partial^{2}\phi+\\{\partial_{w}\phi,\partial_{u}\phi\\},\theta_{n-1}\\}=0\,,$
(49)

or in terms of Hamiltonian vector fields

$\displaystyle\Phi_{\partial^{2}\theta_{n}+\\{\partial_{w}\phi,\partial_{u}\theta_{n}\\}+i\\{\partial_{w}\theta_{n},\partial_{u}\phi\\}}=[\Phi_{\partial^{2}\phi+\\{\partial_{w}\phi,\partial_{u}\phi\\}},\Phi_{\theta_{n-1}}]=0\,.$
(50)

##  3 Moyal deformations in gauge theory and gravity

In the previous section, we reviewed the known double copy at the level of
equations of motion in self-dual Yang-Mills theory and gravity, and also drew
attention to integrability aspects that have not been previously considered in
a double copy context. Our overall aim is to generalise this construction, and
thus we shall proceed by studying the kinds of generalisation of the above
theories that are possible. We will first look at Moyal deformations of our
usual self-dual theories, a concept which first arose in alternative
formulations of quantum mechanics [101, 102] as well as in the study of field
theories in non-commutative spacetimes [103, 104]. The main idea is to replace
the standard product on functions with the star product $\star$, defined in
our present case as follows:

$f\star g\equiv
f\exp\left(\frac{i\hbar}{2}\buildrel{\leftrightarrow}\over{P}\right)g,$ (51)

where we have defined the operator:

$\displaystyle\buildrel{\leftrightarrow}\over{P}\;\equiv\;\buildrel{\leftarrow}\over{\partial}_{w}\buildrel{\rightarrow}\over{\partial}_{u}-\buildrel{\leftarrow}\over{\partial}_{u}\buildrel{\rightarrow}\over{\partial}_{w}$
(52)

that appears in the Poisson bracket of eq. (20):

$\\{f,g\\}\equiv f\buildrel{\leftrightarrow}\over{P}g.$ (53)

The star product is associative and non-commutative, and contains a
deformation parameter $\hbar$, whose notation stems from the original context
in (non-commutative) quantum theories, and whose interpretation will be
clarified below. Its use in defining deformed equations of motion usually
proceeds by defining the so-called Moyal bracket

$\\{f,g\\}^{M}\equiv\frac{1}{i\hbar}(f\star g-g\star f),$ (54)

such that in the limit $\hbar\rightarrow 0$ we have

$\lim_{\hbar\rightarrow 0}f\star g=fg,\qquad\lim_{\hbar\rightarrow
0}\\{f,g\\}^{M}=\\{f,g\\}.$ (55)

That is, the star product of two functions reduces to the conventional
(commutative) product of functions, and the Moyal bracket to the Poisson
bracket. By substituting eq. (51) into eq. (54) and Taylor expanding in
$\hbar$, we may write a general expression for the Moyal bracket in terms of
the Poisson bracket, namely

$\displaystyle\\{f,g\\}^{M}$ $\displaystyle=$
$\displaystyle\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s}(-1)^{j}\binom{2s}{j}\\{\partial_{w}^{2s-j}\partial_{u}^{j}f,\partial_{w}^{j}\partial_{u}^{2s-j}g\\},$
(56)

or equivalently

$\displaystyle\\{f,g\\}^{M}=\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s+1}(-1)^{j}\binom{2s+1}{j}(\partial_{w}^{j}\partial_{u}^{2s+1-j}f)(\partial_{w}^{2s+1-j}\partial_{u}^{j}g).$
(57)

In a similar fashion to the previous section, we can explore the algebraic
properties of this object. First, one can show that for any three functions,
the Moyal bracket satisfies the Jacobi identity

$\\{\\{f,g\\}^{M},h\\}^{M}+\\{\\{g,h\\}^{M},f\\}^{M}+\\{\\{h,g\\}^{M},f\\}^{M}=0.$
(58)

Then, we consider the particular relation

$\\{e^{-ip_{1}x},e^{-ip_{2}x}\\}^{M}=-X^{M}(p_{1},p_{2})e^{-i(p_{1}+p_{2})x},$
(59)

where the structure constant $X^{M}(p_{1},p_{2})$ is given by

$\displaystyle X^{M}(p_{1},p_{2})\equiv
X(p_{1},p_{2})\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s}(-1)^{j}\binom{2s}{j}(p_{1w}p_{2u})^{2s-j}(p_{1u}p_{2w})^{j}.$
(60)

This is a generalisation of the quantity $X(p_{1},p_{2})$ encountered in eq.
(14), and reduces smoothly to the latter in the appropriate limit:

$\lim_{\hbar\rightarrow 0}X^{M}(p_{1},p_{2})=X(p_{1},p_{2}).$ (61)

Furthermore, using eqs. (58) and (60), it is easy to see that the structure
constant $X^{M}$ also satisfies the Jacobi identity

$X^{M}(p_{1},p_{2})X^{M}(p_{1}+p_{2},p_{3})+X^{M}(p_{2},p_{3})X^{M}(p_{2}+p_{3},p_{1})+X^{M}(p_{3},p_{1})X^{M}(p_{3}+p_{1},p_{2})=0.$
(62)

We can therefore use the deformed structure constant $X^{M}(p_{1},p_{2})$ as a
building block in constructing generalised field theories that obey double
copy relations, and will see a number of examples below.

Another useful realisation is the Moyal deformation of the Lie bracket.
Naively, we could define this as

$[V,W]^{M}\equiv\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s}(-1)^{j}\binom{2s}{j}[\partial_{w}^{2s-j}\partial_{u}^{j}V,\partial_{w}^{j}\partial_{u}^{2s-j}W].$
(63)

For vector fields $V$ and $W$, the derivatives inside the Lie bracket are
understood as acting as $\partial_{u}V\equiv[\partial_{u},V]$, etc.
Generically, this idea is naive, because such a Moyal bracket does not in
general satisfy the Jacobi identity. A particular case where it does indeed
satisfy it is that of Hamiltonian vector fields,

$\displaystyle[\Phi_{f},\Phi_{g}]^{M}$ $\displaystyle=$
$\displaystyle\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s}(-1)^{j}\binom{2s}{j}[\partial_{w}^{2s-j}\partial_{u}^{j}\Phi_{f},\partial_{w}^{j}\partial_{u}^{2s-j}\Phi_{g}]$
(64) $\displaystyle=$
$\displaystyle\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s}(-1)^{j}\binom{2s}{j}\Phi_{\\{\partial_{w}^{2s-j}\partial_{u}^{j}f,\partial_{w}^{j}\partial_{u}^{2s-j}g\\}}$
$\displaystyle=$ $\displaystyle\Phi_{\\{f,g\\}^{M}},$

such that, analogously to the Jacobi identity (58),

$[[\Phi_{f},\Phi_{g}]^{M},\Phi_{h}]^{M}+[[\Phi_{g},\Phi_{h}]^{M},\Phi_{f}]^{M}+[[\Phi_{h},\Phi_{f}]^{M},\Phi_{g}]^{M}=0.$
(65)

###  3.1 Deformed self-dual gravity

As a first application of the above ideas, we will consider a Moyal
deformation of the Plebański equation, which was investigated in [106, 117,
118]. The deformed Plebański (second heavenly) equation is

$\partial^{2}\phi+\\{\partial_{w}\phi,\partial_{u}\phi\\}^{M}=0.$ (66)

In momentum space, this deformed equation can be expressed as

$\phi(k)=\frac{1}{2}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}X^{M}(p_{1},p_{2})X(p_{1},p_{2})\phi(p_{1})\phi(p_{2}),$
(67)

and it is clear from eq. (61) that the usual Plebański equation both in
coordinate space (23) and in momentum space eq. (22) are recovered in the
limit $\hbar\rightarrow 0$.

The form of eq. (67) is particularly appealing from a double copy point of
view. That is, all of the theories considered in this paper have momentum-
space integral equations whose kernels involve products of structure
constants, and the present case can be obtained by a process analogous to the
double copy of eq. (21), making instead the replacement

$f^{abc}\rightarrow X^{M}(p_{1},p_{2}),$ (68)

in eq. (15). Below, we will extend this philosophy to obtain a web of theories
with certain desirable properties. Before doing so, however, it is interesting
to examine the integrability of the theory of eq. (66). This has been
investigated in [119, 118], where as usual one may address this question by
formulating a Lax pair. There are two possible ways to do this, namely we may
consider a Lax pair consisting of undeformed vectors satisfying a deformed
bracket, or ‘deformed vectors’ satisfying an undeformed bracket. The two
options are related to attributing $X(p_{1},p_{2})$ or $X^{M}(p_{1},p_{2})$ in
eq. (67) to the Lax compatibility condition or to the Lax pair. For the
undeformed Lax pair, we can take the results of eq. (33) and use (64) to show
that they satisfy the deformed compatibility condition

$[L,M]^{M}=0$ (69)

if $\phi$ satisfies the deformed Plebański eq. (66).

The argument when using a deformed pair (but undeformed bracket) is more
involved. First, let us explain the deformation of the vectors as follows. We
can rewrite the Hamiltonian vector field $\Phi_{f}$ in the $(w,u)$ plane,
given in eq. (35), using the Poisson operator (52) in the form

$\Phi_{f}\equiv
f\buildrel{\leftrightarrow}\over{P}=f_{w}\partial_{u}-f_{u}\partial_{w}.$ (70)

Clearly, this vector satisfies the relation

$\Phi_{f}(g)=f\buildrel{\leftrightarrow}\over{P}g=\\{f,g\\}.$ (71)

Now, following the Poisson bracket case, eq. (71), we can define (in a slight
abuse of language) a deformed Hamiltonian vector field, $\Phi^{M}_{f}$, by

$\Phi^{M}_{f}(g)\equiv\\{f,g\\}^{M}.$ (72)

Then we have [119, 120]

$\displaystyle\Phi^{M}_{f}$
$\displaystyle=\frac{2}{\hbar}f\sin\left(\frac{\hbar}{2}\buildrel{\leftrightarrow}\over{P}\right)$
$\displaystyle=\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s+1}(-1)^{j}\binom{2s+1}{j}(\partial_{w}^{j}\partial_{u}^{2s+1-j}f)\partial_{w}^{2s+1-j}\partial_{u}^{j}.$
(73)

Note that in the $\hbar\rightarrow 0$ limit, we recover a conventional
Hamiltonian vector field, i.e. $\lim_{\hbar\rightarrow
0}\Phi^{M}_{f}=\Phi_{f}$. It follows from the above that the Lie bracket of
deformed Hamiltonian vector fields is

$[\Phi^{M}_{f},\Phi^{M}_{g}]=\Phi^{M}_{\\{f,g\\}^{M}}.$ (74)

Note that, following the Jacobi identity (58), the deformed Hamiltonian vector
fields satisfies the Jacobi identity

$[[\Phi^{M}_{f},\Phi^{M}_{g}],\Phi^{M}_{h}]+[[\Phi^{M}_{g},\Phi^{M}_{h}],\Phi^{M}_{f}]+[[\Phi^{M}_{h},\Phi^{M}_{f}],\Phi^{M}_{g}]=0.$
(75)

Taking the ‘deformed vectors’

$L^{M}=\partial_{u}-\lambda\left(\partial_{\bar{w}}+\Phi_{\phi_{u}}^{M}\right),\qquad\qquad
M^{M}=\partial_{w}-\lambda\left(\partial_{v}+\Phi_{\phi_{w}}^{M}\right),$ (76)

we can check that they satisfy the condition

$[L^{M},M^{M}]=0$ (77)

if $\phi$ satisfies the deformed Plebański eq. (66). To see this, we note that
only the $\lambda^{2}$ term does not vanish trivially, taking the form

$\displaystyle[\partial_{v}+\Phi_{\phi_{w}}^{M},\partial_{\bar{w}}+\Phi_{\phi_{u}}^{M}]$
$\displaystyle=$
$\displaystyle\partial^{2}\Phi_{\phi}^{M}+[\Phi_{\phi_{w}}^{M},\Phi_{\phi_{u}}^{M}]$
(78) $\displaystyle=$
$\displaystyle\Phi_{\partial^{2}\phi+\\{\phi_{w},\phi_{u}\\}^{M}}^{M}$
$\displaystyle=$ $\displaystyle 0.$

In this case we have the deformed vectors

$\displaystyle U$ $\displaystyle=\partial_{u},$ $\displaystyle V$
$\displaystyle=\partial_{v}+\Phi_{\phi_{w}}^{M},$ $\displaystyle W$
$\displaystyle=\partial_{w},$ $\displaystyle\bar{W}$
$\displaystyle=\partial_{\bar{w}}+\Phi_{\phi_{u}}^{M}.$ (79)

The two equivalent approaches mentioned above, of deforming either the bracket
or the Lax pair, are represented by (69) and (77).

We now proceed in analogy to section 2.3 to obtain an infinite tower of
conserved currents for the deformed theory. First, the linearised equation of
motion is obtained from the deformed Plebański equation (66) via the
replacement $\Phi^{M}_{\phi}\rightarrow\Phi^{M}_{\phi}+\Phi^{M}_{\theta}$, is

$\partial^{2}\Phi^{M}_{\theta}+[\partial_{w}\Phi^{M}_{\phi},\partial_{u}\Phi^{M}_{\theta}]+[\partial_{w}\Phi^{M}_{\theta},\partial_{u}\Phi^{M}_{\phi}]=0.$
(80)

Now, let us consider $\Phi^{M}_{\theta}$ of the form

$\Phi^{M}_{\theta}=\sum^{\infty}_{n=0}\Phi^{M}_{\theta_{n}}\lambda^{n}.$ (81)

We can express the infinite tower of linearised solutions
$\\{\Phi^{M}_{\theta_{n}}\\}$ to eq. (80) recursively via

$\displaystyle\partial_{u}\Phi^{M}_{\theta_{n}}=\partial_{\bar{w}}\Phi^{M}_{\theta_{n-1}}+[\partial_{u}\Phi^{M}_{\phi},\Phi^{M}_{\theta_{n-1}}],\qquad\partial_{w}\Phi^{M}_{\theta_{n}}=\partial_{v}\Phi^{M}_{\theta_{n-1}}+[\partial_{w}\Phi^{M}_{\phi},\Phi^{M}_{\theta_{n-1}}],$
(82)

with $\Phi^{M}_{\theta_{-1}}\equiv 0$. This $\Phi^{M}_{\theta_{n}}$ satisfies
the compatibility condition
$\partial_{u}\partial_{w}\Phi^{M}_{\theta_{n}}=\partial_{w}\partial_{u}\Phi^{M}_{\theta_{n}}$
and it is a solution of the linearised equation (80), using the Jacobi
identity (75). Alternatively, the same infinite tower of linearised solutions
can be obtained from Hamiltonian vector fields but using the deformed bracket.
In this notation, we have the recursive equations

$\displaystyle\partial_{u}\Phi_{\theta_{n}}=\partial_{\bar{w}}\Phi_{\theta_{n-1}}+[\partial_{u}\Phi_{\phi},\Phi_{\theta_{n-1}}]^{M},\qquad\partial_{w}\Phi_{\theta_{n}}=\partial_{v}\Phi_{\theta_{n-1}}+[\partial_{w}\Phi_{\phi},\Phi_{\theta_{n-1}}]^{M},$
(83)

with $\Phi_{\theta_{-1}}\equiv 0$. As in the deformed Hamiltonian vector
fields, this $\Phi_{\theta_{n}}$ satisfies the compatibility condition
$\partial_{u}\partial_{w}\Phi_{\theta_{n}}=\partial_{w}\partial_{u}\Phi_{\theta_{n}}$
and it is a solution of the linearised equation

$\partial^{2}\Phi_{\theta}+[\partial_{w}\Phi_{\phi},\partial_{u}\Phi_{\theta}]^{M}+[\partial_{w}\Phi_{\theta},\partial_{u}\Phi_{\phi}]^{M}=0.$
(84)

using the Jacobi identity (65).

The Moyal-deformed self-dual gravity shares features and properties with the
undeformed version. One such property, as discussed above, is integrability.
In terms of the language of the first heavenly equation, eq. (26), this
property can be expressed in a concise geometric way with an associated 2-form
$\mathbf{\Omega}$, which satisfies the equations $d\mathbf{\Omega}=0$ and
$\mathbf{\Omega}\wedge\mathbf{\Omega}=0$ [119]. This is the Kähler form, which
is related to the Kähler potential $\Omega$ as
$\mathbf{\Omega}=\Omega_{\mu\nu}dx^{\mu}\wedge d\tilde{x}^{\nu}$, where
$x^{\mu}=\\{w,u\\}$ and $\tilde{x}^{\nu}=\\{y,z\\}$. In [121], a Moyal
deformation of the first heavenly equation was considered, and a four-
dimensional Moyal deformed integrable Kähler manifold was constructed by
imposing in a consistent way that the deformed 2-form $\mathbf{\Omega}$ is
closed, Hermitian and has unit determinant. This procedure provides an
alternative but equivalent approach to equation (66), based on the second
heavenly equation.

####  3.1.1 Deformed self-dual gravity as undeformed self-dual Yang-Mills

In this subsection, we take an alternative approach to the Moyal deformation
of self-dual gravity, which leads directly to undeformed self-dual Yang-Mills.
Instead of the deformation $X(p_{1},p_{2})\to X^{M}(p_{1},p_{2})$ considered
above, we will see that we can think of $X(p_{1},p_{2})\to f^{abc}$ as a Moyal
deformation, at least in a particular construction.

In order to do that we follow now Ref. [122], which considered the case of
SDYM in $\mathbb{R}^{1,D-1}$ with su$(N)$ or sl$(N,\mathbb{C})$ Lie algebra
valued fields.777By SDYM beyond $D=4$, we mean simply the straightforward
extension of (11), where the wave operator is $D$-dimensional. The
dimensionality is not relevant for our purposes. In Ref. [122], a novel
version of the basis for these algebras was found, in terms of a double index
notation $m=(m_{1},m_{2})$. In more detail, the Lie algebra generators can be
written as

$L_{m}\equiv-{N\over 2\pi}\omega^{m_{1}m_{2}\over 2}S^{m_{1}}T^{m_{2}},$ (85)

where $\omega=\exp\big{(}{2\pi i\over N}\big{)}$, and $S$ and $T$ are $N\times
N$ matrices such that $S^{N}=T^{N}=-1$ and $T\cdot S=\omega\,S\cdot T$. These
generators satisfy

$[L_{m},L_{n}]=i{\cal F}_{m,n}^{(m+n)}L_{m+n},$ (86)

where ${\cal F}_{m,n}^{(m+n)}={N\over\pi}\sin\big{(}{\pi\over N}m\times
n\big{)}$. Thus, the analogue of eq. (15) is given by

$\displaystyle\Phi^{m+n}(k)=\frac{1}{2}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}X(p_{1},p_{2}){N\over\pi}\sin\bigg{(}{\pi\over
N}m\times n\bigg{)}\Phi^{m}(p_{1})\Phi^{n}(p_{2}).$ (87)

In this basis, scalars $\Phi$ are expanded as
$\Phi(x)=\sum_{m}\Phi^{m}(x)L_{m}$.

In the limit $N\to\infty$ for the Lie algebra in eq. (86), we get the
sdiff$(T^{2})$ algebra

$\\{e_{m},e_{n}\\}_{T}=(m\times n)e_{m+n},$ (88)

where $m\times n=m_{1}n_{2}-m_{2}n_{1}$, and the Poisson bracket on $T^{2}$ is
$\\{f,g\\}_{T}\equiv f_{x_{2}}g_{x_{1}}-f_{x_{1}}g_{x_{2}}$. The generators of
sdiff($T^{2}$) with local coordinates $(x_{1},x_{2})$ are

$e_{m}\equiv\exp i(m_{1}x_{1}+m_{2}x_{2}).$ (89)

Furthermore there is an isomorphism which maps the Lie bracket (86) into the
Moyal bracket for $e_{m}$ :

$\\{e_{m},e_{n}\\}_{T}^{M}={2\over\hbar}\sin\bigg{(}{\hbar\over 2}m\times
n\bigg{)}e_{m+n},$ (90)

where we have made the identification

$\hbar=\frac{2\pi}{N}.$ (91)

Thus the large-$N$ limit corresponds to the limit $\hbar\to 0$.

Then, the equation of motion on $\mathbb{R}^{1,D-3}\times T^{2}$ describes an
integrable Moyal deformation of self-dual gravity equation known as the
$\star$-SDYM system [123, 124] given by

$\partial^{2}\phi+\\{\partial_{w}\phi,\partial_{u}\phi\\}_{T}^{M}=0,$ (92)

for $\phi=\sum_{m}\phi^{m}(\hbar;\mbox{\boldmath$x$},x_{1},x_{2})e_{m}$ and
$\phi^{m}=\sum_{n=0}^{\infty}\hbar^{n}\phi^{m}_{n}$. Equation (92) can then be
written as

$\displaystyle\phi^{m+n}(k)={1\over\hbar}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}X(p_{1},p_{2})\sin\bigg{(}{\hbar\over
2}m\times n\bigg{)}\phi^{m}(p_{1})\phi^{n}(p_{2}).$ (93)

The master equation (92) is proven to correspond to an integrable system
[123], by showing the existence of Lax pairs, an infinite hierarchy of
conserved quantities, and a twistor construction. Moreover in Ref. [124] some
explicit examples are given. The direct map between (93) and the SDYM equation
(87) is clear.

Taking the limit $\hbar\to 0$ in eq. (93), we get the Plebański-Przanowski
equation [117]

$\displaystyle\phi^{m+n}_{0}(k)={1\over 2}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}X(p_{1},p_{2})(m\times
n)\phi^{m}_{0}(p_{1})\phi^{n}_{0}(p_{2}).$ (94)

Equivalently, one can take the limit $N\rightarrow\infty$ in eq. (87) on
$\mathbb{R}^{1,D-3}$. For instance, the $D=4$ case corresponds to
$\mathbb{R}^{1,1}\times T^{2}$. In this case the gauge theory is the principal
chiral model, which is an integrable two-dimensional reduction of SDYM
equations on $\mathbb{R}^{1,3}$. This model leads to the Husain-Park heavenly
equation which was discussed in [125, 124].

It is worth mentioning that this procedure can be employed to interpolate
between all theories shown in figure 1. For instance, we can start from the
biadjoint scalar theory and take in eq. (25) two times the representation of
the Lie algebra in the trigonometric basis [122] i.e.
$f^{abc}\to{2\over\hbar}\sin\big{(}{\hbar\over 2}m\times n\big{)}$ and
$\widetilde{f}^{a^{\prime}b^{\prime}c^{\prime}}\to{2\over\hbar^{\prime}}\sin\big{(}{\hbar^{\prime}\over
2}m^{\prime}\times n^{\prime}\big{)}$. Taking the limit, for instance,
$\hbar\to 0$ this represents the inverse zeroth copy and we get self-dual
Yang-Mills equation (15). A further limit $\hbar^{\prime}\to 0$ leads
straightforwardly to the double copy, i.e. the self-dual gravity equation
(22). It is also possible to start from the self-dual gravity equation (22)
and take its single copy as a Moyal deformation of gravity with
$\hbar^{\prime}\not=0$. A further Moyal deformation with $\hbar\not=0$ in the
gauge theory leads directly to the zeroth copy giving the biadjoint scalar
theory. Thus, this procedure allows us to interpolate with two continuous
parameters $\hbar$ and $\hbar^{\prime}$ among the set of theories mentioned in
figure 1. Of course, this particular construction relied on periodic
identifications of coordinates, so that a $T^{2}$ arises, and the Fourier
modes are discrete. If they are not discrete, then we will have structure
constants labeled by continuous indices like $X^{M}(p_{1},p_{2})$ in the
earlier discussion.

###  3.2 Deformed Self-Dual Yang-Mills

We will now explore the Moyal deformation of self-dual Yang-Mills theory. We
will see, however, that the deformation that fits in with the colour-
kinematics duality does not coincide with the most commonly considered Moyal
deformation of self-dual Yang-Mills – whereas the latter deformation is
integrable [126, 127], the former does not preserve it.

Starting from the deformed Plebański equation for self-dual gravity in
momentum space (67), we have two different single copies. One of them,
replacing $X^{M}(p_{1},p_{2})\rightarrow f^{abc}$, is the usual self-dual
Yang-Mills theory. On the other hand, we can take $X(p_{1},p_{2})\rightarrow
f^{abc}$ and we obtain a deformed version of self-dual Yang-Mills. The
equation of motion for this gauge theory in momentum space is

$\displaystyle\Phi^{a}(k)=\frac{1}{2}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}X^{M}(p_{1},p_{2})f^{abc}\Phi^{b}(p_{1})\Phi^{c}(p_{2}).$
(95)

In the limit $\hbar\rightarrow 0$, we recover the usual SDYM equation (15).
Furthermore, one may show that eq. (95) is equivalent to the position space
equation

$\partial^{2}\Phi+i[\partial_{w}\Phi,\partial_{u}\Phi]^{M}=0,$ (96)

where we are using the Moyal bracket (63). This theory differs from the usual
Moyal deformation of self-dual Yang-Mills theory that is considered in the
literature, which can be written as

$\partial^{2}\Phi+i[\partial_{w}\Phi,\partial_{u}\Phi]_{\star}=0,\qquad[A,B]_{\star}\equiv
A\star B-B\star A.$ (97)

This latter equation has a very specific interpretation: it is the theory one
obtains upon introducing self-dual Yang-Mills theory in a non-commutative
spacetime. To our knowledge, no such interpretation exists for eq. (96), which
is the unique theory one obtains upon single copying the deformed Plebański
equation in the above-mentioned fashion. Both eq. (96) and eq. (97) reduce to
the usual self-dual Yang-Mills theory in the limit $\hbar\rightarrow 0$. The
relation between these two distinct deformations is not obvious. In fact, the
colour structure of eq. (97) is very different from the one typically found in
gauge theories. For instance, a calculation shows that

$\displaystyle
i[\partial_{w}(\Phi^{a}T^{a}),\partial_{u}(\Phi^{b}T^{b})]_{\star}=$
$\displaystyle-\frac{1}{2}f^{abc}T^{c}\,\frac{d}{d\hbar}\left(\hbar\,\\{\Phi^{a},\Phi^{b}\\}^{M}\right)-\hbar\,T^{(a}T^{b)}\,\\{\partial_{w}\Phi^{a},\partial_{u}\Phi^{b}\\}^{M}\,$
$\displaystyle=$
$\displaystyle-\frac{1}{2}f^{abc}T^{c}\,\\{\Phi^{a},\Phi^{b}\\}-\hbar\,T^{(a}T^{b)}\,\\{\partial_{w}\Phi^{a},\partial_{u}\Phi^{b}\\}+{\mathcal{O}}(\hbar^{2}).$
(98)

In the right-hand side of the first line, the first/second term has only
odd/even powers of $\hbar$. We can see right away that $\Phi$ is generically
not a colour Lie algebra element in the usual Moyal deformation (97), i.e.,
$\Phi\neq\Phi^{a}T^{a}$; because if you start with $\Phi=\Phi^{a}T^{a}$ in the
left-hand side of eq. (97), a distinct colour structure is generated,
$T^{(a}T^{b)}$, which is not an element of the Lie algebra. Instead, $\Phi$
lives in the enveloping algebra generated by elements
$T^{(a_{1}}T^{a_{2}}\cdots T^{a_{n})}$. This contrasts with the single copy
theory case of eq. (96), where $\Phi$ lives in the Lie algebra. The question
of whether there is a more direct physical interpretation of eq. (96) is
interesting.888There are several instances in the double copy literature where
one theory of no obvious relevance of its own is very useful for providing
building blocks, via the double copy, for more physically relevant theories.
For our present purposes, we shall simply examine the question of
integrability, by attempting to construct a Lax pair following an analogous
process to that of SDYM. We will take a naive approach in order to see where
it fails. Applying the Moyal bracket (63) for a gauge field, one may first
look at the components of the deformed field strength tensor

$F_{\mu\nu}^{M}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-i[A_{\mu},A_{\nu}]^{M},$
(99)

and the deformed self-duality conditions take the form

$\displaystyle F_{uw}^{M}=0,$ $\displaystyle F_{v\bar{w}}^{M}=0,$
$\displaystyle F_{uv}^{M}-F_{w\bar{w}}^{M}=0.$ (100)

These equations arise as the condition $[L,M]^{M}=0$ for the two operators
defined in eq. (30). However, this does not by itself guarantee integrability,
which may be seen upon trying to construct a hierarchy of linearised
solutions. The replacement $\Phi\rightarrow\Phi+\Psi$ results in the
linearised equation

$\partial^{2}\Psi+i[\partial_{w}\Psi,\partial_{u}\Phi]^{M}+i[\partial_{w}\Phi,\partial_{u}\Psi]^{M}=0.$
(101)

Expanding $\Psi$ in powers of the spectral parameter $\lambda$, we can express
the infinite tower $\\{\Psi_{n}\\}$ recursively via

$\displaystyle\partial_{u}\Psi_{n}=\partial_{\bar{w}}\Psi_{n-1}+i[\partial_{u}\Phi,\Psi_{n-1}]^{M},\qquad\partial_{w}\Psi_{n}=\partial_{v}\Psi_{n-1}+i[\partial_{w}\Phi,\Psi_{n-1}]^{M},$
(102)

with $\Psi_{-1}\equiv 0$. One may be tempted to conclude that the theory is
integrable. However, this type of construction relies crucially on the Jacobi
identity for the bracket, which fails in the present case of the Moyal bracket
(63).999We made comments regarding this property near (63), where we noted
that, when the Moyal-deformed bracket was applied to vector fields, the Jacobi
identity did not apply in general, but did apply to Hamiltonian vector fields.
It relies on the Jacobi identity in order to prove that this infinite tower is
a solution of the linearised equation (101) – without the Jacobi identity, one
cannot show that $\Psi_{n}$ is a symmetry if $\Psi_{n-1}$ is. Even before
that, the Jacobi identity is crucial in order to relate the linearised
equation to the presumptive Lax pair. To see in detail how the Jacobi identity
fails, let us take three gauge fields $V=V^{a}T^{a}$, $W=W^{b}T^{b}$ and
$U=U^{c}T^{c}$. First, the Moyal bracket of $V$ and $W$ is explicitly

$\displaystyle[V,W]^{M}$ $\displaystyle=$ $\displaystyle
i\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s}(-1)^{j}\binom{2s}{j}f^{abd}(\partial_{w}^{2s-j}\partial_{u}^{j}V^{a})(\partial_{w}^{j}\partial_{u}^{2s-j}W^{b})T^{d}$
(103) $\displaystyle=$ $\displaystyle
i\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s}(-1)^{j}\binom{2s}{j}f^{abd}V^{a}_{2s-j,j}W^{b}_{j,2s-j}T^{d}$
$\displaystyle=$ $\displaystyle
if^{abd}V^{a}W^{b}T^{d}+{\mathcal{O}}(\hbar^{2}),$

using the notation $\partial_{w}^{n}\partial_{u}^{m}F=F_{n,m}$. Now, for three
gauge fields, we have

$[[V,W]^{M},U]^{M}=-f^{abd}f^{dce}T^{e}\\\
\times\sum_{s,s^{\prime}=0}^{\infty}\frac{(-1)^{s+s^{\prime}}\hbar^{2(s+s^{\prime})}}{2^{2(s+s^{\prime})}(2s+1)!(2s^{\prime}+1)!}\sum_{j=0}^{2s}\sum_{j^{\prime}=0}^{2s^{\prime}}(-1)^{j+j^{\prime}}\binom{2s}{j}\binom{2s^{\prime}}{j^{\prime}}(V^{a}_{2s-j,j}W^{b}_{j,2s-j})_{2s^{\prime}-j^{\prime},j^{\prime}}U^{c}_{j^{\prime},2s^{\prime}-j^{\prime}}\\\
=-f^{abd}f^{dce}T^{e}\left[V^{a}W^{b}U^{c}-\frac{\hbar^{2}}{24}\sum_{j=0}^{2}(-1)^{j}\binom{2}{j}\left(V^{a}_{2-j,j}W^{b}_{j,2-j}U^{c}+(V^{a}W^{b})_{2-j,j}U^{c}_{j,2-j}\right)\right]+{\mathcal{O}}(\hbar^{4}).$
(104)

Finally, the Jacobi type equation gives

$[[V,W]^{M},U]^{M}+[[W,U]^{M},V]^{M}+[[U,V]^{M},W]^{M}\\\
=-(f^{abd}f^{dce}+f^{bcd}f^{dae}+f^{cad}f^{dbe})T^{e}V^{a}W^{b}U^{c}+{\mathcal{O}}(\hbar^{2}).$
(105)

The leading order term, ${\mathcal{O}}(\hbar^{0})$, vanishes due to the colour
Jacobi identity. However, the first subleading term is proportional to

$\sum_{j=0}^{2}(-1)^{j}\binom{2}{j}\left[f^{abd}f^{dce}\left(V^{a}_{2-j,j}W^{b}_{j,2-j}U^{c}+(V^{a}W^{b})_{2-j,j}U^{c}_{j,2-j}\right)\right.\\\
\left.+f^{bcd}f^{dae}\left(W^{b}_{2-j,j}U^{c}_{j,2-j}V^{a}+(W^{b}U^{c})_{2-j,j}V^{a}_{j,2-j}\right)+f^{cad}f^{dbe}\left(U^{c}_{2-j,j}V^{a}_{j,2-j}W^{b}+(U^{c}V^{a})_{2-j,j}W^{b}_{j,2-j}\right)\right]\\\
\neq 0,$ (106)

that is, it does not vanish generically. Hence,
$[[V,W]^{M},U]^{M}+[[W,U]^{M},V]^{M}+[[U,V]^{M},W]^{M}\neq 0$. The
integrability property relies implicitly on the Jacobi identity for the
appropriate bracket, which fails in this case. Therefore, the deformation (96)
of the SDYM equation of motion, which we arrived at by taking the single copy
of Moyal-deformed SDG that keeps the factor $X^{M}(p_{1},p_{2})$, and changes
$X(p_{1},p_{2})\rightarrow f^{abc}$, is not integrable. This is because the
integrability of Moyal-deformed SDG relies precisely on the factor
$X(p_{1},p_{2})$, which we discarded here.

###  3.3 Doubly deformed self-dual gravity

We have so far been constructing various theories from known self-dual
integral equations, by replacing different structure constants by their
deformed counterparts. Continuing this procedure, we can consider the theory
one obtains by taking two copies of the Moyal-deformed kinematic algebra,
whose structure constant is $X^{M}(p_{1},p_{2})$. This can be obtained e.g. by
taking the deformed self-dual Yang-Mills theory of eq. (95), and replacing
$f^{abc}\rightarrow X^{M^{\prime}}(p_{1},p_{2})$. The resulting equation of
motion in momentum space then takes the form

$\phi(k)=\frac{1}{2}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}X^{M}(p_{1},p_{2})X^{M^{\prime}}(p_{1},p_{2})\phi(p_{1})\phi(p_{2}),$
(107)

where

$X^{M^{\prime}}(p_{1},p_{2})=X(p_{1},p_{2})\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{\prime
2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s}(-1)^{j}\binom{2s}{j}(p_{1w}p_{2u})^{2s-j}(p_{1u}p_{2w})^{j},$
(108)

and we have introduced a second deformation parameter $\hbar^{\prime}$ to be
associated with $M^{\prime}$. In position space this equation is

$\partial^{2}\phi+\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s}(-1)^{j}\binom{2s}{j}\\{\partial_{w}^{2s-j+1}\partial_{u}^{j}\phi,\partial_{w}^{j}\partial_{u}^{2s-j+1}\phi\\}^{M^{\prime}}=0.$
(109)

If we take either of the limits $\hbar\rightarrow 0$ or
$\hbar^{\prime}\rightarrow 0$, we recover the deformed Plebański equation
(66). Furthermore, it is straightforward to verify that eq. (109) arises from
the (doubly deformed) Lax Pair condition

$[L^{M^{\prime}},M^{M^{\prime}}]^{M}=0.$ (110)

To see this, one may first note that only the $\lambda^{2}$ does not vanish
trivially, which in turn leads explicitly to

$\displaystyle[\partial_{v}+\Phi_{\phi_{w}}^{M^{\prime}},\partial_{\bar{w}}+\Phi_{\phi_{u}}^{M^{\prime}}]^{M}$
$\displaystyle=$
$\displaystyle\partial^{2}\,\Phi^{M^{\prime}}_{\phi}+[\Phi^{M^{\prime}}_{\phi_{w}},\Phi^{M^{\prime}}_{\phi_{u}}]^{M}$
(111) $\displaystyle=$
$\displaystyle\Phi^{M^{\prime}}_{\partial^{2}\phi+\sum_{s=0}^{\infty}\frac{(-1)^{s}\hbar^{2s}}{2^{2s}(2s+1)!}\sum_{j=0}^{2s}(-1)^{j}\binom{2s}{j}\\{\partial_{w}^{2s+1-j}\partial_{u}^{j}\phi,\partial_{w}^{j}\partial_{u}^{2s+1-j}\phi\\}^{M}}$
$\displaystyle=$ $\displaystyle 0.$

Moreover, we can obtain the same equation with $[L^{M},M^{M}]^{M^{\prime}}=0$.
For this double deformed Plebański equation (109), we obtain a linearised
equation of motion obtained from the replacement
$\Phi^{M^{\prime}}_{\phi}\rightarrow\Phi^{M^{\prime}}_{\phi}+\Phi^{M^{\prime}}_{\theta}$.
Expanding $\Phi^{M^{\prime}}_{\theta}$ in powers of the spectral parameter
$\lambda$, the infinite tower $\\{\Phi^{M^{\prime}}_{\theta_{n}}\\}$ can be
expressed recursively as

$\displaystyle\partial_{u}\Phi^{M^{\prime}}_{\theta_{n}}=\partial_{\bar{w}}\Phi^{M^{\prime}}_{\theta_{n-1}}+[\partial_{u}\Phi^{M^{\prime}}_{\phi},\Phi^{M^{\prime}}_{\theta_{n-1}}]^{M},\qquad\partial_{w}\Phi^{M^{\prime}}_{\theta_{n}}=\partial_{v}\Phi^{M^{\prime}}_{\theta_{n-1}}+[\partial_{w}\Phi^{M^{\prime}}_{\phi},\Phi^{M^{\prime}}_{\theta_{n-1}}]^{M},$
(112)

with $\Phi^{M^{\prime}}_{\theta_{-1}}=0$. Similar to the deformed self-dual
Yang-Mills infinity tower (102), the infinity tower (112) satisfies the
compatibility condition
$\partial_{w}\partial_{u}\Phi^{M^{\prime}}_{\theta_{n}}=\partial_{u}\partial_{w}\Phi^{M^{\prime}}_{\theta_{n}}$.
However, also similarly to the deformed self-dual Yang-Mills theory, the
integrability construction fails because the Jacobi identity of the Moyal
bracket applied to deformed vector fields fails.

In this section, we have demonstrated the existence of a family of generalised
self-dual gauge and gravity theories obeying double copy relationships. The
key ingredients were the expression of such theories as integral equations
whose kernels manifestly contain products of structure constants, and the
introduction of new such constants – e.g., those of Moyal-deformed
diffeomorphism algebra, $X^{M}(p_{1},p_{2})$ – that could be used as
appropriate building blocks. In all cases, we could show that the position
space equations of motion resulted from a Lax pair type of condition, although
this was not always sufficient to guarantee integrability. The Moyal
deformation considered above is in fact only one possible generalisation of
the heavenly double copy between self-dual theories: we examine another in the
following section.

##  4 Generalising to the diffeomorphism algebra

We will now, operating with a similar motive as in the previous section,
consider a second – independent – generalisation of our usual self-dual
theories. That is, we will consider a less restrictive gauge group, by
relaxing the requirement that the kinematic group (SDiff) preserves the volume
form, in this case the $(u,w)$-area form. This gives instead the group of full
two-dimensional diffeomorphisms (Diff) in the $(u,w)$ plane. As we will see,
this Diff group is in turn related to hyper-Hermitian manifolds, thus going
beyond the hyper-Kähler structures in conventional self-dual gravity,
discussed here in section 2.2.

Let us take a vector field $\Psi_{f}$ in the $(w,u)$ plane:

$\Psi_{f}\equiv f^{A}\partial_{A}=f^{w}\partial_{w}+f^{u}\partial_{u},$ (113)

where $A\in\\{w,u\\}$ and $f^{A}=f^{A}(u,v,w,\bar{w})$ depend on all the
coordinates. In this form, $\Psi_{f}$ is an element of diff, the Lie algebra
of Diff. The vector $\Psi_{f}$ satisfies the Jacobi identity

$[[\Psi_{f},\Psi_{g}],\Psi_{h}]+[[\Psi_{g},\Psi_{h}],\Psi_{f}]+[[\Psi_{h},\Psi_{f}],\Psi_{g}]=0,$
(114)

for the functions $f^{A},g^{A},h^{A}$. To make the connection with the
previous sections, we need the kinematic object associated with the Lie
algebra diff. To this end, let us define a vector in the direction $A$ and
momentum $p$ by $\Psi_{p,A}\equiv e^{-ipx}\partial_{A}$. The commutator
between two of such vectors is

$[\Psi_{p,A},\Psi_{q,B}]=iY^{C}(q_{A},p_{B})\Psi_{p+q,C},$ (115)

where we define the kinematic object

$\displaystyle Y^{A}(p_{1C},p_{2B})\equiv
p_{2B}\delta_{C}^{A}-p_{1C}\delta_{B}^{A}.$ (116)

This satisfies $Y^{A}(p_{1C},p_{2B})=-Y^{A}(p_{2B},p_{1C})$ and also, as we
will see shortly, a Jacobi identity. As a consequence, it will be interpreted
as a structure constant of the two-dimensional diffeomorphism algebra (diff).
Following eq. (114), we have the Jacobi relation

$Y^{D}(q_{A},p_{B})Y^{E}(k_{D},(p+q)_{C})+Y^{D}(k_{B},q_{C})Y^{E}(p_{D},(q+k)_{A})+Y^{D}(p_{C},k_{A})Y^{E}(q_{D},(p+k)_{B})=0,$
(117)

for any $E$. The kinematic factor $Y^{A}(p_{1C},p_{2B})$, which is the
structure constant for diff, will be our focus in this section.

###  4.1 Hyper-Hermitian manifold

The notion of a hyper-Hermitian manifold is a generalisation of the hyper-
Kähler case, where the volume preserving condition
$\mathcal{L}_{L}\upsilon=\mathcal{L}_{M}\upsilon=0$ is relaxed on the
gravitational Lax pair (31). Taking the Lax pair (31), we can rewrite the
condition $[L,M]=0$ as

$[U,W]=0,\quad[U,V]+[\bar{W},W]=0,\quad[\bar{W},V]=0.$ (118)

Then, ${\mathbf{V}}=\\{U,V,W,\bar{W}\\}$ forms a null tetrad for a hyper-
Hermitian metric on a four-dimensional complex manifold ${\cal M}$. Following
[128, 129, 114], we can define the vectors

$\displaystyle U$ $\displaystyle=\partial_{u},$ $\displaystyle W$
$\displaystyle=\partial_{w},$ $\displaystyle V$
$\displaystyle=\partial_{v}+\partial_{w}\Psi_{f},$ $\displaystyle\bar{W}$
$\displaystyle=\partial_{\bar{w}}+\partial_{u}\Psi_{f}.$ (119)

As in previous sections, for notational brevity, we denote
$\partial_{w}\Psi_{f}\equiv[\partial_{w},\Psi_{f}]$ and
$\partial_{u}\Psi_{f}\equiv[\partial_{u},\Psi_{f}]$. The tetrad ${\mathbf{V}}$
satisfies eq. (118) if and only if $f^{A}$ satisfies

$\partial^{2}f^{A}+\\{f^{B},\partial_{B}f^{A}\\}=0.$ (120)

It was shown in ref. [114] that this system describes an hyper-Hermitian
manifold. In momentum space, eq. (120) implies the integral equation

$f^{A}(k)=\frac{1}{2}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}X(p_{1},p_{2})Y^{A}(p_{1C},p_{2B})f^{B}(p_{1})f^{C}(p_{2}),$
(121)

thus justifying our above remark relating the Diff group to hyper-Hermitian
manifolds.

Before proceeding, let us point out that Hyper-Hermitian geometries can be
more formally defined as follows. Let $\mathcal{M}$ be a four-dimensional
manifold and $g$ be a Riemannian metric on $\mathcal{M}$. If $\mathcal{M}$ is
equipped with three complex structures $I,J,K$ satisfying the algebra of
quaternions, i.e., $IJ=-JI=K$, and $g$ is a Hermitian metric for the three
complex structures, then $\mathcal{M}$ is hyper-Hermitian. If, besides this,
the three Kähler forms are closed, $d\Omega_{I}=d\Omega_{J}=d\Omega_{K}=0$,
then $\mathcal{M}$ is hyper-Kähler. A hyper-Hermitian manifold has a self-dual
Weyl tensor. On the other hand, a hyper-Kähler manifold has a self-dual Weyl
tensor and also a vanishing Ricci tensor.101010These statements apply to our
convention on (anti-)self-duality. Literature following different conventions
may take the Weyl tensor for these manifolds to be anti-self-dual instead.

Similarly to the arguments in section 3, the fact that the kernel of the
integral equation of eq. (121) involves a product of structure constants
immediately furnishes it with a double copy interpretation. In particular, we
note that eq. (121) can be obtained via the double-copy-like replacement

$\displaystyle f^{abc}\rightarrow Y^{A}(p_{1C},p_{2B}),$ (122)

in the structure constants of the colour Lie algebra in eq. (15). Furthermore,
eq. (121) then shows that hyper-Hermitian manifolds are governed by a product
of kinematic Lie algebras, namely

$\text{sdiff}\;\times\;\text{diff}.$

Now, the Lax pair $L$ and $M$ associated with the vectors (119) for an hyper-
Hermitian manifold is given by

$L=\partial_{u}-\lambda(\partial_{\bar{w}}+\partial_{u}\Psi_{f}),\qquad\qquad
M=\partial_{w}-\lambda(\partial_{v}+\partial_{w}\Psi_{f}),$ (123)

and the eq. (120) arises from the condition $[L,M]=0$. We can recover
conventional self-dual gravity if we impose the condition
$\mathcal{L}_{L}\upsilon=\mathcal{L}_{M}\upsilon=0$ on the Lax pair (123):
there must then exist a function $\phi$ such that $f^{w}=-\partial_{u}\phi$
and $f^{u}=\partial_{w}\phi$, such that eq. (120) reduces to the single
Plebański equation (23).

Similarly to the previous cases, we can construct a hierarchy of linearised
solutions. The linearised hyper-Hermitian equation, for
$\Psi_{f}\rightarrow\Psi_{f}+\Psi_{g}$, is

$\partial^{2}\Psi_{g}+[\partial_{w}\Psi_{f},\partial_{u}\Psi_{g}]+[\partial_{w}\Psi_{g},\partial_{u}\Psi_{f}]=0.$
(124)

Expanding $\Psi_{g}$ in powers of $\lambda$, the infinite tower of linearised
solutions $\\{\Psi_{g_{n}}\\}$ to (124) is recursively expressed as

$\partial_{u}\Psi_{g_{n}}=\partial_{\bar{w}}\Psi_{g_{n-1}}+[\partial_{u}\Psi_{f},\Psi_{g_{n-1}}],\qquad\partial_{w}\Psi_{g_{n}}=\partial_{v}\Psi_{g_{n-1}}+[\partial_{w}\Psi_{f},\Psi_{g_{n-1}}],$
(125)

with $\Psi_{g_{-1}}\equiv 0$. This $\Psi_{g_{n}}$ satisfies the compatibility
condition
$\partial_{u}\partial_{w}\Psi_{g_{n}}=\partial_{w}\partial_{u}\Psi_{g_{n}}$,
and the tower solves eq. (124). This relies on the Jacobi identity (114).

In addition to the structure constants of diff considered above, we could also
introduce a Moyal deformation of sdiff.111111Seen in a different way, we could
start from eq. (95) and make the replacement (122). This results in a deformed
version of the hyper-Hermitian equation (120). The integral form of the
equation of motion reads

$f^{A}(k)=\frac{1}{2}\int
dp_{1}dp_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}X^{M}(p_{1},p_{2})Y^{A}(p_{1C},p_{2B})f^{B}(p_{1})f^{C}(p_{2}),$
(126)

and it is straightforward to obtain the equation in position space,

$\partial^{2}f^{A}+\\{f^{B},\partial_{B}f^{A}\\}^{M}=0.$ (127)

Using the Lax pair eq. (123), this deformed equation arises from the condition
$[L,M]^{M}=0$. The linearised deformed hyper-Hermitian equation, for
$\Psi_{f}\rightarrow\Psi_{f}+\Psi_{g}$, is

$\partial^{2}\Psi_{g}+[\partial_{w}\Psi_{f},\partial_{u}\Psi_{g}]^{M}+[\partial_{w}\Psi_{g},\partial_{u}\Psi_{f}]^{M}=0.$
(128)

Expanding $\Psi_{g}$ in power of $\lambda$ and using the Lax equation, the
infinite tower $\\{\Psi_{g_{n}}\\}$ is recursively expressed as

$\partial_{u}\Psi_{g_{n}}=\partial_{\bar{w}}\Psi_{g_{n-1}}+[\partial_{u}\Psi_{f},\Psi_{g_{n-1}}]^{M},\qquad\partial_{w}\Psi_{g_{n}}=\partial_{v}\Psi_{g_{n-1}}+[\partial_{w}\Psi_{f},\Psi_{g_{n-1}}]^{M},$
(129)

with $\Psi_{g_{-1}}\equiv 0$. In this naive hierarchy we are using the Moyal
bracket on non-Hamiltonian vector fields, and similarly to the deformed self-
dual Yang-Mills and doubly deformed Plebański cases, the Jacobi identity
fails, and hence integrability is lost.121212A different version of a deformed
hyper-Hermitian structure is presented in [130]. That particular example
formally retains integrability of the theory, although a twistor description
is missing. Following [131], they make a deformation of the standard
homomorphism between diff($S^{1}$) and the Poisson algebra on the cotangent
bundle $T^{\ast}S^{1}$. It would be interesting to know whether these ideas
fit into an expanded double copy web of theories.

###  4.2 Other Diff theories

Motivated by the appearance of the Diff group, a natural question is: what is
the ‘gauge theory’ with kinematic algebra diff? This gauge theory is the
single copy of the hyper-Hermitian eq. (121) (or its deformed version (126)),
via the replacement $X(p_{1},p_{2})\rightarrow f^{abc}$. The ‘Diff-gauge’
theory is described by the equation

$\Phi^{aA}(k)=\frac{1}{2}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}f^{abc}Y^{A}(p_{1C},p_{2B})\Phi^{bB}(p_{1})\Phi^{cC}(p_{2}),$
(130)

for $A=u,w$. It is straightforward to express it in position space as

$\partial^{2}\Phi^{A}+i[\Phi^{B},\partial_{B}\Phi^{A}]=0.$ (131)

We can recover the SDYM eq. (11) if we impose on $\Phi^{A}$ the condition
$\Phi^{w}=-\partial_{u}\Phi$ and $\Phi^{u}=\partial_{w}\Phi$. Of course, this
condition is similar to imposing the volume preserving condition in the hyper-
Hermitian case.

Finally, we can make the replacement (122) in eq. (130) and obtain a theory
with two copies of the diff algebra. In momentum space these equations are

$f^{AB}(k)=\frac{1}{2}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-k)}{k^{2}}Y^{A}(p_{1E},p_{2C})Y^{B}(p_{1F},p_{2D})f^{CD}(p_{1})f^{EF}(p_{2}),$
(132)

for $A,B,C,D=u,w$, whereas in position space, we have

$\partial^{2}f^{AB}+f^{CD}(\partial_{C}\partial_{D}f^{AB})-(\partial_{C}f^{AD})(\partial_{D}f^{CB})=0.$
(133)

Equations (130) and (132) complete our web of theories, by combining all of
the modifications of self-dual Yang-Mills that we have considered. Again, the
physical relevance of the new theories on their own – outside the remit of the
double copy – is not clear, and it would be interesting to explore them
further.

We pointed out early on the well-known fact that it is possible to recover
self-dual gravity (hyper-Kähler manifold) as a symmetry reduction of the self-
dual Yang-Mills theory, considering the gauge group as the area-preserving
diffeomorphisms group on the two-surface $\Sigma$, SDiff($\Sigma$) [106, 107,
108], and that this can be used to build hyper-Kähler metrics from solutions
of two-dimensional reduced SDYM equations [109, 110, 111]. In a similar form,
as seen in this section, we can obtain the hyper-Hermitian equations taking
the gauge group as the area-preserving diffeomorphisms group in the Diff-gauge
equations. It would be interesting to see if it is possible to build hyper-
Hermitian metrics from solutions of two-dimensional reduced Diff-gauge
equations.

##  5 Summary of results

$\times$ |  $f^{abc}$ (G) |  $X(p_{1},p_{2})$ (sdiff) |  $X^{M}(p_{1},p_{2})$ (Moyal) |  $Y^{A}(p_{1B},p_{2C})$ (diff)
---|---|---|---|---
$f^{abc}$ (G) |  Biadjoint scalar |  SDYM |  Moyal-deformed SDYM |  Diff-Gauge
$X(p_{1},p_{2})$ (sdiff) |  – |  SDG Plebański (Hyper-Kähler) |  Moyal-deformed SDG (Hyper-Kähler) |  Hyper-Hermitian
$X^{M}(p_{1},p_{2})$ (Moyal) |  – |  – |  Moyal$\times$Moyal SDG |  Moyal-deformed Hyper-Hermitian
$Y^{A}(p_{1B},p_{2C})$ (diff) |  – |  – |  – |  Diff$\times$Diff Gravity
Table 1: The web of theories studied in this paper. Each theory has an
integral equation containing two structure constants, appearing on the left-
hand and upper entries of the table. We represented in grey the theories that
are integrable.

Throughout this paper, we have established a set of theories that generalise
self-dual Yang-Mills and gravity. The momentum space equation for each theory
contains a manifest product of structure constants associated with colour and
/ or kinematic algebras, thus allowing theories within the set to be related
via the double copy. We summarise the complete set in table 1, where the first
column (consisting of choosing two potentially distinct colour algebras)
corresponds to biadjoint scalar theory. The second column contains the the
usual SDYM and SDGR theories (for a hyper-Kähler manifold), whose double copy
was discussed in detail in ref. [81]. The third column corresponds to our
first generalisation of the double copy procedure (section 3), in which one of
the structure constants is taken to be a Moyal deformation of the kinematic
structure constant. This leads to single Moyal deformations of SDYM and SDGR,
and also a doubly deformed gravity theory, whose equation of motion contains a
product of two Moyal kinematic factors. We should point out the (non-
integrable) Moyal-deformed SDYM case considered in the table, which is the one
that straightforwardly matches the double copy structure, is not the same as
the more conventional Moyal deformation of SDYM commonly considered in the
literature, as discussed in section 3.2. The second generalisation considers
the kinematic algebra diff, where the area preserving condition has been
relaxed. A product with sdiff results in a Hyper-Hermitian manifold. In
section 4, we have studied this as well as its Moyal deformation, a Diff-gauge
theory, and a double-Diff gravity, as recapped in the fourth column of Table
1.

The entries of Table 1 identified in grey correspond to integrable theories.
All such theories have $X(p_{1},p_{2})$ as one of the double copy factors, and
this factor is at the origin of their integrability. The equation of motion in
momentum space reads

$\Upsilon_{\rho}(p)=\frac{1}{2}\int
d\\!\\!{}^{-}\\!p_{1}d\\!\\!{}^{-}\\!p_{2}\frac{\delta\\!\\!\\!{}^{-}\\!(p_{1}+p_{2}-p)}{p^{2}}\,X(p_{1},p_{2})\;\Pi_{{1}\,{2}}\;\Upsilon_{\rho_{1}}(p_{1})\Upsilon_{\rho_{2}}(p_{2}),$
(134)

where $\Pi_{{1}\,{2}}$ provides the other structure constant. In coordinate
space, we have

$\partial^{2}\Upsilon_{\rho}+[\partial_{w}\Upsilon_{\rho},\partial_{u}\Upsilon_{\rho}]=0.$
(135)

Pleasingly, all the integrable theories can be written in a similar manner,
where $\Upsilon_{\rho}$ is $\Phi=\Phi^{a}T^{a}$ for self-dual Yang-Mills, is
$\Phi_{\phi}$ for self-dual gravity, is $\Phi_{\phi}^{M}$ for singly Moyal-
deformed self-dual gravity, and is $\Psi_{f}$ for hyper-Hermitian gravity.
Their Lax pairs can be expressed as

$L=\partial_{u}-\lambda(\partial_{\bar{w}}+\partial_{u}\Upsilon_{\rho}),\qquad
M=\partial_{w}-\lambda(\partial_{v}+\partial_{w}\Upsilon_{\rho}),$ (136)

and the equation of motion follows from the condition $[L,M]=0$. Integrability
follows from the existence of an infinite hierarchy of conserved currents,
associated to an infinite hierarchy of solutions to the linearised equation of
motion. The latter is obtained by making the replacement
$\Upsilon_{\rho}\rightarrow\Upsilon_{\rho}+\Upsilon_{\alpha}$, in eq. (135):

$\partial^{2}\Upsilon_{\alpha}+[\partial_{w}\Upsilon_{\rho},\partial_{u}\Upsilon_{\alpha}]+[\partial_{w}\Upsilon_{\alpha},\partial_{u}\Upsilon_{\rho}]=0.$
(137)

The hierarchy of solutions $\\{\Upsilon_{\alpha_{n}}\\}$ arises via

$\partial_{u}\Upsilon_{\alpha_{n}}=\partial_{\bar{w}}\Upsilon_{\alpha_{n-1}}+[\partial_{u}\Upsilon_{\rho},\Upsilon_{\alpha_{n-1}}],\qquad\partial_{w}\Upsilon_{\alpha_{n}}=\partial_{v}\Upsilon_{\alpha_{n-1}}+[\partial_{w}\Upsilon_{\rho},\Upsilon_{\alpha_{n-1}}],$
(138)

where $\Upsilon_{\alpha_{-1}}\equiv 0$. In this hierarchy,
$\Upsilon_{\alpha_{n}}$ is guaranteed to satisfy the linearised equation of
motion if $\Upsilon_{\alpha_{n-1}}$ does.

Several non-integrable theories in Table 1 naively share some of these
features, at least those which possess $X^{M}(p_{1},p_{2})$ as one of the
double copy factors (we exclude here the case where $X(p_{1},p_{2})$ is the
other factor, where the theory is integrable). Then eq. (135) changes only by
employing a Moyal-deformed bracket, that is, we make the substitution
$[\cdot,\cdot]\,\to\,[\cdot,\cdot]^{M}$. The following equations also appear
to tell an analogous story with the same substitution. However, crucially, the
Moyal-deformed bracket fails to obey the Jacobi identity, and the steps
leading to the conclusion of integrability fail.

##  6 Conclusions

In this paper, we have constructed new examples of gauge and gravity theories
satisfying manifest double copy relations, motivated by previous work in the
self-dual sector of Yang-Mills theory and gravity [81]. Each of our theories
has an integral momentum-space equation containing a product of two
(potentially different) sets of structure constants, associated with colour
and / or kinematic degrees of freedom. We have considered two generalisations
of ref. [81], namely the use of Moyal deformations of the kinematic sector,
and also the use of a full two-dimensional diffeomorphism group rather than
its more restricted area-preserving counterpart. This gives new kinematic
structure constants, and our web of theories – summarised in table 1 – consist
of all possible combinations of these various building blocks.

We have also studied the integrability properties of the web of theories. This
aspect was not considered in ref. [81], and we discussed how the well-known
integrability constructions (e.g., Lax pairs) in self-dual Yang-Mills theory
and self-dual gravity are related in a manner expected by the double copy.
More generally, for the web of theories summarised in table 1, the only
integrable theories are those for which at least one of the two sets of
structure constants is $X(p_{1},p_{2})$, associated to sdiff, i.e., the Lie
algebra of area preserving diffeomorphisms. These include self-dual Yang-Mills
theory, self-dual gravity and its single Moyal deformation, and hyper-
Hermitian gravity (an extension of self-dual gravity associated to hyper-
Hermitian manifolds instead of hyper-Kähler manifolds). In those cases, it is
possible to construct a Lax pair, and to relate it to an infinite hierarchy of
linearised symmetries, expressed as conserved currents. When none of the two
sets of structure constants was associated to sdiff, the integrability
construction failed. In these cases, we may still have a presumptive Lax pair
interpretation, and a bracket based on a deformation of the standard Lie or
Poisson brackets, but this bracket fails to obey the Jacobi relation. This
agrees with previous literature, e.g., ref. [119] regarding Moyal-deformed
brackets. We have the interesting consequence that singly Moyal-deformed self-
dual gravity can be thought of as “inheriting” its integrability from one of
its constituent gauge theories under the double copy, i.e., from the one with
kinematic algebra $X(p_{1},p_{2})$ and not the one with kinematic algebra
$X^{M}(p_{1},p_{2})$. The picture that emerges is that, when integrability
exists, it does so because at least one of the sides of the double copy is
integrable. In the cases at hand, all integrable theories are related to self-
dual Yang-Mills theory by substitution of $f^{abc}$ by other structure
constants, which can be thought of as arising from a ‘symmetry reduction’.
Hence, the class of integrable theories considered here is consistent with
Ward’s conjecture [132, 133].

There are a number of possible avenues for further work. Firstly, it would
useful to establish the possible physical interpretation and applications of
the theories that have been introduced in order to complete the web obtained
via the double copy. This includes the question of whether or not they can be
furnished with a geometric interpretation. Finally, our hope is that the
results of our study may stimulate the construction of yet more double copy
examples, which may possibly in turn enhance our understanding of conventional
Yang-Mills and gravity beyond the self-dual sector.

## Acknowledgements

We are grateful to Maciej Dunajski and Maciej Przanowski for useful
discussions. This work has been supported by the UK Science and Technology
Facilities Council (STFC) Consolidated Grant ST/P000754/1 “String theory,
gauge theory and duality”, and by the European Union Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No.
764850 “SAGEX”. EC is supported by the National Council of Science and
Technology (CONACYT). AL is supported by the Department of Energy under Award
Number DE-SC0009937. RM is supported by a Royal Society University Research
Fellowship.

## References

  * [1] H. Kawai, D. Lewellen, and S. Tye, “A Relation Between Tree Amplitudes of Closed and Open Strings,” Nucl.Phys. B269 (1986) 1\.
  * [2] Z. Bern, J. Carrasco, and H. Johansson, “New Relations for Gauge-Theory Amplitudes,” Phys.Rev. D78 (2008) 085011, 0805.3993.
  * [3] Z. Bern, J. J. M. Carrasco, and H. Johansson, “Perturbative Quantum Gravity as a Double Copy of Gauge Theory,” Phys.Rev.Lett. 105 (2010) 061602, 1004.0476.
  * [4] Z. Bern, T. Dennen, Y.-t. Huang, and M. Kiermaier, “Gravity as the Square of Gauge Theory,” Phys.Rev. D82 (2010) 065003, 1004.0693.
  * [5] R. Monteiro, D. O’Connell, and C. D. White, “Black holes and the double copy,” JHEP 1412 (2014) 056, 1410.0239.
  * [6] A. Luna, R. Monteiro, D. O’Connell, and C. D. White, “The classical double copy for Taub-NUT spacetime,” Phys. Lett. B750 (2015) 272–277, 1507.01869.
  * [7] A. Luna, R. Monteiro, I. Nicholson, D. O’Connell, and C. D. White, “The double copy: Bremsstrahlung and accelerating black holes,” 1603.05737.
  * [8] M. Carrillo-González, R. Penco, and M. Trodden, “The classical double copy in maximally symmetric spacetimes,” JHEP 04 (2018) 028, 1711.01296.
  * [9] N. Bahjat-Abbas, A. Luna, and C. D. White, “The Kerr-Schild double copy in curved spacetime,” JHEP 12 (2017) 004, 1710.01953.
  * [10] D. S. Berman, E. Chacón, A. Luna, and C. D. White, “The self-dual classical double copy, and the Eguchi-Hanson instanton,” 1809.04063.
  * [11] I. Bah, R. Dempsey, and P. Weck, “Kerr-Schild Double Copy and Complex Worldlines,” 1910.04197.
  * [12] M. Carrillo González, B. Melcher, K. Ratliff, S. Watson, and C. D. White, “The classical double copy in three spacetime dimensions,” JHEP 07 (2019) 167, 1904.11001.
  * [13] A. Banerjee, E. Colgáin, J. A. Rosabal, and H. Yavartanoo, “Ehlers as EM duality in the double copy,” 1912.02597.
  * [14] A. Ilderton, “Screw-symmetric gravitational waves: a double copy of the vortex,” Phys. Lett. B 782 (2018) 22–27, 1804.07290.
  * [15] R. Monteiro, I. Nicholson, and D. O’Connell, “Spinor-helicity and the algebraic classification of higher-dimensional spacetimes,” 1809.03906.
  * [16] A. Luna, R. Monteiro, I. Nicholson, and D. O’Connell, “Type D Spacetimes and the Weyl Double Copy,” Class. Quant. Grav. 36 (2019) 065003, 1810.08183.
  * [17] K. Lee, “Kerr-Schild Double Field Theory and Classical Double Copy,” 1807.08443.
  * [18] W. Cho and K. Lee, “Heterotic Kerr-Schild Double Field Theory and Classical Double Copy,” JHEP 07 (2019) 030, 1904.11650.
  * [19] K. Kim, K. Lee, R. Monteiro, I. Nicholson, and D. Peinador Veiga, “The Classical Double Copy of a Point Charge,” 1912.02177.
  * [20] L. Alfonsi, C. D. White, and S. Wikeley, “Topology and Wilson lines: global aspects of the double copy,” 2004.07181.
  * [21] N. Bahjat-Abbas, R. Stark-Muchão, and C. D. White, “Monopoles, shockwaves and the classical double copy,” 2001.09918.
  * [22] C. D. White, “Exact solutions for the biadjoint scalar field,” Phys. Lett. B763 (2016) 365–369, 1606.04724.
  * [23] P.-J. De Smet and C. D. White, “Extended solutions for the biadjoint scalar field,” Phys. Lett. B775 (2017) 163–167, 1708.01103.
  * [24] N. Bahjat-Abbas, R. Stark-Muchão, and C. D. White, “Biadjoint wires,” Phys. Lett. B788 (2019) 274–279, 1810.08118.
  * [25] G. Elor, K. Farnsworth, M. L. Graesser, and G. Herczeg, “The Newman-Penrose Map and the Classical Double Copy,” 2006.08630.
  * [26] M. K. Gumus and G. Alkac, “More on the Classical Double Copy in Three Spacetime Dimensions,” 2006.00552.
  * [27] C. Keeler, T. Manton, and N. Monga, “From Navier-Stokes to Maxwell via Einstein,” 2005.04242.
  * [28] N. Arkani-Hamed, Y.-t. Huang, and D. O’Connell, “Kerr black holes as elementary particles,” JHEP 01 (2020) 046, 1906.10100.
  * [29] Y.-T. Huang, U. Kol, and D. O’Connell, “The Double Copy of Electric-Magnetic Duality,” 1911.06318.
  * [30] R. Alawadhi, D. S. Berman, B. Spence, and D. Peinador Veiga, “S-duality and the Double Copy,” 1911.06797.
  * [31] N. Moynihan, “Kerr-Newman from Minimal Coupling,” JHEP 01 (2020) 014, 1909.05217.
  * [32] R. Alawadhi, D. S. Berman, and B. Spence, “Weyl doubling,” 2007.03264.
  * [33] D. A. Easson, C. Keeler, and T. Manton, “The classical double copy of regular non-singular black holes,” 2007.16186.
  * [34] E. Casali and A. Puhm, “A Double Copy for Celestial Amplitudes,” 2007.15027.
  * [35] A. Cristofoli, “Gravitational shock waves and scattering amplitudes,” 2006.08283.
  * [36] A. Luna, R. Monteiro, I. Nicholson, A. Ochirov, D. O’Connell, N. Westerberg, and C. D. White, “Perturbative spacetimes from Yang-Mills theory,” JHEP 04 (2017) 069, 1611.07508.
  * [37] W. D. Goldberger and A. K. Ridgway, “Radiation and the classical double copy for color charges,” Phys. Rev. D95 (2017), no. 12, 125010, 1611.03493.
  * [38] W. D. Goldberger, S. G. Prabhu, and J. O. Thompson, “Classical gluon and graviton radiation from the bi-adjoint scalar double copy,” Phys. Rev. D96 (2017), no. 6, 065009, 1705.09263.
  * [39] W. D. Goldberger and A. K. Ridgway, “Bound states and the classical double copy,” Phys. Rev. D97 (2018), no. 8, 085019, 1711.09493.
  * [40] W. D. Goldberger, J. Li, and S. G. Prabhu, “Spinning particles, axion radiation, and the classical double copy,” Phys. Rev. D97 (2018), no. 10, 105018, 1712.09250.
  * [41] C.-H. Shen, “Gravitational Radiation from Color-Kinematics Duality,” 1806.07388.
  * [42] M. Carrillo-Gonzalez, R. Penco, and M. Trodden, “Radiation of scalar modes and the classical double copy,” 1809.04611.
  * [43] J. Plefka, J. Steinhoff, and W. Wormsbecher, “Effective action of dilaton gravity as the classical double copy of Yang-Mills theory,” 1807.09859.
  * [44] J. Plefka, C. Shi, J. Steinhoff, and T. Wang, “Breakdown of the classical double copy for the effective action of dilaton-gravity at NNLO,” 1906.05875.
  * [45] W. D. Goldberger and J. Li, “Strings, extended objects, and the classical double copy,” 1912.01650.
  * [46] A. PV and A. Manu, “Classical double copy from Color Kinematics duality: A proof in the soft limit,” 1907.10021.
  * [47] A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes, and S. Nagy, “Yang-Mills origin of gravitational symmetries,” Phys. Rev. Lett. 113 (2014), no. 23, 231606, 1408.4434.
  * [48] L. Borsten and M. J. Duff, “Gravity as the square of Yang–Mills?,” Phys. Scripta 90 (2015) 108012, 1602.08267.
  * [49] A. Anastasiou, L. Borsten, M. J. Duff, M. J. Hughes, A. Marrani, S. Nagy, and M. Zoccali, “Twin supergravities from Yang-Mills theory squared,” Phys. Rev. D96 (2017), no. 2, 026013, 1610.07192.
  * [50] G. L. Cardoso, S. Nagy, and S. Nampuri, “A double copy for $\mathcal{N}=2$ supergravity: a linearised tale told on-shell,” JHEP 10 (2016) 127, 1609.05022.
  * [51] L. Borsten, “On $D=6$, $\mathcal{N}=(2,0)$ and $\mathcal{N}=(4,0)$ theories,” 1708.02573.
  * [52] A. Anastasiou, L. Borsten, M. J. Duff, A. Marrani, S. Nagy, and M. Zoccali, “The Mile High Magic Pyramid,”  2017\.  1711.08476.
  * [53] A. Anastasiou, L. Borsten, M. J. Duff, S. Nagy, and M. Zoccali, “Gravity as Gauge Theory Squared: A Ghost Story,” Phys. Rev. Lett. 121 (2018), no. 21, 211601, 1807.02486.
  * [54] G. Lopes Cardoso, G. Inverso, S. Nagy, and S. Nampuri, “Comments on the double copy construction for gravitational theories,” in 17th Hellenic School and Workshops on Elementary Particle Physics and Gravity (CORFU2017) Corfu, Greece, September 2-28, 2017.  2018\.  1803.07670.
  * [55] A. Luna, S. Nagy, and C. White, “The convolutional double copy: a case study with a point,” 2004.11254.
  * [56] L. Borsten and S. Nagy, “The pure BRST Einstein-Hilbert Lagrangian from the double-copy to cubic order,” 2004.14945.
  * [57] L. Borsten, B. Jurco, H. Kim, T. Macrelli, C. Saemann, and M. Wolf, “BRST-Lagrangian Double Copy of Yang-Mills Theory,” 2007.13803.
  * [58] A. Luna, I. Nicholson, D. O’Connell, and C. D. White, “Inelastic Black Hole Scattering from Charged Scalar Amplitudes,” JHEP 03 (2018) 044, 1711.03901.
  * [59] D. A. Kosower, B. Maybee, and D. O’Connell, “Amplitudes, Observables, and Classical Scattering,” JHEP 02 (2019) 137, 1811.10950.
  * [60] B. Maybee, D. O’Connell, and J. Vines, “Observables and amplitudes for spinning particles and black holes,” 1906.09260.
  * [61] Y. F. Bautista and A. Guevara, “On the Double Copy for Spinning Matter,” 1908.11349.
  * [62] Y. F. Bautista and A. Guevara, “From Scattering Amplitudes to Classical Physics: Universality, Double Copy and Soft Theorems,” 1903.12419.
  * [63] C. Cheung, I. Z. Rothstein, and M. P. Solon, “From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion,” 1808.02489.
  * [64] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, “Black Hole Binary Dynamics from the Double Copy and Effective Theory,” JHEP 10 (2019) 206, 1908.01493.
  * [65] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, “Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order,” Phys. Rev. Lett. 122 (2019), no. 20, 201603, 1901.04424.
  * [66] Z. Bern, A. Luna, R. Roiban, C.-H. Shen, and M. Zeng, “Spinning Black Hole Binary Dynamics, Scattering Amplitudes and Effective Field Theory,” 2005.03071.
  * [67] G. Kälin and R. A. Porto, “From Boundary Data to Bound States,” JHEP 01 (2020) 072, 1910.03008.
  * [68] G. Kälin and R. A. Porto, “Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics,” 2006.01184.
  * [69] G. L. Almeida, S. Foffa, and R. Sturani, “Classical Gravitational Self-Energy from Double Copy,” 2008.06195.
  * [70] F. Cachazo, S. He, and E. Y. Yuan, “Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM,” JHEP 07 (2015) 149, 1412.3479.
  * [71] H. Johansson and A. Ochirov, “Pure Gravities via Color-Kinematics Duality for Fundamental Matter,” JHEP 11 (2015) 046, 1407.4772.
  * [72] H. Johansson and A. Ochirov, “Color-Kinematics Duality for QCD Amplitudes,” JHEP 01 (2016) 170, 1507.00332.
  * [73] M. Chiodaroli, M. Gunaydin, H. Johansson, and R. Roiban, “Spontaneously Broken Yang-Mills-Einstein Supergravities as Double Copies,” 1511.01740.
  * [74] M. Chiodaroli, M. Gunaydin, H. Johansson, and R. Roiban, “Complete construction of magical, symmetric and homogeneous N=2 supergravities as double copies of gauge theories,” Phys. Rev. Lett. 117 (2016), no. 1, 011603, 1512.09130.
  * [75] J. J. M. Carrasco, C. R. Mafra, and O. Schlotterer, “Abelian Z-theory: NLSM amplitudes and $\alpha$’-corrections from the open string,” JHEP 06 (2017) 093, 1608.02569.
  * [76] J. J. M. Carrasco, C. R. Mafra, and O. Schlotterer, “Semi-abelian Z-theory: NLSM$+?^{3}$ from the open string,” JHEP 08 (2017) 135, 1612.06446.
  * [77] A. Anastasiou, L. Borsten, M. J. Duff, A. Marrani, S. Nagy, and M. Zoccali, “Are all supergravity theories Yang-Mills squared?,” 1707.03234.
  * [78] T. Azevedo, M. Chiodaroli, H. Johansson, and O. Schlotterer, “Heterotic and bosonic string amplitudes via field theory,” JHEP 10 (2018) 012, 1803.05452.
  * [79] Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban, “The Duality Between Color and Kinematics and its Applications,” 1909.01358.
  * [80] L. Borsten, “Gravity as the square of gauge theory: a review,”.
  * [81] R. Monteiro and D. O’Connell, “The Kinematic Algebra From the Self-Dual Sector,” JHEP 1107 (2011) 007, 1105.2565.
  * [82] A. Parkes, “A Cubic action for selfdual Yang-Mills,” Phys.Lett. B286 (1992) 265–270, hep-th/9203074.
  * [83] J. Plebański, “Some solutions of complex Einstein equations,” J. Math. Phys. 16 (1975) 2395–2402.
  * [84] N. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard, and P. Vanhove, “Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes,” JHEP 1006 (2010) 003, 1003.2403.
  * [85] C. R. Mafra, O. Schlotterer, and S. Stieberger, “Explicit BCJ Numerators from Pure Spinors,” JHEP 1107 (2011) 092, 1104.5224.
  * [86] N. Bjerrum-Bohr, P. H. Damgaard, R. Monteiro, and D. O’Connell, “Algebras for Amplitudes,” JHEP 1206 (2012) 061, 1203.0944.
  * [87] C.-H. Fu, Y.-J. Du, and B. Feng, “An algebraic approach to BCJ numerators,” JHEP 1303 (2013) 050, 1212.6168.
  * [88] R. Monteiro and D. O’Connell, “The Kinematic Algebras from the Scattering Equations,” JHEP 1403 (2014) 110, 1311.1151.
  * [89] C. R. Mafra and O. Schlotterer, “Multiparticle SYM equations of motion and pure spinor BRST blocks,” JHEP 1407 (2014) 153, 1404.4986.
  * [90] S. G. Naculich, “Scattering equations and virtuous kinematic numerators and dual-trace functions,” JHEP 1407 (2014) 143, 1404.7141.
  * [91] C. R. Mafra and O. Schlotterer, “Berends-Giele recursions and the BCJ duality in superspace and components,” JHEP 03 (2016) 097, 1510.08846.
  * [92] N. Bjerrum-Bohr, J. L. Bourjaily, P. H. Damgaard, and B. Feng, “Manifesting Color-Kinematics Duality in the Scattering Equation Formalism,” JHEP 09 (2016) 094, 1608.00006.
  * [93] Y.-J. Du and F. Teng, “BCJ numerators from reduced Pfaffian,” JHEP 04 (2017) 033, 1703.05717.
  * [94] A. Edison and F. Teng, “Efficient Calculation of Crossing Symmetric BCJ Tree Numerators,” 2005.03638.
  * [95] C. Cheung and C.-H. Shen, “Symmetry for Flavor-Kinematics Duality from an Action,” Phys. Rev. Lett. 118 (2017), no. 12, 121601, 1612.00868.
  * [96] C.-H. Fu and K. Krasnov, “Colour-Kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms,” JHEP 01 (2017) 075, 1603.02033.
  * [97] G. Chen, H. Johansson, F. Teng, and T. Wang, “On the kinematic algebra for BCJ numerators beyond the MHV sector,” JHEP 11 (2019) 055, 1906.10683.
  * [98] S. Mizera, “Kinematic Jacobi Identity is a Residue Theorem: Geometry of Color-Kinematics Duality for Gauge and Gravity Amplitudes,” Phys. Rev. Lett. 124 (2020), no. 14, 141601, 1912.03397.
  * [99] H. Frost and L. Mason, “Lie Polynomials and a Twistorial Correspondence for Amplitudes,” 1912.04198.
  * [100] J. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Phil. Soc. 45 (1949) 99–124.
  * [101] H. Groenewold, “On the Principles of elementary quantum mechanics,” Physica 12 (1946) 405–460.
  * [102] G. A. Baker, “Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space,” Phys. Rev. 109 (1958) 2198–2206.
  * [103] C. K. Zachos, D. B. Fairlie, and T. L. Curtright, Quantum Mechanics in Phase Space.  WORLD SCIENTIFIC, 2005.
  * [104] M. Li and Y. Wu, Physics in Non-commutative World: Field theories. I.  Physics in Non-commutative World. Rinton Press, 2002.
  * [105] P. Fletcher, “The Uniqueness of the Moyal algebra,” Phys. Lett. B 248 (1990) 323–328.
  * [106] J. Plebański, M. Przanowski, B. Rajca, and J. Tosiek, “The Moyal deformation of the second heavenly equation,” Acta Phys. Polon. B 26 (1995) 889–902.
  * [107] J. Plebański, M. Przanowski, and H. García-Compeán, “From selfdual Yang-Mills fields to selfdual gravity,” Acta Phys. Polon. B25 (1994) 1079–1085.
  * [108] Q.-H. Park, “2d sigma model approach to 4d instantons,” International Journal of Modern Physics A 7 (1992), no. 07, 1415–1447.
  * [109] E. Chacón and H. García-Compeán, “Self-dual gravity via Hitchin’s equations,” J. Math. Phys. 60 (2019), no. 5, 052502, 1812.08962.
  * [110] M. Dunajski, L. J. Mason, and N. M. J. Woodhouse, “From 2d integrable systems to self-dual gravity,” Journal of Physics A: Mathematical and General 31 (jul, 1998) 6019–6028.
  * [111] J. F. Plebański, M. Przanowski, and H. García-Compeán, “From principal chiral model to self-dual gravity,” Modern Physics Letters A 11 (1996), no. 08, 663–673.
  * [112] Q. Park, “Self-dual gravity as a large-n limit of the 2d non-linear sigma model,” Physics Letters B 238 (1990), no. 2, 287 – 290.
  * [113] L. Mason and E. Newman, “A Connection Between the Einstein and Yang-Mills Equations,” Commun. Math. Phys. 121 (1989) 659–668.
  * [114] M. Dunajski, “The Twisted photon associated to hyperHermitian four manifolds,” J. Geom. Phys. 30 (1999) 266–281, math/9808137.
  * [115] M. Dunajski, Solitons, instantons, and twistors.  2010\.
  * [116] D. Cangemi, “Selfdual Yang-Mills theory and one loop like - helicity QCD multi - gluon amplitudes,” Nucl. Phys. B 484 (1997) 521–537, hep-th/9605208.
  * [117] J. F. Plebański and M. Przanowski, “The Lagrangian of a selfdual gravitational field as a limit of the SDYM Lagrangian,” Phys. Lett. A212 (1996) 22–28, hep-th/9605233.
  * [118] K. Takasaki, “Nonabelian kp hierarchy with moyal algebraic coefficients,” Journal of Geometry and Physics 14 (1994), no. 4, 332 – 364.
  * [119] I. Strachan, “A Geometry for multidimensional integrable systems,” J. Geom. Phys. 21 (1997) 255–278, hep-th/9604142.
  * [120] I. Strachan, “The Moyal algebra and integrable deformations of the self-dual Einstein equations,” Physics Letters B 283 (1992), no. 1, 63 – 66\.
  * [121] M. Maceda and D. Martínez-Carbajal, “A Kähler Compatible Moyal Deformation of the First Heavenly Equation,” Symmetry, Integrability and Geometry: Methods and Applications (Sep, 2019).
  * [122] D. B. Fairlie, P. Fletcher, and C. K. Zachos, “Infinite‐dimensional algebras and a trigonometric basis for the classical lie algebras,” Journal of Mathematical Physics 31 (1990), no. 5, 1088–1094, https://doi.org/10.1063/1.528788.
  * [123] S. Formański and M. Przanowski, “$\ast$-SDYM fields and heavenly spaces: I. $\ast$-SDYM equations as an integrable system,” Journal of Physics A: Mathematical and General 38 (may, 2005) 4399–4418.
  * [124] S. Formański and M. Przanowski, “$\ast$-SDYM fields and heavenly spaces: II. reductions of the $\ast$-SDYM system,” Journal of Physics A: Mathematical and General 38 (oct, 2005) 9371–9385.
  * [125] M. Przanowski, S. Formański, and F. J. Turrubiates, “NONLINEAR GRAVITON AS A LIMIT OF sl(N;C) CHIRAL FIELDS AS N$\rightarrow\infty$,” Modern Physics Letters A 13 (1998), no. 40, 3193–3200, https://doi.org/10.1142/S0217732398003405.
  * [126] K. Takasaki, “Anti-self-dual Yang-Mills equations on noncommutative space-time,” Journal of Geometry and Physics 37 (2001), no. 4, 291 – 306.
  * [127] M. Hamanaka, “Non-commutative Ward’s conjecture and integrable systems,” Nuclear Physics B 741 (2006), no. 3, 368 – 389.
  * [128] J. D. E. Grant and I. A. B. Strachan, “Hypercomplex integrable systems,” Nonlinearity 12 (aug, 1999) 1247–1261.
  * [129] J. D. Finley and J. F. Plebański, “Further heavenly metrics and their symmetries,” Journal of Mathematical Physics 17 (1976), no. 4, 585–596, https://aip.scitation.org/doi/pdf/10.1063/1.522947.
  * [130] M. Dunajski, J. D. E. Grant, and I. A. B. Strachan, “Multidimensional integrable systems and deformations of lie algebra homomorphisms,” Journal of Mathematical Physics 48 (2007), no. 9, 093502, https://doi.org/10.1063/1.2777008.
  * [131] V. Ovsienko and C. Roger, “Deforming the Lie Algebra of Vector Fields on $S^{1}$ Inside the Poisson Algebra on ˙$T^{*}S^{1}$,” Communications in Mathematical Physics 198 (1998) 97–110.
  * [132] R. Ward, “Integrable and solvable systems, and relations among them,” Phil. Trans. Roy. Soc. Lond. A 315 (1985) 451–457.
  * [133] L. Mason and N. Woodhouse, Integrability, selfduality, and twistor theory.  1991\.