QMUL-PH-23-16

Self-Dual Fields on Self-Dual Backgrounds and the Double Copy

 Graham R. Brown, Joshua Gowdy, Bill Spence

Centre for Theoretical Physics
Department of Physics and Astronomy
Queen Mary University of London
Mile End Road, London E1 4NS, United Kingdom

Abstract

We explore the double copy for self-dual gauge and gravitational fields on
self-dual background spacetimes. We consider backgrounds associated to
solutions of the second Plebanski equation and describe results with different
gauge-fixing conditions. Finally we discuss the kinematic and $w$-algebras and
the double copy, identifying modified Poisson structures and kinematic
structure constants in the presence of the self-dual background. The self-dual
plane wave and Eguchi-Hanson spacetimes are studied as examples and their
respective $w$-algebras derived.

{graham.brown, j.k.gowdy<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Flat background
    1. 2.1 Self-dual Yang-Mills
    2. 2.2 Self-dual gravity
  3. 3 General self-dual YM backgrounds
  4. 4 General self-dual background spacetimes
    1. 4.1 Self-dual Yang-Mills
    2. 4.2 Self-dual gravity
  5. 5 The self-dual plane wave spacetime
    1. 5.1 Self-dual Yang-Mills
    2. 5.2 Self-dual gravity
  6. 6 The Eguchi-Hanson spacetime
    1. 6.1 Self-dual Yang-Mills
    2. 6.2 Self-dual gravity
  7. 7 Conclusions

##  1 Introduction

The study of self-dual gauge and gravitational fields has provided a fertile
source of ideas and results in physics and mathematics. Work some time ago
showed that self-dual theories in a light-cone gauge could be described by
equations for scalar fields [1, 2, 3, 4, 5, 6, 7, 8, 9] corresponding to the
positive helicity sectors of Yang-Mills and gravity. The self-dual sector
allows for a simplified study of many features of the full theories. An area
of recent interest is the investigation of self-dual fields in order to learn
more about the structure of the double copy111The classical double copy for
solutions of the equations of motion was first explored in [10, 11] and
kinematic algebras (see [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26] for recent more general work on the double copy). In [27] it was shown
that self-dual Yang-Mills and gravity have manifest colour-kinematics duality,
and the kinematic algebra was identified as that of area-preserving
diffeomorphisms of the plane. In the context of celestial holography[28, 29,
30, 31, 32, 33, 34] this algebra was shown to manifest itself through the soft
and collinear limit of positive helicity gravitons as (the wedge subalgebra
of) $w_{1+\infty}$ [35]. This link between kinematic algebras and OPEs in
celestial holography [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]
was recently discussed in detail in [50].

A natural generalisation of this is to study self-duality conditions on non-
flat spacetimes. Some progress has been made in understanding how the double
copy can be applied to curved backgrounds [51, 52, 53, 54, 55, 56, 57, 58,
59], with the case of AdS receiving particular attention [60, 61, 62, 63, 64,
65, 66, 67, 68, 69, 70, 71, 72, 73].

In this letter we would like to study self-duality for the case of self-dual
spacetime backgrounds (see [58, 59, 74, 75, 76] for some recent work on this
topic). We will work with the spacetime metrics defined by solutions of the
second Plebanski equation, and study the conditions for the existence of self-
dual Yang-Mills fields, and self-dual metric variations, on these backgrounds.
In the flat space case, dealt with in section 2, there are two formulations of
the self-duality conditions which are related by a simple relabelling of
coordinates. A general self-dual background (Yang-Mills backgrounds are dealt
with in section 3 and gravity backgrounds in section 4) does not have this
symmetry, and we find that these two formulations generalise quite
differently. The first class of solutions in gravity backgrounds, which we
call a ‘matched’ gauge, can be seen as generalising the flat space solution to
curved self-dual backgrounds by linearly perturbing the Plebanski scalar. We
also find a second class of solutions, which we call a ‘flipped’ gauge, which
requires a Kerr-Schild condition on the background and leads to a modified
Poisson structure coming from the Plebanski equation in this gauge. We then
describe aspects of the double copy, and kinematic and $w$-algebras revealed
by these results.

This general formulation is discussed in detail in two examples - the self-
dual plane wave spacetime in section 5 and the Eguchi-Hanson (EH) metric in
section 6. These case studies connect with some of the results developed
recently in twistor space in [75] and [76]. For the self-dual plane wave
background we find that a natural definition of a ‘plane wave’-like solution
to the wave/Plebanski equation in that spacetime leads to a kinematic algebra
with modified structure constants when compared to the flat background.
Nevertheless these structure constants match the flat-space case in the
holomorphic collinear limit of the two ‘plane wave’ solutions and so generate
the standard flat-space $w$-algebra. The soft generators generating the
algebra are however altered and correspond to the expansion of the particular
‘plane wave’ solutions adapted to the self-dual plane wave background. The
double copy in our formulation replaces the Lie algebra commutators with
Poisson brackets and leads to a so-called double bracket in the Plebanski
equation. When acting on ‘plane wave’ solutions, in flipped gauge, we show
this procedure replaces colour structure constants with those from the
kinematic algebra $X_{PW}(k_{1},k_{2})$ and so gives the expected squaring
relation of the single copy, i.e. $X_{PW}(k_{1},k_{2})^{2}$.

In the Eguchi-Hanson background, we express the more complicated solutions of
the wave equation discussed in [76] in spacetime coordinates. We then show
that the Poisson bracket of two of these ‘plane waves’ gives an expression for
the deformed kinematic structure constants $X_{EH}(k_{1},k_{2})$, which we
define in the holomorphic collinear limit. The double bracket of two ‘plane
waves’ is then shown to give the square of this expression, but with
additional terms, demonstrating that in the Eguchi-Hanson background the
kinematic algebra squaring relations are modified by curvature terms. Since
even in the holomorphic collinear limit $X_{EH}(k_{1},k_{2})$ differs from the
flat-space and self-dual plane wave cases, we then expect a completely
different ‘$w$-algebra’ of soft generators. We derive this algebra of soft
generators following the same method as in the previous cases, by expanding
the ‘plane wave’ solutions, giving a spacetime realisation of the results of
[76] coming from twistor space.

##  2 Flat background

We will start by setting notation, briefly recalling the standard results for
self-dual Yang-Mills (YM) and self-dual gravity in a flat background. We will
generally follow the discussion in [27] in this section. The spacetime
coordinates are taken to be $(u,v,X,Y)$, with the metric

$ds^{2}=2du\,dv-2dXdY.$ (2.1)

For real coordinates, this will mean $(2,2)$ signature. The coordinates
$(u,v,X,Y)$ are related to the usual $(t,x,y,z)$ as follows

$\begin{split}u=\frac{t+z}{\sqrt{2}},\quad v=\frac{t-z}{\sqrt{2}},\quad
X=\frac{x+y}{\sqrt{2}},\quad Y=\frac{x-y}{\sqrt{2}}\end{split}$ (2.2)

and in terms of the coordinates $(t,x,y,z)$ the metric signature is
$(+,-,+,-)$.

###  2.1 Self-dual Yang-Mills

A gauge field $A_{\mu}=(A_{u},A_{v},A_{X},A_{Y})$ on flat space with metric
(2.1) is self-dual if its field strength222In our conventions
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+[A_{\mu},A_{\nu}]$.
satisfies

$F_{\mu\nu}=\frac{\sqrt{g}}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}\,.$
(2.3)

where $g$ is the determinant of the metric (2.1). Imposing the gauge-fixing
condition $A_{v}=0$, the self-duality condition above can be satisfied by
setting $A_{X}=0$ and

$\begin{split}A_{u}&=\phi_{X},\\\ A_{Y}&=\phi_{v},\end{split}$ (2.4)

for a (Lie algebra valued) function $\phi(u,v,X,Y)$ satisfying the self-dual
Yang-Mills equation

$\Box\phi-2[\phi_{v},\phi_{X}]=0,$ (2.5)

where the scalar Laplacian is
$\Box=2(\partial_{u}\partial_{v}-\partial_{X}\partial_{Y})$. We will be using
a notation where subscripts on scalar fields such as $\phi$ signify partial
derivatives. For example,
$\phi_{v}=\partial_{v}\phi,\phi_{Xv}=\partial_{X}\partial_{v}\phi$. If we
introduce the Poisson bracket

$\\{f,g\\}=\partial_{v}f\partial_{X}g-\partial_{X}f\partial_{v}g,$ (2.6)

then the self-dual Yang-Mills equation becomes

$\Box\phi-[\\{\phi,\phi\\}]=0,$ (2.7)

where we have used a notation suggestive of colour-kinematics duality, as used
in [73],

$[\\{f,g\\}]:=[f_{v},g_{X}]-[f_{X},g_{v}]\,.$ (2.8)

For the covector $k_{\mu}=(k_{u},k_{v},k_{X},k_{Y})$ and coordinate vector
$x^{\mu}=(u,v,X,Y)$, with $k\cdot x:=k_{\mu}x^{\mu}$, the plane wave
$e^{ik\cdot x}$ satisfies

$\Box\,e^{ik\cdot x}=0$ (2.9)

if $k_{\mu}$ is a null vector. In momentum space, the cubic coupling arising
from the self-dual YM equation (2.7) involves the kinematic structure
constants

$X(k_{1},k_{2})=k_{1X}k_{2v}-k_{1v}k_{2X},$ (2.10)

along with the Lie algebra constants $f^{abc}$. Explicitly, the bracket (2.8)
of two plane waves and Lie algebra generators satisfies[27]

$[\\{T^{a}e^{ik_{1}\cdot x},T^{b}e^{ik_{2}\cdot
x}\\}]=X(k_{1},k_{2})f^{abc}\,T^{c}e^{i(k_{1}+k_{2})\cdot x}.$ (2.11)

There is also an alternative gauge-fixing condition $A_{u}=0$, for which the
self-duality condition can be satisfied by setting $A_{Y}=0$ and

$\begin{split}A_{v}&=\phi_{Y},\\\ A_{X}&=\phi_{u},\end{split}$ (2.12)

for a function $\phi(u,v,X,Y)$ satisfying the self-dual YM equation (2.5) but
with the coordinates $u\leftrightarrow v$ and $Y\leftrightarrow X$ exchanged,
that is

$\Box\phi-2[\phi_{u},\phi_{Y}]=0.$ (2.13)

The flat metric is invariant under this exchange and so results obtained in
this new gauge are trivially related to the previous gauge by a simple
interchange of coordinates. This is not the case when we consider self-dual
backgrounds in the sections below, since these backgrounds have no such
symmetry, and we will describe the two different gauges separately.

###  2.2 Self-dual gravity

Now we recall the analogous construction for self-dual gravity, where the
metric is taken to be the following variation from the flat metric:

$ds^{2}=g_{\mu\nu}(\Psi)dx^{\mu}dx^{\nu}=2dudv-2dXdY+\Psi_{XX}du^{2}+\Psi_{vv}dY^{2}+2\Psi_{Xv}dudY,$
(2.14)

with some function $\Psi(u,v,X,Y)$. Define the expression

${\rm
Pleb}_{0}(\Psi):=2(\Psi_{uv}-\Psi_{XY})-\Psi_{XX}\Psi_{vv}+(\Psi_{vX})^{2},$
(2.15)

where the subscript $0$ indicates the flat background, and define the operator
$\Delta_{0}(\Psi)$ by the variation of this expression as

$\Delta_{0}(\Psi)(\delta\Psi):=\delta({\rm Pleb}_{0}(\Psi)).$ (2.16)

Explicitly

$\Delta_{0}(\Psi)=2(\partial_{uv}-\partial_{XY})-\Psi_{XX}\partial_{vv}+2\Psi_{vX}\partial_{vX}-\Psi_{vv}\partial_{XX}.$
(2.17)

Then the anti-self-dual part of the Weyl tensor is zero except for the
component $C^{-}_{uYuY}$ (and components related to this by the symmetries of
the tensor), and we find

$C^{-}_{uYuY}=-\frac{1}{4}\Delta_{0}(\Psi){\rm Pleb}_{0}(\Psi).$ (2.18)

The non-vanishing components of the Ricci tensor are given by

$R_{ab}=-\frac{1}{2}\,\partial_{\bar{a}}\partial_{\bar{b}}{\rm
Pleb}_{0}(\Psi).$ (2.19)

where $a,b=(u,Y)$ and $\bar{u}=X,\bar{Y}=v$. Thus the metric
$g_{\mu\nu}(\Psi)$ given by (2.14) is Ricci-flat and has self-dual Weyl tensor
if the scalar field $\Psi$ satisfies the gravitational Plebanski equation

${\rm
Pleb}_{0}(\Psi)=2(\Psi_{uv}-\Psi_{XY})-\Psi_{XX}\Psi_{vv}+(\Psi_{vX})^{2}=0.$
(2.20)

Defining the following gravitational bracket $\\{\\{\cdot,\cdot\\}\\}$ using
(2.6)

$\\{\\{f,g\\}\\}=\frac{1}{2}\Big{(}\\{\partial_{v}f,\partial_{X}g\\}-\\{\partial_{X}f,\partial_{v}g\\}\Big{)}\,,$
(2.21)

the Plebanski equation (2.20) can be written

$\Box\Psi-\\{\\{\Psi,\Psi\\}\\}=0,$ (2.22)

revealing the double copy relation [27] $\phi\rightarrow\Psi,$
$[\\{\cdot,\cdot\\}]\rightarrow\\{\\{\cdot,\cdot\\}\\}$ compared with (2.7).
Furthermore, we can consider the gravitational bracket acting on a pair of
plane wave solutions in flat space and we find the following double copy
structure

$\\{\\{e^{ik_{1}\cdot x},e^{ik_{2}\cdot
x}\\}\\}=\frac{1}{2}e^{i(k_{1}+k_{2})\cdot x}X(k_{1},k_{2})^{2}\,,$ (2.23)

where the colour structure constants in (2.11) have been replaced by
additional kinematic ones. Alternatively, from (2.11) one may strip off the
colour structure and isolate the kinematic algebra as the Poisson bracket of
two plane waves

$\\{e^{ik_{1}\cdot x},e^{ik_{2}\cdot x}\\}=e^{i(k_{1}+k_{2})\cdot
x}X(k_{1},k_{2})\,.$ (2.24)

As explained in [50], in the context of celestial holography the appearance of
the ‘left structure constants’ $X(k_{1},k_{2})$ in both YM and gravity implies
the chirality of the operator product expansion in both cases. The second
‘right structure constants’ , $f^{abc}$ in the YM case and the second copy of
$X(k_{1},k_{2})$ in the gravity case, correspond to the structure constants of
the OPEs. The soft expansion of the latter may be explored by noting that the
null vector condition $k^{2}=2(k_{u}k_{v}-k_{X}k_{Y})=0$ implies that we may
set

$\frac{k_{u}}{k_{X}}=\frac{k_{Y}}{k_{v}}=\rho$ (2.25)

for some $\rho$. Thus we can write

$k\cdot x=(\rho Y+v)k_{v}+(\rho u+X)k_{X}\,.$ (2.26)

The soft limit of the momentum $k$ then corresponds to
$(k_{v},k_{x})\rightarrow 0$ at fixed $\rho$. Expanding $e^{ik\cdot x}$ in
this limit gives

$e^{ik\cdot
x}=\sum_{a,b=0}^{\infty}\frac{(ik_{v})^{a}(ik_{X})^{b}}{a!b!}e_{ab},$ (2.27)

where the ‘soft mode generators’ are given by $e_{ab}=(\rho Y+v)^{a}(\rho
u+X)^{b}$. To make contact with the algebras appearing in celestial
holography, we now need to take the collinear limit of the two momentum
$k_{1},$ $k_{2}$ appearing in the algebra (2.24). To do this we use the
holomorphic collinear limit where $(\rho_{1}-\rho_{2})\rightarrow 0$. This
makes $k_{1}$ and $k_{2}$ collinear since

$k_{1}\cdot k_{2}=(\rho_{1}-\rho_{2})X(k_{1},k_{2})\,.$ (2.28)

At leading order in the holomorphic collinear limit (corresponding to the
first term in the OPE expansion in the celestial holography context) we may
set $\rho_{1}=\rho_{2}=\rho$ and substitute the expansion (2.27) into the
kinematic algebra (2.24) to obtain

$\\{e_{a,b},e_{c,d}\\}=(ad-bc)e_{a+c-1,b+d-1}.$ (2.29)

Defining the conventional generators $w^{p}_{m}=\frac{1}{2}e_{p-1+m,p-1-m}$ we
then find the wedge sub-algebra of the $w_{1+\infty}$ algebra

$\\{w_{m}^{p},w_{n}^{q}\\}=\Big{(}m(q-1)-n(p-1)\Big{)}w_{m+n}^{p+q-2}.$ (2.30)

The conditions that $a$ and $b$ are integers greater than or equal to zero
translates to the conditions that $p,m$ are half-integers and satisfy $1-p\leq
m\leq p-1$ and $p\geq 1$ (similarly for $q,n$). This algebra has been studied
in the celestial holography context [43, 35, 77] where it is generated by the
commutation relations of operators inserting soft gravitons.

As in the YM case, there is another gauge-fixing condition related to the
above by interchanging $u\leftrightarrow v$ and $Y\leftrightarrow X$, with
(trivially equivalent) consequent equations. We reiterate that these different
types of gauge will not be as trivially related once we consider self-dual
backgrounds in the next section.

##  3 General self-dual YM backgrounds

Our first exploration of self-dual perturbations of self-dual backgrounds will
begin with YM backgrounds in flat space. To begin with we consider a
background self-dual gauge field $A(\chi)$ in the gauge (2.4)

$A(\chi)=(\chi_{X},0,0,\chi_{v})\,,\quad\Box\chi-[\\{\chi,\chi\\}]=0\,.$ (3.1)

Since the gauge field is linear in the scalar $\chi$ we can write a self-dual
perturbation on this background as

$A(\chi+\psi)=(\chi_{X}+\psi_{X},0,0,\chi_{v}+\psi_{v})=A(\chi)+A(\psi)\,.$
(3.2)

where $A(\psi)$ is the perturbation. The total gauge field $A(\chi+\psi)$ must
then satisfy the self-dual YM equation (2.5)

$\displaystyle\Box(\chi+\psi)-[\\{\chi+\psi,\chi+\psi\\}]$
$\displaystyle=\widetilde{\Box}_{\chi}(\psi)-[\\{\psi,\psi\\}]=0\,,$ (3.3)

where we have used (3.1) and defined a “deformed” scalar Laplacian

$\widetilde{\Box}_{\chi}=\Box-2[\\{\chi,\cdot\\}]\,.$ (3.4)

Equation (3) is the analogue of (2.5) in a self-dual background YM field.

We will see shortly that the discussion above can be double copied in two
ways. First, we can just double copy the background gauge field to obtain
equations of motion for self-dual YM on a self-dual gravitational background
(Sec. 4.1). Second, we can double copy both the background and the
perturbation, to obtain self-dual gravity perturbations on a self-dual
background (Sec. 4.2). In either case the operator (3.4) will double copy to a
more familiar object.

##  4 General self-dual background spacetimes

We now turn to generalising the above results to the case of self-dual
background metrics, rather than a flat background metric. We will consider
self-dual metrics of the form

$ds^{2}=g_{\mu\nu}(\Phi)dx^{\mu}dx^{\nu}=2dudv-2dXdY+\Phi_{XX}du^{2}+\Phi_{vv}\,dY^{2}+2\Phi_{Xv}\,du\,dY$
(4.1)

for a scalar function $\Phi(u,v,X,Y)$ satisfying the Plebanksi equation ${\rm
Pleb}_{0}(\Phi)=0$.

Given a co-vector $k_{\mu}=(k_{u},k_{v},k_{X},k_{Y})$ we have

$\begin{split}k^{\mu}k_{\mu}&=2k_{u}k_{v}-2k_{X}k_{Y}-k_{v}^{2}\Phi_{XX}-k_{X}^{2}\Phi_{vv}+2k_{v}k_{X}\Phi_{Xv}\\\
&=2\hat{k}_{u}\hat{k}_{v}-2\hat{k}_{X}\hat{k}_{Y}\,,\end{split}$ (4.2)

where it will prove useful to define

$\begin{split}&\hat{k}_{u}:=k_{u}-\frac{1}{2}\Big{(}k_{v}\Phi_{XX}-k_{X}\Phi_{Xv}\Big{)},\quad\hat{k}_{v}:=k_{v},\\\
&\hat{k}_{X}:=k_{X},\quad\hat{k}_{Y}:=k_{Y}-\frac{1}{2}\Big{(}k_{v}\Phi_{Xv}-k_{X}\Phi_{vv}\Big{)}\,.\end{split}$
(4.3)

The hatted momenta are those in the tangent space - if we write the vierbein

$\begin{split}e_{u}^{a}&=(1,\frac{1}{2}\Phi_{XX},-\frac{1}{2}\Phi_{Xv},0),\quad
e_{v}^{a}=(0,1,0,0),\\\ e_{X}^{a}&=(0,0,1,0),\quad
e_{Y}^{a}=(0,\frac{1}{2}\Phi_{Xv},-\frac{1}{2}\Phi_{vv},1),\end{split}$ (4.4)

satisfying $e_{\mu}^{a}e_{\nu a}=g_{\mu\nu}(\Phi)$, with $g_{\mu\nu}(\Phi)$
the metric in (4.1), then $\hat{k}_{a}=e_{a}^{\mu}k_{\mu}$.

###  4.1 Self-dual Yang-Mills

With a gauge field $A_{\mu}=(A_{u},A_{v},A_{X},A_{Y})$ on this spacetime we
can choose the gauge-fixing condition $n^{\mu}A_{\mu}=0$, with null vector
$n^{\mu}=(0,1,0,0)$, thus setting

$A_{v}=0.$ (4.5)

Now we require that the field strength $F_{\mu\nu}$ is self-dual, i.e. that
the anti-self-dual components $F^{-}_{\mu\nu}$ vanish. This imposes three
independent conditions. Two of these are satisfied if we set

$\begin{split}A_{X}&=0,\\\ A_{u}&=\phi_{X},\\\ A_{Y}&=\phi_{v},\end{split}$
(4.6)

for a scalar field $\phi(u,v,X,Y)$, and the final self-duality condition
imposes the equation

$\Box_{\Phi}\phi-2[\phi_{v},\phi_{X}]=0,$ (4.7)

where $\Box_{\Phi}$ is the Laplacian in the metric (4.1). This gauge matches
the choice made for the self-dual background metric, wherein the components of
the metric $g_{\mu\nu}$ satisfy $g_{vv}=g_{vX}=g_{XX}=0$; we will call this
gauge the ‘matched gauge’. Equation (4.7) is thus the generalisation of the
self-dual Yang-Mills equation (2.5) to the background (4.1). If we define the
Poisson bracket as in the flat space case

$\\{f,g\\}=\partial_{v}f\partial_{X}g-\partial_{v}g\partial_{X}f\,,$ (4.8)

then the Plebanski equation (4.7) can be written as

$\Box_{\Phi}\phi-[\\{\phi,\phi\\}]=0\,,$ (4.9)

with $[\\{\phi,\phi\\}]$ defined in (2.8). As mentioned earlier, the equation
above can be viewed as the double copy of (3) where we only double copy the
background gauge field. Explicitly, performing the double copy on only the
background $\chi$ using $[\\{\chi,\cdot\\}]\rightarrow\\{\\{\Phi,\cdot\\}\\}$,
the deformed Laplacian $\widetilde{\Box}_{\chi}$ (3.4) becomes

$\widetilde{\Box}_{\chi}=\Box-2[\\{\chi,\cdot\\}]\rightarrow\Box-2\\{\\{\Phi,\cdot\\}\\}=\Box_{\Phi}$
(4.10)

where in the last equality we combined $-2\\{\\{\Phi,\cdot\\}\\}$ with the
flat scalar Laplacian to give us the curved Laplacian.

As we have noted earlier, there is also a different gauge choice which reduces
to the gauge-fixing condition $A_{u}=0$ in the flat space case, and which has
a quite different structure. We can find self-dual Yang-Mills fields in this
gauge which satisfy a generalised Plebanski equation, for background metrics
which are of course self-dual themselves, namely

${\rm Pleb}_{0}(\Phi)=0,$ (4.11)

but also are of the Kerr-Schild form and so satisfy

$\Phi_{Xv}^{2}-\Phi_{XX}\Phi_{vv}=0.$ (4.12)

The above conditions imply that the flat Laplacian acting on $\Phi$ vanishes
i.e. $\Phi_{uv}=\Phi_{XY}$. The self-dual gauge field in this case is given by

$\begin{split}A_{u}&=\frac{1}{2}\Big{(}\phi_{Y}\Phi_{XX}\
-\phi_{u}\Phi_{Xv}\Big{)},\\\
A_{v}&=\hat{k}_{Y}(\Phi)(\phi):=\phi_{Y}-\frac{1}{2}\Big{(}\phi_{v}\Phi_{Xv}\
-\phi_{X}\Phi_{vv}\Big{)},\\\
A_{X}&=\hat{k}_{u}(\Phi)(\phi):=\phi_{u}-\frac{1}{2}\Big{(}\phi_{v}\Phi_{XX}\
-\phi_{X}\Phi_{vX}\Big{)},\\\
A_{Y}&=\frac{1}{2}\Big{(}\phi_{Y}\Phi_{Xv}-\phi_{u}\Phi_{vv}\Big{)},\end{split}$
(4.13)

where the previously defined $\hat{k}_{u}$ and $\hat{k}_{Y}$ are now regarded
as differential operators defined by replacing the unhatted $k$’s in their
expression by the corresponding derivatives. The perturbation field $\phi$
then satisfies a generalised Plebanski equation in the background $\Phi$ given
by

$\Box_{\Phi}\phi-2\Big{[}\hat{k}_{u}(\Phi)(\phi),\hat{k}_{Y}(\Phi)(\phi)\Big{]}=0.$
(4.14)

The gauge field above is not adapted to the background metric in the same
fashion as the previous gauge, instead it features the non-trivial components
of the background metric. Since in the flat space limit it is related to the
previous gauge by the coordinate exchange $u\leftrightarrow v$ and
$Y\leftrightarrow X$ we call it the ‘flipped gauge’.

The commutator term in (4.14) in this ‘flipped gauge’ reveals a different
algebraic structure connected with the fact that one can define a curved space
Poisson bracket for this spacetime [78, 79]. We can define this by considering
the expression $\hat{k}_{1u}\hat{k}_{2Y}-\hat{k}_{2u}\hat{k}_{1Y}$ and as
before replacing the unhatted $k$’s in this expression by coordinate
derivatives with respect to the two functions in the Poisson bracket, i.e.

$\\{f,g\\}_{\Phi}=\hat{k}_{u}(\Phi)(f)\hat{k}_{Y}(\Phi)(g)-\hat{k}_{Y}(\Phi)(f)\hat{k}_{u}(\Phi)(g)\,,$
(4.15)

or

$\begin{split}\\{u,v\\}_{\Phi}&=-\frac{1}{2}\Phi_{Xv},\quad\\{u,X\\}_{\Phi}=\frac{1}{2}\Phi_{vv},\quad\\{u,Y\\}_{\Phi}=1,\\\
\\{v,X\\}_{\Phi}&=\frac{1}{4}\Big{(}\Phi_{Xv}^{2}-\Phi_{XX}\Phi_{vv}\Big{)},\quad\\{v,Y\\}_{\Phi}=-\frac{1}{2}\Phi_{XX},\quad\\{X,Y\\}_{\Phi}=\frac{1}{2}\Phi_{Xv}.\end{split}$
(4.16)

The Jacobi identity for the Poisson bracket $\\{\,,\,\\}_{\Phi}$ is satisfied
since the self-dual background $\Phi$ satisfies the Plebanski equation ${\rm
Pleb}_{0}(\Phi)=0$. Furthermore, since the Kerr-Schild condition
$\Phi_{Xv}^{2}-\Phi_{XX}\Phi_{vv}=0$ is satisfied the bracket $\\{v,X\\}$
vanishes. The symplectic form connected with the Poisson bracket (4.15) is

$\omega=\frac{1}{2}\Phi_{Xv}(du\wedge dv-dX\wedge
dY)-\frac{1}{2}\Phi_{vv}\,dv\wedge dY+\frac{1}{2}\Phi_{XX}du\wedge dX+du\wedge
dY\,.$ (4.17)

We observe that $\omega^{2}=0$ and is closed, $d\omega=0$ (c.f. [79]) when the
Kerr-Schild condition and background Plebanski equation are satisfied.

Using the notation

$[\\{f,g\\}]_{\Phi}:=\Big{[}\hat{k}_{u}(\Phi)(f),\hat{k}_{Y}(\Phi)(g)\Big{]}-\Big{[}\hat{k}_{Y}(\Phi)(f),\hat{k}_{u}(\Phi)(g)\Big{]},$
(4.18)

the condition (4.14) on the field $\phi$ may then be written

$\Box_{\Phi}\phi-[\\{\phi,\phi\\}]_{\Phi}=0.$ (4.19)

###  4.2 Self-dual gravity

We now consider self-dual gravity perturbations on the background metric in
(4.1). That is we simply consider the shifted metric $g_{\mu\nu}(\Phi+\Psi)$
given by (4.1) with $\Phi$ replaced by $\Phi+\Psi$. We will take the metric
$g_{\mu\nu}(\Phi)$ to be the background self-dual spacetime, with ${\rm
Pleb}_{0}(\Phi)=0$. This setup corresponds to the so called ‘matched gauge’
for the gravity perturbation.

We can then define the gravitational Plebanski function in the background
metric $g_{\mu\nu}(\Phi)$ by

${\rm
Pleb}_{\Phi}(\Psi)\coloneqq\Box_{\Phi}\Psi+\Psi_{Xv}^{2}-\Psi_{XX}\Psi_{vv},$
(4.20)

with $\Box_{\Phi}$ the scalar Laplacian in the background metric. Once again
this can be written in terms of the double bracket notation (2.21)

${\rm Pleb}_{\Phi}(\Psi)=\Box_{\Phi}\Psi-\\{\\{\Psi,\Psi\\}\\}\,,$ (4.21)

illustrating the double copy structure compared with eqn. (4.9).
Alternatively, (4.21) can be viewed as the double copy of (3), where we double
copy both the YM background and the perturbation. Now one can check that the
Plebanski equation satisfies the following identity

${\rm Pleb}_{0}(\Phi+\Psi)={\rm Pleb}_{0}(\Phi)+{\rm Pleb}_{\Phi}(\Psi).$
(4.22)

This immediately gives the gravitational Plebanski equation in the background
metric as simply

${\rm Pleb}_{\Phi}(\Psi)=0.$ (4.23)

This follows since the identity (4.22) shows that if $\Phi$ leads to a self-
dual metric then $\Phi+\Psi$ does as well if the Plebanski equation for $\Psi$
in a $\Phi$ metric background (4.23) is satisfied. The above conclusions can
be confirmed explicitly. The relevant non-trivial component of the anti-self-
dual part of the Weyl tensor for the metric $g_{\mu\nu}(\Phi+\Psi)$ is given
by

$\begin{split}C^{-}_{uYuY}(g_{\mu\nu}(\Phi+\Psi))&=-\frac{1}{4}\Delta_{0}(\Phi+\Psi){\rm
Pleb}_{0}(\Phi+\Psi))\\\ &=-\frac{1}{4}\Delta_{0}(\Phi+\Psi)({\rm
Pleb}_{0}(\Phi)+{\rm Pleb}_{\Phi}(\Psi))\\\
&=-\frac{1}{4}\Delta_{0}(\Phi+\Psi)({\rm Pleb}_{\Phi}(\Psi))\\\
&=0\end{split}$ (4.24)

where we have used the self-duality of the background metric, with ${\rm
Pleb}_{0}(\Phi)=0$, and imposed the condition (4.23). It is also immediate
that the variations of the Plebanski function (4.20) are related to variations
of the flat Plebanski function - if we define the variation

$\Delta_{\Phi}(\Psi)(\delta\Psi):=\delta_{\Psi}({\rm Pleb}_{\Phi}(\Psi))\,,$
(4.25)

then as differential operators

$\Delta_{\Phi}(\Psi)=\Delta_{0}(\Phi+\Psi)\,,$ (4.26)

as expected. A similar argument, based on (2.19), shows Ricci-flatness of the
shifted metric.

In summary, we have shown the following commuting triangle of double copy
relations for equations of motion in the matched gauge:

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The diagonal arrow above is just the usual self-dual flat space double copy
applied to the sum of the background and perturbation fields $\chi+\psi$ in
(3.2). The double copy properties of backgrounds and perturbations have been
studied beyond the self dual context in [53].

We can also consider the ‘flipped gauge’ for which the natural double copy of
the bracket in (4.18) replaces the YM commutator with the Poisson brackets
$\\{\,,\,\\}_{\Phi}$ of (4.15)333Other double brackets may be defined by
dropping the $\Phi$ terms inside the brackets in (4.28) and/or using the flat
space Poisson bracket.

$\\{\\{f,g\\}\\}_{\Phi}:=\frac{1}{2}\bigg{(}\Big{\\{}\hat{k}_{u}(\Phi)(f),\hat{k}_{Y}(\Phi)(g)\Big{\\}}_{\Phi}-\Big{\\{}\hat{k}_{Y}(\Phi)(f),\hat{k}_{u}(\Phi)(g)\Big{\\}}_{\Phi}\bigg{)}\,.$
(4.28)

These double brackets have a related curved space Plebanski equation of the
form

$\Box_{\Phi}\Psi-\\{\\{\Psi,\,\Psi\\}\\}_{\Phi}=0,$ (4.29)

which may be regarded as the double copy of (4.19). We will discuss these
brackets further in the examples below. It would be interesting to know if
these equations are related to the conditions required for the self-duality of
the curvature of metrics on self-dual backgrounds. One might also study self-
dual backgrounds satisfying the Kerr-Schild condition
$\Phi_{Xv}^{2}=\Phi_{XX}\Phi_{vv}$ more generally. Whilst we have not found
answers to these questions in the general case, the study of interesting
examples reveals more structure, as we will see in the following.

##  5 The self-dual plane wave spacetime

Plane wave backgrounds have been the object of some interest recently in the
area of amplitudes, kinematic algebras and the double copy (see, for example,
[51, 80, 52, 81, 74, 82, 75] and references therein). Here we will study the
self-dual plane wave metric

$ds^{2}_{PW}=2dudv-2dXdY+2F(v)dY^{2},$ (5.1)

where $F(v)$ is a function related to the wave profile.

This metric is an example of the general form (4.1) considered earlier and is
also Kerr-Schild, we simply set $\Phi=\Phi(v)$ with $\Phi_{vv}=2F(v)$. The
self-dual plane wave metric is Ricci-flat and has self-dual Weyl tensor; the
only non-vanishing components of the self-dual part of the Weyl tensor being
$C_{vYvY}=-2F^{\prime\prime}[v]$ and those related to this by the symmetries
of this tensor.

###  5.1 Self-dual Yang-Mills

A self-dual gauge field in the ‘matched gauge’ on this spacetime is given by

$A_{\mu}=(A_{u},A_{v},A_{X},A_{Y})=(\phi_{X},0,0,\phi_{v}),$ (5.2)

where, in order to solve the self-duality conditions, the scalar field
$\phi(u,v,X,Y)$ must satisfy the plane wave background Plebanski equation

$\Box_{PW}\phi-2[\phi_{v},\phi_{X}]=0,$ (5.3)

with $\Box_{PW}$ the Laplacian in the metric (5.1). Using the Poisson bracket
$\\{f,g\\}=f_{v}g_{X}-f_{X}g_{v}$ which is the same as the flat space case, we
can write (5.3) as

$\Box_{PW}\phi-[\\{\phi,\phi\\}]=0,$ (5.4)

where the double bracket notation (2.8) is defined as usual.

There is also the ‘flipped’ self-dual gauge field solution in this background,
from (4.13) which we will use to elucidate the algebraic structure of self-
dual perturbations on the self-dual plane wave background. We find

$A_{\mu}=\big{(}0,\phi_{Y}+F(v)\phi_{X},\phi_{u},-F(v)\phi_{u}\big{)},$ (5.5)

where the field $\phi$ satisfies

$\Box_{PW}\phi-2[\phi_{u},\phi_{Y}+F(v)\phi_{X}]=0.$ (5.6)

This leads us to the modified Poisson bracket in the plane wave background

$\\{f,g\\}_{PW}=f_{u}\big{(}g_{Y}+F(v)g_{X}\big{)}-\big{(}f_{Y}+F(v)f_{X}\big{)}g_{u},$
(5.7)

and the re-writing of (5.6) as

$\Box_{PW}\phi-[\\{\phi,\phi\\}]_{PW}=0.$ (5.8)

We are now tasked with finding the analogue of plane wave solutions to the
wave equation in flat space, but for solutions to the wave equation in the
background (5.1). Such solutions will then act as generators of our kinematic
Poisson algebra. We begin by constructing a null vector in flat space
$k_{\mu}$ satisfying $k_{u}k_{v}-k_{X}k_{Y}=0$ so that (as before) $k_{u}=\rho
k_{X},k_{Y}=\rho k_{v}$ for some $\rho$. Then for the function $G(v)$ given by
the indefinite integral of $F(v)$, ie $G^{\prime}=F$, we may define the
quantity

$\begin{split}Q_{k}(u,v,X,Y)&\coloneqq(\rho Y+v)k_{v}+(\rho
u+X)k_{X}+\frac{1}{\rho}G(v)k_{X}\\\ &=k\cdot
x+\frac{1}{\rho}G(v)k_{X}\,.\end{split}$ (5.9)

Then one can show that the vector $K_{\mu}=\nabla_{\mu}Q_{k}$ is null,
$K^{\mu}K_{\mu}=0$, divergence free, $\nabla^{\mu}K_{\mu}=0$ (which is just
the wave equation on $Q_{k}$), and geodesic, $K^{\nu}\nabla_{\nu}K_{\mu}=0$,
where $\nabla_{\mu}$ is the covariant derivative in the plane wave metric. One
consequence is that any function of $Q_{k}$ is annihilated by the Laplacian,
in particular

$\Box_{PW}e^{iQ_{k}(u,v,X,Y)}=0.$ (5.10)

Whence the function $e^{iQ_{k}(u,v,X,Y)}$ satisfies the wave equation in the
plane wave background and furthermore reduces to the usual plane wave
$e^{ik\cdot x}$ in the flat space limit. The Poisson bracket of two of these
solutions is

$\begin{split}\\{e^{iQ_{1}},e^{iQ_{2}}\\}&=e^{i(Q_{1}+Q_{2})}(k_{1X}k_{2v}-k_{1v}k_{2X}+k_{1X}k_{2X}\frac{(\rho_{1}-\rho_{2})}{\rho_{1}\rho_{2}}F(v))\\\
&=:e^{i(Q_{1}+Q_{2})}X_{PW}(k_{1},k_{2}),\end{split}$ (5.11)

leading to a modification of the structure constants defining the kinematic
algebra compared to the flat space case. This modification is however sub-
leading in the holomorphic collinear limit so we expect it to not alter the
$w$-algebra, which we confirm in the next section.

This result, and hence also the $w$-algebra in (5.20), also holds if one uses
the flipped gauge Poisson bracket (5.7), although in that case it is more
natural to write the function (5.9) in terms of $k_{u}$ and $k_{Y}$ as follows

$Q_{k}(u,v,X,Y)=(Y+\tilde{\rho}v)k_{Y}+(u+\tilde{\rho}X)k_{u}+\tilde{\rho}^{2}G(v)k_{u}$
(5.12)

where $\tilde{\rho}\coloneqq 1/\rho$. The flipped Poisson bracket of two plane
waves is then

$\begin{split}\\{e^{iQ_{1}},e^{iQ_{2}}\\}_{PW}&=e^{i(Q_{1}+Q_{2})}(k_{1Y}k_{2u}-k_{1u}k_{2Y}+k_{1u}k_{2u}(\tilde{\rho}_{1}-\tilde{\rho}_{2})F(v))\\\
&=e^{i(Q_{1}+Q_{2})}(-\rho_{1}\rho_{2})X_{PW}(k_{1},k_{2})\,.\end{split}$
(5.13)

###  5.2 Self-dual gravity

The gravitational analogue of the discussion above in the ‘matched gauge’ is
based on the metric

$ds^{2}_{PWG}=2dudv-2dXdY+2F(v)dY^{2}+\Psi_{vv}dY^{2}+\Psi_{XX}du^{2}+2\Psi_{vX}dudY.$
(5.14)

This metric has vanishing Ricci tensor and self-dual Weyl tensor if the
following Plebanski equation is satisfied:

$\Box_{PW}\Psi+\Psi_{vX}^{2}-\Psi_{vv}\Psi_{XX}=0,$ (5.15)

where $\Box_{PW}$ is the Laplacian in the self-dual background. Employing the
double bracket notation (2.21) as usual we have

$\Box_{PW}\Psi-\\{\\{\Psi,\Psi\\}\\}=0,$ (5.16)

revealing the double copy structure compared with (5.3).

If we apply the double bracket to two of the ‘plane waves’ (5.9) we find

$\\{\\{e^{iQ_{1}},e^{iQ_{2}}\\}\\}=e^{i(Q_{1}+Q_{2})}\frac{1}{2}\bigg{(}X_{PW}(k_{1},k_{2})^{2}-\frac{ik_{1X}k_{2X}(k_{1X}\rho_{1}+k_{2X}\rho_{2})F^{\prime}(v)}{\rho_{1}\rho_{2}}\bigg{)}$
(5.17)

which is not just the simple square of the relation (5.11). Despite this, we
can still derive a $w-$algebra as follows. Similarly to the flat space case,
we may expand the above solutions to the wave equation in powers of soft
momenta variables $k_{v},k_{X}$ to find

$e^{iQ_{k}(u,v,X,Y)}=\sum_{a,b=0}^{\infty}\frac{(ik_{v})^{a}(ik_{X})^{b}}{a!b!}e_{ab}\,,$
(5.18)

where we have defined $e_{ab}=(\rho Y+v)^{a}(\rho u+X+\frac{1}{\rho}G(v))^{b}$
in the self-dual plane wave background. Defining the modified $w$ generators

$w^{p}_{m}\coloneqq\frac{1}{2}e_{p-1+m,p-1-m}=\frac{1}{2}(\rho
Y+v)^{p-1+m}(\rho u+X+\frac{1}{\rho}G(v))^{p-1-m}\,,$ (5.19)

in analogy with the flat space case. We recover the standard
$w_{1+\infty}$-algebra for these modified generators, working to leading order
in the holomorphic collinear limit

$\\{w_{m}^{p},w_{n}^{q}\\}=\Big{(}m(q-1)-n(p-1)\Big{)}w_{m+n}^{p+q-2}\,.$
(5.20)

We may also consider the ‘flipped gauge’ with its modified Poisson bracket
(5.7) which satisfies a double copy relation analogous to (2.23) acting on two
solutions $e^{iQ_{k}(u,v,X,Y)}$. First we define a modified double bracket

$\\{\\{f,g\\}\\}_{PW}=\frac{1}{2}\Big{(}\\{f_{u},g_{Y}+F(v)g_{X}\\}_{PW}-\\{f_{Y}+F(v)f_{X},g_{u}\\}_{PW}\Big{)}\,,$
(5.21)

then we find the expected double copy of (5.13), that is

$\\{\\{e^{iQ_{1}},e^{iQ_{2}}\\}\\}_{PW}=\frac{1}{2}e^{i(Q_{1}+Q_{2})}(-\rho_{1}\rho_{2}X_{PW}(k_{1},k_{2}))^{2}.$
(5.22)

Interestingly, in contrast to the matched double bracket (5.17) of plane
waves, the above does exhibit a simple squaring relation when compared to the
single bracket (5.13). As mentioned before, we can also define analogous soft
generators $\tilde{w}_{m}^{p}$ in the flipped gauge, now as coefficients of
$k_{u}^{a}$ and $k_{Y}^{b}$. One can then show that these generators also
satisfy the $w_{1+\infty}$ algebra (5.20), but now with the bracket (5.7).

##  6 The Eguchi-Hanson spacetime

We now move on to consider a more complicated example, the Eguchi-Hanson
space-time. This is self-dual, and in the form (4.11) has the scalar function

$\Phi_{EH}=\frac{mv^{2}}{2Y^{2}(uv-XY)},$ (6.1)

with $m$ a constant, satisfying the Plebanski equation in flat space

${\rm Pleb}_{0}(\Phi_{EH})=0.$ (6.2)

The full metric is then

$\begin{split}ds^{2}_{EH}&=g_{\mu\nu}(\Phi_{EH})dx^{\mu}dx^{\nu}\\\
&=2dudv-2dXdY+\frac{mv^{2}}{(uv-XY)^{3}}du^{2}+\frac{mX^{2}}{(uv-
XY)^{3}}dY^{2}-\frac{2mvX}{(uv-XY)^{3}}dudY\end{split}$ (6.3)

and satisfies the Kerr-Schild condition.

We now repeat the methods laid out for the general case and the plane wave
example but now with the function $\Phi_{EH}$. We will encounter a much richer
algebraic structure than was found in the self-dual plane wave background,
reproducing in spacetime some of the results recently described via twistor
space in [76].

###  6.1 Self-dual Yang-Mills

Consider firstly self-dual Yang-Mills in an Eguchi-Hanson background. From the
results earlier, a gauge field $A_{\mu}$ in the ‘matched gauge’ $A_{v}=0$ has
self-dual field strength if in addition $A_{X}=0,A_{u}=\phi_{X}$ and
$A_{Y}=\phi_{v}$, with $\phi$ satisfying the Plebanski equation in the EH
background

$\Box_{EH}\phi-2[\phi_{v},\phi_{X}]=0,$ (6.4)

where $\Box_{EH}$ is the Laplacian in the metric (6.3). In the case at hand,
the EH Plebanski equation can be written in terms of the flat space Poisson
bracket (2.6) as

$\Box_{EH}\phi-[\\{\phi,\phi\\}]=0\,.$ (6.5)

The alternative ‘flipped gauge‘ (4.13) in the Eguchi-Hanson case comes from
the null vector $m^{\mu}=(1,0,0,v/X)$ and gauge-fixing condition
$m^{\mu}A_{\mu}=0$ and sets

$\begin{split}A_{u}&=\frac{mv}{2(uv-
XY)^{3}}\Big{(}v\phi_{Y}+X\phi_{u}\Big{)},\\\
A_{v}&=\hat{k}_{Y}(\Phi_{EH})(\phi)=\phi_{Y}+\frac{mX}{2(uv-
XY)^{3}}\Big{(}X\phi_{X}+v\phi_{v}\Big{)},\\\
A_{X}&=\hat{k}_{u}(\Phi_{EH})(\phi)=\phi_{u}-\frac{mv}{2(uv-
XY)^{3}}\Big{(}X\phi_{X}+v\phi_{v}\Big{)},\\\ A_{Y}&=-\frac{mX}{2(uv-
XY)^{3}}\Big{(}v\phi_{Y}+X\phi_{u}\Big{)}.\end{split}$ (6.6)

This gauge field has self-dual field strength if the scalar field $\phi$
satisfies

$\Box_{EH}\phi-2\Big{[}\hat{k}_{u}(\Phi_{EH})(\phi)\,,\,\hat{k}_{Y}(\Phi_{EH})(\phi)\Big{]}=0.$
(6.7)

Using the notation

$[\\{f,g\\}]_{EH}:=\Big{[}f_{u}-\frac{mv}{2(uv-
XY)^{3}}\big{(}Xf_{X}+vf_{v}\big{)}\,,\,g_{Y}+\frac{mX}{2(uv-
XY)^{3}}\big{(}Xg_{X}+vg_{v}\big{)}\Big{]}+(f\leftrightarrow g),$ (6.8)

equation (6.7) may be written (c.f. (4.9))

$\Box_{EH}\phi-[\\{\phi,\phi\\}]_{EH}=0.$ (6.9)

The deformed Poisson bracket (4.15) in the Eguchi-Hanson metric is then

$\begin{split}\\{f,g\\}_{EH}&=\hat{k}_{u}(\Phi_{EH})(f)\,\hat{k}_{Y}(\Phi_{EH})(g)-\hat{k}_{Y}(\Phi_{EH})(f)\,\hat{k}_{u}(\Phi_{EH})(g)\,,\\\
\end{split}$ (6.10)

and we note that the terms quadratic in $m$ in the above Poisson bracket in
fact drop out.

To find the equivalent of plane wave solutions in the EH background we
introduce a null co-vector $k_{\mu}=(k_{u},k_{v},k_{X},k_{Y})$ whose
components satisfy $k_{u}k_{v}=k_{X}k_{Y}$ so that as before we may write

$\frac{k_{u}}{k_{X}}=\frac{k_{Y}}{k_{v}}=\rho$ (6.11)

for some parameter $\rho$. As was the case for the self-dual plane wave, we
look for solutions to the EH wave equation which are of an exponential form
and return the usual $e^{ik\cdot x}$ plane wave in the flat space limit
$m\rightarrow 0$. Following [76], we define the function

$\begin{split}R_{k}(u,v,X,Y)&\coloneqq(k\cdot
x)^{2}-\frac{m(vk_{v}+Xk_{X})^{2}}{2(uv-XY)^{2}}\\\ &=((\rho Y+v)k_{v}+(\rho
u+X)k_{X})^{2}-\frac{m(vk_{v}+Xk_{X})^{2}}{2(uv-XY)^{2}}.\end{split}$ (6.12)

Then the vector $K_{\mu}=\nabla_{\mu}R_{k}$ is null, $K^{\mu}K_{\mu}=0$,
divergence free, $\nabla^{\mu}K_{\mu}=0$ (which is just the wave equation on
$R_{k}$), and geodesic, $K^{\nu}\nabla_{\nu}K_{\mu}=0$, where $\nabla_{\mu}$
is the covariant derivative in the EH metric. One consequence is that any
function of $R_{k}$ is annihilated by the Laplacian, in particular

$\Box_{\Phi}e^{i\sqrt{R_{k}(u,v,X,Y)}}=0,$ (6.13)

where $e^{i\sqrt{R_{k}(u,v,X,Y)}}$ gives the standard plane wave $e^{ik\cdot
x}$ in the flat space limit $m\rightarrow 0$. Note the qualitative difference
between the Eguchi-Hanson function $R_{k}$, which is quadratic in the null
momenta $k_{\mu}$, versus $Q_{k}$ in the self-dual plane wave background which
is linear in $k_{\mu}$.

We can now perform the Poisson bracket of two of the solutions
$e^{i\sqrt{R_{k}}}$ to the wave equation with momenta $k_{1},k_{2}$, using the
form of $R_{k}$ on the second line of (6.12). We work in the holomorphic
collinear limit $\rho_{1}=\rho_{2}=\rho$ which is all that will be needed to
recover a $w$-algebra. This gives

$\\{e^{i\sqrt{R_{1}}},e^{i\sqrt{R_{2}}}\\}=e^{i(\sqrt{R_{1}}+\sqrt{R_{2}})}X_{EH}(k_{1},k_{2})$
(6.14)

where here

$X_{EH}(k_{1},k_{2})=\frac{1}{\sqrt{R_{1}R_{2}}}\Big{(}k_{1X}k_{2v}-k_{1v}k_{2X}\Big{)}\Big{(}(k_{1}\cdot
x)(k_{2}\cdot x)-\frac{m(vk_{1v}+Xk_{1X})(vk_{2v}+Xk_{2X})}{2(uv-
XY)^{2}}\Big{)},$ (6.15)

and dot products $k\cdot x$ here mean $(\rho Y+v)k_{v}+(\rho u+X)k_{X}$. The
final factor may be compared to the right-hand side of (6.12). Eqn (6.14) may
be viewed as the Eguchi-Hanson background version of the expression in
equation (2.11). We note that the kinematic algebra has modified kinematic
structure ‘constants’ compared to the flat-space case and the modification
survives in the holomorphic collinear limit so we expect the $w$-algebra to
also be modified. As in the plane wave case, the Poisson bracket relation
(6.14) also holds if we use the Eguchi-Hanson flipped bracket (6.10), up to an
overall factor which also appeared in (5.13).

###  6.2 Self-dual gravity

For the case of self-dual gravity, a perturbation of the EH metric in the
matched gauge is given by
$g_{\mu\nu}(\Phi_{EH}+\Psi)=g_{\mu\nu}(\Phi_{EH})+g_{\mu\nu}(\Psi)$ has
vanishing anti-self-dual components of the Weyl tensor except for

$C^{EH-}_{uYuY}=-\frac{1}{4}\Delta_{\Phi_{EH}}(\Psi)({\rm
Pleb}_{\Phi_{EH}}(\Psi)).$ (6.16)

Thus, the perturbed EH metric has self-dual Weyl tensor if the EH Plebanski
equation is satisfied. The EH Plebanski equation for self-dual gravity in this
case is given by

${\rm
Pleb_{\Phi_{EH}}}(\Psi)=\Box_{EH}\Psi-\Psi_{XX}\Psi_{vv}+(\Psi_{vX})^{2}=\Box_{EH}\Psi-\\{\\{\Psi,\Psi\\}\\}=0\,,$
(6.17)

using the double bracket (2.21). Similarly for the Ricci tensor one finds that
its components vanish except for $R^{EH}_{ab}$ with $a,b\in(u,Y)$ and for
these components

$R^{EH}_{ab}=-\frac{1}{2}\partial_{\bar{a}}\partial_{\bar{b}}{\rm
Pleb}_{\Phi_{EH}}(\Psi),$ (6.18)

where $\bar{u}=X,\bar{Y}=v$.

The non-trivial form of the single bracket (6.14) suggests that the double
copy, realised by using a double bracket, may involve more than just the
square of $X(k_{1},k_{2})$. This proves to be the case - the double brackets
of two plane wave solutions $e^{i\sqrt{R_{k}}}$ in the EH background in the
holomorphic collinear limit give a double copy-type formula

$\\{\\{e^{i\sqrt{R_{1}}},e^{i\sqrt{R_{2}}}\\}\\}=\frac{1}{2}e^{i(\sqrt{R_{1}}+\sqrt{R_{2}})}X_{EH}(k_{1},k_{2})^{2}+\dots,$
(6.19)

(c.f (2.23) in the flat space case) where $X_{EH}(k_{1},k_{2})$ is given in
(6.15) and the terms indicated by dots are more complicated expressions which
multiply $(R_{1})^{-1/2},(R_{2})^{-1/2}$ and $(R_{1}R_{2})^{-1/2}$ and are of
order $m,m^{2}$ or $m^{3}$ and hence vanish in the flat space limit
$m\rightarrow 0$. These results suggest that in general the double copy and
related kinematic algebra on curved space backgrounds are not just given by a
simple squaring operation of the relevant curved space term, as seen in the
first term on the right-hand side of eqn. (6.19), but can involve other
curvature corrections.

We now consider the soft expansion of the solution
$e^{i\sqrt{R_{k}(u,v,X,Y)}}$ in powers of the soft momentum variables
$k_{Y},k_{u}$ and once again work in the holomorphic collinear limit where
$\rho_{1}=\rho_{2}=\rho$. We define functions $X_{g},Y_{g},Z_{g}$ which give
the coefficients of $k_{v}^{2},k_{X}^{2}$ and $k_{v}k_{X}$ in the function
$R_{k}$

$\begin{split}X_{g}&=(\rho Y+v)^{2}-\frac{mv^{2}}{2(uv-XY)^{2}}\,,\\\
Y_{g}&=(\rho u+X)^{2}-\frac{mX^{2}}{2(uv-XY)^{2}}\,,\\\ Z_{g}&=(\rho Y+v)(\rho
u+X)-\frac{mXv}{2(uv-XY)^{2}}\,,\\\ \end{split}$ (6.20)

which satisfy

$X_{g}k_{v}^{2}+Y_{g}k_{X}^{2}+2Z_{g}k_{v}k_{X}=(k\cdot
x)^{2}-\frac{m(vk_{v}+Xk_{X})^{2}}{2(uv-XY)^{2}}\,,$ (6.21)

and the discriminant constraint

$X_{g}Y_{g}-Z_{g}^{2}=-\frac{m\rho^{2}}{2}\,.$ (6.22)

The quantities $X_{g},Y_{g},Z_{g}$ correspond to the $X,Y,Z$ of [76]. The
parameter $c^{2}(\lambda)$ in that reference is related to ours by
$c^{2}(\lambda)=\frac{m^{2}\rho^{2}}{2}$.

One can then expand the ‘plane wave’ $e^{i\sqrt{R_{k}}}$ in powers of the
variables $k_{v},k_{X}$444The authors of [76] consider the quantity
$\cos(\sqrt{R_{k}})$ since they also impose the $Z_{2}$ symmetry required by
global considerations. This involves the same basic generators. and the
Poisson bracket of the coefficients in this expansion generates a $w$-type
algebra. Due to the constraint (6.22) one can define a new basis of generators
$V_{2p,2q}:=X_{g}^{p}Y_{g}^{q},\,V_{2p+1,2q+1}:=X_{g}^{p}Y_{g}^{q}Z_{g}$, and
the Poisson brackets of these generates the underlying algebra

$\begin{split}\\{V_{2p,2q},V_{2r,2s}\\}&=4(ps-qr)V_{2p+2r-1,2q+2s-1}\,,\\\
\\{V_{2p,2q},V_{2r+1,2s+1}\\}&=2(p(2s+1)-q(2r+1))V_{2p+2r,2q+2s}+2m\rho^{2}(ps-
qr)V_{2p+2r-2,2q+2s-2}\,,\\\
\\{V_{2p+1,2q+1},V_{2r+1,2s+1}\\}&=((2p+1)(2s+1)-(2q+1)(2r+1))V_{2p+2r+1,2q+2s+1}\\\
&\qquad\qquad\qquad+2m\rho^{2}(ps-qr)V_{2p+2r-1,2q+2s-1}\,.\end{split}$ (6.23)

The full celestial chiral algebra of self-dual gravity on an Eguchi-Hanson
background can then be written in terms of sums of these generators (see
[76]).

We can also consider the double brackets of the flipped gauge (4.28) in the EH
background which are given by

$\\{\\{f,g\\}\\}_{EH}:=\frac{1}{2}\Big{\\{}f_{u}-\frac{mv}{2(uv-
XY)^{3}}\big{(}vf_{v}\ +Xf_{X}\big{)}\,,\,g_{Y}+\frac{mX}{2(uv-
XY)^{3}}\big{(}vg_{v}+Xg_{X}\big{)}\Big{\\}}_{EH}+(f\leftrightarrow g)\,,$
(6.24)

and using these in the holomorphic collinear limit we find a double copy-type
formula like (6.19) with the same leading term, but with different sub-leading
terms. As in the plane wave case, we could expand the solution (6.12) in terms
of $k_{u}^{2}$, $k_{Y}^{2}$ and $k_{u}k_{Y}$ instead to define analogous soft
generators $\tilde{X}_{g}$, $\tilde{Y}_{g}$ and $\tilde{Z}_{g}$. These then
satisfy the same algebra as (6.23) but with the flipped Poisson bracket
(6.10).

##  7 Conclusions

We have studied the self-duality of gauge and gravitational fields on the
self-dual background spacetimes defined by solutions of Plebanski’s second
equation. In light-cone gauges we showed that the conditions for self-duality
could be reduced to second order scalar equations generalising the flat space
equations. We found two classes of general solutions. One, which we called a
‘matched’ gauge, was a direct generalisation of the flat space solutions to
the curved self-dual backgrounds under consideration. The other involves a
Kerr-Schild condition on the background, which we called the ‘flipped’ gauge,
and can be seen as the curved space versions of ‘flipped’ flat space
solutions. We discussed the double copy and kinematic algebra in these two
cases. Finally, we studied two examples in more detail - the self-dual plane
wave spacetime and the Eguchi-Hanson (EH) metric - connecting with some recent
results from [75] and [76], and noting that in the EH background the kinematic
algebra squaring relations are modified by curvature terms.

There are a number of avenues of research which follow from this. It would be
interesting to explore more examples in detail, and investigate perturbative
solutions to the equations where direct solutions prove difficult.
Gravitational analogues of the ‘flipped’ gauge self-dual YM solution, eqn.
(4.13) could be studied further, in general and in particular examples.
Plebanski-type conditions of the generic form
$\Box\phi-\\{\\{\phi,\phi\\}\\}=0$ for the different double brackets given
above would be expected to feature. In radiative spacetimes this should
connect with the very recent analysis of self-dual deformations in [75], which
relates these to twistor sigma models and MHV generating functionals. It would
also be interesting to explore applications to known deformations of the
Plebanski equations such as those involving Moyal brackets (c.f. [12, 76] and
references therein). The application of the formalism used recently for self-
dual YM in [83] could also be explored in self-dual backgrounds.

Acknowledgements: We would like to thank Ricardo Monteiro, Chris White and Sam
Wikeley for helpful comments. This work was supported by the Science and
Technology Facilities Council (STFC) Consolidated Grants ST/P000754/1 “String
theory, gauge theory and duality” and ST/T000686/1 “Amplitudes, strings and
duality”. The work of GRB and JG is supported by STFC quota studentships.

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