compat=1.1.0 bigblob/.style= shape=circle, draw=blue, fill=red
11institutetext: Theoretical Physics, Blackett Laboratory, Imperial College,
London, SW7 2AZ, United Kingdom 22institutetext: CEICO, Institute of Physics
of the Czech Academy of Sciences, Na Slovance 2, 182 21 Praha 8, Czech
Republic 33institutetext: Higgs Centre for Theoretical Physics, School of
Physics and Astronomy, The University of Edinburgh, EH9 3FD, Scotland, United
Kingdom 44institutetext: Department of Physics and Astronomy, Queen Mary
University of London, 327 Mile End Road, London, E1 4NS, United Kingdom
55institutetext: Department of Physics and Center for Theoretical Physics,
National Taiwan University, Taipei 10617, Taiwan

# Mini-twistors and the Cotton Double Copy

Mariana Carrillo González 2 William T. Emond 3 Nathan Moynihan 5 Justinas
Rumbutis 4 and Chris D. White<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The double copy relates quantities in gauge, gravity and related theories. A
well-known procedure for relating exact classical solutions is the Weyl double
copy in four spacetime dimensions, and a three-dimensional analogue of this –
the Cotton double copy – has recently been found for topologically massive
gauge theory and gravity. In this paper, we use twistor methods to provide a
derivation of the position-space Cotton double copy, where this is seen to
arise from combining appropriate data in so-called minitwistor space. Our
methods rely on a massive generalisation of the Penrose transform linking
spacetime fields with cohomology classes in minitwistor space. We identify the
relevant transform from the twistor literature, but also show that it
naturally arises from considering scattering amplitudes in momentum space. We
show that the Cotton double copy in position space is only valid for type N
solutions, but that a simple twistor space double copy is possible for non-
type N solutions, where we use anyons to illustrate our arguments.

††preprint: QMUL-PH-22-38, Imperial/TP/2022/MC/06

##  1 Introduction

In recent years, a correspondence known as the double copy has generated a
great deal of interest. Inspired by previous work in string theory
Kawai:1985xq , its original incarnation stipulates that scattering amplitudes
in gauge theory can be straightforwardly turned into gravity amplitudes
Bern:2008qj ; Bern:2010ue ; Bern:2010yg . To do so, one must substitute the
appropriate coupling constants, as well as replace colour charge information
with additional kinematic factors. This works for a wide variety of gauge and
gravity theories, both with and without supersymmetry. Furthermore, one may
also start with gauge amplitudes and go the other way, replacing kinematic
with colour information. This is called the zeroth copy, and generates
amplitudes in a scalar theory with two distinct types of colour charge, which
has become known as biadjoint scalar theory. Whilst not a physical theory by
itself, its dynamics is at least partially inherited by gauge and gravity
theories. Furthermore, this ladder of theories includes a wide variety of
examples (e.g. both with and without supersymmetry), and is itself part of a
wider web of theories known to exhibit such correspondences: see e.g. refs.
Bern:2019prr ; Borsten:2020bgv ; Adamo:2022dcm ; Bern:2022wqg  for recent
reviews. In the past few years, it has become increasingly recognised that the
double copy applies beyond fixed-order scattering amplitudes, in particular to
all-order perturbative information Oxburgh:2012zr ; Vera:2012ds ;
Johansson:2013nsa ; Saotome:2012vy , exact classical solutions
Monteiro:2014cda ; Luna:2015paa ; Ridgway:2015fdl ; Bahjat-Abbas:2017htu ;
Carrillo-Gonzalez:2017iyj ; CarrilloGonzalez:2019gof ; Bah:2019sda ;
Alkac:2021seh ; Alkac:2022tvc ; Luna:2018dpt ; Sabharwal:2019ngs ;
Alawadhi:2020jrv ; Godazgar:2020zbv ; White:2020sfn ; Chacon:2020fmr ;
Chacon:2021wbr ; Chacon:2021hfe ; Chacon:2021lox ; Dempsey:2022sls ;
Easson:2022zoh ; Chawla:2022ogv ; Han:2022mze ; Armstrong-Williams:2022apo ;
Han:2022ubu  (see also refs. Didenko:2008va ; Didenko:2009td  for related work
in a different context, and Didenko:2022qxq  for a recent overview of how this
is related), perturbative classical solutions Elor:2020nqe ;
Farnsworth:2021wvs ; Anastasiou:2014qba ; LopesCardoso:2018xes ;
Anastasiou:2018rdx ; Luna:2020adi ; Borsten:2020xbt ; Borsten:2020zgj ;
Goldberger:2017frp ; Goldberger:2017vcg ; Goldberger:2017ogt ;
Goldberger:2019xef ; Goldberger:2016iau ; Prabhu:2020avf ; Luna:2016hge ;
Luna:2017dtq ; Cheung:2016prv ; Cheung:2021zvb ; Cheung:2022vnd ;
Cheung:2022mix , and potential non-perturbative aspects Monteiro:2011pc ;
Borsten:2021hua ; Alawadhi:2019urr ; Banerjee:2019saj ; Huang:2019cja ;
Berman:2018hwd ; Alfonsi:2020lub ; Alawadhi:2021uie ; White:2016jzc ;
DeSmet:2017rve ; Bahjat-Abbas:2018vgo ; Cheung:2022mix ; Borsten:2022vtg .

The double copy offers new calculational tools for General Relativity and
related theories, and indeed has already been used to generate new results
needed for gravitational wave experiments Bern:2019nnu ; Bern:2019crd .
However, it also offers new conceptual insights not only about gravity, but
also the very foundations of field theory itself. It is then important to find
explanations of where the double copy comes from, particularly in those cases
in which exact statements can be made. In four spacetime dimensions, a well-
known exact classical double copy is the Weyl double copy of ref. Luna:2018dpt
. Written using the spinorial formalism of field theory (see e.g. refs.
Penrose:1985bww ; Penrose:1986ca ; Stewart:1990uf  for reviews), it relates
spacetime fields in biadjoint, gauge and gravity theories directly in position
space111More specifically, it relates the Abelian versions of these theories.
For explorations where both Abelian and non-Abelian solutions map to the same
gravitational object see Bahjat-Abbas:2020cyb ; Oxburgh:2012zr .. This is at
odds with the original double copy for scattering amplitudes Bern:2008qj ;
Bern:2010ue ; Bern:2010yg , which is naturally formulated in momentum space.
Nevertheless, it is argued to be exact for certain gravity solutions, namely
those vacuum solutions which are of type D in the well-known Petrov
classification. Exact type N cases are also known Godazgar:2020zbv , as well
as other Petrov types at linearised level only White:2020sfn ; Chacon:2021wbr
.

In order to explain the above results, refs. White:2020sfn ; Chacon:2021wbr
found a derivation of the Weyl double copy using twistor theory Penrose:1967wn
; Penrose:1968me ; Penrose:1972ia , a decades-old set of mathematical ideas
linking field theory, complex analysis and algebraic geometry (see e.g. refs.
Penrose:1986ca ; Huggett:1986fs ; Adamo:2017qyl  for pedagogical reviews). In
a nutshell, twistor theory maps points in spacetime non-locally to an abstract
twistor space, such that certain quantities in the former show up as
mathematically convenient data in the latter. In particular, an integral
formula known as the Penrose transform Penrose:1969ae  relates certain
“functions” in twistor space to spacetime solutions of the massless free field
equations, where a given spin of the spacetime field translates to a given
homogeneity of the twistor function under rescalings of its argument. By
multiplying together functions of different homogeneity in an appropriate
manner, refs. White:2020sfn ; Chacon:2021wbr  showed that one could derive the
Weyl double copy in position space. Furthermore, the twistor approach provided
a geometric interpretation of certain aspects of the Weyl double copy that had
previously been obscure, such as the inverse zeroth copy that takes one from
biadjoint scalar to gauge theory. Applications of the twistor approach include
generalising the Weyl double copy away from type D or N solutions (albeit at
linearised level only), and also showing that physical properties such as
multipoles can be straightforwardly mapped between different theories
Chacon:2021hfe .

Despite the above successes, the twistor double copy is not without its
conceptual problems. Chief among these is the fact that the quantities
entering the Penrose transform are not, strictly speaking, functions. Rather,
they may be redefined by equivalence transformations, which do not change the
result of the Penrose transform integral. Mathematically speaking, such
quantities are representatives of cohomology classes, as identified in ref.
Eastwood:1981jy . This then poses a puzzle, in that the non-linear product of
twistor functions needed to obtain the Weyl double copy in position space is
clearly inconsistent with the ability to first perform equivalence
transformations. It thus seems that special representatives of each cohomology
class must be chosen in each theory in order to make the double copy manifest,
and it is not obvious a priori how to achieve this. A number of papers have
subsequently addressed this point. First was ref. Adamo:2021dfg , which
considered radiative spacetimes, and showed that data at past or future or
null infinity could be used to pick out special cohomology representatives in
twistor space. These were defined in terms of so-called Dolbeault cohomology
groups, as distinct from the C̆ech cohomology groups that enter the original
formulation of the Penrose transform. Reference Chacon:2021lox  also
considered the Dolbeault language, and argued that one can use known methods
in Euclidean signature Woodhouse:1985id  to pick out special twistor
representatives for each spacetime field, so that a product structure is
manifest in twistor space. It is not clear how the procedures of refs.
Adamo:2021dfg ; Chacon:2021lox  are related, if at all.

More recently, ref. Guevara:2021yud  considered the relationship between
scattering amplitudes, twistor space, and classical solutions. It is known
that certain classical solutions can be obtained as inverse Fourier transforms
of momentum-space amplitudes. Reference Guevara:2021yud  craftily split this
inverse Fourier transform into two steps, where the first maps the amplitudes
into quantities in twistor space. The second step then corresponds to the
Penrose transform from twistor to position space, and it is therefore the case
that scattering amplitudes themselves can be used to pick out cohomology
representatives for certain classical fields in twistor space. As has been
made clear elsewhere Luna:2022dxo , the representatives picked out by the
relevant amplitudes in gauge, gravity or scalar theory are precisely those
entering the original twistor double copy of refs. White:2020sfn ;
Chacon:2021wbr . Not only does this fix the cohomological ambiguities in the
twistorial approach, it also establishes a very strong link between the double
copies in momentum, twistor and position space.

Equally important as explaining the origin of known double copies is to
continue to generalise this correspondence to novel theories or situations. In
this spirit, refs. Gonzalez:2022otg ; Emond:2022uaf  recently proposed a new
exact classical double copy for topologically massive gauge theory and
gravity, called the Cotton double copy. Like the Weyl double copy in four
dimensions, it uses the spinorial formalism of field theory, and expresses a
precise relationship between scalar, gauge and gravity fields in position
space. Indeed, this relationship is analogous to its four-dimensional
counterpart, although appears to hold for a more restricted class of solutions
than in the Weyl case. That is, both refs. Gonzalez:2022otg ; Emond:2022uaf
only found Cotton double copy examples in position space of Petrov type N,
rather than the more general type D, and this fact demands a further
explanation. It is also natural to ask whether there is a twistorial
justification for the Cotton double copy, that mirrors its four-dimensional
counterpart. Constructing such an argument should itself settle the issue of
how general the Cotton double copy is, and this paper will show that this is
indeed possible. We will use the language of minitwistors in three spacetime
dimensions, and the presence of a topological mass means that we will have to
consider an alternative to the usual Penrose transform. Just such a transform
has been provided before in the mathematical literature tsai_1996 . It is
formulated by considering the most general possible cohomology classes in
minitwistor space. This involves introducing an extra parameter in twistor
space relative to the conventional four-dimensional case, whose presence
corresponds to the presence of the topological mass in spacetime. Armed with
this minitwistor transform, we will show explicitly that appropriately
combining particular minitwistor representatives allows us to derive the
position-space Cotton double copy.

Our minitwistor derivation of the Cotton double copy will suffer from similar
conceptual issues to its four-dimensional counterpart. Namely, the form of the
double copy in minitwistor space involves products of “functions”, which
should properly be interpreted as representatives of cohomology classes, with
an appropriate procedure for picking them. However, the ideas of ref.
Guevara:2021yud  will once again come to the rescue: we will show that they
can be generalised to the three-dimensional case, such that the minitwistor
double copy follows as a consequence of the known double copy for scattering
amplitudes in topologically massive theories. We will explicitly consider
amplitudes corresponding to point-like sources emitting gauge bosons, which
correspond to type D classical solutions222Strictly speaking we only consider
linearized solutions, not exact solutions which are usually described within
the Petrov classification.. These will allow us to independently validate the
form of the massive Penrose transform, where particular cohomology
representatives are necessarily picked out. Interestingly, we will find that a
simple twistor-space double copy occurs even in the type D case. However, a
simple position-space double copy is restricted to type N only, as a direct
consequence of the form of the massive Penrose transform. Our results provide
a firm foundation for the Cotton double copy, whilst also providing an
interesting counterpoint for the four-dimensional twistor double copy. This in
turn suggests the use of twistor methods more widely in the study of
(non-)exact classical double copies, including in higher dimensions where
applicable.

The structure of our paper is as follows. In section 2, we review relevant
properties of topologically massive theories in three dimensions, including
the Cotton double copy. We also introduce the concept of minitwistors, and
their associated Penrose transform, following ref. tsai_1996 . In section 3,
we provide a twistorial derivation of the Cotton double copy, emphasising the
similarities and differences with the four-dimensional twistor double copy of
refs. White:2020sfn ; Chacon:2021wbr . In section 4, we show how the ideas of
ref. Guevara:2021yud  can be adapted to three dimensions, and use anyon
solutions to illustrate our general arguments. Finally, we discuss our results
and conclude in section 5.

##  2 Review of necessary concepts

In this section, we review salient material for the rest of the paper, both in
order to set up our notation and conventions, and also to make the
presentation relatively self-contained. We begin by introducing the spinorial
formalism for field theories.

###  2.1 Spinors in (2+1) dimensions

Our first encounter with relativistic field theories in four spacetime
dimensions typically involves the use of 4-vectors and tensors. As is well-
known, however, it is possible to recast all relevant field equations into an
alternative language, namely that of 2-component spinors Witten:1959zza ;
Penrose:1960eq  (see e.g. refs. Penrose:1985bww ; Penrose:1986ca ;
Stewart:1990uf  for pedagogical reviews). Similar ideas occur in (2+1)
dimensions Milson:2012ry ; castillo20033 ; doi:10.1063/1.1592611 , which we
now briefly review.

We will be concerned with (dual) spinors $\lambda^{A}$ ($\lambda_{A}$), whose
indices $A\in\\{0,1\\}$ may be raised and lowered using the two-dimensional
Levi-Civita symbol:

$\lambda_{B}=\epsilon_{BA}\lambda^{A},\quad\lambda^{B}=\epsilon^{BA}\lambda_{A},\quad\epsilon_{AB}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}=-\epsilon^{AB}.$ (1)

Note that, in contrast to the well-known four-dimensional case, only one type
of spinor index $A$ occurs. This is because the Lorentz group in (2+1)
dimensions is covered by a single SL(2,C) group. In four dimensions, on the
other hand, the Lorentz group is covered by two distinct SL(2,C) groups,
leading to the presence of spinors $\lambda_{A}$ and conjugate spinors
$\pi_{A^{\prime}}$, where the prime is used to differentiate which SL(2,C)
group acts on which index.

We can convert (2+1)-dimensional tensors into spinors (and vice versa) using
the Infeld-van der Waerden symbols, whose explicit form depends upon the
chosen basis in spinor space. It is in fact possible to choose them to be
real, so that we will adopt the $SL(2,\mathbb{R})$ representation

$\sigma^{\mu}_{AB}=\left\\{\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\right\\}.$ (2)

A given tensor index is thus converted into a pair of spinor indices, and one
may also verify the following useful identities:

$\eta^{\mu\nu}=\frac{1}{2}\sigma^{\mu}_{AB}\sigma^{\nu
AB},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \sigma^{\mu}_{AB}\sigma_{\mu
GD}=-(\epsilon_{AG}\epsilon_{BD}+\epsilon_{AD}\epsilon_{BG}).$ (3)

As an example, a single 4-vector with real components has the spinorial
translation

$\displaystyle
p_{AB}=p_{\mu}\sigma^{\mu}_{AB}=\begin{pmatrix}p_{0}+p_{2}&p_{1}\\\
p_{1}&p_{0}-p_{2}\end{pmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\det(p_{\mu}\sigma^{\mu}_{AB})=-p_{\mu}p^{\mu},$ (4)

where the matrix thus obtained is referred to as a bispinor. As may be
verified by direct computation, one can always decompose a bispinor in (2+1)
dimensions into the outer product of two complex spinors

$\displaystyle p_{AB}=\lambda_{(A}\bar{\lambda}_{B)},$ (5)

where the latter are given by

$\displaystyle\lambda_{A}=\frac{1}{\sqrt{p_{2}-p_{0}}}\begin{pmatrix}p_{2}-p_{0}\\\
p_{1}-im\end{pmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\bar{\lambda}_{A}=\frac{1}{\sqrt{p_{2}-p_{0}}}\begin{pmatrix}p_{2}-p_{0}\\\
p_{1}+im\end{pmatrix}.$ (6)

In the scattering amplitudes literature, it is common to introduce a Dirac
notation for (dual) spinors:

$\ket{\lambda}\equiv\lambda_{A},\quad\bra{\lambda}\equiv\lambda^{A}.$ (7)

Then we can define spinor helicity variables in the usual way as

$\displaystyle\braket{\lambda^{i}\lambda^{j}}\equiv\epsilon^{AB}\lambda^{i}_{B}\lambda^{j}_{A},$
(8)

where we note in particular the identities

$\displaystyle\braket{\bar{\lambda}\lambda}=2im,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \braket{\lambda}{\gamma^{\mu}}{\bar{\lambda}}=-2p^{\mu}.$
(9)

###  2.2 Topologically massive theories and their double copy

Having reviewed the language of two-spinors in (2+1) dimensions, let us now
introduce the theories that we will encounter throughout the paper. First up
is topologically massive Yang-Mills theory, which is described by the action

$S_{TMYM}=\int
d^{3}x\Bigg{(}-\frac{1}{4}F^{a\mu\nu}F_{a\mu\nu}+\epsilon_{\mu\nu\rho}\frac{m}{12}\left(6A^{a\mu}\partial^{\nu}A^{\rho}_{a}+g\sqrt{2}f_{abc}A^{a\mu}A^{b\nu}A^{c\rho}\right)\Bigg{)},$
(10)

implying the equation of motion

$D_{\mu}F^{\mu\nu}+\frac{m}{2}\varepsilon^{\nu\rho\gamma}F_{\rho\gamma}=0\ .$
(11)

Physically, this describes a gauge boson with mass $m$ and a single helicity
$h=\frac{m}{|m|}$. The mass term in the action is only possible in three
spacetime dimensions, due to the presence of the three-dimensional Levi-Civita
tensor. Furthermore, unlike conventional mass terms in arbitrary spacetime
dimension, one may show that the mass term introduced here is manifestly
gauge-invariant. The mass is topological in the sense that it is independent
of the local metric.

We will also be concerned with topologically massive gravity, whose action is

$S_{TMG}=\frac{1}{\kappa^{2}}\int
d^{3}x\sqrt{-g}\left(-R-\frac{1}{2m}\epsilon^{\mu\nu\rho}\left(\Gamma^{\alpha}_{\mu\sigma}\partial_{\nu}\Gamma^{\sigma}_{\alpha\rho}+\frac{2}{3}\Gamma^{\alpha}_{\mu\sigma}\Gamma^{\sigma}_{\nu\beta}\Gamma^{\beta}_{\rho\alpha}\right)\right)\
,$ (12)

and leads to the equation of motion

$G_{\mu\nu}+\frac{1}{m}C_{\mu\nu}=0\ ,$ (13)

where $C_{\mu\nu}$ is a symmetric tensor known as the Cotton tensor:

$\displaystyle
C_{\mu\nu}=\epsilon_{\mu\rho\sigma}D^{\rho}\left(R^{\sigma}_{\nu}-\frac{1}{4}\delta^{\sigma}_{\nu}R\right)\
.$ (14)

We can think of this as a $(2+1)$-dimensional analogue of the Weyl tensor in
four (or higher) dimensions, where the latter is what the Riemann curvature
reduces to in the case of vacuum solutions of the Einstein equations. Like the
Weyl tensor, the Cotton tensor vanishes for conformally flat spacetimes. As
for the Yang-Mills case discussed above, the second term in the action of eq.
(12) is impossible to write down in four spacetime dimensions. It is a
correction to the pure Einstein-Hilbert action, and generates a mass for the
graviton that is invariant under diffeomorphisms.

As discussed in the introduction, it is by now very well-known that Yang-Mills
theory and gravity (plus their generalisations) are related by the double
copy, which applies to both scattering amplitudes and classical solutions. It
was recently also conjectured that the topologically massive gauge and gravity
theories considered here are related by a similar double copy Moynihan:2020ejh
, evidence for which has been presented in a number of non-trivial scenarios
Burger:2021wss ; Gonzalez:2021bes ; Moynihan:2021rwh ; Emond:2021lfy ;
Hang:2021oso ; Gonzalez:2021ztm . Important for this paper is the Cotton
double copy Gonzalez:2022otg ; Emond:2022uaf 333Note that the Cotton tensor
has appeared in a different off-shell double copy construction in Ben-
Shahar:2021zww ., which directly relates classical solutions of the above
equations of motion, expressed in the spinorial formalism. As reviewed in
refs. Gonzalez:2022otg ; Emond:2022uaf , the spinorial translation of the free
field equation for Abelian topologically massive gauge theory is

${\partial^{G}}_{A}\Phi_{GB}=m\Phi_{AB},$ (15)

where $\partial_{AB}$ is the spinorial translation of the partial derivative
operator $\partial_{\mu}$, and we have defined444In order to verify eq. (15),
one must also use the relation
$\epsilon^{\alpha\beta}\left(\partial_{\beta}^{\gamma}\varphi_{\gamma\alpha}-\partial_{\alpha}^{\gamma}\varphi_{\gamma\beta}\right)=0$,
which follows from the well-known Bianchi identity for the field strength
tensor.

$\displaystyle\Phi_{AB}=\frac{1}{2}\sigma^{\mu}_{AB}\epsilon_{\mu\nu\rho}F^{\nu\rho}.$
(16)

Similarly, the free-field equations of topologically massive gravity take the
form

${\partial^{E}}_{A}C_{EBGD}=mC_{ABGD},$ (17)

where

$\displaystyle C_{ABGD}=C_{\mu\nu}\sigma^{\mu}_{AB}\sigma^{\nu}_{GD}.$ (18)

It is instructive to contrast these equations with their natural counterparts
in four-dimensional gauge and gravity theory, namely the massless free field
equation

${\partial}^{A_{1}A^{\prime}}\Phi_{A_{1}A_{2}\ldots A_{2n}}=0.$ (19)

Here $\Phi_{A_{1}\ldots A_{2n}}$ is a multi-index symmetric spinor
corresponding to a single polarisation state of the field555The other
polarisation state obeys a similar equation to eq. (19), but with (un)primed
indices interchanged., and the derivative operator is now the appropriate
four-dimensional spinorial translation of the partial derivative operator in
spacetime. We have written eq. (19) for a general spin-$n$, from which we may
note that there are $2n$ spinor indices for a spin-$n$ field. Apart from the
slight difference in derivatives, we see that eqs. (15, 17) differ from eq.
(19) due to the presence of the mass term on the right-hand side.

Solutions of eq. (19) of different spin can be related to each other by the
Weyl double copy Luna:2018dpt , which has been shown to work for certain
algebraically special spacetimes.

$\Phi_{ABCD}=\frac{\Phi_{(AB}\Phi_{CD)}}{\Phi}.$ (20)

Here $\Phi$ is a field satisfying the massless Klein-Gordon equation in
spacetime, and to clarify where this applies, we may note that a consequence
of the limited range of spinor indices is that an arbitrary multi-index
symmetric spinor can be decomposed in terms of single-index principal spinors,
such that we have

$\Phi_{ABCD}=\alpha_{(A}\beta_{B}\gamma_{C}\delta_{D)}.$ (21)

So-called Petrov type D solutions are those for which there are two distinct
principal spinors, each of double multiplicity. Type N solutions have a single
principal spinor of multiplicity four. These are the two cases of
algebraically special solutions for which the Weyl double copy is known to be
exact.

Motivated by the Weyl double copy, refs. Gonzalez:2022otg ; Emond:2022uaf
considered whether a similar relation can be written for topologically massive
Yang-Mills and gravity theories. Indeed it can, provided one replaces the Weyl
tensor with the Cotton tensor, and instead considers $\Phi$ to be a solution
of the massive Klein-Gordon equation:

$C_{ABGD}=\frac{\Phi_{(AB}\Phi_{GD)}}{\Phi}\ .$ (22)

This is the Cotton double copy formula alluded to above, and is known to apply
at least for type N solutions. In order to see whether it is in fact more
general than this (as is the Weyl double copy), it is fruitful to seek a more
underlying explanation of where the Cotton double copy comes from. In the case
of the Weyl double copy, refs. White:2020sfn ; Chacon:2021wbr  provided a
derivation of the position-space formula using the techniques of twistor
theory. This suggests that similar techniques could prove useful in deriving
the Cotton double copy. Before we can do this, however, we must first
familiarise ourselves with twistor techniques in (2+1) dimensions. This is the
subject of the following section.

###  2.3 Minitwistor theory

In this section, we give a brief introduction to the subject of twistors in
three-dimensional space. Pedagogical reviews of four-dimensional twistor
theory can be found in e.g. refs. Penrose:1986ca ; Huggett:1986fs ;
Adamo:2017qyl . The subject of three-dimensional twistor theory is less well-
known, and thus our aim is to collect a number of useful results from the
literature in one place Ward:1989vja ; tsai_1996 . The relevant concepts are
similar to the case of the four-dimensional twistor double copy defined in
refs. White:2020sfn ; Chacon:2021wbr : given flat spacetime, one may construct
an abstract twistor space, such that points in spacetime are mapped non-
locally to the latter and vice versa. Solutions of the massless free field
equation of eq. (19) can be obtained as a certain contour integral in twistor
space, which is known as the Penrose transform. In order to apply these same
ideas to (2+1) dimensions, we must first define the relevant twistor space,
and then arrive at the necessary Penrose transform, which must somehow take
into account the presence of the mass in topological gauge theory or gravity.
Let us take each of these topics in turn.

####  2.3.1 Minitwistor geometry

Let us first consider complexified Minkowski space ${\cal M}=\mathbb{C}^{3}$,
with line element

$ds^{2}=-dt^{2}+dx^{2}+dy^{2},\quad t,x,y\in\mathbb{C}.$ (23)

Using the Infeld-van-der-Waerden symbols of eq. (2), a point $x\in{\cal M}$
has a spinorial translation as a symmetric $2\times 2$ matrix:

$x^{AB}=\left(\begin{array}[]{cc}-t-y&-x\\\ -x&-t+y\end{array}\right).$ (24)

We may then define minitwistor space $\mathbb{MT}$ as the two-dimensional set
of null planes in ${\cal M}$. Any such plane is defined by a null three-
dimensional normal vector $n^{\mu}$, such that

$n_{\mu}x^{\mu}=u,\quad n^{2}=0,\quad u\in\mathbb{C}.$ (25)

Nullity of $n_{\mu}$, and the fact that it is defined only up to arbitrary
scalings, implies that its spinorial translation factorises as follows:

$n_{AB}\equiv n_{\mu}\sigma^{\mu}_{AB}=\lambda_{A}\lambda_{B},$ (26)

where $\lambda_{A}$ is itself only defined up to an overall complex scale:

$n^{\mu}\rightarrow\alpha^{2}n^{\mu}\quad\Rightarrow\quad\lambda_{A}\rightarrow\alpha\lambda_{A},\quad\alpha\in\mathbb{C}.$
(27)

The first condition in eq. (25) then implies

$u=x^{AB}\lambda_{A}\lambda_{B}\ ,$ (28)

such that a given point in minitwistor space (representing a particular null
plane) is described by coordinates

$Z^{\alpha}=(u,\lambda_{A})\ ,$ (29)

satisfying the incidence relation of eq. (28). Equations (25, 27) then imply
that the coordinates appearing in eq. (29) are defined only up to the scalings

$\left(u,\lambda_{A}\right)\sim\left(r^{2}u,r\lambda_{A}\right)\ ,$ (30)

for $r\in\mathbb{C}^{*}$666Here and in what follows, $\mathbb{C}^{*}$ denotes
the set of non-zero complex numbers.. A spinor $\lambda_{A}$ has two complex
components, which reduces to one if an overall complex scale is removed. Thus,
$\lambda_{A}$ defines a point on the Riemann sphere $\mathbb{CP}^{1}$. In
general, we need two coordinate patches to cover the sphere, which we may
choose as

$\displaystyle U_{0}:\quad\lambda_{A}=(1,z)$ (31) $\displaystyle
U_{1}:\quad\lambda_{A}=(w,1).$ (32)

Given that $\lambda_{A}$ is defined only up to rescalings, we may identify
$z=w^{-1}$ on the overlap $U_{0}\cap U_{1}$. The single complex coordinate $u$
is defined at each point on the Riemann sphere, and thus we may think of
$\mathbb{MT}$ as a fibre bundle, with $\mathbb{CP}^{1}$ as the base space.
Formally speaking, it is the holomorphic tangent bundle $T\mathbb{CP}^{1}$ of
the Riemann sphere. In particular, a given point in minitwistor space assigns
a holomorphic tangent vector to each point on the Riemann sphere associated
with $\lambda_{A}$. To see this, note that a general holomorphic vector field
on $\mathbb{CP}^{1}$ may be written as

$f(z)\partial_{z}=\left(\sum_{n=0}^{\infty}a_{n}z^{n}\right)\partial_{z}=-\left(\sum_{n=0}^{\infty}a_{n}w^{2-n}\right)\partial_{w},$
(33)

where we have used $\partial_{z}=-w^{2}\partial_{w}$ in the second equality.
Holomorphicity in both coordinate choices (in particular the absence of poles)
then implies $a_{n}=0$ for $n>2$, such that a general holomorphic vector field
on $U_{0}\cap U_{1}$ may be written as

$(a_{0}+a_{1}z+a_{2}z^{2})\partial_{z}.$ (34)

The incidence relation of eq. (28) can be expanded in $U_{0}$ using eq. (24)
as

$u=(-t+y)z^{2}-2xz-(y+t),$ (35)

From eq. (34), this defines a holomorphic vector field

$u(z)\partial_{z}$

on $\mathbb{CP}^{1}$, as required777To fully identify $\mathbb{MT}$ with the
holomorphic tangent bundle of $\mathbb{CP}^{1}$, one must show that all
possible holomorphic vector fields can be obtained from the incidence
relation. We will not formally prove this here.. As described above and as is
hopefully clear from eq. (35), a fixed spacetime point $x$ and picks out a
specific vector at each point on the Riemann sphere associated with $\lambda$.
Thus, the coordinate $u$ defines a section

$u:\quad\mathbb{CP}^{1}\rightarrow T\mathbb{CP}^{1}$ (36)

of the holomorphic tangent bundle of the Riemann sphere. Thus, a point in
spacetime corresponds to a section of the holomorphic tangent bundle of a
Riemann sphere $X$. As explained in ref. Ward:1989vja , we may visualise this
in twistor space as shown in figure 1. The horizontal axis shows the Riemann
sphere $X$ represented as a complex line. The vertical axis then denotes the
value of $u$ at each point on $X$, such that we may visualise this as a curve.

Figure 1: Points $p$ and $q$ in spacetime can be visualised as curves in
minitwistor space, where the coordinate $u$ is defined at each point on the
Riemann spheres $X_{p}$ and $X_{q}$ corresponding to $p$ and $q$. Curves
associated with different spacetime points can intersect in at least two
places.

So far we have worked in complexified Minkowski space. If we wish to use real
coordinates in Lorentzian signature, then the matrix given in eq. (24) will be
real; that is, we choose to impose the reality condition
$(u,\lambda_{A})\sim(\bar{u},\bar{\lambda}_{A})$. Now, the incidence relation
of eq. (28) defines a real null-plane or a timelike line, depending on whether
$u$ and $\lambda_{0}/\lambda_{1}$ are (non-)real.

####  2.3.2 Dimensional reduction

In what follows, we will have to obtain the relevant Penrose transform that
converts data in minitwistor space into solutions of the topological gauge and
gravity equations in spacetime. In doing so, we will find it useful to rely on
some alternative ways of thinking about $\mathbb{MT}$. The first of these
relies on the more well-known concept of twistor space for four-dimensional
complexified Minkowski spacetime ${\cal M}_{4}=\mathbb{C}^{4}$. As remarked
already above, we must consider two types of spinor in four spacetime
dimensions, given that the Lorentz group is covered by two distinct
SL(2,$\mathbb{C})$ groups, which we may refer to as SL(2,$\mathbb{C})_{L}$ and
SL(2,$\mathbb{C})_{R}$. These act on (conjugate) spinors, which carry
(un-)primed indices respectively. To convert a given tensor or 4-vector into
the spinorial language, we can again contract with the relevant Infeld-van-
der-Waerden symbols, for which a suitable choice is

$\sigma^{\mu}_{AA^{\prime}}=\left\\{\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}\right\\}.$ (37)

Comparing with eq. (2), we see that the Infeld-van-der-Waerden symbols carry a
spinor index associated with each of the groups SL$(2,\mathbb{C})_{L,R}$, and
we have also appended the ”missing” Pauli matrix to be the third component
$\sigma^{3}_{AA^{\prime}}$, where now $\mu\in\\{0,1,2,3\\}$. With these
conventions, one has

$x_{AA^{\prime}}=\left(\begin{array}[]{cc}-t+y&x-iz\\\
x+iz&-t-y\end{array}\right),\quad
x^{AA^{\prime}}=\left(\begin{array}[]{cc}-t-y&-x-iz\\\
-x+iz&y-t\end{array}\right).$ (38)

The twistor space corresponding to four-dimensional Minkowski spacetime can
also be identified with the space of certain null 2-planes. It turns out that
these can be parameterised by twistor coordinates888For want of a better
notation, we will use (non-)calligraphic symbols to refer to (three-) four-
dimensional twistors respectively.

${\cal Z}^{\alpha}=(\mu^{A^{\prime}},\lambda_{A}),$ (39)

subject to the incidence relation

$\mu^{A^{\prime}}=x^{AA^{\prime}}\lambda_{A}.$ (40)

Twistors satisfying this relation are defined only up to an overall rescaling

${\cal Z}^{\alpha}\sim r{\cal
Z}^{\alpha},\quad\Rightarrow\quad(\mu^{A^{\prime}},\lambda_{A})\sim(r\mu^{A^{\prime}},r\lambda_{A}),$
(41)

and are said to live in projective twistor space $\mathbb{PT}$. We may now
obtain minitwistor space of $\mathbb{C}^{3}$ by dimensionally reducing four-
dimensional twistor space. After dimensional reduction, one may choose to work
with real coordinates in a specific signature by picking a reality condition
as in the previous section. To see how this works note that, in our above
examples, we can isolate the three-dimensional coordinates from
$x^{AA^{\prime}}$ by introducing a constant vector with spinorial translation

$T_{BA^{\prime}}=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right),$ (42)

and forming the combination

$x_{AA^{\prime}}{T_{B}}^{A^{\prime}}=x_{AA^{\prime}}\epsilon^{A^{\prime}C^{\prime}}T_{BC^{\prime}}=x_{AB}+\left(\begin{array}[]{cc}0&-iz\\\
iz&0\end{array}\right),$ (43)

as follows from explicit computation. Then we may write

$x^{AB}={x^{(A}}_{A^{\prime}}T^{B)A^{\prime}},$ (44)

whose geometric interpretation is that we are ignoring translations along the
direction of $T^{\mu}$ in spacetime. Furthermore, SL(2,$\mathbb{C}$)
covariance of eq. (44) means that we can pick any direction in spacetime in
order to perform the dimensional reduction. Removing the symmetrising brackets
in eq. (44) will generate an antisymmetric contribution on the left-hand side,
which for a two-dimensional matrix must be proportional to the Levi-Civita
symbol. Thus, on general grounds we may write Hitchin:1982gh ; Jones_1985 ;
Adamo:2017xaf

${x^{A}}_{A^{\prime}}T^{BA^{\prime}}=x^{AB}+b\epsilon^{AB}\ ,$ (45)

where taking the determinant of both sides can be used to infer the relation

$b=i\sqrt{x_{\text{4d}}^{2}-x_{\text{3d}}^{2}}.$ (46)

Using eq. (44), we can recover the minitwistor incidence relation in eq. (28)
from the 4d incidence relation of eq. (40). To do so, one may define

$u\equiv\mu_{A^{\prime}}T^{AA^{\prime}}\lambda_{A}\ .$ (47)

This can be shown to be invariant under the equivalent of translations along
the vector field $T^{\mu}$ in twistor space Adamo:2017xaf . Also, the four-
dimensional twistor scaling property of eq. (41) implies $u\rightarrow
r^{2}u$, as required in eq. (30). Combining eqs. (40, 45, 47), we find

$u=(x^{AB}\lambda_{A}+b\lambda^{B})\lambda_{B},$ (48)

where the second term in the brackets corresponds to the effect of a
translation in the $T^{\mu}$ direction. This vanishes after contracting with
the spinor $\lambda_{B}$ outside the brackets, but suggests that the most
general equivalence relation in the three-dimensional twistor coordinates is

$(x^{AB}\lambda_{A}\lambda_{B},\lambda_{A})\sim(r^{2}(x^{AB}\lambda_{A}+b\lambda^{B})\lambda_{B},r\lambda_{A}).$
(49)

Indeed, this motivates another way to define coordinates on minitwistor space,
as

$Z^{A}=(\mu^{A},\lambda_{A})\ ,\quad\mu^{A}=x^{AB}\lambda_{B}\ ,$ (50)

which arise more naturally from the dimensional reduction point of view. In
this case, the coordinates are defined up to the following equivalence:

$(\mu^{A},\lambda_{A})\sim(r(\mu^{A}+b\lambda^{A}),r\lambda_{A})\ .$ (51)

In discussing the Penrose transform we need for topological gauge and gravity
theory, it is useful to discuss yet another, and rather more formal, way to
describe minitwistor space. Recall that $\mathbb{MT}$ consists of the set of
null two-planes in $\mathbb{C}^{3}$. Reference tsai_1996  considers Euclidean
signature for the latter, and points out that the complex Euclidean group of
transformations that define the space (rotations plus translations) is covered
by the group ESL(2,$\mathbb{C}$). If we then quotient this group by the group
of isometries of null planes, we will obtain the group that acts on
minitwistor space. The elements of the former group can be written as

$\mathrm{ESL}(2,\mathbb{C})=\\{(A,B)|A\in\mathrm{SL}(2,\mathbb{C}),B\in\\{\text{2$\times$2
complex trace-free matrices}\\}\\},$

subject to the composition law

$(A,B)\circ(A^{\prime},B^{\prime})=(AA^{\prime},AB^{\prime}A^{-1}+B).$ (52)

We thus see that $A$ is an SL(2,$\mathbb{C}$) transformation associated with
rotations, and $B$ is associated with translations. The relevant closed
subgroup that we must quotient out is given by tsai_1996

$Q=\left\\{\left(R=\left(\begin{array}[]{cc}r&t\\\
0&r^{-1}\end{array}\right),S=\left(\begin{array}[]{cc}b&a\\\
0&-b\end{array}\right)\right)\ ;\ a,b,r,t\in\mathbb{C},r\neq 0\right\\}\ ,$
(53)

and to show that this is correct, we can simply apply the equivalence relation
$g\sim gq$ (with $g\in\mathrm{ESL}(2,\mathbb{C})$, $q\in Q$), and show that
this corresponds to the equivalence relation of eq. (51) when acting on
minitwistor coordinates. One may then parametrise

$g=(A_{AB}=(\lambda_{A}\,\chi_{B})\ ,\ X=x^{AB})\ ,$ (54)

where $\chi_{B}$ is an arbitrary spinor. From eq. (52), $Q$ then acts on $g$
as

$gq=(AR\ ,\ X+ASA^{-1})\ .$ (55)

Expanding appropriately and using eq. (53), one obtains the correspondence
tsai_1996

$g\rightarrow gq:\quad\lambda_{A}\rightarrow r\lambda_{A}\ ,\quad
x^{AB}\lambda_{A}\rightarrow r(x^{AB}\lambda_{A}+b\lambda_{A})\ .$ (56)

We can then interpret $\lambda_{A}$ and $\mu^{B}=x^{AB}\lambda_{A}$ as
coordinates on minitwistor space, subject to equivalence relations which
indeed match those found by dimensional reduction in eq. (51).

####  2.3.3 The Penrose transform for massless free fields

Having introduced minitwistor space in various ways, our next task is to find
the appropriate Penrose transform that expresses spacetime fields as contour
integrals in minitwistor space. To this end, let us first recall the Penrose
transform in four-dimensional twistor theory Penrose:1969ae . Solutions of the
four-dimensional massless free field equation of eq. (19) can be expressed via
the following contour integral:

$\phi_{A_{1}A_{2}\ldots A_{2n}}=\frac{1}{2\pi i}\oint_{\Gamma}\braket{\lambda
d\lambda}\lambda_{A_{1}}\lambda_{A_{2}}\ldots\lambda_{A_{2n}}\rho_{x}[f({\cal
Z}^{\alpha})],\quad\braket{\lambda d\lambda}=\lambda_{E}d\lambda^{E}.$ (57)

Here the contour $\Gamma$ lies on the Riemann sphere $X$ associated with a
given spacetime point $x$, and $\lambda_{A}$ is the spinor that forms half of
the twistor components of eq. (39). There is then a holomorphic function
$f({\cal Z}^{\alpha})$ of twistor coordinates, where the symbol $\rho_{x}$
denotes restriction to the Riemann sphere $X$, such that all twistors obey the
incidence relation of eq. (40). The contour $\Gamma$ must be such that it
separates any poles of $f({\cal Z}^{\alpha})$, and for there to be a non-zero
answer, there must be at least two poles, one on either side of $\Gamma$. We
may take the latter to correspond to the equator of the Riemann sphere without
loss of generality.

As is well-known Eastwood:1981jy , the “functions” $f({\cal Z}^{\alpha})$ are
not unique, but can be subjected to equivalence transformations that do not
affect the result of the contour integral:

$f({\cal Z}^{\alpha})\sim f({\cal Z}^{\alpha})+f_{N}({\cal
Z}^{\alpha})+f_{S}({\cal Z}^{\alpha}),$ (58)

where $f_{N}({\cal Z}^{\alpha})$ ($f_{S}({\cal Z}^{\alpha})$) has poles only
in the northern (southern) hemisphere of $X$ respectively. Substituting eq.
(58) into eq. (57), we may evaluate the additional contributions by simply
closing the contour in the opposite side to where the poles are, giving rise
to a zero result, as required. In more formal mathematical terms, we say that
the quantities $f({\cal Z}^{\alpha})$ are representatives of (C̆ech)
cohomology classes, and a fuller exposition of this in the present context can
be found in ref. Chacon:2021lox . There is, however, a further restriction on
$f({\cal Z}^{\alpha})$, arising from the fact that twistors obeying the
incidence relation constitute points in projective twistor space
$\mathbb{PT}$, and thus are only defined up to the rescalings of eq. (41). If
the integrand and measure in eq. (57) are to be invariant under ${\cal
Z}^{\alpha}\rightarrow r{\cal Z}^{\alpha}$, then it must be the case that
$f({\cal Z})$ is a homogeneous function of degree $(-2n-2)$, for a spin-$n$
spacetime field:

$f(r{\cal Z}^{\alpha})=r^{-2n-2}f({\cal Z}^{\alpha}).$ (59)

Denoting holomorphic functions on minitwistor space of homogeneity $N$ by
${\cal O}(N)$, we would then say in formal language that the Penrose transform
is an isomorphism between spacetime fields of spin $n$, and elements of the
C̆ech cohomology group999Strictly speaking, the Penrose transform of eq. (57)
relates to sheaf cohomology groups, where ${\cal O}(N)$ then denotes the sheaf
of holomorphic functions of homogeneity $N$. However, C̆ech cohomology
provides a suitable approximation to sheaf cohomology for all practical
purposes here. See e.g. ref. Huggett:1986fs  for a pedagogical discussion of
this point. $H^{1}(\mathbb{PT},{\cal O}(-2n-2))$.

It is straightforward to write down a Penrose transform for solutions of the
three-dimensional massless free field equation. The latter is given by

$\partial^{A_{1}B}\phi_{A_{1}\cdots A_{2n}}(x)=0\ ,$ (60)

and the Penrose transform itself by Ward:1989vja

$\phi_{A_{1}\ldots A_{2n}}=\frac{1}{2\pi i}\oint_{\Gamma}\braket{\lambda
d\lambda}\lambda_{A_{1}}\ldots\lambda_{A_{2n}}\rho_{x}[f(Z^{\alpha})].$ (61)

This is directly analogous to eq. (57), where $\Gamma$ is again a contour on
the Riemann sphere $X$ associated with the spacetime point $x$, and
$f(Z^{\alpha})$ a holomorphic function of the minitwistor coordinates of eq.
(29).

Above, we have presented the Penrose transforms for massless free fields in
the language of C̆ech cohomology, in which they take the form of contour
integrals in (mini-)twistor space. An alternative approach exists, in which
twistor integrands are interpreted using differential forms, and the freedom
to redefine twistor integrands is interpreted using Dolbeault cohomology (see
e.g. ref. Chacon:2021lox  for a recent detailed comparison of the two
approaches). We will remain with the C̆ech approach in what follows, which
will turn out to be more convenient for our purposes. However, an obvious
deficiency of eq. (61) is that it only works for massless free fields, and
thus is inapplicable to topologically massive gauge and gravity theory. That
it is possible to generalise the three-dimensional Penrose transform to
incorporate (topological) mass is possible on very general grounds, which we
review in the following section.

####  2.3.4 The Penrose transform for massive free fields

A three-dimensional Penrose transform for massive fields has been presented in
the twistor literature by Tsai tsai_1996 , whose starting point is to consider
the above construction of minitwistor space $\mathbb{MT}$ as the quotient
space $G/H$, where $G$=ESL(2,$\mathbb{C})$ is the universal cover of the
complex Euclidean group that generates all points in $\mathbb{C}^{3}$ and
$H=Q$ is the group of isometries of null planes. To construct a Penrose
transform, we must consider defining functions on $\mathbb{MT}$. Functions at
a point form a vector space $V$ under addition, and we must therefore consider
a mathematical structure consisting of a copy of $V$ associated with all
points in $G/H$, such that one has a type of fibre bundle. In fact, this
structure is known as a homogeneous vector bundle, where the word
“homogeneous” refers to the fact that the base space is itself a quotient
space. There is a canonical way to construct homogeneous vector bundles (see
e.g. ref. Ward:1990vs ), as follows. We can first think of constructing a
conventional vector bundle on $G$ by placing a copy of $V$ (the “fibre”) above
each point of $G$, and then letting a representation $\rho(g)$ of each element
$g\in G$ act on vectors $v\in V$. Then, given $g\in G$ and $v\in V$, we may
identify points in this vector bundle by asserting the equivalence

$(g,v)\sim(gh,\rho(h^{-1})v),h\in H.$ (62)

The first component of this relation tells us that $gh$ is to be identified
with $g$, which is simply the action of quotienting out $G$ by the closed
subgroup $H$. The second component then implements the fact that the vectors
in the fibres above $g$ and $gh$ must be identified101010To see why $h^{-1}$
rather than $h$ occurs in the second relation in eq. (62), one may demand that
the effect of acting on both components with a group element $h_{1}h_{2}\in H$
is the same as acting first with $h_{2}$, then with $h_{1}$. The group $H$
acts towards the left on elements of $G$, but towards the right on elements of
$V$, such that the inverse ensures that the ordering of successive
transformations acting on $V$ is correct..

Returning to the specific case of $G$=ESL(2,$\mathbb{C})$ and $H=Q$, we will
be considering scalar functions, which must then be acted on by one-
dimensional representations of $Q$. From eqs. (52, 53), one finds

$(R_{1},S_{1})\circ(R_{2},S_{2})=\left(\left(\begin{array}[]{cc}r_{1}r_{2}&\ast\\\
0&(r_{1}r_{2})^{-1}\end{array}\right),\left(\begin{array}[]{cc}b_{1}+b_{2}&\ast\\\
0&-b_{1}-b_{2}\end{array}\right)\right),$ (63)

and thus one sees that the $r$ parameters are multiplicative, whereas the $b$
parameters are additive. Physically, this is related to the fact that the
former are associated with rotations, and the latter with translations. A one-
dimensional representation that embodies these properties can be easily
written down as

$\rho((R,S))=r^{-N}e^{Mb},\quad N\in\mathbb{Z},\quad M\in\mathbb{C}.$ (64)

Indeed, this represents an infinite family of one-dimensional representations,
one for each combination $(N,M)$, and our reason for restricting $N$ to be an
integer will be clarified below. We have seen that, acting on minitwistor
coordinates $(\mu^{A},\lambda_{A})$, transformations $h\in Q$ act according to
eq. (51). Thus, functions acted on by the representation of eq. (64) must
satisfy

$\check{f}_{M}(r(\mu^{A}+b\lambda^{A}),r\lambda_{A})=r^{N}e^{-Mb}\check{f}_{M}(\mu^{A},\lambda_{A})\
.$ (65)

To make sense of this condition, we can consider the case $M=0$, for which
there will no $b$ parameter in eq. (64). Then eq. (65) reduces to

$\check{f}_{0}(r\mu^{A},r\lambda_{A})=r^{N}\check{f}_{0}(\mu^{A},\lambda_{A}),$

which is merely the requirement that the function $f_{0}$ be homogeneous with
degree $N$. Such functions enter the massless Penrose transforms in three and
four dimensions, where $N=-2n-2$ is related to the spin $n$ of the spacetime
field. For non-zero $M$, the parameter $b$ in eq.(65) corresponds to the
additional freedom to redefine minitwistor coordinates, as in eq. (51). Given
this more general class of functions, we can construct a generalised Penrose
transform. First, by setting $r=1$ in Eq. (65), differentiating this equation
with respect to $b$, and evaluating it at $b=0$ we get that $\check{f}_{M}$
must obey:

$\lambda_{A}\frac{\partial\check{f}_{M}(Z)}{\partial\mu_{A}}=-M\check{f}_{M}(Z),$
(66)

A solution of this equation can be written as:

$\check{f}_{M}(Z)=e^{-M\frac{\braket{a\mu}}{\braket{a\lambda}}}\check{g}(\lambda_{A},\braket{\mu\lambda}).$
(67)

where $a_{A}$ is an arbitrary spinor. To be compatible with (65) $\check{g}$
must be homogeneous of degree $N$, which we will choose as above to be
$N=-2n-2$ (it is for this reason that we have chosen $N\in\mathbb{Z}$ in eq.
(64)). We may then consider the contour integral

$\phi_{A_{1}\ldots A_{2n}}=\frac{1}{2\pi i}\oint_{\Gamma}\braket{\lambda
d\lambda}\lambda_{A_{1}}\ldots\lambda_{A_{2n}}\rho_{x}[\check{f}_{M}(Z^{\alpha})],$
(68)

which consists of simply replacing the “function” in the massless three-
dimensional Penrose transform of eq. (61) with one of the more general types
of function defined above. Acting on both sides with a derivative operator, we
find

$\displaystyle{\nabla^{B}}_{A_{1}}\phi_{BA_{2}\ldots A_{2n}}(x)$
$\displaystyle=-\frac{1}{2\pi i}\oint_{\mathcal{C}}\langle\lambda
d\lambda\rangle
2\lambda_{B}\lambda_{A_{2}}\ldots\lambda_{A_{2n}}\lambda^{(B}\rho_{x}\left[\frac{\partial\check{f}_{m}(Z)}{\partial\mu^{A_{1})}}\right]$
$\displaystyle=\frac{M}{2\pi i}\oint_{\mathcal{C}}\langle\lambda
d\lambda\rangle\lambda_{A_{1}}\lambda_{A_{2}}\ldots\lambda_{A_{2n}}\rho_{x}[\check{f}_{M}(Z)]$
$\displaystyle=M\phi_{A_{1}\ldots A_{2n}}(x)\ ,$ (69)

where in the first line we have used

${\nabla^{B}}_{A_{1}}f={(\sigma^{\mu})^{B}}_{A_{1}}\sigma_{\mu}^{DC}\lambda_{C}\frac{\partial
f}{\partial\mu^{D}},$

together with eq. (3). This tells us two things: (i) $\phi_{A_{1}\ldots
A_{2n}}$ constructed in this manner satisfies the massive free field equation
of topologically massive gauge theory and gravity (eqs. (15, 17)); (ii) the
parameter $M$, which arose above in classifying the most general type of
functions that can be defined on minitwistor space, can be identified with the
topological mass $m$. We will thus make this identification in what follows.

Note that a more general solution to eq. (65) can be constructed as a sum over
different arbitrary spinors $a_{A}$ and homogeneous functions $g_{i}$:

$\check{f}_{m}(Z)=\sum_{a,i}e^{-m\frac{\braket{a\mu}}{\braket{a\lambda}}}\check{g}_{i}(\lambda_{\alpha},\braket{\mu\lambda})\
.$ (70)

That this satisfies the general massive free field equation can be verified by
explicit calculation, but anyway follows from linearity of the field equation.

As in the massless case, the twistor functions $\check{f}_{m}(Z)$ entering the
Penrose transform of eq. (68) are not actually functions but defined only up
to equivalence transformations, in this case of the form

$\check{f}_{m}(Z)\sim\check{f}_{m}(Z)+e^{-m\frac{\braket{a_{N}\mu}}{\braket{a_{N}\lambda}}}\check{g}_{N}^{\prime}(\lambda_{\alpha},\braket{\mu\lambda})+e^{-m\frac{\braket{a_{S}\mu}}{\braket{a_{S}\lambda}}}\check{g}^{\prime}_{S}(\lambda_{\alpha},\braket{\mu\lambda})\
,$ (71)

where $\check{g}^{\prime}_{N,S}$ has poles only in the northern and southern
hemispheres of $X$ respectively, and $\braket{a_{N(S)}\lambda}\neq 0$ in the
southern (northern) hemisphere. In formal mathematical language, we would say
that the quantity $\check{f}_{M}(Z^{\alpha})$ is a representative of a
cohomology class, which is itself a member of the C̆ech cohomology group
$H^{1}(\mathbb{MT},{\cal O}(N,M))$, where ${\cal O}(N,M)$ denotes the
functions acted on by eq. (64). Note that our above arguments merely show that
a cohomology class in mini-twistor space gives a solution of the field
equations in spacetime. However, ref. tsai_1996  proves (see Proposition 2.10)
that all possible solutions can be obtained in this way, so that the
relationship between fields and cohomology classes is formally an isomorphism.

In this section, we have reviewed a particular generalised Penrose transform
on minitwistor space, whose “functions” correspond to cohomology classes
labelled by two parameters $(N,M)$. The first of these represents the
homogeneity of the cohomology representative, and the second turns out to
correspond to the mass in the field equations of topologically massive gauge
theory and gravity in three spacetime dimensions. The latter are special cases
of eq. (69), but we will also need the spinless case, for which one may verify
that the spacetime field $\phi$ satisfies the massive Klein-Gordon equation

$(\partial^{2}+m^{2})\phi=0.$ (72)

In four spacetime dimensions, the Penrose transform may be used to show that
the position-space double copy for massless free fields can be derived from a
certain procedure in twistor space White:2020sfn ; Chacon:2021wbr . Now that
we have identified the appropriate Penrose transform for minitwistor space, we
can perform a similar analysis for topologically massive gauge and gravity
theory in three spacetime dimensions.

##  3 A minitwistor derivation of the Cotton double copy

We have seen that the generalised Penrose transform of eq. (68) identifies
solutions of the massive field equation of eq. (69) with holomorphic twistor
“functions” (cohomology class representatives) having the form of eq. (67).
The homogeneity of a spin-$n$ field was found above to be $N=-2n-2$, and thus
a scalar, gauge and gravity field will be associated with twistor
representatives of homogeneity $-2$, $-4$ and $-6$ respectively. Let us
introduce a scalar representative $\check{f}_{-2}(Z^{\alpha})$, and a pair of
gauge theory representatives $\check{f}^{(i)}_{-4}(Z^{\alpha})$
($i\in\\{1,2\\}$):

$\check{f}_{-2}(Z^{\alpha})=e^{-m\frac{\braket{a\mu}}{\braket{a\lambda}}}\check{g}_{-2}(u,\lambda_{A}),\quad\check{f}_{-4}(Z^{\alpha})=e^{-m\frac{\braket{a\mu}}{\braket{a\lambda}}}\check{g}_{-4}(u,\lambda_{A})\
,$ (73)

where $\check{g}_{N}$ is an homogeneous “function” of degree $N$. It follows
that one may construct a gravitational twistor representative by forming the
product

$\check{f}_{-6}(Z^{\alpha})=\frac{\check{f}^{(1)}_{-4}(Z^{\alpha})\check{f}^{(2)}_{-4}(Z^{\alpha})}{\check{f}(Z^{\alpha})}=e^{-m\frac{\braket{a\mu}}{\braket{a\lambda}}}\check{g}_{-6}(u,\lambda_{A}),$
(74)

with

$\check{g}_{-6}(u,\lambda_{A})=\frac{\check{g}^{(1)}_{-4}(u,\lambda_{A})\check{g}^{(2)}_{-4}(u,\lambda_{A})}{\check{g}_{-2}(u,\lambda_{A})}.$
(75)

In the four-dimensional case of refs. White:2020sfn ; Chacon:2021wbr , it was
argued that choosing certain representatives allows to derive the Weyl double
copy in position space. We may do something very similar here, in order to
obtain the Cotton double copy. To see how this works, we may first recall that
for twistor representatives with at most two poles on the Riemann sphere $X$,
a $p-$fold pole gives rise to a $(2n-p+1)-$fold principal spinor of the
corresponding spacetime field, at the point $x$ (see e.g. ref. Penrose:1986ca
, and ref. Chacon:2021wbr  for a more recent discussion of this point). We may
then consider the representatives

$\check{f}_{-2-2n}(Z^{\alpha})=e^{-m\frac{\braket{a\mu}}{\braket{a\lambda}}}\frac{G(u,\lambda_{A})}{(\chi(u,\lambda_{A}))^{p}}\
.$ (76)

Here $G(u,\lambda_{A})$ and $\chi(u,\lambda_{A})$ are homogeneous and
holomorphic minitwistor functions, such that $\chi(u,\lambda_{A})$ has $q\leq
2n$ simple zeros, corresponding to poles in $\check{f}_{-2-2n}(Z^{\alpha})$
enclosed by the contour $\mathcal{C}$. Furthermore, $G(u,\lambda_{A})$ is
regular at the $p$-fold pole given by the zero of $\chi(u,\lambda_{A})$. For
Type N solutions in which the field has only one $2n$-fold principal spinor,
$\chi(u,\lambda_{A})$ has a simple zero and $p=1$. For Type D solutions which
have 2 different $n$-fold principal spinors, $\chi(u,\lambda_{A})$ has two
simple zeros and $p=n+1$. In refs. Gonzalez:2022otg ; Emond:2022uaf , the
Cotton double copy was explicitly argued to hold in position space for type N
solutions only. Thus, we will shortly show how the type N Cotton double copy
can indeed be obtained from representatives of the form of eq. (76).

Before moving on, however, some comments are in order regarding the product of
twistor functions in eq. (74). As has been made clear repeatedly above, these
are representatives of cohomology classes, and thus – in a given theory – can
be subjected to equivalence transformations of the form of eq. (58). However,
the non-linear product that is needed to generate gravitational solutions in
the twistor space double copy of eq. (74) (likewise in the four-dimensional
case of refs. White:2020sfn ; Chacon:2021wbr ) is clearly incompatible with
the ability to first perform equivalence transformations. This is not actually
a problem if all one wants to do is to derive the Cotton double copy in
position space: one merely regards the product as only being true for certain
representatives in twistor space, such that any representatives which yield
the correct double copy structure in position space (if it exists) will do.
Nevertheless, it is desirable to have some motivation a priori for picking out
certain representatives, where this would ideally relate to the physics of the
double copy. Reference Adamo:2021dfg  was the first to consider this point,
using the language of Dolbeault rather than C̆ech cohomology. The authors
considered certain radiative solutions, and showed that data at null infinity
could be used to uniquely fix twistor representatives in the various theories
entering the double copy. Reference Chacon:2021lox  took a different approach,
by looking at spacetime fields in Euclidean signature, and using existing the
ideas of ref. Woodhouse:1985id  to argue that upon choosing special cohomology
representatives in twistor space (corresponding to harmonic differential
forms), a product structure in twistor space is naturally obtained.
Unfortunately, neither of these procedures is obviously related to the other,
nor to the original BCJ double copy for scattering amplitudes. Reference
Guevara:2021yud , however, provided a much better motivation for the formula
of eq. (74), at least in principle, by showing that special twistor
representatives can be defined by a certain integral transform acting on
momentum-space amplitudes. Indeed, ref. Luna:2022dxo  showed that these
representatives are precisely those entering the twistor double copy of refs.
White:2020sfn ; Chacon:2021wbr . Thus, the twistor double copy can indeed be
viewed as arising from the BCJ double copy for three-point scattering
amplitudes. Similar arguments can be used in the present context of solutions
of topologically massive gauge theory and gravity, and we return to this in
section 4.

###  3.1 Cotton double copy for Type N

Let us now see how the type N Cotton double copy arises from twistor space. In
line with our comments above, a type N solution should be generated by eq.
(74), provided the gravity twistor representative has a simple pole in twistor
space. We may thus choose representatives

$\check{f}_{-2}(Z^{\alpha})=e^{-m\frac{\braket{a\mu}}{\braket{a\lambda}}}\frac{G_{0}(u,\lambda_{A})}{\chi_{1}(u,\lambda_{A})\xi_{1}(u,\lambda_{A})},\quad\check{f}_{-4}^{(1,2)}(Z^{\alpha})=e^{-m\frac{\braket{a\mu}}{\braket{a\lambda}}}\frac{G_{0}(u,\lambda_{A})}{\chi_{1}(u,\lambda_{A})(\xi_{1}(u,\lambda_{A}))^{3}},$
(77)

where $\chi_{1}(u,\lambda_{A})$ and $\xi_{1}(u,\lambda_{A})$ are homogeneous
of degree $1$ and have simple zeros and $G_{0}(u,\lambda_{A})$ is homogeneous
of degree $0$ and has no poles, such that one finds

$\check{f}_{-6}(Z^{\alpha})=e^{-m\frac{\braket{a\mu}}{\braket{a\lambda}}}\frac{G_{0}(u,\lambda_{A})}{\chi_{1}(u,\lambda_{A})(\xi_{1}(u,\lambda_{A}))^{5}}.$
(78)

Upon substituting this into the Penrose transform of eq. (68), we may carry
out the latter by choosing the patch $U_{0}$ in eq. (32), so that
$\lambda_{A}=(1,z)$. We will assume without loss of generality that only the
simple pole, which arises from $\xi_{1}$, of each cohomology representative
lies in $U_{0}$. On general grounds, we may further define

$\displaystyle\rho_{x}\left[\frac{\braket{a\mu}}{\braket{a\lambda}}\right]=q(x;z),\quad\rho_{x}\left[G_{0}(u,\lambda_{A})\right]=G(x;z),$
$\displaystyle\rho_{x}\left[\chi_{1}(u,\lambda_{A})\right]=\frac{(z-z_{0})}{N_{1}(x)},\quad\rho_{x}\left[\xi_{1}(u,\lambda_{A})\right]=\frac{(z-z_{1})}{N_{2}(x)}\
,$ (79)

where $z_{0}$ is the position of the simple zero in $\chi_{1}(u,\lambda_{A})$,
in terms of the parameter $z$, and the position dependence of each quantity
arises upon imposing the incidence relation in eq. (28). Equation (68) then
becomes

$\displaystyle\phi_{AB\ldots D}(x)=$ $\displaystyle\frac{1}{2\pi
i}\int_{\Gamma}\mathrm{d}z(1,z)_{A}(1,z)_{B}\cdots(1,z)_{D}G(x;z)e^{-m\
q(x;z)}$
$\displaystyle\quad\times\frac{N_{1}(x)}{z-z_{0}}\left(\frac{N_{2}(x)}{z-z_{1}}\right)^{2n+1}$
$\displaystyle=$
$\displaystyle(1,z_{0})_{A}(1,z_{0})_{B}\cdots(1,z_{0})_{D}\,N_{1}(x)\,G(x;z_{0})\,\left(\frac{N_{2}(x)}{z_{0}-z_{1}}\right)^{2n+1}e^{-m\
q(x;z_{0})}\ ,$ (80)

where we have carried out the contour integral in the second line by assuming
that $q(x,z)$ is non-singular at $z=z_{0}$. The fields of eq. (80) clearly
satisfy the Cotton double copy of Eq. (22), when taking $C_{ABGD}=\phi_{ABGD}$
and $\Phi_{AB}=\phi_{AB}$. Thus, the Cotton double copy indeed emerges from a
product in twistor space, as claimed.

To give an explicit example of the above construction, let us examine pp-wave
solutions, for which the following representatives in twistor space can be
constructed for a spin-$n$ field:

$\check{f}_{-2-2n}=e^{-m\frac{\braket{a\mu}}{\braket{a\lambda}}}\frac{1}{\braket{o\lambda}}\left(\frac{1}{\braket{s\lambda}}\right)^{2n+1}g\left(\frac{u}{(s^{A}\lambda_{A})^{2}}\right)\
.$ (81)

Here we have introduced the constant spinors $a^{A}=(1,0)$, $o^{A}=(0,1)$, and
$s^{A}=(1,c)$, where $c\ \epsilon\ \mathbb{C}$. Comparing to eq. (79) we have

$\displaystyle\rho_{x}\left[G_{0}(u,\lambda_{A})\right]=\rho_{x}\left[g\left(\frac{u}{(s^{A}\lambda_{A})^{2}}\right)\right]=g\left(\frac{(-t+y)z^{2}-2xz-
y-t}{(1+cz)^{2}}\right)$
$\displaystyle\rho_{x}\left[\chi_{1}(u,\lambda_{A})\right]=\rho_{x}\left[\braket{o\lambda}\right]=z,\quad\rho_{x}\left[\xi_{1}(u,\lambda_{A})\right]=\rho_{x}\left[\braket{s\lambda}\right]=1+cz\
.$ (82)

where the pole $z_{0}=0$ is in $U_{0}$ and $z_{1}=-1/c$ is in $U_{1}$.
Carrying out the relevant Penrose transforms as in eq.(80), one finds

$\displaystyle C_{ABCD}=\phi(y_{+},x)\
\alpha_{A}\alpha_{B}\alpha_{C}\alpha_{D}\ ,\quad f_{AB}=\phi(y_{+},x)\
\alpha_{A}\alpha_{B}\ ,\quad\phi(y_{+},x)=g(y_{+})e^{-mx}\ ,$ (83)

with $y_{\pm}=t\pm y$ and $\alpha_{A}=(1,0)$ the principal spinor of the plane
wave solutions. These are indeed the pp-wave solutions for topologically
massive gravity, topologically massive electrodynamics, and a massive scalar
field Chow:2009km .

###  3.2 Beyond type N solutions

Having reproduced the Cotton double copy for type N solutions of refs.
Gonzalez:2022otg ; Emond:2022uaf , it is natural to ask whether or not the
arguments can be extended to type D solutions. The latter indeed double copy
in the four-dimensional Weyl double copy, whose twistorial incarnation has
been presented in refs. White:2020sfn ; Chacon:2021wbr . The twistor
description allows us to address this directly, and in fact shows that type D
solutions do not obey a simple position space double copy in general. To see
this, we may write the explicit formula

$\check{f}_{-2n-2}(Z^{\alpha})=e^{-m\frac{\braket{a\mu}}{\braket{a\lambda}}}\frac{G_{0}(u,\lambda_{A})}{(\chi_{1}(u,\lambda_{A})\xi_{1}(u,\lambda_{A}))^{n+1}}\
,$ (84)

where as before, $\chi_{1}(u,\lambda_{A})$ and $\xi_{1}(u,\lambda_{A})$ are
homogeneous of degree $1$ and have simple zeros, and $G_{0}(u,\lambda_{A})$ is
homogeneous of degree $0$. We can further define

$\rho_{x}\left[\frac{G_{0}(u,\lambda_{A})}{(\chi_{1}(u,\lambda_{A})\xi_{1}(u,\lambda_{A}))^{n+1}}\right]=\frac{N(x)}{(z-z_{0})^{n+1}(z-z_{1})^{n+1}},$
(85)

where we have again imposed the incidence relation on the coordinate patch
$U_{0}$ and we assume that only the pole at $z_{0}$ is in $U_{0}$. Upon
substituting this into the Penrose transform, one finds

$\displaystyle\phi_{AB\ldots D}(x)=$ $\displaystyle\frac{1}{2\pi
i}\int_{\Gamma}\mathrm{d}z(1,z)_{A}(1,z)_{B}\cdots(1,z)_{D}\,G(x;z)e^{-m\
q(x;z)}$
$\displaystyle\quad\times\left(\frac{N(x)}{(z-z_{0})^{n+1}(z-z_{1})^{n+1}}\right).$
(86)

For type D solutions, poles of second order or higher will be present in the
integrand which, upon taking residues, will generate terms involving
derivatives of the combination

$G(x;z)e^{-mq(x;z)}.$

While we have shown this explicitly for type D solutions, this behaviour will
hold for all non-type N solutions. Thus, rather than a single term in position
space, one will obtain a sum of terms, such that a simple product of spacetime
fields is not obtained in general. One way of simplifying matters is to only
consider twistor representatives such that the function $G(x;z)$ is constant.
Indeed, all of the type D representatives considered in the four-dimensional
twistor double copy of refs. White:2020sfn ; Chacon:2021wbr  were of this
form. However, this will not suffice in the present context, due to the
exponential factor $e^{-mq(x;z)}$, whose presence is an unavoidable
consequence of considering topologically massive gauge and / or gravity
theory. We therefore conclude that, unlike the case of the Weyl double copy in
four spacetime dimensions, the exact position-space Cotton double copy will be
restricted purely to type N solutions111111Note that we have assumed that the
factor in the exponential has no poles in $U_{0}$. Below we will see that for
linearized solutions that can be constructed from three-point amplitudes, this
is not the case. In such cases, we will again find that the position space
Cotton double copy does not hold.. This is in stark contrast to the case of
the Weyl double copy in four spacetime dimensions, where some of the simplest
relevant solutions – consisting of simple point-like objects at the origin –
are of type D. We have thus explained why refs. Gonzalez:2022otg ;
Emond:2022uaf  only succeeded in finding Cotton double copies for type N
solutions. Our results are also interesting in that they show that, even for
non-type N solutions, there can still be a simple product-like double copy
structure in twistor space. The lack of a double copy in position space is a
consequence of the generalised Penrose transform, and thus ultimately due to
the presence of the topological mass. It is instructive to illustrate the
general discussion of this section with a concrete example. This is the
subject of the following section.

##  4 From scattering amplitudes to cohomology representatives

In the previous section, we have seen that Cotton double copy follows
naturally from minitwistor space, analogous to how the Weyl double copy in
four spacetime dimensions can be derived using twistor methods White:2020sfn ;
Chacon:2021wbr . Until recently, quite how the Weyl (position-space) and
twistor double copies related to the BCJ double copy for scattering amplitudes
remained mysterious. This was first settled in refs. Monteiro:2020plf ;
Monteiro:2021ztt , which showed that the Weyl and BCJ double copy for
scattering amplitudes are equivalent, where they overlap, by using the so-
called KMOC formalism Kosower:2018adc  that expresses classical solutions as
inverse on-shell Fourier transforms of scattering amplitudes. Reference
Luna:2022dxo  investigated this further, by using methods developed in ref.
Guevara:2021yud  to show that one may split the inverse Fourier transform from
momentum to position space into two stages. The first takes momentum-space
scattering amplitudes into twistor space, thereby picking out a particular
cohomology representative. The second comprises the Penrose transform from
twistor to position space, and ref. Luna:2022dxo  thus makes clear that the
amplitude, twistor and Weyl double copies are precisely equivalent where they
overlap. A canonical example is that of a point mass or charge in gravity /
gauge theory respectively, corresponding to the well-known Schwarzschild and
Coulomb solutions. Similar solutions exist in topologically massive theories,
namely gravitational and gauge theory anyons, whose double copy properties
have been explored in refs. Burger:2021wss . Such solutions are not type N,
such that we do not expect them to possess a simple position-space double
copy, according to the arguments of the previous section. However, we do
expect to see a simple product-like twistor-space double copy, where the
relevant cohomology representatives are picked out by scattering amplitudes in
momentum space. It is interesting to confirm this by seeing what actually
happens if we take the relevant scattering amplitudes, and generalise the
arguments of refs. Guevara:2021yud ; Luna:2022dxo  to three-dimensional
topologically massive gauge theories and gravity.

Let us begin by developing the necessary ideas from the KMOC formalism of ref.
Kosower:2018adc , which must be adapted to the present context (see also refs.
Monteiro:2020plf ; Monteiro:2021ztt ; Emond:2022uaf  for relevant
ingredients). We will first focus on a scalar field, which we can mode expand
in the usual way as

$\displaystyle\phi(x)=\int d\Phi(q)\left[a(q)e^{-iq\cdot
x}+a^{\dagger}(q)e^{iq\cdot x}\right],$ (87)

where

$\mathrm{d}\Phi(q)=\frac{d^{3}q}{(2\pi)^{3}}\hat{\delta}(q^{2}+m^{2})\Theta(q_{0})$
(88)

is the three-dimensional on-shell measure and $\hat{\delta}(x)\equiv
2\pi\delta(x)$. We are interested in the field generated by a static particle
of mass $M$, which we take to be described by an initial state

$\displaystyle\ket{\psi}=\int\mathrm{d}\Phi(p)\psi(p)\ket{p},$ (89)

where $\psi(p)$ is a wavefunction in momentum space, corresponding to a
wavepacket sharply peaked around the classical momentum $p^{\mu}=Mu^{\mu}$,
with $u^{\mu}$ the 4-velocity. Evolving this state into the far future using
the S-matrix, the classical field is given by the expectation value

$\displaystyle\varphi(x)=\braket{\psi}{S^{\dagger}\phi(x)S}{\psi},$ (90)

which in turn yields

$\displaystyle\varphi(x)=\int\mathrm{d}\Phi(q)\left[\braket{\psi}{S^{\dagger}a(q)S}{\psi}e^{-iq\cdot
x}+h.c.\right]$ (91)

Next, we can adopt the conventional expansion of the $S$-matrix:

$S=1+iT,$ (92)

and note that

$\braket{\psi}{a}{\psi}=0,$ (93)

given that there are no $\phi$ excitations in the initial state. We thus get

$\displaystyle\varphi(x)$
$\displaystyle=2\operatorname{Re}i\int\mathrm{d}\Phi(q)\mathrm{d}\Phi(p)\mathrm{d}\Phi(p^{\prime})\psi(p)^{*}\psi(p^{\prime})\left[\braket{p^{\prime}}{a(q)T}{p}e^{-iq\cdot
x}\right]$ (94)
$\displaystyle=2\operatorname{Re}i\int\mathrm{d}\Phi(q)\mathrm{d}\Phi(p)\hat{\delta}(2p\cdot
q+q^{2})\psi(p)^{*}\psi(p+q)\left[\mathcal{A}^{(3)}(q)e^{-iq\cdot x}\right],$

where in the second line we have introduced the three-point amplitude for the
emission of the $\phi$ field by the source:

$\braket{p^{\prime}}{a(q)T}{p}=\mathcal{A}^{(3)}(q)\hat{\delta}(p+p^{\prime}-q).$
(95)

To understand this equation, note that the $a(q)$ operator acts as a creation
operator on the left, creating a quantum of the $\phi$ field. The expectation
value of the T-matrix is then, by definition, the three-point amplitude
multiplied by a momentum-conserving delta function. As shown in ref.
Kosower:2018adc , by carefully accounting for factors of $\hbar$ (absent in
natural units), one can neglect the shift by $q$ in the wavefunction, and also
the $q^{2}$ term in the delta function. One may then integrate out the
momentum $p$ by assuming that the wavefunction $|\psi(p)|^{2}$ is
appropriately normalised to find

$\displaystyle\varphi(x)$
$\displaystyle=\frac{1}{M}\operatorname{Re}i\int\mathrm{d}\Phi(q)\hat{\delta}(u\cdot
q)\left[\mathcal{A}^{(3)}(q)e^{-iq\cdot x}\right].$ (96)

In words: the classical field is obtained as an on-shell inverse Fourier
transform of the three-point amplitude. Following refs. Guevara:2021yud ;
Luna:2022dxo , we can split this transform into two stages as follows. First,
we introduce spinor variables by appealing to eq. (5):

$\displaystyle
q_{AB}=\omega(\lambda_{A}\bar{\lambda}_{B}+\lambda_{B}\bar{\lambda}_{A}).$
(97)

Here $\omega$ has units of energy, so that the spinors
$(\lambda_{A},\bar{\lambda}_{B})$ are dimensionless, and defined only up to
the little group scalings

$\lambda_{A}\rightarrow\xi\lambda_{A},\quad\bar{\lambda}_{B}\rightarrow\frac{1}{\xi}\bar{\lambda}_{B}.$
(98)

Transforming to the new variables, we find that

$\displaystyle
d^{3}q=2\omega^{2}d\omega|\braket{\lambda\bar{\lambda}}|\braket{\lambda
d\lambda}\braket{\bar{\lambda}d\bar{\lambda}}.$ (99)

Furthermore, the three-point amplitude for a scalar field emitted by a scalar
source is simply given by a coupling constant, which we set to unity in what
follows. Equation (96) then becomes

$\displaystyle\varphi(x)=\operatorname{Re}\frac{i}{2\pi M}\int
d\omega\braket{\lambda
d\lambda}\braket{\bar{\lambda}d\bar{\lambda}}|\omega|\,|\langle\lambda\bar{\lambda}\rangle|\,\delta(\braket{\lambda}{u}{\bar{\lambda}})\,\delta\left(\omega^{2}\braket{\lambda\bar{\lambda}}^{2}+m^{2}\right)e^{-i\omega\braket{\lambda}{x}{\bar{\lambda}}},$
(100)

where we have chosen to work in the rest frame where $u^{\mu}=(1,0,0)$ and
thus $q_{0}=0$, and we have used the fact that $\Theta(q_{0})=\Theta(0)=1/2$.
The first delta function then implies that

$\displaystyle\ket{\bar{\lambda}}\propto u\ket{\lambda},$ (101)

which we may write as an equality by relying upon the little group rescaling
of eq. (98) (see also ref. Luna:2022dxo  for a discussion of this point).
Performing the $\bar{\lambda}$ integral then yields

$\phi=\operatorname{Re}\frac{i}{2\pi M}\int d\omega\braket{\lambda
d\lambda}|\omega|\,|\langle\lambda|u|\lambda\rangle|\,\delta(\omega^{2}\langle\lambda|u|\lambda\rangle^{2}+m^{2})e^{-i\omega\langle\lambda|xu|\lambda\rangle}.$
(102)

To make further progress, we note that in our original metric signature
$(-,+,+)$ for $m\in\mathbb{R}$, it is impossible to simultaneously solve the
dual kinematic conditions

$u\cdot q=0,\quad q^{2}+m^{2}=0.$ (103)

To get around this problem, we may instead analytically continue to $(+,+,-)$
signature, by setting

$(q_{0},q_{2})=i(\tilde{q}_{0},\tilde{q}_{2}),\quad(t,y)=i(\tilde{t},\tilde{y}).$
(104)

From eq. (5) applied to $q^{\mu}$ with $\tilde{q}_{0}=0$, we may rescale
according to eq. (98) to write

$\lambda_{A}=(1,z),\quad z=\frac{q_{1}-im}{i\tilde{q}_{2}}.$ (105)

The analytically continued kinematic constraint

$\tilde{q}_{1}^{2}-\tilde{q}_{2}^{2}+m^{2}=0$ (106)

then implies $|z|=1$, and we also have

$\langle\lambda|u|\lambda\rangle\rightarrow
i\langle\lambda|u|\lambda\rangle,\quad\langle\lambda|xu|\lambda\rangle\rightarrow
i\langle\lambda|xu|\lambda\rangle,$

so that eq. (102) becomes

$\phi=\operatorname{Re}\frac{i}{4\pi Mm}\int d\omega\braket{\lambda
d\lambda}|\omega|\left[\delta\left(\omega-\frac{m}{\langle\lambda|u|\lambda\rangle}\right)+\delta\left(\omega+\frac{m}{\langle\lambda|u|\lambda\rangle}\right)\right]e^{\omega\langle\lambda|xu|\lambda\rangle)},$
(107)

where we have used the standard identity

$\delta(x^{2}-\alpha^{2})=\frac{1}{2|\alpha|}\Big{[}\delta(x-\alpha)+\delta(x+\alpha)\Big{]}.$
(108)

It turns out that both delta function contributions are the same, such that we
may take only the first with a factor of two. We then arrive at

$\varphi(x)=\operatorname{Re}\frac{i}{M}\int_{\rm\Gamma}\frac{\langle\lambda
d\lambda\rangle}{2\pi}\frac{e^{m\frac{\langle\lambda|xu|\lambda\rangle}{\langle\lambda|u|\lambda\rangle}}}{\langle\lambda|u|\lambda\rangle},$
(109)

where $\Gamma$ is the appropriate integration contour. Recognising
$\langle\lambda|x=\langle\mu|$, eq. (109) has precisely the form of the
Penrose transform integrand of eq. (67), where in this case
$|a\rangle=u|\lambda\rangle$, and a specific function of $\lambda_{A}$ occurs,
due to having transformed a particular momentum-space amplitude into
(mini-)twistor space. This is a highly useful validation that eq. (67) is the
correct Penrose transform integrand for topologically massive theories. But it
also forms the basis for calculating similar results for gauge and gravity
anyon solutions, and examining their double copy properties.

Explicit calculation of the various spinor products in eq. (109) yields

$\displaystyle\phi(x)=\operatorname{Re}\frac{i}{M}\int_{\Gamma}\frac{dz}{2\pi}\frac{e^{-m\frac{-2i\tilde{y}z+x(1-z^{2})}{1+z^{2}}}}{1+z^{2}},$
(110)

such that transforming to the variable $z=e^{i\alpha}$ gives

$\phi(x)=-\frac{1}{4\pi
M}\operatorname{Re}\int_{-\pi/2}^{\pi/2}\frac{d\alpha}{\cos\alpha}\exp\left(im\frac{\tilde{y}-x\sin\alpha}{\cos\alpha}\right),$
(111)

We can use the rotational symmetry of the scalar solution to set $\tilde{y}=0$
and $x=r$, such that eq. (111) becomes

$\displaystyle\phi=-\frac{1}{4\pi
M}\operatorname{Re}\int_{-\pi/2}^{\pi/2}d\alpha\frac{e^{-imr\tan\alpha}}{\cos\alpha}=-\frac{1}{2\pi
M}K_{0}(mr),$ (112)

where we have used a known integral representation for the modified Bessel
function of the second kind 121212Strictly speaking, the argument of the
$K_{0}$ function should be $|mr|$, but $m$ and $r$ are both positive in our
case. This justifies the remark made above that the contributions from both
delta functions in eq. (107) give the same result.. Finally we may
analytically continue back to the metric signature $(-,+,+)$, which does not
change the form of the result, but ensures that $r=\sqrt{x^{2}+y^{2}}$.

###  4.1 Topologically massive gauge theory

In the previous section, we have seen that a classical scalar field can be
obtained as an inverse on-shell Fourier transform of a three-point amplitude,
such that splitting this transform into two stages allows us to recognise the
Penrose transform from twistor to position space. Similar arguments apply to
topologically massive gauge theory and gravity, and thus allow us to examine
how to Cotton double copy does or does not work for pointlike solutions. In
the gauge theory, we are concerned with the relevant curvature spinor, derived
from the dual field strength in the case of topologically massive
electromagnetism. This is in turn related to the three-point amplitude for
emission of a photon by a source Emond:2022uaf , which we again take to be a
scalar particle of mass $M$.

We can mode expand the dual field strength on-shell in the usual way131313We
note however that there is no sum over helicities here, since topologically
massive theories propagate only a single helicity.

$\displaystyle\tilde{F}^{\mu}(x)=\frac{1}{2}\epsilon^{\mu\nu\rho}F_{\nu\rho}=-im\int\mathrm{d}\Phi(q)\left[a(q)\epsilon^{\mu}(q)e^{-iq\cdot
x}+a^{\dagger}(q)\epsilon^{*\mu}(q)e^{iq\cdot x}\right].$ (113)

Then, analogously to the scalar case of eq. (90), the curvature spinor is
related to the expectation value of the field strength evolved into the far
future by the $S$-matrix

$\displaystyle\varphi_{AB}(x)=\braket{\psi}{S^{\dagger}\tilde{F}^{\mu}(x)\sigma_{\mu
AB}S}{\psi},$ (114)

where $\ket{\psi}$ is the initial one-particle state defined in eq. (89), and
where we now assume this particle is charged such that it may emit photons.
The electromagnetic curvature spinor is then given by

$\displaystyle\varphi_{AB}(x)=-im\int\mathrm{d}\Phi(q)\left[\braket{\psi}{S^{\dagger}a(q)S}{\psi}\epsilon_{AB}(q)e^{-iq\cdot
x}+h.c.\right],$ (115)

such that upon following similar steps to those leading to eq. (96), we arrive
at

$\displaystyle\varphi_{AB}(x)$
$\displaystyle=-\frac{m}{M}\operatorname{Re}\int\mathrm{d}\Phi(q)\hat{\delta}(u\cdot
q)\left[\mathcal{A}_{\rm gauge}^{(3)}(q)\epsilon_{AB}(q)e^{-iq\cdot
x}\right],$ (116)

where ${\cal A}_{\rm gauge}$ is the appropriate three-point amplitude for the
emission of a (topologically massive) gauge boson from a scalar. As in the
scalar case, we may transform to spinor coordinates according to eq. (97). A
suitable choice for the polarization vector is

$\displaystyle\epsilon^{\mu}=\omega\frac{\braket{\lambda}{\gamma^{\mu}}{\lambda}}{2m}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \implies\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\epsilon_{AB}=\frac{\omega}{m}\lambda_{A}\lambda_{B},$ (117)

Furthermore, the amplitude ${\cal A}^{(3)}_{\rm gauge}$ can be fixed by
dimensional analysis and little group scaling Moynihan:2020ejh :

$\displaystyle\mathcal{A}^{(3)}=2eM(u\cdot\epsilon(q)).$ (118)

Repeating similar arguments to those leading to eq. (109), we ultimately find

$\displaystyle\varphi_{AB}(x)=-em\operatorname{Re}i\int_{\Gamma}\frac{\braket{\lambda\mathrm{d}\lambda}}{2\pi}\lambda_{A}\lambda_{B}\frac{e^{m\frac{\braket{\mu}{u}{\lambda}}{\braket{\lambda}{u}{\lambda}}}}{\braket{\lambda}{u}{\lambda}^{2}}.$
(119)

This looks almost identical to the scalar case of eq. (96), but such that the
integrand now contains additional powers of spinor variables, as is
appropriate for a spin-1 field. Again, we have obtained a specific example of
the massive Penrose transform integrand of eq. (67).

Equation (119) will be useful for examining double copy properties of anyon
solutions, but let us first carry out the Penrose transform to position space.
To do this, we may choose the parametrisation $\lambda_{A}=(1,z)$, and define
the master integral

$I_{p,q}(x,y)=\operatorname{Re}i\int_{\Gamma}\frac{dz}{2\pi}\frac{z^{p}\,e^{\frac{-m[-2yz+x(1-z^{2})]}{1+z^{2}}}}{(1+z^{2})^{q+1}}.$
(120)

In terms of this integral, the scalar field of eq. (110) can be written as

$\phi=\frac{1}{M}I_{0,0},$ (121)

after analytically continuing back to the mostly plus metric signature, from
which we find

$I_{0,0}=AK_{0}(mr),\quad A=-\frac{1}{\pi}.$ (122)

Similarly, the field strength spinor of eq. (119) has the form

$\varphi_{00}(x)=-emI_{0,1},\quad\varphi_{01}=\varphi_{10}=-emI_{1,1},\quad\varphi_{11}=-emI_{2,1}.$
(123)

Given the result of eq. (122), we may carry out the integrals on the right-
hand side of eq. (123) without performing any further explicit integrals. To
see how, note that we can differentiate eq. (120) to obtain the recurrence
relations

$\partial_{x}I_{p,q}=-m[I_{p,q+1}-I_{p+2,q+1}],\quad\partial_{y}I_{p,q}=2mI_{p+1,q+1}.$
(124)

A further recurrence relation can be obtained by substituting

$1=\frac{1+z^{2}}{1+z^{2}}$

into the integrand of eq. (120), yielding

$I_{p+2,q+1}=I_{p,q}-I_{p,q+1}.$ (125)

We thus find that all integrals entering the field strength spinor of eq.
(123) can be expressed in terms of derivatives of eq. (122), leading to the
explicit results

$\displaystyle I_{0,1}$
$\displaystyle=\frac{A}{2}\left[K_{0}(mr)+\frac{x}{r}K_{1}(mr)\right];$
$\displaystyle I_{1,1}$ $\displaystyle=-\frac{Ay}{2r}K_{1}(mr);$
$\displaystyle I_{2,1}$
$\displaystyle=\frac{A}{2}\left[K_{0}(mr)-\frac{x}{r}K_{1}(mr)\right].$ (126)

We thus find

$\displaystyle\varphi_{00}$
$\displaystyle=\frac{em}{2\pi}\left[K_{0}(mr)+\frac{x}{r}K_{1}(mr)\right];$
$\displaystyle\varphi_{01}$
$\displaystyle=-\frac{em}{2\pi}\frac{y\,K_{1}(mr)}{r}$
$\displaystyle\varphi_{11}$
$\displaystyle=\frac{em}{2\pi}\left[K_{0}(mr)-\frac{x}{r}K_{1}(mr)\right].$
(127)

###  4.2 Topologically massive gravity

A similar analysis to the previous section may be carried out for
topologically massive gravity, where we may consider a scalar particle
emitting (massive) gravitons. The classical result for the Cotton spinor of
eq. (18) is then given by

$C_{ABCD}(x)=\langle\psi|S^{\dagger}C_{ABCD}S|\psi\rangle,$ (128)

such that the analogue of eq. (116) is, following Emond:2022uaf

$\displaystyle C_{ABCD}(x)=-\kappa\frac{m}{2M}\operatorname{Re}i\int d\omega$
$\displaystyle|\braket{\lambda\bar{\lambda}}|\frac{\braket{\lambda
d\lambda}\braket{\bar{\lambda}d\bar{\lambda}}}{2\pi}\delta(\braket{\lambda}{u}{\bar{\lambda}})\delta(\omega^{2}\braket{\lambda\bar{\lambda}}^{2}+m^{2})|\omega|^{3}$
$\displaystyle\times\left[{\cal A}_{\rm
grav.}(q)\lambda_{A}\lambda_{B}\lambda_{C}\lambda_{D}e^{-iq\cdot x}\right].$
(129)

In eq. (129), ${\cal A}_{\rm grav.}$ is the three-point amplitude for emission
of a graviton by a scalar. To find this, we may quote a general result for a
spin-$s$ field coupled to two scalars:

$\displaystyle\mathcal{A}^{(3)}_{+s}=g_{s}M^{s}(2u\cdot\epsilon(q))^{s},$
(130)

where $g_{s}$ is some coupling. By similar arguments to the gauge theory case,
this simply evaluates to a constant, such that the analogue of eq. (119) turns
out to be

$C_{ABCD}(x)=-\frac{\kappa^{2}m^{3}M}{2}\operatorname{Re}i\int_{\Gamma}\frac{\langle\lambda
d\lambda\rangle}{2\pi}\lambda_{A}\lambda_{B}\lambda_{C}\lambda_{D}\frac{e^{m\frac{\mu|u|\lambda\rangle}{\langle\lambda|u|\lambda\rangle}}}{\langle\lambda|u|\lambda\rangle^{3}}.$
(131)

In terms of the basis of integrals defined in eq. (120), we then have

$C_{ABCD}=-\frac{\kappa^{2}m^{3}M}{2}I_{n_{1}(ABCD),2},$ (132)

where $n_{1}(ABCD)$ is the number of $1$ indices (rather than $0$ indices) in
the string $ABCD$. By use of the above recurrence relations we find

$\displaystyle C_{0000}$
$\displaystyle=\frac{\kappa^{2}m^{3}M}{8\pi}\left[\left(2-\frac{y^{2}}{r^{2}}\right)K_{0}(mr)+\left(\frac{2x}{r}+\frac{(r^{2}-2y^{2})}{mr^{3}}\right)K_{1}(mr)\right];$
$\displaystyle C_{1000}$
$\displaystyle=\frac{\kappa^{2}m^{3}M}{8\pi}\left[-\frac{xy}{r^{2}}K_{0}(mr)+\left(-\frac{y}{r}-\frac{2xy}{mr^{3}}\right)K_{1}(mr)\right];$
$\displaystyle C_{1100}$
$\displaystyle=\frac{\kappa^{2}m^{3}M}{8\pi}\left[\frac{y^{2}}{r^{2}}K_{0}(mr)+\left(-\frac{1}{mr}+\frac{2y^{2}}{mr^{3}}\right)K_{1}(mr)\right];$
$\displaystyle C_{1110}$
$\displaystyle=\frac{\kappa^{2}m^{3}M}{8\pi}\left[\frac{xy}{r^{2}}K_{0}(mr)+\left(-\frac{y}{r}+\frac{2xy}{mr^{3}}\right)K_{1}(mr)\right];$
$\displaystyle C_{1111}$
$\displaystyle=\frac{\kappa^{2}m^{3}M}{8\pi}\left[\frac{(2r^{2}-y^{2})}{r^{2}}K_{0}(mr)+\left(-\frac{2x}{r}+\frac{1}{mr}-\frac{2y^{2}}{mr^{3}}\right)K_{1}(mr)\right].$
(133)

We have checked explicitly that the results of eqs. (127, 133) agree with the
known anyon solutions in topologically massive gauge and gravity theory
Deser:1989ri , once these are translated into the spinorial language, and up
to an overall normalisation constant (which we have defined differently in our
choice of constant amplitudes above).

###  4.3 Double copy properties of anyon solutions

In the previous sections, we have seen that the solution for spin-$n$ field
for a pointlike source, in a (topologically) massive theory, takes the general
form

$\psi_{AB\ldots D}(x)=K_{n}\int_{\Gamma}\frac{\langle\lambda
d\lambda\rangle}{2\pi}\lambda_{A}\lambda_{B}\ldots\lambda_{D}\frac{e^{m\frac{\mu|u|\lambda\rangle}{\langle\lambda|u|\lambda\rangle}}}{\langle\lambda|u|\lambda\rangle^{n+1}},$
(134)

where $K_{n}$ is a normalisation constant. Each solution is a special case of
the massive Penrose transform of eq. (86), and we thus see that there is a
multiplicative double-copy structure in twistor space. That is, upon picking
out a cohomology representative $f_{n}(\lambda_{A},u)$ for each spin-$n$ field
(scalar, gauge and gravity) by transforming amplitudes into twistor space,
these representatives are related by a simple multiplicative rule

$f_{2}(\lambda_{A},u)=\frac{f_{1}(\lambda_{A},u)f_{1}(\lambda_{A},u)}{f_{0}(\lambda_{A},u)}.$
(135)

The same result is obtained in four spacetime dimensions Luna:2022dxo .
However, unlike in that case, the simple multiplicative nature of the double
copy in twistor space does not correspond to a simple structure in position
space, as comparison of eqs. (127, 133) makes clear. This thus provides an
explicit illustration of the general discussion in section 3.2, namely that
the presence of the exponential factor in the massive Penrose transform
disrupts the simple nature of the position-space double copy. Given that
fields generated by a pointlike source are perhaps the simplest static
solutions one can imagine, this bolsters the conclusions of ref. Luna:2022dxo
, that exact position space double copies are rather special and restricted in
nature, and that the double copy prefers to live in momentum space. Another
interesting feature of the three-dimensional solutions considered here is
that, in contrast to their four-dimensional counterparts, the twistor space
representatives have essential singularities, rather than poles. Again this is
due to the presence of the exponential factor, and guarantees that one is able
to reconstruct the relevant transcendental functions entering the spacetime
field (i.e. modified Bessel functions) upon taking residues in the Penrose
transform integral.

##  5 Conclusion

In this paper, we have examined the Cotton double copy recently presented in
refs. Gonzalez:2022otg ; Emond:2022uaf , that relates solutions of
topologically massive gauge and gravity theories. It is a three dimensional
(massive) counterpart of the Weyl double copy for certain exact solutions in
four spacetime dimensions Luna:2018dpt . However, whereas the latter is known
to apply to arbitrary Petrov type D vacuum solutions, the former is apparently
restricted solely to the type N case. In order to clarify this issue, we have
here used twistor methods, which have previously been useful in examining the
origin (and special nature) of the Weyl double copy White:2020sfn ;
Chacon:2021wbr ; Chacon:2021hfe ; Luna:2022dxo .

In three spacetime dimensions, the relevant twistor space is called
minitwistor space, and we have reviewed known results from the twistor
literature tsai_1996  that provide a massive generalisation of the well-known
Penrose transform relating classical fields in spacetime with cohomology
classes in twistor space. Armed with this Penrose transform, one may show that
although it is possible to construct gravitational cohomology representatives
by combining representatives from (massive) scalar and gauge theories, this
leads to a simple position-space double copy only in the case of type N
solutions. We thus confirm the results of refs. Gonzalez:2022otg ;
Emond:2022uaf , and further clarify our results by considering arguably the
simplest possible static solutions, corresponding to a pointlike source. The
relevant classical fields can be expressed as on-shell inverse Fourier
transforms of three-point amplitudes, following the methods of ref.
Kosower:2018adc . By splitting this transform into two stages, we first
transform amplitudes into minitwistor space, revealing that the remaining to
spacetime takes precisely the form of the massive Penrose transform mentioned
above. Although we find a simple multiplicative double copy in twistor space,
this fails to translate to a simple relationship in spacetime, thus validating
our more general analysis.

The emerging picture from this and similar recent studies Luna:2022dxo  is
that exact position-space double copies are rare. However, knowing that they
exist – and what their limitations are – is undoubtedly useful. Methods for
elucidating the landscape of exact double copies are a necessary part of this
ongoing effort, and we hope that the twistor methods developed in this paper
may prove of further use in this regard.

## Acknowledgments

We are grateful to Tim Adamo and Graham Brown for useful discussions. This
work has been supported by the UK Science and Technology Facilities Council
(STFC) Consolidated Grant ST/P000754/1 “String theory, gauge theory and
duality”.
MCG is supported by the European Union’s Horizon 2020 Research Council grant
724659 MassiveCosmo ERC–2016–COG and the STFC grants ST/P000762/1 and
ST/T000791/1. JR is supported by the National Science and Technology Council
of Taiwan grant NSTC 111-2811-M-002-125. NM is supported by STFC grant
ST/P0000630/1 and the Royal Society of Edinburgh Saltire Early Career
Fellowship. WTE is supported by the Czech Science Foundation GACR, project
20-16531Y.

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