QMUL-PH-22-15

Non-perturbative aspects of the self-dual double copy

Kymani<EMAIL_ADDRESS>Chris D.
<EMAIL_ADDRESS>and Sam<EMAIL_ADDRESS>
Centre for Theoretical Physics, Department of Physics and Astronomy,

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

The double copy is by now a firmly-established correspondence between
amplitudes and classical solutions in biadjoint scalar, gauge and gravity
theories. To date, no strongly coupled examples of the double copy in four
dimensions have been found, and previous attempts based on exact non-linear
solutions of biadjoint theory in Lorentzian signature have failed. In this
paper, we instead look for biadjoint solutions in Euclidean signature, which
may be relatable to Yang-Mills or gravitational instantons. We show that
spherically symmetric power-like Euclidean solutions do not exist in precisely
four spacetime dimensions. The explanation for why this is the case turns out
to involve the Eguchi-Hanson instanton, whose single copy structure is found
to be more complicated (and interesting) than previously thought. We provide a
more general prescription for double-copying instantons, and explain how our
results provide a higher-dimensional complement to a recently presented non-
perturbative double copy of exact solutions in two spacetime dimensions. In
doing so, we demonstrate how the replacement of colour by kinematic Lie
algebras operates at the level of exact classical solutions.

##  1 Introduction

In recent years, a correspondence known as the double copy has received a
great deal of attention. Originating in the study of field theory scattering
amplitudes [1, 2], and inspired by earlier work in string theory [3], it
states that quantities in non-abelian gauge theories can be straightforwardly
mapped to counterparts in gravity theories. Similarly, one can take objects in
gauge theory, and translate them to a so-called biadjoint scalar theory, in
which a scalar field carries two types of colour charge. This is known as the
zeroth copy, and it is common to depict the ladder of correspondences between
these different theories as in figure 1, although this is itself a subset of a
much wider web of theories (see e.g. [4] for a review).

Figure 1: Correspondences between various types of field theory.

Much of the work in recent years has focused on how generally we are to
interpret the scheme of figure 1. We unfortunately lack a complete
understanding of the double copy at the level of e.g. Lagrangians or equations
of motion (although see refs. [2, 5, 6, 7, 8, 9, 10, 11, 12] for interesting
developments), so that it is not known whether figure 1 applies to the
complete theories. If it does, it suggests a profound and previously hidden
commonality between our theories of nature, that traditional ways of thinking
have obscured. Furthermore, there is a practical way to test how generally
figure 1 applies. Namely, to take different objects in each of the theories,
and to find rules for matching these up that can be seen to generalise the
original double copy for scattering amplitudes. The first such work in this
regard was ref. [13], which extended the double and zeroth copies to a special
family of exact classical solutions of the relevant theories: those associated
with Kerr-Schild metrics in General Relativity (see also refs. [14, 15] for
related work in a different context). Follow-up work has considered the
implications for specific solutions, including in different numbers of
spacetime dimension [16, 17, 18, 19, 20, 21, 22, 23]. A second exact double
copy procedure is the Weyl double copy introduced in ref. [24], and studied
further in refs. [25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. This is
complementary to the Kerr-Schild double copy in that it relies on the
spinorial formulation of field theory rather than the tensorial approach.
Nevertheless, it agrees where overlap exists. In addition to exact solutions,
one may double-copy classical solutions order-by-order in perturbation theory.
For a variety of classical double copy approaches, see e.g. refs. [35, 36, 37,
38, 39, 40, 9, 10, 11, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53].

In all of the above double copies, solutions of linearised biadjoint scalar
theory play a crucial role. They emerge as the denominators in scattering
amplitudes, or as intermediate quantities in classical double copies. To date,
there has been no concrete realisation of figure 1 for genuinely non-
perturbative / strongly coupled solutions in four dimensions, involving fully
non-linear solutions of each theory, including the biadjoint scalar case. A
number of ideas have been proposed for exploring non-perturbative aspects of
the double copy, including the study of how symmetries in different theories
can be mapped [5, 11, 54, 55, 56], or non-trivial geometric / topological
properties [19, 57, 58]. Recently, a proposal for a non-perturbative double
copy in two spacetime dimensions has been made [52], and we return to this
below. First, however, we continue a programme of work initiated in ref. [59],
which found exact non-linear solutions of biadjoint scalar theory. Further
solutions were found in refs. [60, 61], and the hope is that by assembling a
catalogue of such solutions, it might be possible to identify their
counterparts in gauge or gravity theory, thus providing a non-perturbative
realisation of the double copy and related correspondences.

The simplest non-linear solutions of biadjoint theory consist of spherically-
symmetric monopole-like objects, involving a singularity at the origin. It is
then natural to propose that these might be associated with singular monopoles
in Yang-Mills theory, namely the Wu-Yang monopoles of ref. [62]. Indeed, this
conjecture was tentatively made in ref. [59], but subsequent work has shown
that it cannot be true [63, 57]. The Wu-Yang monopole turns out to be related
(by a singular gauge transformation) to a non-abelian version of the well-
known Dirac magnetic monopole, whose double and zeroth copies are already
known: in gravity, it corresponds to a so-called NUT charge [64, 65], as first
shown in ref. [16]. Thus, there is no room left for the “biadjoint monopole”
to correspond to an obvious gauge theory solution, and it remains unclear how
to proceed.

Given the above uncertainty regarding monopole solutions – which are in
standard Lorentzian signature – we take here a different approach. Besides
monopoles, some of the most well-known non-perturbative solutions of gauge and
gravity theories are instantons, which are solutions of the field equations in
Euclidean signature. There is thus a clear motivation for trying to find non-
linear solutions of Euclidean biadjoint scalar theory, in the hope that they
may be relatable to known instantons. That this should be possible is further
motivated by the fact that instanton solutions, as commonly referred to in
both Yang-Mills (YM) theory and gravity, are (anti)-self-dual. It is known, at
least in principle, that the self-dual sectors of YM and gravity can be
written as manifest double copies of each other [5], where the zeroth copy to
biadjoint theory also takes a simple form. Generalisations of this idea to
more exotic theories are also known [66], although it is not clear what the
precise implications are for classical solutions rather than scattering
amplitudes.

In this paper, we will start by trying to find simple power-like spherically
symmetric solutions of Euclidean biadjoint scalar field theory, analogous to
the Lorentzian solutions found in ref. [59]. Curiously, we will see that
whilst non-linear solutions do indeed exist in $d\neq 4$ spacetime dimensions,
they are entirely absent in $d=4$. This may at first seem surprising, but is
in fact easily explainable: the power-like solutions one obtains for general
dimensions solve the linearised biadjoint equation in $d=4$, and thus cannot
be solutions of the non-linear equation. Based on previous insights [19, 24],
we are able to identify these linear solutions with the zeroth copy of the
Eguchi-Hanson (gravitational) instanton [67, 68, 69]. However, in doing so, we
find that the single copy of the Eguchi-Hanson instanton is more intricate
than previously thought: one may provide a fully non-abelian single copy of
this solution, in addition to its previously understood abelian counterpart.
This mirrors the situation found for monopoles in refs. [63, 57], where either
an abelian or non-abelian monopole are found to double copy to the same
gravity solution (a NUT charge). We will also be able to write a more general
ansatz for double-copying instantons than has previously been used. However,
ultimately we find that it applies only to those gauge or gravity solutions
which linearise the equations of motion.

As mentioned above, ref. [52] recently proposed a non-perturbative double copy
procedure for exact solutions in a variety of theories in two spacetime
dimensions. The role of the gravity theory is played by Special Galileon (SG)
theory, and that of the gauge theory by so-called Zakharov-Mikhailov (ZM)
theory. The equations of motion for these theories, which we write explicitly
in section 5, bear a strong resemblance to those of self-dual Yang-Mills
theory and gravity in four spacetime dimensions. This will allow us to
interpret the results of ref. [52] as a close counterpart of the known four-
dimensional self-dual double copy. Conversely, the ideas of ref. [52] will
also clarify aspects of our four-dimensional results. In particular, the
original double copy for scattering amplitudes relies on a phenomenon known as
BCJ duality [70], which states that there is a kinematic algebra underlying
gauge theory amplitudes, mirroring the Lie algebra describing the colour
degrees of freedom. The gravity theory has no colour algebra, but instead has
two copies of the kinematic algebra. The nature of this algebra has remained
mysterious in general, but it can be made explicit at the level of equations
of motion in the (anti-)self-dual sector, where it is known to correspond to a
certain infinite-dimensional Lie algebra of area-preserving diffeomorphisms
[5]. Until now, it has remained unclear how this algebra relates to properties
of (exact) classical solutions, but ref. [52] provides an answer to this
question in two spacetime dimensions. In turn, this allows us to interpret how
the kinematic algebra is made manifest in the case of four-dimensional
(anti-)self-dual solutions, thus resolving a long-standing conceptual issue in
the double copy literature.

The structure of our paper is as follows. In section 2, we detail our attempts
to find power-like solutions of Euclidean biadjoint scalar field theory in
various numbers of spacetime dimension, showing that non-trivial examples are
absent in $d=4$. In section 3, we interpret this result as being due to known
properties of the Eguchi-Hanson instanton. In section 4, we present a more
general ansatz for double-copying instanton solutions. In section 5, we
explain how our results relate to the recent two-dimensional non-perturbative
double copy proposal of ref. [52], and clarify certain conceptual issues in
the four-dimensional non-perturbative double copy. Finally, we discuss our
results and conclude in section 6.

##  2 Solutions of Euclidean biadjoint scalar theory

In Lorentzian $(-,+,+,+)$ signature, the equation of motion for biadjoint
scalar theory is as follows:

$\partial^{2}\Phi^{aa^{\prime}}+yf^{abc}\tilde{f}^{a^{\prime}b^{\prime}c^{\prime}}\Phi^{bb^{\prime}}\Phi^{cc^{\prime}}=0.$
(1)

Here $\Phi^{aa^{\prime}}$ is a scalar field with two colour indices in the
adjoint representation of two global gauge groups, whose structure constants
are $f^{abc}$ and $\tilde{f}^{a^{\prime}b^{\prime}c^{\prime}}$ respectively.
Furthermore, $y$ is a coupling constant. The quadratic interaction term in eq.
(1) arises from a cubic Lagrangian, such that the energy of biadjoint scalar
theory is unbounded from below. This in turn makes it a not particularly
physical theory, given that its solutions will be dynamically unstable.
However, this does not prevent us from looking for such solutions, as in
previous instances of figure 1 for both scattering amplitudes and classical
solutions, quantities in biadjoint theory are indeed related to physically
well-behaved objects in gauge and gravity theory. Thus, it is worthwhile and
meaningful to classify solutions in biadjoint theory, regardless of whether or
not they make sense if considered in isolation.

We may proceed to Euclidean signature in eq. (1) by analytically continuing
the time coordinate $t\rightarrow\tau=it$, such that one has

$\partial^{2}\rightarrow\Delta,$ (2)

where $\Delta$ is the Laplacian operator, and we work in $d$ spacetime
dimensions in general. We then have

$\Delta\Phi^{aa^{\prime}}+yf^{abc}\tilde{f}^{a^{\prime}b^{\prime}c^{\prime}}\Phi^{bb^{\prime}}\Phi^{cc^{\prime}}=0,$
(3)

for which we may attempt to find solutions as in the Lorentzian case of ref.
[59]. First, we take the gauge groups to coincide, such that
$f^{abc}=\tilde{f}^{abc}$. We may also write

$f^{abc}f^{a^{\prime}bc}=T_{A}\delta^{aa^{\prime}},$ (4)

where the constant $T_{A}$ depends on the common gauge group $G$, and the
normalisation of the generators. Next, we restrict to spherically symmetric
solutions via the ansatz

$\Phi^{aa^{\prime}}=\frac{\delta^{aa^{\prime}}}{yT_{A}}f(r),\quad
r^{2}=x_{\mu}x^{\mu}.$ (5)

Substituting this into eq. (3), one obtains

$\frac{1}{r^{d-1}}\frac{d}{dr}\left(r^{d-1}\frac{df(r)}{dr}\right)+f^{2}(r)=0,$
(6)

where we have used the known form of the Laplacian in $d$-dimensional
spherical polar coordinates:

$\Delta f=\frac{1}{r^{d-1}}\frac{\partial}{\partial
r}\left(r^{d-1}\frac{\partial f}{\partial
r}\right)+\frac{1}{r^{2}}\Delta_{S^{d-1}}f.$ (7)

The second term contains the Laplace-Beltrami operator $\Delta_{S^{d-1}}$,
which depends only upon angular coordinates. It thus does not contribute to
eq. (6) given the dependence upon only the radial coordinate.

Let us now look for pure power-like solutions by stipulating

$f(r)=Ar^{\alpha},$ (8)

for some constants $A$ and $\alpha$. Substituting this into eq. (6) yields

$A\alpha(d+\alpha-2)r^{\alpha-2}+A^{2}r^{2\alpha}=0.$ (9)

If this is to be true for all $r>0$, then we must have

$\alpha=-2\quad\Rightarrow\quad A[A-2(d-4)]=0.$ (10)

We thus find $A=0$ or $A=2(d-4)$, such that translating back to eq. (5)
implies that there are two power-like solutions in general. The first is not
interesting – it is the trivial (vacuum) solution $\Phi^{aa^{\prime}}=0$. The
second is a non-trivial power-like solution

$\Phi^{aa^{\prime}}=\frac{2\delta^{aa^{\prime}}}{yT_{A}}\frac{d-4}{r^{2}}.$
(11)

Interestingly, the power of $r^{-2}$ is common to all spacetime dimensions,
which can be confirmed from dimensional analysis: the dimensions of the field
$\Phi^{aa^{\prime}}$ and the coupling constant $y$ both vary with the number
of dimensions, in just such a way as to fix the power of radial distance for
solutions involving an inverse power of the coupling. This is in contrast to
solutions of the linearised field equation, whose power of distance must vary
in order to maintain the correct dimensions of the field. A consistency check
of eq. (11) is that it reproduces the Lorentzian monopole solutions of ref.
[59] for $d=3$. These are static solutions, and thus the field equation of eq.
(1) reduces to that of eq. (3), with a three-dimensional Laplacian involving
the spatial coordinates.

Arguably the most curious feature of eq. (11) is the presence of $(d-4)$ in
the numerator, which tell us that the non-trivial power-like solution of the
full Euclidean biadjoint scalar field equation is absent in four spacetime
dimensions. To gain more insight into what is going on, it is instructive to
examine more general spherically symmetric solutions, as was done for the
Lorentzian case in ref. [60]. That paper looked for solutions in which the
divergence of the biadjoint field at the origin was (partially)
screened.444Finite energy static solutions of biadjoint scalar theory are
impossible, as a consequence of Derrick’s theorem [71]. This motivates the
form

$f(r)=\frac{K(r)-1}{r^{2}},$ (12)

which we are always entitled to write, and for which $K(r)\rightarrow 1$
(everywhere) constitutes the trivial solution. By further introducing the
variable $\xi$ via

$r=e^{-\xi},\quad-\infty<\xi<\infty,$ (13)

one may show that eq. (6) amounts to

$\frac{\partial^{2}K}{\partial\xi^{2}}-(d-6)\frac{\partial
K}{\partial\xi}+(K-1)(K-2d+7)=0.$ (14)

This is a non-linear second-order differential equation, which cannot be
solved analytically in general.555By a further transformation, one may recast
eq. (14) as an Abel equation of the second kind, albeit not one that has a
tractable solution in terms of known functions. However, we may visualise
solutions as follows. Defining $\psi\equiv\partial K/\partial\xi$, we may
write eq. (14) as two coupled first-order equations:

$\left(\frac{\partial
K}{\partial\xi},\frac{\partial\psi}{\partial\xi}\right)=\Big{(}\psi,(d-6)\psi-(K-1)(K-2d+7)\Big{)}.$
(15)

This defines a vector field in the $(K,\psi)$ plane, whose integral curves
correspond to solutions of eq. (14).

Figure 2: Integral curves of the vector field of eq. (15) in the $(K,\psi)$
plane, corresponding to solutions of eq. (14). Shown are the cases $d=2$, 3,
4, and 5 respectively. The red dots correspond to the fixed point solutions
$K=1$ and $K=2d-7$.

We show such curves for the cases of $d=2$, 3, 4 and 5 in figure 2. For
general $d$, there are fixed points for

$K\in\\{1,2d-7\\},$ (16)

as is evident from eq. (14). Indeed, these correspond to the trivial solution
and the non-trivial power-like solution of eq. (11) respectively. As the
number of dimensions increases from $d<4$, the non-trivial solution moves to
the right in the $(K,\psi)$ plane. For precisely $d=4$, the two fixed points
coincide, so that there is only the trivial solution, as found above.

If we want to find solutions that partially screen the divergence at the
origin, we must look for bounded curves in the $(K,\psi)$ plane i.e. those
that correspond to finite numerators in eq. (12). For $d\neq 4$, there is
always precisely one such bounded curve, connecting the points $(1,0)$ and
$(2d-7,0)$. Thus, there is a single extended spherically symmetric solution
that corresponds to a screened charge. This broadens the implications of the
absence of a second fixed point in the $d=4$ case: not only is there no non-
trivial power-like solution, but there are are no non-trivial extended
solutions of the type in eq. (12) either (with bounded numerators).

In this section, we have undertaken a first investigation of the spectrum of
solutions of Euclidean biadjoint scalar field theory. We find non-trivial
power-like solutions in all spacetime dimensions $d\neq 4$, with concomitant
extended solutions. These solutions deserve further study, but for the
remainder of this paper we will explain fully why there are no non-trivial
power-like solutions for $d=4$, and examine related implications.

##  3 The Eguchi-Hanson instanton revisited

In the previous section, we saw that there are no non-trivial power-like
solutions (or spherically symmetric solutions) of Euclidean biadjoint theory
in $d=4$. There is in fact a very simple reason why this is the case. First,
we may recall the observation made above that non-linear power-like solutions
always have a radial dependence $\sim r^{-2}$, where this power can be fixed
by dimensional analysis for solutions that involve an inverse power of the
coupling constant $y$. However, in $d=4$, $r^{-2}$ is a harmonic function,
whose Laplacian vanishes for $r\neq 0$. To see this, note that in $d$
dimensions, eq. (7) implies

$\Delta r^{-n}=n(n+2-d)r^{-n-2},$ (17)

which indeed vanishes if $r>0$ for $n=2$ and $d=4$ (at $r=0$, there is a
singularity, leading to an appropriately normalised delta function on the
right-hand side of eq. (17)). If $r^{-2}$ is harmonic in $d=4$, this means
that it solves the linearised biadjoint equation of eq. (3). There is thus no
room left for it to solve the non-linear equation, which is why there is no
non-trivial power-like solution of Euclidean biadjoint theory in four
spacetime dimensions. Indeed, we could simply have started with this
observation, and not bothered with the analysis of the previous section at
all. We maintain, however, that the results of the previous section remain
useful: there are indeed non-trivial power-like solutions in other numbers of
dimensions. Furthermore, the observation that there are no bounded extended
solutions of Euclidean biadjoint theory, as well as power-like forms, is
itself interesting.

As is well-known [13], one can turn harmonic functions into solutions of the
full biadjoint scalar theory by dressing them with constant colour vectors
$\\{c^{a},\tilde{c}^{a^{\prime}}\\}$, which in the case of our power-like
solution in $d=4$ becomes

$\Phi^{aa^{\prime}}=\frac{\alpha c^{a}\tilde{c}^{a^{\prime}}}{r^{2}},$ (18)

where we have included an arbitrary constant $\alpha$ that remains unfixed by
the requirement that the kinematic dependence is harmonic. It is
straightforward to check that, upon substitution into eq. (3), the non-linear
term vanishes, leaving only the linear term as required. Given that previous
examples of the classical double copy have focused on solutions that linearise
biadjoint theory, we can then ask if it is possible to identify the gauge and
gravity solutions for which eq. (18) constitutes the zeroth copy. Indeed, the
solution turns out to be already known: it is related to the Eguchi-Hanson
(EH) solution in gravity. First derived and discussed in refs. [67, 68, 69],
this is a solution whose finite energy, self-dual nature and asymptotically
Euclidean character lead to its interpretation as a gravitational instanton.
As pointed out in e.g. refs. [72, 19, 24], it is particularly convenient to
express the EH solution in (2,2) signature, using the coordinate system

$u=\frac{\tau-iz}{\sqrt{2}},\quad v=\frac{\tau+iz}{\sqrt{2}},\quad
X=\frac{ix-y}{\sqrt{2}},\quad Y=\frac{ix+y}{\sqrt{2}},$ (19)

in terms of Euclidean Cartesian coordinates666As is common in the instanton
literature, we let Greek indices take the values $\mu,\nu,...=1,2,3,4$, with
$x_{4}=\tau$.

$x_{\mu}=(x,y,z,\tau).$ (20)

Then the EH metric may be written as

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},$ (21)

where $h_{\mu\nu}$ is the graviton field

$h_{\mu\nu}=\phi k_{\mu}k_{\nu},\quad\phi=\frac{\lambda}{(uv-XY)},\quad
k_{\mu}=\frac{1}{(uv-XY)}(v,0,0,-X).$ (22)

The metric of eqs. (21, 22) is in so-called Kerr-Schild form, where the vector
$k_{\mu}$ satisfies the null and geodesic conditions

$k^{2}=0\quad k\cdot\partial k_{\mu}=0.$ (23)

This in turn means that its single and zeroth copies may be straightforwardly
taken, using the general procedure defined in ref. [13]. One simply writes

$A_{\mu}^{a}=c^{a}\phi
k_{\mu},\quad\phi^{aa^{\prime}}=c^{a}\tilde{c}^{a^{\prime}}\phi,$ (24)

where $c^{a}$ and $\tilde{c}^{a}$ are constant colour vectors. Then, the gauge
and biadjoint fields thus constructed are guaranteed to solve the Yang-Mills
and biadjoint equations, which happen to linearise in both cases. Upon
translating to (Euclidean) Cartesian coordinates in eq. (22), one finds

$\phi=\frac{2\lambda}{r^{2}},$ (25)

so that the zeroth copy of the Eguchi-Hanson solution in eq. (24) precisely
matches the power-like solution of eq. (18) as claimed, provided we
identify777Alternatively, one can absorb the arbitrary constant $\alpha$ into
the additional colour vector $\tilde{c}^{a^{\prime}}$ that appears in the
biadjoint field. $\alpha=2\lambda$.

There is another way to interpret the above results. First writing the gauge
field of eq. (24) in terms of an abelian gauge field $A_{\mu}$:

$A_{\mu}^{a}=c^{a}A_{\mu},\quad A_{\mu}=\frac{\lambda}{(uv-
XY)^{2}}\left(v,0,0,-X\right),$ (26)

we may recognise the latter as

$A_{\mu}=\hat{k}_{\mu}\phi,\quad\hat{k}_{\mu}=-(\partial_{u},0,0,\partial_{Y}),$
(27)

such that the single copy is given by the action of a differential operator on
$\phi$. It is easily checked that the Eguchi-Hanson graviton is given in terms
of the same operator:

$h_{\mu\nu}=\hat{k}_{\mu}\hat{k}_{\nu}\phi,$ (28)

such that the double copy is formulated as a product in momentum, rather than
position, space. A similar idea has occurred in the literature before.
Reference [13] pointed out that setting up the double copy in terms of
differential operators reproduces known descriptions of (anti-)self-dual Yang-
Mills theory and gravity. That is, for a $\hat{k}_{\mu}$ satisfying the two
conditions

$\hat{k}^{2}=0,\quad\partial\cdot\hat{k}=0,$ (29)

substituting the ansatz of eq. (28) into the Einstein equations reduces the
latter to the Plebanski equation of self-dual gravity. For a suitable choice
of $\hat{k}_{\mu}$, this can be written in the lightcone coordinate system as

$\partial^{2}\phi+\kappa\left\\{\partial_{Y}\phi,\partial_{u}\phi\right\\}=0,$
(30)

where the Poisson bracket is defined by

$\\{A,B\\}=(\partial_{Y}A)(\partial_{u}B)-(\partial_{u}A)(\partial_{Y}B).$
(31)

Alternatively, one may write the Plebanski equation directly in terms of a
general operator $\hat{k}_{\mu}$ satisfying eq. (29), as [13]888Following
convention, we do not raise or lower indices given that we are in Euclidean
signature.

$\partial^{2}\phi-\frac{1}{2}(\hat{k}_{\mu}\hat{k}_{\nu}\phi)(\partial_{\mu}\partial_{\nu}\phi)=0.$
(32)

Similarly, substituting the ansatz

$A_{\mu}^{a}=\hat{k}_{\mu}\Phi^{a},$ (33)

into the Yang-Mills equations, for $\hat{k}_{\mu}$ satisfying eq. (29), leads
to a known formulation of self-dual Yang-Mills theory [73]999Our eq. (34) can
be obtained from eq. (28) of ref. [13]. Our conventions differ in that we are
in Euclidean signature. We have also set coupling constants to unity in both
eqs. (32) and (34), and normalised the vector $\hat{k}_{\mu}$ such that
numerical constants are the same in all theories, for reasons that will become
clear later on.

$\partial^{2}\Phi^{a}-\frac{1}{2}\epsilon^{abc}(\hat{k}_{\mu}\Phi^{b})(\partial_{\mu}\Phi^{c})=0,$
(34)

where the anti-self-dual sectors of both gauge and gravity theory can be
obtained similarly (i.e. by a different choice of $\hat{k}_{\mu}$). The
(anti-)self dual sectors constitute explicit cases in which the double copy
can be made manifest at the level of equations of motion [5], and
generalisations of this construction to exotic deformed theories are also
known [66]. The Eguchi-Hanson instanton is a special case in which the fields
are of Kerr-Schild form. In the gauge theory, this specialises the ansatz of
eq. (33) to

$\Phi^{a}=c^{a}\phi,$ (35)

which linearises the Yang-Mills equations, such that one may consider the
abelian field $A_{\mu}$ of eq. (26). As noted in ref. [19], the observation
that (anti-)self-dual gravity solutions can be defined in terms of
differential operators, and associated with (null) electromagnetic fields, was
made long ago in ref. [72]. The double copy reinterprets this observation, and
provides a framework for potential generalisations.

Returning to the present study, it is not obvious that the double copy between
eqs. (26, 27) and eq. (28) is a special case of the known self-dual double
copy of ref. [24], as the $\hat{k}_{\mu}$ operator of eq. (27) does not
satisfy both conditions in eq. (29). For our purposes, it will be useful to
transform the operator to Cartesian coordinates, in which it takes the
form101010In an abuse of notation, we will refer to the operator as
$\hat{k}_{\mu}$ in both the $(x,y,z,\tau)$ and $(u,v,X,Y)$ coordinate systems,
given that the explicit coordinates that appear in any given equation imply no
ambiguity.

$\hat{k}_{\mu}=-\frac{1}{2}\left(\partial_{x}+i\partial_{y},\partial_{y}-i\partial_{x},\partial_{z}-i\partial_{\tau},\partial_{\tau}+i\partial_{z}\right),$
(36)

which may be written more compactly as

$\hat{k}_{\mu}=-\frac{1}{2}\left(\delta_{\mu\nu}+i\bar{\eta}^{3}_{\mu\nu}\right)\partial_{\nu}.$
(37)

Here $\bar{\eta}^{3}_{\mu\nu}$ is a special case of the ’t Hooft symbols
$\\{\bar{\eta}^{a}_{\mu\nu}\\}$, which arise in the study of instantons. For
convenience, we review the properties of these symbols, as well as useful
identities, in appendix A. More briefly, the ’t Hooft symbols
$\\{\bar{\eta}^{a}_{\mu\nu}\\}$ form a representation of an SU(2) subalgebra
of SO(4), where the latter group is equivalent to the Lorentz group in
Euclidean signature. Thus, the presence of the ’t Hooft symbol in eq. (37)
means that the operator $\hat{k}_{\mu}$ involves a particular “rotation” of
the derivative operator $\partial_{\nu}$. Using eq. (123), it is then
straightforward to verify that

$\hat{k}^{2}=0,\quad\partial\cdot\hat{k}=-\frac{1}{2}\Delta,$ (38)

and thus that the second condition in eq. (29) is not satisfied. However, one
may instead consider the alternative differential operator

$\hat{k}^{\prime}_{\mu}=(0,\partial_{Y},\partial_{u},0),$ (39)

In Cartesian coordinates, this translates as

$\hat{k}^{\prime}_{\mu}=\frac{1}{2}\left(\bar{\eta}^{2}_{\mu\nu}-i\bar{\eta}^{1}_{\mu\nu}\right)\partial_{\nu}=-\bar{\eta}^{2}_{\mu\nu}\hat{k}_{\nu},$
(40)

which does indeed satisfy both of the properties in eq. (29). Furthermore, we
see that $\hat{k}^{\prime}_{\mu}$ is a coordinate transformation of $k_{\mu}$
to a new frame whose coordinates are

${x^{\prime}}_{\mu}=-\bar{\eta}^{2}_{\mu\nu}x_{\nu}\quad\Rightarrow\quad\left(\begin{array}[]{c}x^{\prime}\\\
y^{\prime}\\\ z^{\prime}\\\
\tau^{\prime}\end{array}\right)=\left(\begin{array}[]{c}z\\\ \tau\\\ -x\\\
-y\end{array}\right).$ (41)

We then have

$(r^{\prime})^{2}={x^{\prime}}_{\mu}\,{x^{\prime}}_{\mu}={x}_{\mu}\,{x}_{\mu}=r^{2},$
(42)

such that one may write the Eguchi-Hanson solution and its single copy in the
primed coordinate system as

$A^{\prime}_{\mu}=c^{a}\hat{k}^{\prime}_{\mu}\phi(r^{\prime}),\quad
h^{\prime}_{\mu\nu}=\hat{k}^{\prime}_{\mu}\hat{k}^{\prime}_{\nu}\phi(r^{\prime}),$
(43)

which is indeed a special case of the general self-dual construction of eqs.
(28–33).

In this section, we have explained the absence of non-linear power-like
solutions in Euclidean biadjoint scalar theory, by pointing out that simple
power-like solutions in four dimensions are in fact already “claimed” by the
zeroth copy of the Eguchi-Hanson instanton, for which the biadjoint field
equations linearise. A natural language for describing the single and zeroth
copies of the EH solution is in terms of differential operators satisfying eq.
(29), such that one obtains a special case of the self-dual double copy
construction proposed in ref. [13]. Particularly compelling in our present
paper is the fact that differential operators satisfying eqs. (29) can be
written in terms of ’t Hooft symbols, as in eq. (40). This suggests a general
ansatz for double-copying certain instanton solutions, that we explore in the
following section. We will also find that one may easily construct non-abelian
single copies of the Eguchi-Hanson instanton, thus making its single copy
structure more intricate than previously thought.

##  4 A general ansatz for double-copying instantons

In the previous section, we discussed the single copy of the Eguchi-Hanson
instanton, which can be taken to be an abelian-like gauge field. In general,
however, there are many instanton solutions of non-abelian gauge theories,
namely (anti-)self-dual classical solutions, of finite energy (see e.g. refs.
[74, 75] for pedagogical reviews). To be concrete, let us consider the case of
SU(2) gauge theory. Finite energy demands that the field become pure gauge at
infinity, and a given solution then constitutes a map from the boundary of
spacetime ($S^{3}$) to the gauge group manifold, which is also $S^{3}$ for
SU(2). Instantons can then be classified by their winding, or instanton
number, which has a simple interpretation as the number of times the first
$S^{3}$ space wraps around the second, in mapping the two
manifolds.111111Mathematically, one talks about the third homotopy group of a
manifold ${\cal M}$, which classifies non-trivial maps from $S^{3}$ to a given
manifold ${\cal M}$. Choosing ${\cal M}$ to be the gauge group manifold for
SU(2), one has $\pi_{3}(S^{3})={\mathbb{Z}}$, meaning that there are
topologically distinct maps labelled by different integers. This is precisely
the winding number mentioned above, where positive (negative) values
correspond to the (anti-)self dual sectors. The winding number is given by the
volume integral

$k=\frac{1}{16\pi^{2}}\int d^{4}x{\rm Tr}\left[\tilde{\bf F}_{\mu\nu}\,{\bf
F}_{\mu\nu}\right],$ (44)

where the field strength ${\bf F}_{\mu\nu}\equiv F_{\mu\nu}^{a}{\bf T}^{a}$
and its dual $\tilde{\bf F}_{\mu\nu}\equiv\tilde{F}_{\mu\nu}^{a}{\bf T}^{a}$
are given respectively by

$F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{c},\qquad\tilde{F}^{a}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{a}_{\rho\sigma}.$
(45)

There will be a non-trivial parameter space of solutions for definite winding
number $k$ in general, with the various parameters representing the width or
size of the solution, its position in space, rotation angles in spacetime or
in the gauge space etc. Some of these parameters are redundant under gauge
transformations or other redundancies, but the set of independent parameters
that label instantons of given $k$ are called moduli, and they form a moduli
space. The metric in this space is not completely smooth, but can be singular
at certain points e.g. for instantons that have zero size, or where the
centres of multiple instantons coincide. Remarkably, it is known how to
classify all possible instanton solutions in pure YM theory [76].

A large family of SU(2) instanton solutions is given by dressing a vector
field $V_{\mu}$ according to the ’t Hooft ansatz [77]

$A_{\mu}^{a}=-\bar{\eta}^{a}_{\mu\nu}V_{\nu},\quad
A_{\mu}^{a}=-\eta^{a}_{\mu\nu}V_{\nu},$ (46)

for the case of self-dual and anti-self dual fields respectively, and where
$\\{\eta_{\mu\nu}^{a},\bar{\eta}_{\mu\nu}^{a}\\}$ are the ’t Hooft symbols
encountered above, and reviewed briefly here in appendix A.121212The minus
signs in eq. (46) are conventional. Also, note that use of the anti-self-dual
matrix $\bar{\eta}^{a}_{\mu\nu}$ actually results in a self-dual field.
Whether or not $\eta^{a}_{\mu\nu}$ or $\bar{\eta}^{a}_{\mu\nu}$ appears in the
gauge field can depend upon the gauge. In our previous use of a ’t Hooft
symbol in eq. (37), this was acting merely as a representation of a particular
spacetime rotation, where the upper index labelled which particular
infinitesimal rotation we were talking about. In eq. (46), however, the upper
indices on the ’t Hooft symbols are to be interpreted as adjoint indices
associated with the SU(2) gauge group. That this is possible is due to the
fact that the $\\{\eta^{a}_{\mu\nu}\\}$ and $\\{\bar{\eta}^{a}_{\mu\nu}\\}$
separately form SU(2) algebras of SO(4) rotations. They can thus be mapped to
the SU(2) gauge algebra. Focusing on the self-dual case, substitution of eq.
(46) into the Yang-Mills equations yields the condition

$\partial_{\mu}V_{\mu}+V_{\mu}V_{\mu}=0,$ (47)

as well as

$\tilde{f}_{\mu\nu}=f_{\mu\nu},\quad
f_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}.$ (48)

Here we can recognise $f_{\mu\nu}$ as being analogous to an abelian-like field
strength tensor. However, $V_{\mu}$ cannot necessarily be interpreted as an
abelian gauge field, given that $f_{\mu\nu}$ is not guaranteed to satisfy the
Maxwell equation

$\partial_{\mu}f_{\mu\nu}=0$ (49)

in general. Equation (47) is usually satisfied by taking $V_{\mu}$ to be the
gradient of the logarithm of a harmonic function:

$V_{\mu}=\partial_{\mu}\log V,\quad\Delta V=0.$ (50)

However, motivated by the discussion in the previous section, there is another
ansatz we can make. Let us construct a vector field according to the
prescription:

$A_{\mu}=\hat{k}_{\mu}\phi,\quad\hat{k}_{\mu}=\left(A\delta_{\mu\nu}+B_{i}\,\bar{\eta}^{i}_{\mu\nu}\right)\partial_{\nu},\quad
i\in\\{1,2,3\\},$ (51)

where the scalar $A$ and three-vector $B_{i}$ are possibly complex constants.
This generalises the definitions of $\hat{k}_{\mu}$ and
$\hat{k}^{\prime}_{\mu}$ given by eqs. (37, 40). Note that, as in eq. (37),
the upper index on each ’t Hooft symbol is not a gauge (adjoint) index, but
merely labels which infinitesimal rotations we are talking about. With this
field we can compute

$f_{\mu\nu}=B_{i}\left(\bar{\eta}^{i}_{\nu\rho}\partial_{\mu}-\bar{\eta}^{i}_{\mu\rho}\partial_{\nu}\right)\partial_{\rho}\phi,$
(52)

from which one finds

$\tilde{f}_{\mu\nu}=f_{\mu\nu}+B_{i}\bar{\eta}^{i}_{\mu\nu}\Delta\phi.$ (53)

Thus, we see that the requirement for $A_{\mu}$ to satisfy the self-dual
equations in eq. (48) is such as to force the field $\phi$ to be a harmonic
function ($\Delta\phi=0$). If we further insist that the double copy of eq.
(51), defined according to eq. (28), be a solution of self-dual gravity, then
the differential operator $\hat{k}_{\mu}$ must satisfy the dual constraints of
eq. (29). For the first, we find

$\hat{k}^{2}=(A^{2}+B^{2})\Delta=0\quad\Rightarrow\quad A^{2}=-B^{2},$ (54)

where $B^{2}=B_{i}B^{i}$ and we have used the identities in appendix A. The
second condition gives

$\partial_{\mu}\hat{k}_{\mu}=A\Delta=0.$ (55)

Thus, the requirement that the abelian self-dual gauge field of eq. (51)
double-copies to a self-dual gravity solution imposes

$A=0,\quad B^{2}=0.$ (56)

The components $\\{B_{i}\\}$ are then required to be complex in general.
Indeed, the abelian single copy of Eguchi-Hanson, with a differential operator
defined as in eq. (40), emerges as a special case. Interestingly, though, eq.
(56) implies that eq. (47) is automatically satisfied: one finds

$\partial_{\mu}A_{\mu}+A_{\mu}A_{\mu}=A\Delta\phi+(A^{2}+B^{2})(\partial_{\mu}\phi)(\partial_{\mu}\phi)=0.$
(57)

This immediately implies that we may dress the solution of eq. (51) to form a
non-abelian SU(2) instanton, as in eq. (46):

$\displaystyle A_{\mu}^{a}$
$\displaystyle=-\bar{\eta}^{a}_{\mu\nu}\hat{k}_{\nu}\phi.$ (58)

A suggestive way to write this is as

$A_{\mu}^{a}=-\hat{k}^{a}_{\mu}\phi,\quad\hat{k}^{a}_{\mu}\equiv\bar{\eta}^{a}_{\mu\nu}\hat{k}_{\nu},$
(59)

i.e. in terms of a “non-abelian” differential operator. Intriguingly, the non-
abelian instanton can be double-copied directly, upon tracing over the colour
indices: using eq. (123), one finds

$\hat{k}_{\mu}^{a}\hat{k}_{\nu}^{a}=-\hat{k}_{\mu}\hat{k}_{\nu}.$ (60)

Hence, if we construct the graviton field

$h_{\mu\nu}=-\hat{k}^{a}_{\mu}\hat{k}^{a}_{\nu}\phi,$ (61)

this yields precisely the same gravity solution as double copying the field of
eq. (51) according to eq. (28). As noted above, eq. (51) is not guaranteed to
be an abelian gauge field in general, as it may not necessarily satisfy the
Maxwell equation of eq. (49). In fact it does, however, due to the consequence
derived above that $\phi$ be harmonic. To see this, note that eq. (52) implies

$\partial_{\mu}f_{\mu\nu}=B_{i}\bar{\eta}^{i}_{\nu\rho}\partial_{\rho}(\Delta\phi)=0,$
(62)

where we have used the antisymmetry property
$\bar{\eta}^{i}_{\mu\nu}=-\bar{\eta}^{i}_{\nu\mu}$. We therefore find that one
may construct either an abelian or non-abelian single copy of certain
(anti-)self-dual gravity solutions, for which the Eguchi-Hanson solution
discussed in the previous section is a special case. Pleasingly, this mirrors
the situation that has been found for magnetic monopole solutions in refs.
[63, 57]. That is, one may regard the single copy of the pure NUT solution in
gravity as an abelian-like (Dirac) magnetic monopole, dressed by a constant
colour vector, or as a genuinely non-abelian Wu-Yang monopole, where there is
a singular gauge transformation that relates the two forms. This scheme is
shown in figure 3, and is interesting to note that a similar idea has occurred
before in the original double copy for scattering amplitudes. For example, the
infrared singularities of both abelian and non-abelian gauge theory map to the
same infrared singularities in gravity, to all orders in perturbation theory
[78]. There is presumably a gauge transformation (that we have not been able
to find) that relates the two forms of the single copy instantons considered
here.

Figure 3: Schematic depiction of the multiple single copies of the
gravitational instanton solutions considered in this paper. The same scheme
has previously been obtained for magnetic monopoles [63, 57], and for infrared
singularities of scattering amplitudes [78].

In this section, we have given a general ansatz for single-copying a class of
gravitational instantons, that allows us to construct non-abelian as well as
abelian single copies. However, all of the solutions thus obtained turn out to
be special, in that they linearise the equations of motion in both gravity and
gauge theory (and hence, by association, biadjoint theory). The question then
arises of whether one can single-copy more general instantons, or more
formally: what portion of the moduli space of gauge theory instantons is
captured by the ansatz of eq. (58)? The answer to the latter question appears
to be rather limited. For example, we may attempt to calculate the winding
number integral of eq. (44), for the single copy of the Eguchi-Hanson
solution. This turns out to yield

${\rm Tr}[\tilde{\bf F}_{\mu\nu}{\bf
F}_{\mu\nu}]\propto\frac{\lambda^{2}B^{2}}{r^{8}},$ (63)

which vanishes due to being proportional to $B^{2}=0$. Thus, the non-abelian
single copy of the Eguchi-Hanson instanton is topologically
trivial.131313Similar conclusions were reached for the abelian single copy in
ref. [19], albeit using a different single copy field. See ref. [24] for a
discussion of the relation between the single copy of ref. [19], and that used
in this paper. Also, as a singular solution whose singularity appears non-
removable by a gauge transformation, it should not properly be regarded as
part of the moduli space of gauge theory instantons. Similar conclusions will
be reached for more general functions $\phi$: the winding number cannot depend
on which particular basis of the rotation generators entering $\hat{k}_{\mu}$
we choose i.e. the ’t Hooft symbols $\\{\bar{\eta}^{i}_{\mu\nu}\\}$. This in
turn implies invariance under rotations of the vector $B_{i}$, such that the
result can only depend on $B^{2}$, which is trivial. Interestingly, though,
the space of harmonic functions $\phi$ in gauge theory still allows for some
potentially interesting gravity solutions. For example, one may choose

$\phi=\sum_{i=1}^{N}\frac{c_{i}}{(x-a_{i})^{2}},$ (64)

which has $N$ point-like disturbances at 4-positions $\\{a_{i}\\}$. The double
copy of this would appear to be a multi-centre generalisation of the Eguchi-
Hanson solution. Given that the single-centre Eguchi-Hanson case is related to
a two-centre Gibbons-Hawking metric [79, 80], the graviton obtained from eqs.
(28, 64) may also be of multi-centre Gibbons-Hawking type. We have not been
able to find an explicit coordinate transformation that realises this, which
by no means rules out that such a transformation is possible.

Of course, we have not considered the most general ansatz for gauge theory
solutions in this paper. Equation (58) relies on a single function $\phi$
which is dressed by additional factors, rather than the full adjoint-valued
field $\phi^{a}$ of eq. (33). However, it is not clear how to generate the
additional information required by eq. (33) upon taking the single copy, i.e.
how to turn the single gravitational function $\phi$ into the multiple
functions $\\{\phi^{a}\\}$.141414Reference [53] recently considered the more
general ansatz of eq. (33) in exploring how asymptotic symmetries in self-dual
Yang-Mills and gravity are related by the double copy. However, replacements
between theories were at the level of commutators, or equivalently structure
constants. A prescription for achieving something similar has recently been
proposed in various two-dimensional theories [52]. Indeed, comparing our
analysis in this paper with the results of ref. [52] yields a number of useful
insights, which we explore in the following section.

##  5 Relation to the two-dimensional non-perturbative double copy

In the previous section, we provided an ansatz for single-copying exact
solutions of (anti-)self-dual gravity. Recently, another non-perturbative
single copy procedure has appeared [52], and the aim of this section is to
compare their approach with our analysis in this paper. As we will see, this
comparison reveals a number of useful insights, that clarify long-standing
questions in the double copy literature, but also generalise the results of
ref. [52] itself. The latter reference considered various field theories in
two spacetime dimensions, including the biadjoint scalar field theory we have
already encountered in this paper. In the conventions of ref. [52], and
allowing for arbitrary gauge groups, this has equation of motion

$\partial^{2}\phi^{aa^{\prime}}-\frac{1}{2}f^{abc}\tilde{f}^{a^{\prime}b^{\prime}c^{\prime}}\phi^{bb^{\prime}}\phi^{cc^{\prime}}=0,$
(65)

in Lorentzian signature. Solutions in this theory were argued to obey a
similar scheme to figure 1, but where the gauge theory is replaced by
Zakharov-Mikhailov (ZM) theory [81]:

$\partial^{2}\phi^{a}-\frac{1}{2}f^{abc}\partial_{\mu}\phi^{a}\tilde{\partial}^{\mu}\phi^{b}=0.$
(66)

Here the field $\phi^{a}$ carries a single adjoint index, and we have
introduced the dual derivative operator

$\tilde{\partial}^{\mu}=\epsilon^{\mu\nu}\partial_{\nu},$ (67)

where $\epsilon^{\mu\nu}$ is the two-dimensional Levi-Civita symbol. Finally,
the gravity theory is replaced by Special Galileon (SG) theory [82, 83, 84],
whose equation of motion is

$\partial^{2}\phi-\frac{1}{2}(\partial_{\mu}\partial_{\nu}\phi)(\tilde{\partial}^{\mu}\tilde{\partial}^{\nu}\phi)=0.$
(68)

There is a clear pattern of replacements in proceeding from eq. (65) through
eqs. (66, 68): adjoint indices are progressively removed, and the number of
spacetime (and dual) derivatives increases. More formally, given two adjoint-
valued fields $V^{a}$ and $W^{a}$, one may define the formal replacement rules

$V^{a}\rightarrow V,\quad
f^{abc}V^{a}W^{b}\rightarrow(\partial_{\mu}V)(\tilde{\partial}^{\mu}W),$ (69)

which can indeed be used to transform between the various theories mentioned
above. The second replacement can be interpreted as replacing the structure
constants of the colour algebra with those of an infinitely dimensional
kinematic algebra. The kinematic structure constants are more easily viewed in
momentum space, and to find them we may Fourier transform the right-hand side
of the second replacement in eq. (69) to get

$\displaystyle\int d^{2}xe^{ip_{1}\cdot
x}\partial_{\mu}V\tilde{\partial}^{\mu}W$ $\displaystyle=\int
d^{2}x\int\frac{d^{2}p_{2}}{(2\pi)^{2}}\int\frac{d^{2}p_{3}}{(2\pi)^{2}}e^{i(p_{1}-p_{2}-p_{3})\cdot
x}[-\epsilon^{\mu\nu}p_{2\mu}p_{3\nu}]\tilde{V}(p_{2})\tilde{W}(p_{3})$
$\displaystyle=\int\frac{d^{2}p_{2}}{(2\pi)^{2}}\int\frac{d^{2}p_{3}}{(2\pi)^{2}}[-\epsilon^{\mu\nu}p_{2\mu}p_{3\nu}\delta^{2}(p_{2}+p_{3}-p_{1})]\tilde{V}(p_{2})\tilde{W}(p_{3}),$
(70)

where we have introduced momentum modes of the spacetime fields, denoted with
tildes. We thus see that eq. (69) replaces adjoint-valued fields with scalars,
which are then combined according to the momentum-space kinematic structure
constant

${f_{p_{2}p_{3}}}^{p_{1}}=X(p_{2},p_{3})\delta^{2}(p_{2}+p_{3}-p_{1}),\quad
X(p_{2},p_{3})=-\epsilon^{\mu\nu}p_{2\mu}p_{3\nu}.$ (71)

As to the nature of this kinematic algebra, it describes area-preserving
diffeomorphisms of the two-dimensional spacetime. To see this, we may write
the general form of an such an infinitesimal diffeomorphism in two-dimensional
spacetime:

${\bf V}=-(\tilde{\partial}^{\mu}V)\partial_{\mu}.$ (72)

According to standard differential geometry results, an infinitesimal
diffeomorphism $f^{\mu}\partial_{\mu}$ is area-preserving (in more than two
dimensions, volume-preserving) provided that $\partial_{\mu}f^{\mu}=0$.
Equation (67) implies

$\partial_{\mu}\tilde{\partial}^{\mu}V=0,$ (73)

so that the diffeomorphism of eq. (72) is area-preserving as claimed. As
explained in ref. [52], evaluating the Lie algebra of the generators in eq.
(72) yields

$[{\bf V},{\bf W}]={\bf Z},\quad{\bf
Z}=-\tilde{\partial}^{\mu}\left(\partial_{\nu}V\tilde{\partial}^{\nu}W\right)\partial_{\mu}.$
(74)

To recognise the above structure constants, we can take generators
corresponding to individual momentum modes:

${\bf V}_{p_{i}}=-\tilde{\partial}^{\mu}(V_{p_{i}})\partial_{\mu},\quad
V_{p_{i}}=e^{ip_{i}\cdot x},$ (75)

from which one finds

$[{\bf V}_{p_{2}},{\bf V}_{p_{3}}]=X(p_{2},p_{3}){\bf
V}_{p_{2}+p_{3}}={f_{p_{2}p_{3}}}^{p_{1}}{\bf V}_{p_{1}},$ (76)

where we have recognised the form of the structure constant of eq. (71).

Above, we have seen that one may transform between the theories of eqs. (65,
66, 68) by successively replacing colour algebras with a Lie algebra of area-
preserving diffeomorphisms. This does not explain why one should make such
replacements, however, and ref. [52] provided the following motivation. It is
a known fact that the group U($N$) becomes isomorphic, at large $N$, to the
group of diffeomorphisms of a torus [85]. To make this precise, one may use
the fact that there is a particular basis $\\{{\bf T}_{p}\\}$ of the
generators of U($N$) (for odd $N$), where $p$ is a 2-vector whose components
are integer modulo $N$, and such that the structure constants are [85, 52]

${f_{p_{2}p_{3}}}^{p_{1}}=-\frac{N}{2\pi}\sin\left(\frac{2\pi}{N}\epsilon^{\mu\nu}p_{2\mu}p_{3\nu}\right).$
(77)

Upon taking the large $N$ limit, one may reinterpret the vectors $\\{p_{i}\\}$
as specifying momentum modes on a torus, and the structure constants of eq.
(77) reproduce precisely those of eq. (71). Thus, in the large $N$ limit, the
three theories of eqs. (65, 66, 68) (for gauge group U($N$), repeated in the
case of biadjoint theory) become mutually isomorphic. Reference [52] then uses
this to argue that the colour-kinematic replacements of eq. (69) should be
made also for finite $N$, and also proposes a scheme for turning non-
perturbative solutions of SG theory into counterparts in ZM and biadjoint
theory, which is accurate up to subleading corrections in $N$.

Regardless of this motivation, a very similar scheme – albeit perhaps not
noticeably so – has appeared in the literature before, namely in the study of
(anti-)self dual Yang-Mills and gravity [5]. We have quoted the relevant field
equations in eqs. (34, 32), where different choices of the differential
operator $\hat{k}_{\mu}$ correspond to the (anti-)self-dual cases
respectively. Comparing these with eqs. (66, 68), we see that four-dimensional
(anti-)self-dual Yang-Mills theory and gravity have precisely the same forms
as two-dimensional ZM and SG theory respectively, but where the dual
derivative operator $\tilde{\partial}^{\mu}$ is replaced by the differential
operator $\hat{k}^{\mu}$.151515It should now hopefully be clear why we have
chosen different conventions in our eqs. (32, 34) relative to those in
existing literature [13]: it is to make the similarity between our four-
dimensional theories and the two-dimensional theories of ref. [52] more
striking. Indeed, as is the role of $\tilde{\partial}^{\mu}$ in two
dimensions, we can associate $\hat{k}_{\mu}$ with area-preserving
diffeomorphisms, where the most general such transformation will be given by

${\bf V}=-(\hat{k}^{\mu}V)\partial_{\mu}.$ (78)

That this is area-preserving follows from our imposition that
$\partial\cdot\hat{k}=0$, and the cases of self-dual and anti-self-dual YM
theory or gravity arise, as explained above, from the general ansätze

$\hat{k}_{\mu}\Big{|}_{\rm
SD}=\frac{1}{2}B_{i}\bar{\eta}^{i}_{\mu\nu}\partial^{\nu},\quad\hat{k}_{\mu}\Big{|}_{\rm
ASD}=\frac{1}{2}B_{i}\eta^{i}_{\mu\nu}\partial^{\nu},\quad$ (79)

respectively. The role of the vector $B_{i}$ is to pick out the two-
dimensional planes in which the area-preserving diffeomorphisms of eq. (78)
act. Specifically, we may write

$B_{i}\bar{\eta}^{i}_{\mu\nu}=b^{(1)}_{[\mu}b^{(2)}_{\nu]},$ (80)

where the explicit forms of the vectors on the right-hand side, as may be
verified using eq. (122), are

$b^{(1)}_{\mu}=\left(B_{1},B_{2},B_{3},0\right)\quad
b^{(2)}_{\mu}=\left(0,\frac{B_{3}}{B_{1}},\frac{-B_{2}}{B_{1}},-1\right),$
(81)

where we have assumed $B_{1}\neq 0$ and $B^{2}=0$ as before. Then

$(\hat{k}^{\mu}\phi)\partial_{\mu}=(b^{(1)[\mu}b^{(2)\nu]}\partial_{\nu}\phi)\partial_{\mu}$
(82)

will generate diffeomorphisms in the plane defined by the bivector
$b^{(1)}_{[\mu}b^{(2)}_{\nu]}$. As an example, we may consider the self-dual
operator of eq. (40), which has

$B_{1}=-i,\quad B_{2}=1,\quad B_{3}=0.$ (83)

This gives (in Cartesian coordinates)

$b^{(1)}_{\mu}=(-i,1,0,0),\quad b^{(2)}_{\mu}=(0,0,-i,-1).$ (84)

Translating to the lightcone coordinate system, one finds diffeomorphisms in
the $(u,Y)$ plane as expected. Note that the vectors $\\{b^{(i)}\\}$ satisfy
the conditions

$b^{(i)}\cdot b^{(j)}=0,\quad\forall i\in\\{1,2\\}.$ (85)

This makes the plane defined by the above bivector an example of a null plane.
Null planes defined by self-dual and anti-self-dual bivectors are called
$\alpha$-planes and $\beta$-planes respectively, and they play a key role in
the study of instantons (see e.g. ref. [86] for a pedagogical review). To
summarise, we have found that (anti-)self-dual YM theory and gravity provide a
four-dimensional generalisation of the non-perturbative double copy
construction of ref. [52], but where the area-preserving diffeomorphisms take
place in $\alpha$\- or $\beta$-planes. To visualise this over the entire
space, we may foliate four-dimensional Euclidean space by a family of parallel
$\alpha$\- or $\beta$-planes, whose orientation is determined by the vector
$B_{i}$. The vector field $A_{\mu}$ will then generate area-preserving
diffeomorphisms in each one, as shown in figure 4.

Figure 4: Foliation of four-dimensional Euclidean space by a family of
$\alpha$\- or $\beta$-planes. The abelian single copy gauge field $A_{\mu}$
generates area-preserving diffeomorphisms in each plane, represented here by
field lines (integral curves).

It was already known that the kinematic algebra of self-dual YM and gravity
consisted of area-preserving diffeomorphisms [5]. New to this paper, however,
are the geometric construction of area-preserving diffeomorphisms in arbitrary
null planes, and the recognition that this provides a four-dimensional
analogue of the two-dimensional non-perturbative double copy proposed in ref.
[52]. Regarding the latter, it would be possible, for example, to argue for
the double copy replacements

$\phi^{a}\rightarrow\phi,\quad
f^{abc}\phi_{1}^{b}\phi_{2}^{c}\rightarrow\partial_{\mu}\phi_{1}\,\hat{k}^{\mu}\phi_{2}$
(86)

on similar grounds to eq. (69). That is, one could take the large $N$ limit of
self-dual U($N$) Yang-Mills theory (or U($N$)$\times$U($N$) biadjoint theory),
consider periodic solutions in the planes associated with $\hat{k}^{\mu}$, and
then argue that the theories become mutually isomorphic.

Although the presence of an area-preserving diffeomorphism algebra in
(anti-)self-dual gauge and gravity theory has been known for some time, it has
not been known how these transformations are visible or relevant when
considering exact classical solutions. For scattering amplitudes, the
situation is much clearer, as first explained in ref. [5]: amplitudes in all
theories defined by eqs. (32, 34, 65) are given by an expansion in cubic
diagrams, each of whose vertices involves a product of two structure constants
appropriate to the theory of interest. Thus, amplitudes in one theory can
simply be obtained from amplitudes in another by replacing the appropriate
structure constants, which gives a direct operational meaning to phrases such
as “replacing the colour algebra with a kinematic algebra”. For classical
solutions, no structure constants manifestly appear, and thus it is not clear
how moving from one theory to another involves a replacement of algebras. This
conceptual problem is especially pronounced given that BCJ duality for
amplitudes is an intrinsically non-linear phenomenon: it crucially relies on
higher orders in perturbation theory, such that products of structure
constants appear. The above ideas, however, indeed allow us to interpret what
is happening, even though our exact classical solutions linearise the
equations of motion. For a given gravity solution of the form of eq. (28), let
us choose its abelian single copy

$A_{\mu}^{a}=c^{a}A_{\mu},\quad A_{\mu}=\hat{k}_{\mu}\phi.$ (87)

Using standard results from differential geometry (see e.g. [87]), we may
regard the vector field $A_{\mu}$ as generating an infinitesimal
diffeomorphism

$A^{\mu}\partial_{\mu},$ (88)

whose physical interpretation is that it performs a simultaneous translation
along the integral curves of the field (figure 5). Vector fields that are the
single copies of gravitational solutions will then generate diffeomorphisms of
form

$(\hat{k}^{\mu}\phi)\partial_{\mu},$ (89)

which, as remarked above, generate area-preserving diffeomorphisms in each of
the null planes associated with the operator $\hat{k}_{\mu}$. Thus, even for
the case of linearised solutions that do not involve higher-order contractions
of structure constants, there is still a well-defined way in which their
properties are governed by the area-preserving diffeomorphism algebra.
Furthermore, translating between biadjoint, gauge and gravity theory entails
replacing the diffeomorphism generators with colour generators, or vice versa.
This is straightforward to see in the case of solutions with abelian-like
single copies, which give rise to the following fields in different theories
(contracted with appropriate generators):

$\Phi=(c^{a}{\bf
T}^{a})\otimes(\tilde{c}^{a^{\prime}}{\bf\tilde{T}}^{a^{\prime}})\phi,\quad{\bf
A}^{\mu}\partial_{\mu}=(c^{a}{\bf
T}^{a})(\hat{k}^{\mu}\phi)\partial_{\mu},\quad
h^{\mu\nu}\partial_{\mu}\partial_{\nu}=(\hat{k}^{\mu}\hat{k}^{\nu}\phi)\partial_{\mu}\partial_{\nu}.$
(90)

Upon proceeding from left to right, we can see directly that colour generators
are replaced by generators of area-preserving diffeomorphisms. There is
nothing particularly profound going on here: the replacements of generators
simply constitute the statement that the field in each theory must be Lie-
algebra valued in two Lie algebras. These will be colourful or kinematical, as
dictated by which theory we are in. Notably, this correspondence works for any
value of $N$, not just the large $N$ limit. But, as explained above, it
applies only to those solutions that can be chosen to linearise their
respective field equations. Interestingly, the non-perturbative double copy
discussed in ref. [52] suffers from a similar specialism, in that SG theory is
ultimately related to a free field theory.

Figure 5: A vector field defines a diffeomorphism consisting of simultaneous
translations along the integral curves of the field. For an area-preserving
diffeomorphism, the areas of the two shapes shown on the left and right will
be the same.

Given the ideas of this section, it is not clear how to fully interpret the
non-abelian single copy of eq. (58), which does not obviously translate to
replacing an area-preserving diffeomorphism generator with a colour
counterpart. It would be interesting to find non-trivial examples of double
copies of the form of eq. (33), which in turn relates to the question of
whether all gauge theory instantons can be double-copied. It may well turn out
that, whilst all solutions of the Plebanksi equation can be single-copied to
make non-abelian instantons, the converse is not possible. Indeed, similar
comments were made in ref. [52] regarding how all solutions of SG theory can
be used to make solutions of ZM theory, but that the opposite is not true.
Complementary statements were made, some time ago and completely independently
of the double copy, in ref. [88]. The authors showed that every gravitational
instanton solution can be mapped to a SU(2) non-abelian instanton, albeit one
that lives in the same Ricci-flat background as that defined by the
gravitational solution i.e. the SU(2) solutions are “self-gravitating”. In
double copy lingo, this corresponds to a so-called type B curved-space double
copy [18], in which a classical solution in gravity can be identified with a
gauge field living on a non-dynamical curved background. Although this is a
different situation to that considered in this paper, it nevertheless has the
property that not all gauge theory solutions can be mapped to those in
gravity. The reason in this case, though, is unique to the particular set-up
considered in ref. [88]: one cannot copy a SU(2) solution on a particular
curved background to obtain a gravity solution corresponding to a different
curved space.

##  6 Discussion

In this paper, we have performed a first investigation of the spectrum of non-
linear solutions of biadjoint scalar field theory in four Euclidean
dimensions. Our motivation stems from the known double copy relationships
between various field theories, summarised here in figure 1. It is not known
how general this scheme is, and finding a genuinely non-perturbative
incarnation would be a big step forward in this regard. We found that the
spectrum of Euclidean solutions is rather rich in dimensions other than four,
mirroring similar results that have been obtained previously in Lorentzian
signature [59, 60, 61]. In precisely four spacetime dimensions, however, there
are no simple power-like solutions, with a consequent absence of dressed
solutions that screen a power-like divergence at the origin. This can be
traced to the fact that the power-like form that is required is a harmonic
function in $d=4$, and thus solves the linearised biadjoint field equation. It
can be identified with the zeroth copy of the Eguchi-Hanson solution, which
has previously been considered from a double copy point of view in refs. [19,
24]. The single and zeroth copies can be formulated in terms of certain
differential operators, obeying conditions that are similar (but not the same)
as the Kerr-Schild conditions underlying the exact classical double copy of
ref. [13].

We have here reinterpreted these differential operators as a special case of a
more general ansatz, that involves the well-known ’t Hooft symbols that arise
in the study of SU(2) gauge theory instantons. We provide a geometric
interpretation of this ansatz, showing that our general (anti-)self dual
operators are associated with area-preserving diffeomorphisms in given
families of null planes. Further use of the ’t Hooft symbols allows us to
construct exact non-abelian single copies of (anti-)self-dual gravity
solutions, living in an SU(2) gauge theory. However, the gauge theory
requirements of (anti-)self-duality restrict the class of solutions to those
that are ultimately linearisable. Nevertheless, the presence of both abelian
and non-abelian single copies realises the same scheme (figure 3) that has
previously arisen in the study of magnetic monopoles [63, 57].

Our results make contact with a recently proposed non-perturbative double copy
in two spacetime dimensions. In particular, the replacement of colour algebras
by area-preserving diffeomorphism algebras in four dimensions is a direct
analogue of similar replacements that relate biadjoint scalar, Zakharov-
Mikhailov and Special Galileon theory in two dimensions. Whilst the presence
of such algebras in the (anti-)self dual sectors of YM theory and gravity have
been known for some time [5], we show for the first time how to interpret the
replacement of colour by kinematic algebras for exact classical solutions,
rather than amplitudes.

Given that the full set of instanton solutions in non-abelian gauge theories
is known [76], the question arises of how general our methods for explicitly
double-copying them are. In more formal language, it remains unclear how the
moduli space of gauge theory instantons maps onto that of gravitational
instantons. Intriguing in this regard is that the explicit examples of
solutions that we have given here seem to be excluded from the moduli space of
SU(2) instanton solutions. Based on similar conclusions in other contexts [52,
88], it seems likely that not all gauge theory instantons can be double-copied
to make gravitational solutions. However, it is correct to say that a full
classification of which solutions can be copied is both useful, and missing.

There are many potential avenues for further work. One might look for non-
spherically symmetric solutions of Euclidean biadjoint theory, and also
interpret the spectrum of existing solutions we have found in various numbers
of dimension, including their relationship with the Lorentzian solutions of
refs. [59, 60, 61]. Examining how general our methods are – in terms of
mapping out the known moduli space of gauge theory instantons – would be
useful, and a first step in this regard would perhaps be to try to make sense
of the more general ansatz of eq. (33), as noted explicitly in ref. [53].
Finally, we note that twistor methods are ubiquitous in the study of
instantons, and have recently arisen in the context of the exact classical
double copy [28, 36, 30, 32, 31, 89, 90]. Some sort of twistorial description
of the (anti-)self-dual double copy is surely possible. We look forward to
reporting on these various topics in the future.

##  Appendix A ’t Hooft symbols and their properties

In this appendix, we introduce and briefly review the properties of ’t Hooft
symbols, which are used in the study of instanton solutions in non-abelian
gauge theories. They are used in two different ways throughout this paper.
First, they arise as infinitesimal generators of Euclidean rotations (the
Euclidean signature equivalents of Lorentz transformations), which constitute
the group SO(4). This group is six-dimensional, comprising three rotations
$\\{J_{i}\\}$ in the $(x_{i},x_{j})$ plane, and three rotations $\\{K_{i}\\}$
in the $(x_{i},x_{4})$ plane, where $i\in\\{1,2,3\\}$, and we have defined
coordinates as in eq. (20). Note that the latter are the analogue of boosts in
Lorentzian signature. One may then form the combinations

$M_{i}=\frac{1}{2}(J_{i}+K_{i}),\quad N_{i}=\frac{1}{2}(J_{i}-K_{i}),$ (91)

which furnish two independent SU(2) subalgebras:

$[M_{i},M_{j}]=-\epsilon_{ijk}M_{k}\quad[N_{i},N_{j}]=-\epsilon_{ijk}N_{k},\quad[M_{i},N_{j}]=0.$
(92)

To interpret these, we may note that a representation of $\\{M_{i},N_{i}\\}$
acting on four-dimensional vectors can be given in terms of the ’t Hooft
symbols

$\displaystyle\eta^{a}_{\mu\nu}$
$\displaystyle=\tensor{\epsilon}{{}^{a}_{\mu\nu
4}}+\delta^{a}_{\mu}\delta_{\nu 4}-\delta^{a}_{\nu}\delta_{\mu 4}$
$\displaystyle\bar{\eta}^{a}_{\mu\nu}$
$\displaystyle=\tensor{\epsilon}{{}^{a}_{\mu\nu
4}}-\delta^{a}_{\mu}\delta_{\nu 4}+\delta^{a}_{\nu}\delta_{\mu 4},$ (93)

satisfying (in matrix notation)

$[\eta^{a},\eta^{b}]=-2\epsilon^{abc}\eta^{c},\quad[\bar{\eta}^{a},\bar{\eta}^{b}]=-2\epsilon^{abc}\bar{\eta}^{c},\quad[\eta^{a},\bar{\eta}^{b}]=0,$
(94)

such that one may set

$M_{i}\rightarrow\frac{1}{2}\bar{\eta}^{i}_{\mu\nu},\quad
N_{i}\rightarrow\frac{1}{2}\eta^{i}_{\mu\nu}.$ (95)

From eq. (93), one may verify that the ’t Hooft symbols satisfy

$\eta^{a}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\eta^{a}_{\rho\sigma},\qquad\bar{\eta}^{a}_{\mu\nu}=-\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\bar{\eta}^{a}_{\rho\sigma},$
(96)

and thus are self-dual and anti-self-dual respectively. Their explicit matrix
representation, from eq. (93) is

$\displaystyle\eta^{1}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&0&0&1\\\ 0&0&1&0\\\ 0&-1&0&0\\\
-1&0&0&0\end{array}\right),\quad\eta^{2}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&-1&0&0\end{array}\right),\quad\eta^{3}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&1&0&0\\\ -1&0&1&0\\\ 0&0&0&1\\\
0&0&-1&0\end{array}\right);$ (109) $\displaystyle\bar{\eta}^{1}_{\mu\nu}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}0&0&0&-1\\\
0&0&1&0\\\ 0&-1&0&0\\\ 1&0&0&0\end{array}\right),\quad\bar{\eta}^{2}_{\mu\nu}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}0&0&-1&0\\\
0&0&0&-1\\\ 1&0&0&0\\\ 0&1&0&0\end{array}\right),\quad\bar{\eta}^{3}_{\mu\nu}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}0&1&0&0\\\
-1&0&1&0\\\ 0&0&0&-1\\\ 0&0&1&0\end{array}\right).$ (122)

Further useful properties of these symbols, that we use throughout the paper,
are

$\displaystyle\bar{\eta}^{a}_{\mu\nu}\,\bar{\eta}^{a}_{\rho\sigma}$
$\displaystyle=\delta_{\mu\rho}\,\delta_{\nu\sigma}-\delta_{\mu\sigma}\,\delta_{\nu\rho}-\epsilon_{\mu\nu\rho\sigma};$
$\displaystyle\bar{\eta}^{a}_{\mu\rho}\,\bar{\eta}^{b}_{\mu\sigma}$
$\displaystyle=\delta^{ab}\,\delta_{\rho\sigma}+\epsilon^{abc}\bar{\eta}^{c}_{\rho\sigma};$
$\displaystyle\epsilon^{abc}\,\bar{\eta}^{b}_{\mu\nu}\,\bar{\eta}^{c}_{\rho\sigma}$
$\displaystyle=\delta_{\mu\rho}\bar{\eta}^{a}_{\nu\sigma}+\delta_{\nu\sigma}\bar{\eta}^{a}_{\mu\rho}-\delta_{\mu\sigma}\bar{\eta}^{a}_{\nu\rho}-\delta_{\nu\rho}\bar{\eta}^{a}_{\mu\sigma}$
$\displaystyle\bar{\eta}^{a}_{\mu\nu}\bar{\eta}^{b}_{\mu\nu}$
$\displaystyle=4\delta^{ab}.$ (123)

These and additional identities may be found in e.g. appendix B of ref. [91].

Thus far, the ’t Hooft symbols have been described as representing particular
rotations in spacetime. However, they have a second use in the study of non-
abelian instantons, namely in embedding vector fields into non-abelian gauge
groups. Considering the case of pure SU(2) Yang-Mills theory, this will have a
matrix-valued gauge field ${\bf A}_{\mu}=A_{\mu}^{a}{\bf T}^{a}$, where
$\\{{\bf T}^{a}\\}$ are the generators of the gauge group. Given a vector
field $V_{\mu}$, a general procedure for turning this into a non-abelian field
with components $A_{\mu}^{a}$ is the ’t Hooft ansatz of eq. (46). In this
equation, the upper index on the ’t Hooft symbol is to be interpreted as an
adjoint index associated with the gauge group, rather than labelling a given
rotation generator. The ’t Hooft ansatz thus represents a map from a selected
SU(2) subalgebra of $\text{SO}(4)$, to the SU(2) colour algebra. Embedding
vector fields into higher gauge groups is also possible, given that the latter
will contain SU(2) subgroups.

## Acknowledgments

We thank Silvia Nagy for many useful discussions, and collaboration on related
topics. We are also grateful to the participants of the Amplitudes meeting at
the Higgs Centre for Theoretical Physics for useful feedback and advice. 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”.
KAW is supported by a studentship from the UK Engineering and Physical
Sciences Research Council (EPSRC).

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