††institutetext: Centre for Theoretical Physics, Department of Physics and
Astronomy,
Queen Mary University of London, E1 4NS, United Kingdom

# From Moyal deformations to chiral higher-spin theories and to celestial
algebras

Ricardo Monteiro

###### Abstract

We study the connection of Moyal deformations of self-dual gravity and self-
dual Yang-Mils theory to chiral higher-spin theories, and also to deformations
of operator algebras in celestial holography. The relation to Moyal
deformations illuminates various aspects of the structure of chiral higher-
spin theories. For instance, the appearance of the self-dual kinematic algebra
in all the theories considered here leads via the double copy to vanishing
tree-level scattering amplitudes. Regarding celestial holography, the Moyal
deformation of self-dual gravity was recently shown to lead to the loop
algebra of $W_{\wedge}$, and we obtain here the analogous deformation of a
Kac-Moody algebra corresponding to Moyal-deformed self-dual Yang-Mills theory.
We also introduce the celestial algebras for various chiral higher-spin
theories.

QMUL-PH-22-39

##  1 Introduction

This paper aims to connect three topics that have ignited interest at
different times among theoretical physicists. The topics are: Moyal
deformations of field theories, chiral higher-spin theories of massless
particles, and chiral algebras that appear in celestial holography.

The Moyal bracket was introduced long ago to provide a geometric understanding
of phase-space non-commutativity in quantum mechanics Moyal:1949sk . It has
since been applied also to describe non-commutativity in a position-space
plane. It is characterised by a deformation parameter, which we will call
$\alpha$, and it turns out to be the most general deformation of a Poisson
bracket involving two coordinates that is still a Lie bracket Fletcher:1990ib
. Because of this property, it is often the case that when a Poisson bracket
arises in an integrable system, the Moyal deformation of the bracket preserves
the integrability. That was the original motivation to introduce a Moyal
deformation of self-dual gravity, whose equation of motion is integrable
STRACHAN199263 . Moyal deformations became the subject of intense interest in
higher-energy theory when they appeared in M/string theory; see e.g.
Connes:1997cr ; Nekrasov:1998ss ; Schomerus:1999ug ; Seiberg:1999vs  as well
as the reviews Douglas:2001ba ; Szabo:2001kg . For instance, it was described
in Seiberg:1999vs  how Yang-Mills theory in a space with a non-commutative
plane arises from the low-energy limit of open strings on a background with a
B-field. The Moyal deformation of self-dual Yang-Mills theory was considered
in Nekrasov:1998ss . Our formulation of the Moyal-deformed self-dual theories,
closer to the works STRACHAN199263 ; Takasaki:2000vs ; Garcia-Compean:2003nix
, focuses on the case where the plane of non-commutativity is null and can be
chosen to align with a particular gauge choice. This case allows for very
simple Moyal-deformed equations of motion of the self-dual theories in light-
cone gauge.

The Moyal-deformed theories are not Lorentz invariant, due to the plane of
non-commutativity. As we will see, however, if we interpret their fields as
generating functions of higher-spin fields, these non-Lorentz-invariant
theories generate Lorentz-invariant theories of massless higher-spin
particles. The latter coincide with the ‘chiral higher-spin theories’
discussed in Ponomarev:2016lrm ; Ponomarev:2017nrr ; Skvortsov:2018jea , based
on the chiral three-point vertices introduced in Metsaev:1991mt ;
Metsaev:1991nb ; see Ponomarev:2022vjb ; Bekaert:2022poo  for related reviews.
These theories are higher-spin extensions of self-dual gravity/Yang-Mills.
Moyal brackets are known to arise in higher-spin theories, and for the
particular chiral theories that we study here, the Moyal bracket was
identified from the vertices in Ponomarev:2017nrr ; see also Skvortsov:2022syz
; Tran:2022tft . We develop further these insights by explicitly connecting
the chiral higher-spin theories to Moyal-deformed self-dual gravity/Yang-
Mills, and exploring some of the implications.

It is natural to expect that 4D-chiral theories lead to 2D-chiral celestial
algebras, by which we mean the operator-product algebras that arise from the
consideration of asymptotic scattering states in celestial holography; see
Raclariu:2021zjz ; Pasterski:2021rjz ; Pasterski:2021raf ; McLoughlin:2022ljp
for recent reviews. Celestial holography builds on the connection between
asymptotic symmetries in asymptotically flat spacetimes and ‘soft theorems’
obeyed by scattering amplitudes Strominger:2013jfa ; He:2014laa ;
Strominger:2017zoo . The soft tower associated to asymptotic symmetries
Donnay:2018neh ; Adamo:2019ipt ; Puhm:2019zbl ; Guevara:2019ypd  admits a very
simple formulation when restricted to the self-dual sectors of gravity/Yang-
Mills Guevara:2021abz . For self-dual gravity, one obtains (the loop algebra
of the wedge subalgebra of) $w_{1+\infty}$ Strominger:2021lvk . This algebra
can be seen to arise in various ways; see Adamo:2021lrv  for a twistorial
approach as well as a historical discussion of related results, or e.g. refs.
Himwich:2021dau ; Jiang:2021csc ; Mago:2021wje ; Freidel:2021ytz ;
Costello:2022wso ; Costello:2022upu ; Ren:2022sws ; Compere:2022lzx ;
Monteiro:2022lwm . Deformations of the chiral algebras of self-dual
gravity/Yang-Mills have also been studied, with a focus on what class of
deformations preserves the associativity of the algebras Himwich:2021dau ;
Mago:2021wje ; Ren:2022sws . It was found in Monteiro:2022lwm ; Bu:2022iak
that the Moyal deformation of self-dual gravity leads to a know deformation of
$w_{1+\infty}$ into $W_{1+\infty}$. The present paper was partly motivated by
Ren:2022sws , where it was observed in examples that the vertices of chiral
higher-spin theories lead to an associative celestial algebra. See refs.
Ponomarev:2022ryp ; Ponomarev:2022qkx  for other work on the celestial
holography of chiral higher-spin theories.

There is a thread running through all these topics that illuminates their
relationship, and that is the double-copy structure of the theories in
question. The double copy is a factorisation of the interactions that most
famously connects the perturbative structure of Yang-Mills theory and gravity;
see recent reviews in Bern:2019prr ; Bern:2022wqg ; Kosower:2022yvp ;
Adamo:2022dcm  and the seminal papers Kawai:1985xq ; Bern:2008qj . For works
relating celestial holography and the double copy, see Pasterski:2020pdk ;
Casali:2020vuy ; Casali:2020uvr ; Campiglia:2021srh ; Godazgar:2021iae ;
Adamo:2021dfg ; Gonzo:2022tjm ; Huang:2019cja ; Alawadhi:2019urr ;
Banerjee:2019saj ; Monteiro:2022lwm ; Guevara:2022qnm ; Nagy:2022xxs . In
particular, refs. Monteiro:2022lwm ; Guevara:2022qnm  identified the property
of associativity in chiral algebras, including the appearance of
$w_{1+\infty}$ in self-dual gravity, with the algebraic structure of the
colour-kinematics duality, which is very simple in the self-dual sector
Monteiro:2011pc . This extends to the Moyal deformations and also applies, as
we will discuss here building on Ponomarev:2017nrr , to the chiral higher-spin
theories.

The structure of this paper is as follows. In section 2, we describe how
Moyal-deformed self-dual gravity/Yang-Mills generate chiral theories of
higher-spin fields. In section 3, we discuss the related double-copy
structures of Moyal deformations and of chiral higher-spin theories, and the
vanishing of the tree amplitudes that is associated to classical
integrability. The celestial chiral algebras arising from the various theories
considered are presented in section 4. Section 5 provides a brief overview of
loop amplitudes in these theories. The paper concludes with a discussion in
section 6.

##  2 Chiral higher-spin theories from Moyal deformations

In this section, we discuss the use of Moyal deformations of self-dual gravity
and self-dual Yang-Mills theory as generating theories for the chiral higher-
spin theories developed in Metsaev:1991mt ; Metsaev:1991nb ; Ponomarev:2016lrm
; Skvortsov:2018jea , and other related chiral higher-spin theories. Our
light-cone-gauge approach is closely related to covariant approaches followed
in Krasnov:2021nsq ; Tran:2021ukl ; Skvortsov:2022syz ; Sharapov:2022faa ;
Adamo:2022lah ; Sharapov:2022wpz ; Sharapov:2022nps ; Hahnel:2016ihf ;
Adamo:2016ple  and especially Tran:2022tft , and builds on observations in
Ponomarev:2017nrr  for the symmetry algebra of the interactions. We follow the
notation of Monteiro:2022lwm , which is useful for connecting the discussion
to the literature on the double copy later in the paper.

###  2.1 chs($\alpha$) from Moyal-SDG

The Moyal deformation of self-dual gravity (Moyal-SDG) STRACHAN199263  is
defined in terms of the light-cone gauge formulation of SDG Plebanski:1975wn ;
Siegel:1992wd . We employ light-cone coordinates $(u,v,w,{\bar{w}})$, with the
wave operator
$\,\square:=2(-\partial_{u}\partial_{v}+\partial_{w}\partial_{\bar{w}})\,$.
The equation of motion of Moyal-SDG is111For notational simplicity, we
suppress the gravitational coupling constant $\kappa$, which multiplies the
interaction term. This is achieved by a rescaling of $\phi$. All the gravity-
like theories in this paper inherit this coupling constant. The analogous
statement is true for the Yang-Mills-like theories to be seen later.

$\displaystyle\square\phi+\\{\partial_{u}\phi,\partial_{w}\phi\\}^{M}=0\,.$
(1)

The difference with respect to SDG is that the latter’s Poisson bracket

$\displaystyle\\{f,g\\}:=f\buildrel{\leftrightarrow}\over{P}g\,,\qquad\buildrel{\leftrightarrow}\over{P}\;=\;\buildrel{\leftarrow}\over{\partial}_{u}\buildrel{\rightarrow}\over{\partial}_{w}-\buildrel{\leftarrow}\over{\partial}_{w}\buildrel{\rightarrow}\over{\partial}_{u}\,,$
(2)

is substituted by its deformation into the Moyal bracket

$\displaystyle\\{f,g\\}^{M}:=\frac{1}{2\alpha}(f\star g-g\star
f)=\frac{1}{\alpha}\,f\sinh(\alpha\\!\buildrel{\leftrightarrow}\over{P})\,g\,,$
(3)

defined from the Moyal product $f\star
g:=f\exp(\alpha\\!\buildrel{\leftrightarrow}\over{P})\,g$ , which is
associative but not commutative. The parameter $\alpha$ characterises the
deformation, such that $\\{f,g\\}^{M}\to\\{f,g\\}$ as $\alpha\to 0$ . The
Moyal bracket is the most general Lie bracket of functions of two variables
Fletcher:1990ib , here $(u,w)$.222Notice that the chosen non-commutative plane
is complex in Lorentzian signature. In fact, so are all the chiral-type
theories we consider in this paper. A more natural setting is split signature,
where the coordinates $w$ and $\bar{w}$ are real and independent, and so are
the two chiralities of the fields, e.g. $\phi_{h}$ and $\phi_{-h}$ to be seen
below, where $h$ is the helicity.

It was suggested in Monteiro:2022lwm  that $\phi$ may be interpreted as a
composite field of chiral higher-spin fields. This is closely related to
discussions in Ponomarev:2017nrr  and also to a large body of work concerning
the symmetries of higher-spin theories. Here, we aim to make that suggestion
more concrete. Suppose that we write

$\displaystyle\phi=\zeta^{2}\sum_{h\in{\mathbb{Z}}}\zeta^{-h}\,\phi_{h}\,,$
(4)

where $\phi_{h}$ is a helicity-$h$ field and $\zeta$ is a parameter that will
allow us to project into specific helicity spaces. Now, consider the Laurent
expansion of (1) in powers of $\zeta$ after substituting
$\alpha\mapsto\alpha\,\zeta\,$,

$\displaystyle 0$
$\displaystyle=\square\phi+\phi\,\frac{\buildrel{\leftrightarrow}\over{P}\sinh(\alpha\,\zeta\\!\buildrel{\leftrightarrow}\over{P})}{\alpha\,\zeta}\,\phi=\square\phi+\phi\,\Bigg{(}\,\sum_{\sigma\geq
1}\frac{(\alpha\,\zeta\buildrel{\leftrightarrow}\over{P})^{2\sigma}}{\alpha^{2}\,\zeta^{2}\,(2\sigma-1)!}\,\Bigg{)}\,\phi$
$\displaystyle=\zeta^{2}\,\sum_{h}\zeta^{-h}\left(\square\phi_{h}+\\!\\!\sum_{\begin{subarray}{c}h_{1},h_{2}\\\
\text{even}\;h_{1}+h_{2}-h>0\end{subarray}}\\!\\!\phi_{h_{1}}\;\frac{(\alpha\buildrel{\leftrightarrow}\over{P})^{h_{1}+h_{2}-h}}{\alpha^{2}\,(h_{1}+h_{2}-h-1)!}\;\phi_{h_{2}}\right)\,.$
(5)

Imposing that the equations of motion hold independently for each coefficient
in the $\zeta$-expansion, we obtain precisely the equations of motion of the
chiral higher-spin theory of Ponomarev:2016lrm ; Metsaev:1991mt ;
Metsaev:1991nb , which we will name chs($\alpha$).333The helicity fields in
these references are translated into our conventions as
$(\partial_{u})^{h}\,\phi_{h}$ in order to simplify the expressions. This
causes the interactions to appear to have higher order in derivatives, e.g.
for self-dual gravity in the first line of (2.1). In fact, we can also obtain
the action for this theory, by introducing a projector
${\mathcal{P}}_{(\zeta)}$ into the helicity-0 part that acts linearly as

$\displaystyle{\mathcal{P}}_{(\zeta)}\,\zeta^{n}=\delta_{n,0}\,\qquad\forall
n\in{\mathbb{Z}}\,,$ (6)

and writing444It would appear to be more convenient here to define
${\mathcal{P}}_{(\zeta)}\,\zeta^{n}=\delta_{n,4}$, but other choices are more
convenient for later examples. So we chose instead to make the definition (6)
and multiply by $\zeta^{-4}$ in the first line. Any such definition can be
enforced via a contour integral.

$\displaystyle S_{\text{chs}(\alpha)}(\phi_{h})$
$\displaystyle={\mathcal{P}}_{(\zeta)}\int
d^{4}x\Bigg{(}\frac{1}{2}\,\phi\,\square\,\phi+\frac{1}{3}\,\frac{\left(\phi\buildrel{\leftrightarrow}\over{P}\sinh(\alpha\,\zeta\\!\buildrel{\leftrightarrow}\over{P})\,\phi\right)}{\alpha\,\zeta}\;\phi\Bigg{)}\zeta^{-4}$
$\displaystyle=\int
d^{4}x\Bigg{(}\frac{1}{2}\sum_{h}\phi_{-h}\,\square\,\phi_{h}\,+\frac{1}{3}\\!\\!\sum_{\begin{subarray}{c}h_{1},h_{2},h_{3}\\\
\text{even}\;h_{1}+h_{2}+h_{3}>0\end{subarray}}\\!\\!\frac{\big{(}\phi_{h_{1}}\big{(}\alpha\buildrel{\leftrightarrow}\over{P}\big{)}^{h_{1}+h_{2}+h_{3}}\phi_{h_{2}}\big{)}}{\alpha^{2}\,(h_{1}+h_{2}+h_{3}-1)!}\;\phi_{h_{3}}\Bigg{)}\,.$
(7)

Simpler versions of a chiral higher-spin theory have been considered in the
literature. It is possible to restrict the theory above to admit only even
spins. Independently, it is possible to simplify the interactions. Suppose
that we apply (4) to SDG, rather than to its Moyal deformation:

$\displaystyle 0$
$\displaystyle=\square\phi+\phi\,\buildrel{\leftrightarrow}\over{P}{}^{2}\,\phi$
$\displaystyle=\zeta^{2}\,\sum_{h}\zeta^{-h}\Bigg{(}\square\phi_{h}+\\!\\!\sum_{\begin{subarray}{c}h_{1},h_{2}\\\
h_{1}+h_{2}-h=2\end{subarray}}\\!\\!\phi_{h_{1}}\,\buildrel{\leftrightarrow}\over{P}{}^{2}\,\phi_{h_{2}}\Bigg{)}\,.$
(8)

This gives a two-derivative chiral higher-spin theory introduced in
Ponomarev:2017nrr  as a ‘contraction’ of the more involved theory of
Ponomarev:2016lrm  we saw above. The action can be obtained analogously to
(2.1), keeping only the interaction terms obeying $h_{1}+h_{2}+h_{3}=2$ . In
fact, naming the theory with action (2.1) chs($\alpha$), this simpler theory
is just chs($0$). So even undeformed SDG generates via (4) a theory of higher
spins.

We can also consider an even simpler chiral higher-spin theory, which was
called ‘higher-spin SDG’ in Krasnov:2021nsq . While the theories above are all
higher-spin generalisations of SDG in some sense, this theory has the closest
structure, because it possesses only $(+^{s_{1}}+^{s_{2}}-^{s_{3}})$ vertices,
for spins $s_{i}\geq 2$; we recall that SDG has a single vertex
$(+^{2}+^{2}-^{2})$. In this case, we work with the generating fields

$\displaystyle\phi=\zeta^{2}\sum_{s\geq
2}\zeta^{-s}\,\phi_{s}\,,\qquad\quad\bar{\phi}=\zeta^{-2}\sum_{s\geq
2}\zeta^{s}\,\phi_{-s}\,,$ (9)

to obtain the action

$\displaystyle S_{\text{hsSDG}}(\phi_{\pm s}|_{s\geq 2})$
$\displaystyle={\mathcal{P}}_{(\zeta)}\int
d^{4}x\;\;\bar{\phi}\,\left(\square\phi+\phi\,\buildrel{\leftrightarrow}\over{P}{}^{2}\,\phi\right)$
$\displaystyle=\int d^{4}x\sum_{s\geq
2}\phi_{-s}\Bigg{(}\square\phi_{s}+\\!\\!\sum_{\begin{subarray}{c}s_{1},s_{2}\geq
2\\\
s_{1}+s_{2}-s=2\end{subarray}}\\!\\!\phi_{s_{1}}\,\buildrel{\leftrightarrow}\over{P}{}^{2}\,\phi_{s_{2}}\Bigg{)}\,.$
(10)

###  2.2 gl-chs($\alpha$) from Moyal-SDYM

Here we consider the ‘gluonic’ chiral higher-spin theory, with U(N)-valued
fields, that is associated to Moyal-deformed self-dual Yang-Mills theory
(Moyal-SDYM). The equation of motion of Moyal-SDYM is Takasaki:2000vs ;
Nekrasov:1998ss

$\displaystyle\square\Psi+[\partial_{u}\Psi,\partial_{w}\Psi]^{M}=0\,.$ (11)

For U(N)-valued $A$ and $B$, we define

$\displaystyle{}^{M}:=A\star B-B\star
A=A\,\exp(\alpha\\!\buildrel{\leftrightarrow}\over{P})\,B-B\,\exp(\alpha\\!\buildrel{\leftrightarrow}\over{P})\,A\;\stackrel{{\scriptstyle\alpha\to
0}}{{\longrightarrow}}\;[A,B]\,.$ (12)

The equation (11) can be written as

$\displaystyle\square\Psi^{\text{a}}+f^{\text{a}\text{b}\text{c}}\,\Psi^{\text{b}}\buildrel{\leftrightarrow}\over{P}\cosh(\alpha\\!\buildrel{\leftrightarrow}\over{P})\,\Psi^{\text{c}}+d^{\text{a}\text{b}\text{c}}\,\Psi^{\text{b}}\buildrel{\leftrightarrow}\over{P}\sinh(\alpha\\!\buildrel{\leftrightarrow}\over{P})\,\Psi^{\text{c}}=0\,,$
(13)

where we denote

$\displaystyle
T^{\text{a}}T^{\text{b}}-T^{\text{b}}T^{\text{a}}=f^{\text{a}\text{b}\text{c}}\,T^{\text{c}}\,,\qquad
T^{\text{a}}T^{\text{b}}+T^{\text{b}}T^{\text{a}}=d^{\text{a}\text{b}\text{c}}\,T^{\text{c}}\,.$
(14)

Analogously to (4), we consider the generating field

$\displaystyle\Psi=\zeta\sum_{h=-\infty}^{\infty}\zeta^{-h}\,\Psi_{h}\,,$ (15)

where $\Psi_{h}$ is a helicity-$h$ field. The Laurent expansion of (1) in
powers of $\zeta$ after substituting $\alpha\mapsto\alpha\,\zeta\,$ is

$\displaystyle 0$
$\displaystyle=\square\Psi^{\text{a}}+f^{\text{a}\text{b}\text{c}}\,\Psi^{\text{b}}\buildrel{\leftrightarrow}\over{P}\cosh(\alpha\,\zeta\\!\buildrel{\leftrightarrow}\over{P})\,\Psi^{\text{c}}+d^{\text{a}\text{b}\text{c}}\,\Psi^{\text{b}}\buildrel{\leftrightarrow}\over{P}\sinh(\alpha\,\zeta\\!\buildrel{\leftrightarrow}\over{P})\,\Psi^{\text{c}}$
$\displaystyle=\zeta\sum_{h}\zeta^{-h}\Bigg{(}\square\Psi_{h}^{\text{a}}+\\!\sum_{\begin{subarray}{c}h_{1},h_{2}\\\
h_{1}+h_{2}-h>0\end{subarray}}\\!\\!t^{\text{a}\text{b}\text{c}}_{h_{1}+h_{2}-h}\,\frac{\Psi_{h_{1}}^{\text{b}}\,\big{(}\alpha\\!\buildrel{\leftrightarrow}\over{P}\big{)}^{h_{1}+h_{2}-h}\,\Psi_{h_{2}}^{\text{c}}}{\alpha\,(h_{1}+h_{2}-h-1)!}\Bigg{)}\,,$
(16)

where we define, for convenience,

$\displaystyle t^{\text{a}\text{b}\text{c}}_{\Sigma
h}=f^{\text{a}\text{b}\text{c}}\;\;\text{for odd}\;\Sigma h\,,\qquad
t^{\text{a}\text{b}\text{c}}_{\Sigma
h}=d^{\text{a}\text{b}\text{c}}\;\;\text{for even}\;\Sigma h\,.$ (17)

We obtain the equations of motion of the gluonic chiral higher-spin theory
considered in Skvortsov:2018jea ; Skvortsov:2020wtf . The action is obtained
from

$\displaystyle S_{\text{gl-chs}(\alpha)}(\Psi_{h})={\mathcal{P}}_{(\zeta)}\int
d^{4}x\;\;\text{tr}\Bigg{(}\frac{1}{2}\,\Psi\,\square\,\Psi+\frac{1}{3}\,\Psi\,[\partial_{u}\Psi,\partial_{w}\Psi]^{M}\Bigg{)}\zeta^{-2}$
$\displaystyle\quad=\int
d^{4}x\;\Bigg{(}\frac{1}{2}\sum_{h}\Psi^{\text{a}}_{-h}\,\square\,\Psi^{\text{a}}_{h}\,+\frac{1}{3}\\!\\!\sum_{\begin{subarray}{c}h_{1},h_{2},h_{3}\\\
h_{1}+h_{2}+h_{3}>0\end{subarray}}\\!\\!t^{\text{a}\text{b}\text{c}}_{h_{1}+h_{2}+h_{3}}\,\frac{\big{(}\Psi_{h_{1}}^{\text{a}}\,\big{(}\alpha\\!\buildrel{\leftrightarrow}\over{P}\big{)}^{h_{1}+h_{2}+h_{3}}\,\Psi_{h_{2}}^{\text{b}}\big{)}}{\alpha\,(h_{1}+h_{2}+h_{3}-1)!}\;\Psi_{h_{3}}^{\text{c}}\Bigg{)}\,.$
(18)

If we apply (15) to SDYM, rather than to its Moyal deformation, we obtain

$\displaystyle 0$
$\displaystyle=\square\Psi^{\text{a}}+f^{\text{a}\text{b}\text{c}}\,\Psi^{\text{b}}\buildrel{\leftrightarrow}\over{P}\,\Psi^{\text{c}}$
$\displaystyle=\zeta\sum_{h}\zeta^{-h}\Bigg{(}\square\Psi_{h}^{\text{a}}+f^{\text{a}\text{b}\text{c}}\\!\\!\sum_{\begin{subarray}{c}h_{1},h_{2}\\\
h_{1}+h_{2}-h>0\end{subarray}}\\!\\!\Psi_{h_{1}}^{\text{b}}\,\buildrel{\leftrightarrow}\over{P}\,\Psi_{h_{2}}^{\text{c}}\Bigg{)}\,,$
(19)

If we call the theory (2.2) gl-chs($\alpha$), then the theory with the
equations of motion given in (2.2) is simply gl-chs(0). It was first
considered in Ponomarev:2017nrr , and it can be truncated to admit only odd
spins, unlike gl-chs($\alpha$).

Finally, again analogously to the gravity case, there is a theory even more
similar to SDYM, which was called ‘higher-spin SDYM’ in Krasnov:2021nsq . The
generating fields are

$\displaystyle\Psi=\zeta\sum_{s\geq
1}\zeta^{-s}\,\Psi_{s}\,,\qquad\quad\bar{\Psi}=\zeta^{-1}\sum_{s\geq
1}\zeta^{s}\,\Psi_{-s}\,,$ (20)

and the action of this theory is

$\displaystyle S_{\text{hsSDYM}}(\Psi_{\pm s}|_{s\geq 1})$
$\displaystyle={\mathcal{P}}_{(\zeta)}\int
d^{4}x\;\;\text{tr}\;\;\bar{\Psi}\left(\square\Psi+[\partial_{u}\Psi,\partial_{w}\Psi]\right)$
$\displaystyle=\int d^{4}x\sum_{s\geq
1}\Psi^{\text{a}}_{-s}\Bigg{(}\square\Psi_{s}^{\text{a}}+f^{\text{a}\text{b}\text{c}}\\!\\!\sum_{\begin{subarray}{c}s_{1},s_{2}\geq
1\\\
s_{1}+s_{2}-s=1\end{subarray}}\\!\\!\Psi^{\text{b}}_{s_{1}}\,\buildrel{\leftrightarrow}\over{P}\,\Psi^{\text{c}}_{s_{2}}\Bigg{)}\,.$
(21)

###  2.3 Useful table of chiral higher-spin theories

For the reader’s convenience, we list the chiral higher-spin theories
discussed above in table 1.

Table 1: Table of chiral higher-spin theories. The generation technique is
described in section 2. The vertices are discussed in section 3. The actions
appear in section 2, but their relevance is discussed in section 5.
Truncations to even spins are allowed in chs($\alpha$), chs($0$) and hsSDG.
Truncations to odd spins are allowed in gl-chs($0$) and hsSDYM.
Chiral higher-spin theory | Vertices | Generating theory
---|---|---
chs($\alpha$) all spins |  $\displaystyle\frac{\big{(}\alpha X(k_{1},k_{2})\big{)}^{h_{1}+h_{2}+h_{3}}}{\alpha^{2}(h_{1}+h_{2}+h_{3}-1)!}$ even $h_{1}+h_{2}+h_{3}>0$ |  Moyal-SDG action($\phi$)
chs($0$) all spins |  $X(k_{1},k_{2})^{2}$ $h_{1}+h_{2}+h_{3}=2$ |  SDG action($\phi$)
hsSDG $s\geq 2$ |  $X(k_{1},k_{2})^{2}$ $s_{1}+s_{2}-s_{3}=2$ |  SDG action($\phi,\bar{\phi}$)
gl-chs($\alpha$) all spins |  $\displaystyle\frac{\big{(}\alpha X(k_{1},k_{2})\big{)}^{h_{1}+h_{2}+h_{3}}\,t^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}_{h_{1}+h_{2}+h_{3}}}{\alpha\,(h_{1}+h_{2}+h_{3}-1)!}$ $h_{1}+h_{2}+h_{3}>0$ |  Moyal-SDYM action($\Psi$)
gl-chs($0$) all spins |  $X(k_{1},k_{2})\,f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}$ $h_{1}+h_{2}+h_{3}=1$ |  SDYM action($\Psi$)
hsSDYM $s\geq 1$ |  $X(k_{1},k_{2})\,f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}$ $s_{1}+s_{2}-s_{3}=1$ |  SDYM action($\Psi,\bar{\Psi}$)

##  3 Double-copy structure and scattering amplitudes

In this section, building on ref. Ponomarev:2017nrr , we will see that the
chiral higher-spin theories considered above inherit their double-copy
structure from the Moyal-deformed theories. This leads to vanishing tree-level
amplitudes beyond 3-point scattering.

Before proceeding, let us recall the double-copy structure of SDG and SDYM
Monteiro:2011pc ; Boels:2013bi . These theories possess only a 3-point vertex,
given by555The equations of motion for each theory were given in the first
lines of (2.1) and (2.2).

$\displaystyle V_{\text{SDG}}=X(k_{1},k_{2})^{2}\,,\qquad
V_{\text{SDYM}}=X(k_{1},k_{2})\,f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}\,,$
(22)

where

$\displaystyle X(k_{1},k_{2}):=k_{1w}k_{2u}-k_{1u}k_{2w}=-X(k_{2},k_{1})\,.$
(23)

These are bosonic vertices: notice that, due to momentum conservation,
$X(k_{i},k_{j})$ is completely antisymmetric in $\\{k_{1},k_{2},k_{3}\\}$. The
double-copy structure of these theories is based on the factorisation of the
vertices into the structure constants of two Lie algebras: the self-dual-type
‘kinematic algebra’ $X(k_{1},k_{2})$, and the colour Lie algebra in the case
of $f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}$. The former is the algebra of
area-preserving diffeomorphisms in the $(u,w)$ plane,

$\displaystyle[L_{k_{1}},L_{k_{2}}]=X(k_{1},k_{2})\,L_{k_{1}+k_{2}}\,,$ (24)

generated by Hamiltonian vector fields

$\displaystyle L_{k}:=\\{e^{i\,k\cdot x},\cdot\\}=i\,e^{i\,k\cdot
x}(k_{u}\partial_{w}-k_{w}\partial_{u})=e^{i\,k\cdot
x}\,k_{u}\,\epsilon^{+}\\!\cdot\partial$ (25)

associated to positive helicity. In particular, the rightmost equality defines
a standard positive-helicity polarisation vector.

For all the theories considered in this paper, we will find the double-copy
structure

$\displaystyle V=X(k_{1},k_{2})\,C_{I_{1}I_{2}}{}^{I_{3}}\,,$ (26)

where $C_{I_{1}I_{2}}{}^{I_{3}}$ are the structure constants of a Lie algebra
specific to the theory. This fact was known for many of the theories studied
Monteiro:2011pc ; Ponomarev:2017nrr ; Chacon:2020fmr , to which we will add
the cases of Moyal-SDYM and the chiral higher-spin theory gl-chs($\alpha$),
related via (2.2). Our approach emphasises how the double-copy structure of
chiral higher-spin theories follows from the Moyal deformation.

###  3.1 Kinematic algebras in Moyal-SDG

The 3-point vertex in Moyal-SDG is

$\displaystyle V_{\text{Moyal-SDG}}=X(k_{1},k_{2})\,X^{M}(k_{1},k_{2})\,,$
(27)

where

$\displaystyle
X^{M}(k_{1},k_{2}):=\frac{1}{\alpha}\sinh\big{(}\alpha\,X(k_{1},k_{2})\big{)}\,.$
(28)

So the double-copy structure is that we have one copy of $X(k_{1},k_{2})$ and
one copy of its Moyal deformation $X^{M}(k_{1},k_{2})$ Chacon:2020fmr . The
former arises from a Poisson bracket and the latter arises from the associated
Moyal bracket.666The counterpart of the generators (25) in the deformed
algebra is ${L_{k}^{M}}:=\\{e^{i\,k\cdot x},\cdot\\}^{M}=\frac{e^{i\,k\cdot
x}}{\alpha}\sinh\big{(}\alpha\,k_{u}\,\epsilon^{+}\\!\cdot\partial\big{)}$ .
Alternatively, the generators ${E_{k}^{M}}:=\frac{e^{i\,k\cdot
x}}{2\alpha}\exp\big{(}\alpha\,k_{u}\,\epsilon^{+}\\!\cdot\partial\big{)}$
lead to the same structure constants. The latter choice is singular if we take
$\alpha\to 0$.

Consider the Jacobi identity for the Moyal Lie bracket,

$\displaystyle\sinh\big{(}\alpha\,X(k_{1},k_{2})\big{)}\sinh\big{(}\alpha\,X(k_{1}+k_{2},k_{3})\big{)}+\text{cyc}(123)=0\,,$
(29)

where $\text{cyc}(123)$ denotes the other two cyclic permutations of legs 123.
Introducing $k_{4}$ such that $\sum_{i=1}^{4}k_{i}=0$, the expansion in
$\alpha$ is

$\displaystyle 0$
$\displaystyle=\sinh\big{(}\alpha\,X(k_{1},k_{2})\big{)}\sinh\big{(}\alpha\,X(k_{3},k_{4})\big{)}+\text{cyc}(123)$
$\displaystyle=\sum_{\begin{subarray}{c}\Lambda\geq 4\\\
\text{even\,}\Lambda\end{subarray}}\alpha^{\Lambda-2}\left(\,\sum_{\begin{subarray}{c}n=1\\\
\text{odd\,}n\end{subarray}}^{\Lambda-2}\,\frac{X(k_{1},k_{2})^{\Lambda-2-n}}{(\Lambda-2-n)!}\;\frac{X(k_{3},k_{4})^{n}}{n!}\;+\;\text{cyc}(123)\right).$
(30)

Now, for the chiral higher-spin theory chs($\alpha$), with action (2.1), the
vertex is

$\displaystyle
V_{\text{chs}(\alpha)}=X(k_{1},k_{2})\cdot\frac{1}{\alpha}\,\frac{\big{(}\alpha
X(k_{1},k_{2})\big{)}^{h_{1}+h_{2}+h_{3}-1}}{(h_{1}+h_{2}+h_{3}-1)!}=X(k_{1},k_{2})\,\tilde{V}_{\text{chs}(\alpha)}\,,$
(31)

where the helicities $h_{i}$ are taken to be incoming with respect to the
vertex. In this expression, we already show the two copies of structure
constants, as originally identified in Ponomarev:2017nrr . This should be
compared to (27). The Jacobi relation for the algebra associated to
$\tilde{V}_{\text{chs}(\alpha)}$ takes the form

$\displaystyle\sum_{h_{I}}{}^{\prime}\;\frac{X(k_{1},k_{2})^{h_{1}+h_{2}+h_{I}-1}}{(h_{1}+h_{2}+h_{I}-1)!}\;\frac{X(k_{3},k_{4})^{h_{3}+h_{4}-h_{I}-1}}{(h_{3}+h_{4}-h_{I}-1)!}\;+\;\text{cyc}(123)$
$\displaystyle=\frac{1}{2\,(\Lambda-2)!}\,\Big{[}\big{(}X(k_{1},k_{2})+X(k_{3},k_{4})\big{)}^{\Lambda-2}-\big{(}X(k_{1},k_{2})-X(k_{3},k_{4})\big{)}^{\Lambda-2}\Big{]}\,+\,\text{cyc}(123)$
$\displaystyle=0\,,$ (32)

where $\Lambda:=\sum_{i=1}^{4}h_{i}$ must be even, and moreover must satisfy
$\Lambda\geq 4$ because the sum over helicities in each vertex must be
positive and, due to Bose symmetry, even. Momentum conservation leads to a
cancellation in pairs in the final equality. The sum over $h_{I}$ in the first
line is restricted (hence the prime) so that both exponents are positive and
odd; so the ‘odd part’ of the binomial formula is used in the first equality.
The point is that the first line of (32) is identical to the coefficient with
$\Lambda=\sum_{i=1}^{4}h_{i}$ in the last line of (30). The Jacobi relation
for $X^{M}(k_{1},k_{2})$ encodes the Jacobi relation for
$\tilde{V}_{\text{chs}(\alpha)}$, as noticed already in Ponomarev:2017nrr .
This follows from

$\displaystyle X^{M}(k_{1},k_{2})=\sum_{\sigma\geq 1}\,\frac{\big{(}\alpha
X(k_{1},k_{2})\big{)}^{2\sigma-1}}{\alpha(2\sigma-1)!}=\sum_{\sigma\geq
1}\,\tilde{V}_{\text{chs}(\alpha)}(h_{1}+h_{2}+h_{3}=2\sigma)\,.$ (33)

The conclusion here is that the double-copy structure of Moyal-SDG encodes the
double-copy structure of the associated chiral higher-spin theory.

For the simpler chiral higher-spin theory chs(0), where we have the
restriction $h_{1}+h_{2}+h_{3}=2$, the double-copy structure of the vertex is
the same as for SDG, in (22). Indeed, the expression for the vertex is the
same, only the particle content is extended. The same conclusion applies to
the theory hsSDG, with action (2.1), which is obtained from chs(0) by keeping
only spins $s_{i}\geq 2$ and vertices of the form
$(+^{s_{1}}+^{s_{2}}-^{s_{3}})$.

###  3.2 Kinematic and colour-kinematic algebras in Moyal-SDYM

The 3-point vertex in Moyal-SDYM, with equation of motion (13), is

$\displaystyle V_{\text{Moyal-
SDYM}}=X(k_{1},k_{2})\Big{(}\\!\cosh\big{(}\alpha
X(k_{1},k_{2})\big{)}\,f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}+\sinh\big{(}\alpha
X(k_{1},k_{2})\big{)}\,d^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}\Big{)}\,.$
(34)

The double-copy structure is similar to the previous cases, with the Jacobi
relation for the ‘right algebra’ taking the form777Generators for the algebra
are provided by ${E_{k}^{\text{a}}{}^{M}}:={e^{i\,k\cdot
x}}T^{\text{a}}\exp\big{(}\alpha\,k_{u}\,\epsilon^{+}\\!\cdot\partial\big{)}$.
Notice that there is a restriction on the colour group, which we take to be
U(N).

$\displaystyle
f^{\text{a}_{1}\text{a}_{2}\text{b}}f^{\text{b}\text{a}_{3}\text{a}_{4}}\cosh\big{(}\alpha
X(k_{1},k_{2})\big{)}\cosh\big{(}\alpha X(k_{3},k_{4})\big{)}$
$\displaystyle+\,$ $\displaystyle
f^{\text{a}_{1}\text{a}_{2}\text{b}}d^{\text{b}\text{a}_{3}\text{a}_{4}}\cosh\big{(}\alpha
X(k_{1},k_{2})\big{)}\sinh\big{(}\alpha X(k_{3},k_{4})\big{)}$
$\displaystyle+\,$ $\displaystyle
d^{\text{a}_{1}\text{a}_{2}\text{b}}f^{\text{b}\text{a}_{3}\text{a}_{4}}\sinh\big{(}\alpha
X(k_{1},k_{2})\big{)}\cosh\big{(}\alpha X(k_{3},k_{4})\big{)}$
$\displaystyle+\,$ $\displaystyle
d^{\text{a}_{1}\text{a}_{2}\text{b}}d^{\text{b}\text{a}_{3}\text{a}_{4}}\sinh\big{(}\alpha
X(k_{1},k_{2})\big{)}\sinh\big{(}\alpha X(k_{3},k_{4})\big{)}$
$\displaystyle+\,$ $\displaystyle\text{cyc}(123)=0\,.$ (35)

This is an interesting instance of the double copy. Instead of the usual
factorisation between colour and kinematics as in (22), one of the algebras
mixes colour and kinematics! In the limit $\alpha\to 0$, we recover the usual
colour Lie algebra. This type of deformation of the colour algebra appears in
non-commutative Yang-Mills theory, e.g. Armoni:2000xr ; see Huang:2010fc  for
its appearance in a context close to ours.888 For a different recent
appearance of $d^{\text{a}\text{b}\text{c}}$ in the double-copy literature,
see Carrasco:2021ptp ; Carrasco:2022jxn .

The associated gluonic chiral higher-spin theory gl-chs($\alpha$), with action
(2.2), has the vertex

$\displaystyle V_{\text{gl-
chs}(\alpha)}=X(k_{1},k_{2})\cdot\frac{t^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}_{h_{1}+h_{2}+h_{3}}\,\big{(}\alpha
X(k_{1},k_{2})\big{)}^{h_{1}+h_{2}+h_{3}-1}}{(h_{1}+h_{2}+h_{3}-1)!}=X(k_{1},k_{2})\,\tilde{V}_{\text{gl-
chs}(\alpha)}\,,$ (36)

Similarly to the previous subsection, the relation (35) can be checked to
encode in its $\alpha$-expansion the Jacobi relation for $\tilde{V}_{\text{gl-
chs}}$.

For the simpler gluonic chiral higher-spin theory gl-chs(0), where we have the
restriction $h_{1}+h_{2}+h_{3}=1$, the double-copy structure of the vertex is
the same as for SDYM, in (22). The same conclusion applies to the theory
hsSDYM, with action (2.2), which is obtained from gl-chs(0) by keeping only
spins $s_{i}\geq 1$ and vertices of the form $(+^{s_{1}}+^{s_{2}}-^{s_{3}})$.

###  3.3 Comment on 3-point scattering amplitudes

We now discuss the interpretation of 3-point scattering amplitudes, focusing
on Moyal-SDG. Consider the chiral higher-spin vertex

$\displaystyle V_{\text{chs}(\alpha)}=\frac{\big{(}\alpha
X(k_{1},k_{2})\big{)}^{h_{1}+h_{2}+h_{3}}}{\alpha^{2}(h_{1}+h_{2}+h_{3}-1)!}\,.$
(37)

In our gauge choices and conventions, the 3-point amplitude is unhelpfully the
same for any triplet of external states with the same $h_{1}+h_{2}+h_{3}$. We
can backtrack slightly, and introduce the two following elements to compute
amplitudes in the spinor helicity-formalism: we define999We follow the
notation of refs. Boels:2013bi ; Monteiro:2022lwm .

$\displaystyle X(k_{1},k_{2})=\langle\eta|k_{1}k_{2}|\eta\rangle\,,$ (38)

and introduce the helicities factor for $n$ external particles

$\displaystyle\prod_{i=1}^{n}\,(-\langle\eta i\rangle^{-2})^{h_{i}}\,.$ (39)

For massless particles, $X(k_{1},k_{2})=\langle\eta 1\rangle[12]\langle
2\eta\rangle$. The rules here reduce to the previous ones for a certain choice
of the spinors. We can exemplify the rules by checking that the 3-point
amplitude is, up to a constant factor,

$\displaystyle-\frac{X(k_{1},k_{2})^{h_{1}+h_{2}+h_{3}}}{\prod_{i=1}^{3}\langle\eta
i\rangle^{2h_{i}}}$
$\displaystyle=-\frac{X(k_{1},k_{2})^{h_{1}+h_{2}-h_{3}}X(k_{2},k_{3})^{h_{2}+h_{3}-h_{1}}X(k_{3},k_{1})^{h_{3}+h_{1}-h_{2}}}{\prod_{i=1}^{3}\langle\eta
i\rangle^{2h_{i}}}$
$\displaystyle=[12]^{h_{1}+h_{2}-h_{3}}[23]^{h_{2}+h_{3}-h_{1}}[31]^{h_{3}+h_{1}-h_{2}}\,,$
(40)

as expected. Notice that, at 3 points, on-shell kinematics requires
complexified momenta, so the amplitude has no support on real momenta in
Lorentzian signature.

Now, the basic observation is that101010See also related discussions in
Ponomarev:2017nrr ; Ponomarev:2022atv ; Monteiro:2022lwm ; Bu:2022iak .

$\displaystyle V_{\text{Moyal-SDG}}=\sum_{\sigma\geq 1}\,\frac{\big{(}\alpha
X(k_{1},k_{2})\big{)}^{2\sigma}}{\alpha^{2}(2\sigma-1)!}=\sum_{\sigma\geq
1}\,V_{\text{chs($\alpha$)}}(h_{1}+h_{2}+h_{3}=2\sigma)\,.$ (41)

This exhibits the breaking of Lorentz symmetry in Moyal-SDG that arises from
the ($u,w$) plane of non-commutativity that is associated to $X(k_{1},k_{2})$.
Notice that each term in the sum over $\sigma$ can give rise to a Lorentz-
invariant 3-point amplitude if multiplied by appropriate external helicity
factors (i.e. such that $h_{1}+h_{2}+h_{3}=2\sigma$). There is no way,
however, for the sum of these terms to exhibit Lorentz invariance — understood
as independence on $|\eta\rangle$ and having homogeneous little-group scaling
in each particle. Still, the Lorentz non-invariance of Moyal-SDG is special,
as the vertex generates vertices of the Lorentz-invariant theory
chs($\alpha$).

###  3.4 Vanishing tree-level amplitudes

All the theories seen here have a 3-point vertex of the form (26). This
subsection has a single piece of information: following the argument given in
Monteiro:2022lwm , based on the double copy, the fact that all these theories
have the same ‘left algebra’ associated to $X(k_{1},k_{2})$ means that
$n$-point tree-level scattering amplitudes for $n>3$ vanish. For clarity, this
is an on-shell statement. In the case of the gluonic higher-spin theory gl-
chs($\alpha$), the vanishing was shown already in Skvortsov:2018jea , based on
Berends-Giele recursion for colour-ordered amplitudes. For the non-gluonic
theory chs($\alpha$), the amplitude was shown to vanish at 4 points in
Ponomarev:2016lrm , and our observation provides an all-multiplicity proof.
These proofs apply for generic kinematics. See Ponomarev:2022atv  for a recent
discussion of amplitudes supported on special (complexified) kinematics with
vanishing Mandelstam variables, which occurs also without higher spins
Witten:2003nn .

##  4 Celestial chiral algebras

Celestial holography aims to interpret scattering amplitudes in four spacetime
dimensions as correlation functions of a conformal-type field theory on the
two-dimensional celestial sphere Cheung:2016iub ; Pasterski:2016qvg ;
Pasterski:2017kqt ; Pasterski:2017ylz . The external states in a scattering
amplitude correspond to operator insertions on the celestial sphere. As the
figure below illustrates, the singular part of the operator product expansion
(OPE) between two such operators is fixed by the collinear behaviour of the
amplitudes Fan:2019emx ; Pate:2019lpp .

In this section, we will use the momentum-space approach (and conventions) of
Monteiro:2022lwm  to obtain the chiral celestial operator product expansions
relevant to the theories considered in this paper. Let us review this approach
and its results. We use the following parametrisation of massless momenta:

$\displaystyle
k_{A\dot{A}}=\lambda_{A}\tilde{\lambda}_{\dot{A}}\,,\quad\lambda_{A}=(1,z)\,,\quad\tilde{\lambda}_{\dot{A}}=(k_{u},k_{w})\,,$
(42)

so that the holomorphic coordinate on the celestial sphere is

$\displaystyle z=\frac{k_{\bar{w}}}{k_{u}}=\frac{k_{v}}{k_{w}}\,.$ (43)

With this choice, we identify the anti-holomorphic spinor bracket with the
self-dual kinematic algebra,

$\displaystyle:=\epsilon^{\dot{A}\dot{B}}\,\tilde{\lambda}_{1\,\dot{A}}\,\tilde{\lambda}_{2\,\dot{B}}=X(k_{1},k_{2})\,,$
(44)

and we can write the Mandelstam variables as

$\displaystyle s_{12}:=(k_{1}+k_{2})^{2}=(z_{1}-z_{2})\,{[12]}\,.$ (45)

Taking a ‘holomorphic collinear limit’ $z_{1}\to z_{2}$, we obtain for the OPE
in SDYM (which matches the $++$ OPE in full Yang-Mills)

$\displaystyle{\mathcal{O}}^{\text{a}_{1}}_{+}(k_{1})\,\,{\mathcal{O}}^{\text{a}_{2}}_{+}(k_{2})\,\sim\,\frac{{X(k_{1},k_{2})}\,{f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}}}{s_{12}}\,\,{\mathcal{O}}^{\text{a}_{3}}_{+}(k_{1}+k_{2})\,=\,\frac{{f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}}}{z_{1}-z_{2}}\,\,{\mathcal{O}}^{\text{a}_{3}}_{+}(k_{1}+k_{2})\,.$
(46)

The gravity counterpart is

$\displaystyle{\mathcal{O}}_{+}(k_{1})\,\,{\mathcal{O}}_{+}(k_{2})\,\sim\,\frac{{X(k_{1},k_{2})^{2}}}{s_{12}}\,\,{\mathcal{O}}_{+}(k_{1}+k_{2})\,=\,\frac{{X(k_{1},k_{2})}}{z_{1}-z_{2}}\,\,{\mathcal{O}}_{+}(k_{1}+k_{2})\,.$
(47)

The double-copy structure matches that of the celestial OPEs Monteiro:2022lwm
; Guevara:2022qnm : the $X$ algebra is associated to the chirality of the
OPEs, while the second algebra (colour for SDYM and $X$ for SDG) provides the
structure constants of the OPE, ensuring its associativity.

The ‘soft-generator’ OPEs are obtained by taking the soft limit at fixed $z$,
corresponding to $\tilde{\lambda}_{\dot{A}}=(k_{u},k_{w})\to(0,0)$. The
operators are expanded as

$\displaystyle{\mathcal{O}}^{\text{a}}_{+}(k)=\sum_{a,b=0}^{\infty}\frac{(ik_{u})^{a}}{a!}\,\frac{(ik_{w})^{b}}{b!}\;{\varpi^{\text{a}}_{a,b}}(z)\,,\quad\>\>{\mathcal{O}}_{+}(k)=\sum_{a,b=0}^{\infty}\frac{(ik_{u})^{a}}{a!}\,\frac{(ik_{w})^{b}}{b!}\;{\varpi_{a,b}}(z)\,.$
(48)

The soft generators lie in the ‘wedge’ $a,b\geq 0$. They lead to the soft OPEs
for SDYM,

$\displaystyle\varpi^{\text{a}_{1}}_{a,b}(z_{1})\,\varpi^{\text{a}_{2}}_{c,d}(z_{2})\,\sim\,\frac{f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}}{z_{1}-z_{2}}\,\,\varpi^{\text{a}_{3}}_{a+c,b+d}(z_{2})\,,$
(49)

and for SDG,

$\displaystyle\varpi_{a,b}(z_{1})\,\varpi_{c,d}(z_{2})\,\sim\,\frac{{ad-
bc}}{z_{1}-z_{2}}\,\,\varpi_{a+c-1,b+d-1}(z_{2})\,.$ (50)

We may also expand the soft generators in Laurent modes,

$\displaystyle\varpi^{\text{a}}_{a,b}(z)=\sum_{n\in{\mathbb{Z}}}\,\frac{\varpi^{\text{a}}_{a,b\,;\,n}}{z^{n+1}}\,,\qquad\varpi_{a,b}(z)=\sum_{n\in{\mathbb{Z}}}\,\frac{\varpi_{a,b\,;\,n}}{z^{n+1}}\,.$
(51)

The SDYM modes satisfy the algebra

$\displaystyle[\,\varpi^{\text{a}_{1}}_{a,b\,;\,n}\,,\,\varpi^{\text{a}_{2}}_{c,d\,;\,m}\,]=f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}\,\varpi^{\text{a}_{3}}_{a+c,b+d\,;\,n+m}\,,$
(52)

while for SDG we have

$\displaystyle[\,\varpi_{a,b\,;\,n}\,,\,\varpi_{c,d\,;\,m}\,]=(ad-
bc)\,\varpi_{a+c-1,b+d-1\,;\,n+m}\,.$ (53)

For SDYM, we have an affine Kac-Moody algebra with level zero, which was
identified in Fan:2019emx ; Pate:2019lpp ; see also He:2015zea  for previous
related work. For SDG, we have the loop algebra of the wedge subalgebra of
$w_{1+\infty}$, which was identified in Strominger:2021lvk 111111The
translation between the conventions in Strominger:2021lvk  and ours is
$w^{p}_{m}(z)=\frac{1}{2}\,\varpi_{p-1+m\,,\,p-1-m}(z)$ . building on
Guevara:2021abz ; see the twistor derivation and historical discussion in
Adamo:2021lrv .121212Despite the simplicity of these algebras of soft
generators, the status of the latter as local operators in a celestial
conformal field theory was questioned recently in Ball:2022bgg .

###  4.1 Moyal-SDG and $LW_{\wedge}$

We review now the case of Moyal-SDG, already analysed in Monteiro:2022lwm ;
Bu:2022iak . From the vertex (27), we have

$\displaystyle{\mathcal{O}}_{+}(k_{1})\,\,{\mathcal{O}}_{+}(k_{2})\,\sim\,\frac{{X(k_{1},k_{2})}X^{M}(k_{1},k_{2})}{s_{12}}\,\,{\mathcal{O}}_{+}(k_{1}+k_{2})\,=\,\frac{{X^{M}(k_{1},k_{2})}}{z_{1}-z_{2}}\,\,{\mathcal{O}}_{+}(k_{1}+k_{2})\,.$
(54)

Expanding as in (48), we obtain

$\displaystyle\varpi_{a,b}(z_{1})\,\varpi_{c,d}(z_{2})\,\sim\frac{1}{z_{1}-z_{2}}\;\times$
$\displaystyle\quad\sum_{s\geq
0}\frac{\alpha^{2s}}{(2s+1)!}\sum_{j=0}^{2s+1}(-1)^{j}\binom{2s+1}{j}\,[a]_{2s+1-j}[b]_{j}[c]_{j}[d]_{2s+1-j}\;{\varpi_{a+c-1-2s,b+d-1-2s}}(z_{2})\,,$
(55)

which is isomorphic to the loop algebra of the wedge subalgebra of
$W_{1+\infty}$. $W_{1+\infty}$, introduced in Pope:1989ew ; Pope:1989sr ;
Pope:1990kc  and reviewed in Pope:1991ig ; Shen:1992dd , is a higher-spin (in
the 2D notion of spin) extension of the Virasoro algebra; the latter is
generated by the Laurent modes of the (spin-2) chiral stress-energy tensor,
whereas $W_{1+\infty}$ is an extension to generators of all spins$\geq 1$. The
isomorphism in the wedge subalgebra between (4.1) and $W_{1+\infty}$ follows
from the results of Fairlie:1990wv , and was well explained in Bu:2022iak ,
where $LW_{\wedge}$ was used to denote the loop algebra in this class of
isomorphic wedge algebras. For $\alpha=0$, we recover the loop algebra of the
wedge subalgebra of $w_{1+\infty}$ as in (50). Refs. Mago:2021wje ;
Ren:2022sws  discussed related deformations of the SDG chiral algebra.

###  4.2 chs($\alpha$)

We introduce now the celestial algebra of the chiral higher-spin theory
chs($\alpha$), which is related to Moyal-SDG. We now have the vertex (31),
which leads to the OPE

$\displaystyle{\mathcal{O}}_{h_{1}}(k_{1})\,\,{\mathcal{O}}_{h_{2}}(k_{2})\,\sim\,\frac{1}{z_{1}-z_{2}}\sum_{\begin{subarray}{c}h_{3}\\\
\text{even}\;\Sigma
h>0\end{subarray}}\\!\frac{\big{(}\alpha\,X(k_{1},k_{2})\big{)}^{\Sigma
h-1}}{\alpha(\Sigma h-1)!}\,\,{\mathcal{O}}_{-h_{3}}(k_{1}+k_{2})\,,$ (56)

where $\Sigma h:=h_{1}+h_{2}+h_{3}$ . The OPE is associative due to the
double-copy structure of the vertex as before, with the Jacobi relation now
corresponding to (32). This proves an observation made in Ren:2022sws , where
it was noticed based on low-spin examples that the vertices of Metsaev that
define chs($\alpha$) lead to associativity.

The soft expansion of the OPE leads to the wedge-type algebra

$\displaystyle\varpi_{a,b}^{h_{1}}(z_{1})\,\varpi_{c,d}^{h_{2}}(z_{2})\,\sim\frac{1}{z_{1}-z_{2}}\sum_{\begin{subarray}{c}h_{3}\\\
\text{even}\;\Sigma h>0\end{subarray}}\\!\frac{\alpha^{\Sigma h-2}}{(\Sigma
h-1)!}\;\times$ $\displaystyle\quad\sum_{j=0}^{\Sigma
h-1}(-1)^{j}\binom{\Sigma h-1}{j}\,[a]_{\Sigma h-1-j}[b]_{j}[c]_{j}[d]_{\Sigma
h-1-j}\;{\varpi_{a+c+1-\Sigma h,b+d+1-\Sigma h}^{-h_{3}}}(z_{2})\,.$ (57)

###  4.3 hsSDG

For the theory hsSDG in (2.1), the vertex matches that of SDG but admits
higher spins ($s\geq 2$) such that $s_{1}+s_{2}-s_{3}=2$. This leads to an OPE
that is a straightforward extension of the SDG case,

$\displaystyle{\mathcal{O}}_{s_{1}}(k_{1})\,\,{\mathcal{O}}_{s_{2}}(k_{2})\,\sim\,\frac{X(k_{1},k_{2})}{z_{1}-z_{2}}\;{\mathcal{O}}_{s_{1}+s_{2}-2}(k_{1}+k_{2})\,.$
(58)

The soft expansion leads to the wedge-type algebra

$\displaystyle\varpi_{a,b}^{s_{1}}(z_{1})\,\varpi_{c,d}^{s_{2}}(z_{2})\,\sim\,\frac{ad-
bc}{z_{1}-z_{2}}\;{\varpi_{a+c-1,b+d-1}^{s_{1}+s_{2}-2}}(z_{2})\,.$ (59)

###  4.4 Moyal-SDYM and deformed Kac-Moody algebra

From the vertex (34) of Moyal-SDYM, we obtain the OPE

$\displaystyle{\mathcal{O}}_{+}^{a_{1}}(k_{1})\,\,{\mathcal{O}}_{+}^{a_{1}}(k_{2})\,\sim\,$
$\displaystyle\frac{1}{z_{1}-z_{2}}\,\Big{(}\\!\cosh\big{(}\alpha
X(k_{1},k_{2})\big{)}\,f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}+\sinh\big{(}\alpha
X(k_{1},k_{2})\big{)}\,d^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}\Big{)}\,{\mathcal{O}}_{+}^{a_{3}}(k_{1}+k_{2})\,.$
(60)

The associated soft OPE is the wedge-type algebra

$\displaystyle\varpi_{a,b}^{\text{a}_{1}}(z_{1})\,\varpi_{c,d}^{\text{a}_{2}}(z_{2})\,\sim\frac{1}{z_{1}-z_{2}}\;\sum_{s=0}^{\infty}\;\frac{\alpha^{s}}{s!}\;t^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}_{s+1}\;\times$
$\displaystyle\qquad\qquad\sum_{j=0}^{s}(-1)^{j}\binom{s}{j}\,[a]_{s-j}[b]_{j}[c]_{j}[d]_{s-j}\;{\varpi_{a+c-s,b+d-s}^{\text{a}_{3}}}(z_{2})\,,$
(61)

where $t^{\text{a}\text{b}\text{c}}_{s^{\prime}}$ was defined in (17). This is
a deformation of a Kac-Moody algebra analogous to the case of $LW_{\wedge}$
for Moyal-SDG. For $\alpha=0$, we recover the Kac-Moody algebra of SDYM in
(49). Refs. Mago:2021wje ; Ren:2022sws  discussed related deformations of the
SDYM chiral algebra.

###  4.5 gl-chs($\alpha$)

For the U(N)-valued chiral higher-spin theory gl-chs($\alpha$), which is
related to Moyal-SDYM, the vertex (36) leads to the OPE

$\displaystyle{\mathcal{O}}_{h_{1}}^{\text{a}_{1}}(k_{1})\,\,{\mathcal{O}}_{h_{2}}^{\text{a}_{2}}(k_{2})\,\sim\,\frac{1}{z_{1}-z_{2}}\,\sum_{\begin{subarray}{c}h_{3}\\\
\Sigma
h>0\end{subarray}}\frac{t^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}_{\Sigma
h}\,\big{(}\alpha X(k_{1},k_{2})\big{)}^{\Sigma h-1}}{(\Sigma
h-1)!}\,\,{\mathcal{O}}_{-h_{3}}^{\text{a}_{3}}(k_{1}+k_{2})\,,$ (62)

where $\Sigma h:=h_{1}+h_{2}+h_{3}$ . As in all the previous cases, the OPE is
associative due to the double-copy structure of the vertex. The soft expansion
of the OPE leads to the wedge-type algebra

$\displaystyle\varpi_{a,b}^{h_{1},\text{a}_{1}}(z_{1})\,\varpi_{c,d}^{h_{2},\text{a}_{2}}(z_{2})\,\sim\frac{1}{z_{1}-z_{2}}\,\sum_{\begin{subarray}{c}h_{3}\\\
\Sigma h>0\end{subarray}}\frac{\alpha^{\Sigma h-1}}{(\Sigma
h-1)!}\;t^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}_{\Sigma h}\;\times$
$\displaystyle\quad\sum_{j=0}^{\Sigma h-1}(-1)^{j}\binom{\Sigma
h-1}{j}\,[a]_{\Sigma h-1-j}[b]_{j}[c]_{j}[d]_{\Sigma
h-1-j}\;{\varpi_{a+c+1-\Sigma h,b+d+1-\Sigma
h}^{-h_{3},\text{a}_{3}}}(z_{2})\,.$ (63)

###  4.6 hsSDYM

Finally, for the theory hsSDYM in (2.2), the vertex matches that of SDYM but
admits higher spins ($s\geq 1$) such that $s_{1}+s_{2}-s_{3}=1$. This leads to
an OPE that is a straightforward extension of the SDYM case,

$\displaystyle{\mathcal{O}}_{s_{1}}^{\text{a}_{1}}(k_{1})\,\,{\mathcal{O}}_{s_{2}}^{\text{a}_{2}}(k_{2})\,\sim\,\frac{f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}}{z_{1}-z_{2}}\;{\mathcal{O}}_{s_{1}+s_{2}-2}^{\text{a}_{3}}(k_{1}+k_{2})\,.$
(64)

The soft expansion leads to the wedge-type algebra

$\displaystyle\varpi_{a,b}^{s_{1},\text{a}_{1}}(z_{1})\,\varpi_{c,d}^{s_{2},\text{a}_{2}}(z_{2})\,\sim\,\frac{f^{\text{a}_{1}\text{a}_{2}\text{a}_{3}}}{z_{1}-z_{2}}\;{\varpi_{a+c,b+d}^{s_{1}+s_{2}-1,\text{a}_{3}}}(z_{2})\,.$
(65)

##  5 Loop level

The loop-level study of chiral higher-spin theories was initiated in
Skvortsov:2018jea ; Skvortsov:2020wtf ; Skvortsov:2020gpn , which focused on
one-loop planar amplitudes in the theory gl-chs($\alpha$) with action (2.2).
In this section, we will discuss loop amplitudes in the various chiral higher-
spin theories we considered. Many statements are explicit or implicit in
Skvortsov:2018jea ; Skvortsov:2020wtf ; Skvortsov:2020gpn  and also in
Adamo:2022lah , so we are just aiming to give a useful overview. The collinear
behaviour of loop amplitudes is relevant to the quantum fate of the tree-level
celestial chiral algebras that we discussed above. Obviously, if the loop
amplitudes vanish, then the chiral algebra is exact. It is sufficient,
however, that the collinear behaviour of a theory is preserved at loop level,
as exemplified in Ball:2021tmb  for SDG, where $w_{1+\infty}$ is a
perturbatively exact symmetry. At one loop, this is implied by the vanishing
of the tree amplitudes under some assumptions, because the one-loop correction
to the collinear splitting function multiplies the tree amplitude; see e.g.
Bu:2022iak  for a discussion concerning Moyal-SDG. Recent results indicate
that, for generic theories, the celestial OPEs acquire loop corrections that
violate the associativity of the chiral algebra Bhardwaj:2022anh ;
Costello:2022upu ; Bittleston:2022jeq . For the chiral theories of the type
studied here, no such violation is known.

Let us start with the theories hsSDG (2.1) and hsSDYM (2.2), which have the
simplest structure of vertices. They possess only vertices of the type
$(+^{s_{1}}+^{s_{2}}-^{s_{3}})$, for spins $s_{i}\geq 2$ in the case of hsSDG,
and $s_{i}\geq 1$ in the case of hsSDYM. As such, like SDG and SDYM, they are
one-loop exact theories, because it is not possible to write down higher-loop
diagrams with only vertices of that type. It may be convenient for some
readers to describe the argument in detail. The number of loops is

$\displaystyle L=I-v+1\,,$ (66)

where $I$ is the number of internal lines and $v$ is the number of vertices.
The argument only cares about the signs of the helicities, so we can base it
on $(++-)$ vertices.

Dressing the diagram with helicity signs as in the figures above, the plus and
minus signs can be counted from either the lines (external and internal) or
the $(++-)$ vertices:

$\displaystyle n_{+}+I=2v\,,\qquad n_{-}+I=v\,.$ (67)

We obtain

$\displaystyle n_{+}=1-L+v\,,\qquad n_{-}=1-L\,.$ (68)

Hence, at tree level we have $n_{-}=1$, at one loop we have $n_{-}=0$, and
there is no solution for $L>1$. Therefore, we need to consider only one loop.
Let us consider first hsSDYM. Since at one loop the number of external
particles is $n=n_{+}=v$, and since the sum over helicities in each vertex is
$\Sigma h=1$, we have that $\sum_{i=1}^{n}h_{i}=n$. This requires that the
external helicities, which must be positive, are $h_{i}=1$ in hsSDYM, just
like in SDYM. For each one-loop diagram, any vertex that is not part of the
loop is of the type $(+^{1}+^{1}-^{1})$ as in SDYM, while the vertices in the
loop are of the type $(+^{1}+^{s}-^{s})$, and there is a sum over spins $s$
running in the loop. Therefore, the one-loop amplitudes in hsSDYM equal the
ones in SDYM multiplied by a factor $\sum_{s\geq 1}1$.131313Notice that a
factor of 2 accounting for $(+^{1}\pm^{s}\mp^{s})$ is already included in
SDYM. Following Skvortsov:2018jea , we use the zeta-regularisation procedure
of Beccaria:2015vaa  to identify that factor with $\zeta(0)=-\frac{1}{2}$.
Hence, the non-vanishing one-loop amplitudes are Adamo:2022lah

$\displaystyle{\mathcal{A}}^{(1)}_{\text{hsSDYM}}(1^{1},2^{1},\cdots,n^{1})=-\frac{1}{2}\,{\mathcal{A}}^{(1)}_{\text{SDYM}}(1^{+},2^{+},\cdots,n^{+})\,,$
(69)

where, on the left-hand side, the superscript indicates the helicity of each
particle. A similar story applies to hsSDG, with $h_{i}=2$ and a factor
$\sum_{s\geq 2}1$. We assume that the latter is regularised as
$\zeta(0)-1=-\frac{3}{2}$. Hence, the non-vanishing one-loop amplitudes are

$\displaystyle{\mathcal{A}}^{(1)}_{\text{hsSDG}}(1^{2},2^{2},\cdots,n^{2})=-\frac{3}{2}\,{\mathcal{A}}^{(1)}_{\text{SDG}}(1^{+},2^{+},\cdots,n^{+})\,.$
(70)

A recent line of work Costello:2021bah ; Costello:2022wso ; Bittleston:2022nfr
; Bittleston:2022jeq ; Monteiro:2022nqt ; Bu:2022dis  has described one-loop
amplitudes in SDYM and SDG as resulting from an anomaly, as first suggested in
Bardeen:1995gk . The anomaly can be cancelled by coupling the gauge field to
an ‘axion’, leading to an anomaly free theory with vanishing loop amplitudes.
The same can be done for hsSDYM and hsSDG, as recently pointed out in
Adamo:2022lah . Our discussion indicates that it is sufficient that the axion
couples to spin 1 in hsSDYM and to spin 2 in hsSDG, since the higher spins do
not contribute as external states to non-vanishing one-loop amplitudes. The
coupling to the axion is then fixed by adjusting the coefficients that apply
in SDYM and SDG to ones consistent with the two expressions for amplitudes
above. Curiously, in the case of hsSDYM, if we truncate the theory to odd
spins, and assume the zeta-function regularisation of $\sum_{\text{odd\,}s\geq
1}1$ is $0$,141414This corresponds to $\sum_{\text{odd\,}s\geq
1}1=\sum_{n=1}^{\infty}(2n-1)^{-s}\to 0$ in the limit $s\to 0$. the one-loop
amplitudes vanish. Following the discussions in Costello:2021bah ;
Costello:2022wso , there should be no loop-level obstruction to directly
uplifting this theory to twistor space. For hsSDG, if we truncate to even
spins, the analogous zeta-function regularisation of $\sum_{\text{even\,}s\geq
2}1$ leads instead to a factor of $-\frac{1}{2}$.151515This corresponds to
$\sum_{\text{even\,}s\geq
2}1=\sum_{n=1}^{\infty}(2n)^{-s}=2^{-s}\zeta(s)\to\zeta(0)=-\frac{1}{2}$ in
the limit $s\to 0$.

Before proceeding to other theories, we note that hsSDG and hsSDYM have a
collinear behaviour analogous to that of SDG and SDYM, and therefore their
celestial algebras are quantum exact, similarly to the SDG case of
$w_{1+\infty}$ Ball:2021tmb .

Let us move now to the chiral higher-spin theories chs(0) and gl-chs(0),
obtained respectively from the actions (2.1) and (2.2) after setting
$\alpha=0$. Consider one loop. The vertices satisfy again $\Sigma h=2$ for
chs(0) and $\Sigma h=1$ for gl-chs(0), so the external states must obey
$\sum_{i=1}^{n}h_{i}=2n$ and $\sum_{i=1}^{n}h_{i}=n$ respectively. However,
the external states do not individually have the same restriction as they did
in hsSDG ($h_{i}=2$) and hsSDYM ($h_{i}=1$), because the vertices are not all
of the type $(+^{s_{1}}+^{s_{2}}-^{s_{3}})$. Nevertheless, given an $m$-gon
loop subdiagram (i.e. $m=2$ is a bubble, $m=3$ is a triangle, etc), its
‘external’ helicities obey the sum rules $\sum_{i=1}^{m}h_{i}=2m$ and
$\sum_{i=1}^{m}h_{i}=m$ for chs(0) and gl-chs(0), respectively; and moreover,
fixing a single helicity along the loop fixes them all, with the sum rule
being the consistency condition. The conclusion is that the possibility of
fixing a single helicity along the loop to any value still leads, for the
amplitudes, to a factor representing the sum over helicities running in the
loop. This factor is

$\displaystyle\sum_{h}1=1+2\sum_{s\geq 1}1=1+2\zeta(0)=0\,,$ (71)

where the isolated 1 is the scalar contribution, and $2\sum_{s\geq 1}1$ is the
contribution from the other helicities. This vanishing factor multiplies in
chs(0) and gl-chs(0) the one-amplitudes of hsSDG and hsSDYM, and there is also
a helicity-dependent overall factor in each case. The point is that the
vanishing factor multiplies a finite contribution, so the one-loop amplitudes
vanish, which is the argument given in Skvortsov:2018jea ; Skvortsov:2020wtf ;
Skvortsov:2020gpn . Remarkably, if we truncate gl-chs(0) to admit only odd
spins, we get zero again from $2\sum_{\text{odd\,}s\geq 1}1=0$; and if we
truncate chs(0) to admit only even spins, we get zero again from
$1+2\sum_{\text{even\,}s\geq 2}1=1+2(-\frac{1}{2})=0$. We conclude that the
theories chs(0) and gl-chs(0) are actually simpler at one loop than hsSDG and
hsSDYM.

At higher loops, however, the amplitudes in hsSDG and hsSDYM manifestly vanish
due to the class of vertices, as we mentioned — an argument that does not
apply to chs(0) and gl-chs(0). To illustrate this point, we can start by
recalling the story for SDYM. Its equation of motion admits two types of cubic
action: one with a single field Leznov:1986mx ; PARK1990287 ,

$\displaystyle S_{\Psi^{3}\,\text{`SDYM'}}(\Psi)=\int
d^{4}x\;\;\text{tr}\left(\frac{1}{2}\,\Psi\square\Psi+\frac{1}{3}\,\Psi[\partial_{u}\Psi,\partial_{w}\Psi]\right)\,,$
(72)

and one with two fields representing opposite helicities that was introduced
later in Chalmers:1996rq ,

$\displaystyle S_{\text{SDYM}}(\Psi,\bar{\Psi})=\int
d^{4}x\;\;\text{tr}\;\bar{\Psi}\left(\square\Psi+[\partial_{u}\Psi,\partial_{w}\Psi]\right)\,.$
(73)

This second action has a ($++-$) vertex and is therefore one-loop exact. This
is the correct action for SDYM as a quantum field theory, and it can be seen
as a sector of the full Yang-Mills theory. The theories hsSDG and hsSDYM are
analogous to this case, as it is clear in the way they are generated from SDG
and SDYM in (2.1) and (2.2). The action (72), on the other hand, admits
diagrams with any number of loops. This is not a Lorentz-invariant theory: the
propagator does not connect fields of opposite helicity, unless the field is a
scalar, but then the vertex would unavoidably break Lorentz invariance because
there is no gauge to be fixed. Surprisingly, this non-Lorentz-invariant theory
does generate the Lorentz-invariant theory gl-chs(0) as seen in (2.2) if we
set $\alpha=0$. Given that the one-loop amplitudes vanish so beautifully due
to a sum over helicities, it is tempting to conclude that the same will happen
at higher loops. In fact, the sum (71) will still arise for an $m$-gon loop
subdiagram with given ‘external’ helicities joining the loop. However, it is
not clear now whether such vanishing factors multiply a finite kinematic
function; it was clear at one loop because of the relation to one-loop
amplitudes in SDG and SDYM. If the kinematic functions diverge, perhaps the
conclusion is still that the loop amplitudes vanish, but there is a worrying
competition of regularisations (states running in the loop versus, say,
dimensional regularisation). We will not pursue this further here.

The ‘oldest’ and most complicated chiral higher-spin theories are
chs($\alpha$) and gl-chs($\alpha$) for $\alpha\neq 0$, with actions (2.1) and
(2.2). The gl-chs($\alpha$) case was studied in Skvortsov:2018jea ;
Skvortsov:2020wtf ; Skvortsov:2020gpn , where it was found that the planar
one-loop amplitudes are again related to the planar SDYM amplitudes, with a
helicities-dependent dressing factor; they still vanish in view of an overall
factor giving the sum over states running in the loop, which vanishes in zeta
regularisation, similarly to gl-chs(0). In fact, the arguments above still
apply, because an $m$-gon loop (sub)diagram, with given ‘external’ helicities
and with given $\Sigma h$ for each vertex, will provide the sum (71) over
states in the loop. Now, however, it is even harder to know whether such
vanishing factors multiply a finite kinematic function, due to the higher
tensorial rank of the numerators in a loop integrand. At one loop, this was
checked explicitly in Skvortsov:2018jea ; Skvortsov:2020wtf ;
Skvortsov:2020gpn  for planar gl-chs($\alpha$). For the non-planar part and
for chs($\alpha$), we still have the unitarity-based argument that the one-
loop amplitudes cannot have branch cuts because the tree amplitudes vanish,
but this does not show that the amplitudes are finite. Beyond one loop, the
status of chs($\alpha$) and gl-chs($\alpha$) is at least as unclear as the
status of their $\alpha=0$ counterparts mentioned earlier: sums over states in
the loops suggest that the amplitudes vanish, but the possibility of
divergences in the loop integration should give us pause.

Finally, one may wonder about the relation to amplitudes in Moyal-SDG and
Moyal-SDYM, seen respectively as theories of spin 2 and spin 1 particles with
Lorentz-breaking higher-derivative corrections. Notice that the Moyal deformed
self-dual equations of motion admit actions analogous to (72) or to (73). In
the latter case, the theories are one-loop exact. In either case, the one-loop
amplitudes cannot have branch cuts because the tree amplitudes vanish. As for
the relation to the chiral higher-spin theories, it arises from actions of the
type (72), as we saw in (2.1) and (2.2). Hence, a priori, the chiral higher-
spin theories could have amplitudes of any loop order. We can consider the
correspondence (41) between Moyal-SDG and chs($\alpha$). It implies that a
loop integrand in chs($\alpha$), with a specific choice of external
helicities, is related to a particular power of $\alpha$ in the analogous
Moyal-SDG integrand, up to the sum over states running in the loop and to an
overall factor fixing the external helicity weights. However, the perturbative
behaviour of theories on non-commutative spaces can be non-analytical in the
deformation parameter Filk:1996dm ; Minwalla:1999px , so this may be a subtle
question, and we will not pursue it here.

##  6 Conclusion

We discussed various connections between chiral-type theories, namely self-
dual gravity or Yang-Mills, their Moyal deformations, and chiral higher-spin
theories. Table 1 summarised some of the main results. We described a notion
of theories ‘generating’ other theories, which is realised in a simple manner
in light-cone gauge. Indeed, there is a wonderful ambiguity in the use of
light-cone gauge: if one is given a gauge-fixed equation, it may prove
difficult to understand basic facts about its constituents, such as the spin
of the fields; and yet, not only does the simplicity of the expressions —
particularly for chiral theories — allow one to obtain crucial insights about
the structure of a theory, but also the ‘ambiguity’ can be fruitful in
suggesting connections between theories. The very existence of chiral higher-
spin theories was noticed first using light-cone-gauge methods Metsaev:1991mt
; Metsaev:1991nb ; Ponomarev:2016lrm . Another example is the straightforward
off-shell realisation of the colour-kinematics duality in self-dual Yang-Mills
theory Monteiro:2011pc . We saw how all the chiral-type theories were akin to
self-dual Yang-Mills, in that the double-copy structure of the vertex has one
copy of the self-dual kinematic algebra and one copy of another (colour,
kinematic or colour-kinematic) algebra. We also mentioned how this is
sufficient to guarantee the vanishing of the tree-level amplitudes in all
these theories, applying an argument from Monteiro:2022lwm  to new examples.
(One may ask, in the context of the double copy, what happens if one uses a
vertex that is the product of, say, two Moyal-deformed self-dual kinematic
algebras. The amplitudes do not vanish, but the theory breaks Lorentz
invariance, apparently without an interesting interpretation.)

Along with making these connections between theories, one of our main goals
was to describe associativity-preserving deformations of OPE algebras arising
in celestial holography, in particular the chiral algebras of soft modes of
gluons and gravitons; see Himwich:2021dau ; Mago:2021wje ; Ren:2022sws  for
related work. The case of Moyal-deformed self-dual gravity, which we reviewed,
was discussed recently in Monteiro:2022lwm ; Bu:2022iak  and is related to
$W_{1+\infty}$. In this paper, we added the case of Moyal-deformed self-dual
Yang-Mills, which gives a related deformation of a U(N) Kac-Moody algebra. We
also presented here the (2D-)chiral algebras for the (4D-)chiral higher-spin
theories.

Finally, we gave an overview of loop amplitudes in these various theories.
Light-cone gauge played a crucial role here again, allowing us to focus our
arguments on physical degrees of freedom, thereby avoiding the need for
Faddeev-Popov ghosts.

There are various open questions. Regarding chiral higher-spin theories, it
would be interesting to consider extensions, such as the theory recently
discussed in Adamo:2022lah  and also supersymmetric extensions.

Regarding the celestial algebras, this paper builds on the results of
Monteiro:2022lwm ; Guevara:2022qnm  connecting the colour-kinematics duality
to the associative structure of the tree-level celestial algebras. Indeed, it
would be surprising if these were two fully distinct algebraic structures. It
is important, however, to extend this connection to the full theories, i.e.
beyond the chiral sector. The simplest next step would be the MHV (next-to-
self-dual) sector, where relevant OPEs have been recently worked out to all
orders in the collinear expansion Adamo:2022wjo . At loop level, the
connection is harder to envisage, but its exploration may lead to lessons on
both sides. We could also study in greater detail the various chiral-type
theories discussed here even at tree level. We came across a deformation of a
U(N) Kac-Moody algebra, and it would be interesting to study it further and to
extend it to central terms and other parameters, similarly to the case of
$W_{1+\infty}$.

As a final open question, the precise relations of some or all of the chiral-
type theories considered here to the ${\mathcal{N}}=2$ strings remain to be
fully uncovered; see e.g. Ooguri:1991fp ; Chalmers:2000bh ; Lechtenfeld:2000nm
.

## Acknowledgements

We are grateful to Alfredo Guevara, Henrik Johansson, Radu Roiban, Evgeny
Skvortsov, Andrew Strominger and Tung Tran for discussions and suggestions.
This work was supported by a Royal Society University Research Fellowship.

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