aainstitutetext: Department of Mathematical Sciences, Durham University,
Durham, DH1 3LE, UKbbinstitutetext: Department of Mathematics and Statistics ,
Concordia University, Montreal, H3G 1M8, Canada

# Infinite-dimensional hierarchy of recursive extensions for all subn-leading
soft effects in Yang-Mills

Silvia Nagy,111Corresponding author. b Javier Peraza a Giorgio Pizzolo
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Building on our proposal in Nagy:2024dme , we present in detail the
construction of the extended phase space for Yang-Mills at null infinity,
containing the asymptotic symmetries and the charges responsible for subn-
leading soft theorems at all orders. The generality of the procedure allows it
to be directly applied to the computation of both tree and loop-level soft
limits. We also give a detailed study of Yang-Mills equations under the radial
expansion, giving a thorough construction of the radiative phase space for
decays compatible with tree-level amplitudes for both light-cone and radial
gauges. This gives rise to useful recursion relations at all orders between
the field strength and the vector gauge coefficients. We construct the subn-
leading charges recursively, and show a hierarchical truncation such that each
charge subalgebra is closed, and their action in the extended phase space is
canonical. We relate these results with the infinite-dimensional algebras that
have been recently introduced in the context of conformal field theories at
null infinity.

##  1 Introduction

The infrared limit of gauge and gravity theories has allowed for the discovery
of deep connections between the soft limits of scattering amplitudes,
asymptotic symmetries, and the memory effect, organised in the so-called IR
triangle Strominger:2017zoo . This has lead to insights which have allowed for
the development of a conjectured holographic principle for asymptotically flat
spacetimes222See reviews Pasterski:2021raf ; Raclariu:2021zjz  and references
within..

In this article we will be focusing specifically on the relation between soft
theorems and asymptotic symmetries. The former, first discovered many years
ago Weinberg1965 ; Low1958 , govern the behaviour of scattering amplitudes in
the limit where one or more of the particles have a vanishingly small energy.
The latter are local symmetries that survive in the limit where we go to the
null boundary of spacetime ($\mathcal{I}$). Then, unlike for standard gauge
transformations, we can construct charges, which, via the Ward identities,
provide proofs for the aforementioned soft theorems. This relies on
constructing a well-defined phase space at $\mathcal{I}$, containing the free
data of the theory, and on which symmetries act canonically. The above can be
set up perturbatively, via an expansion in the small energy parameter. At
leading order, where an elegant universal factorisation rule arises, the
result has by now been established Strominger:2013lka ; Strominger:2013jfa ;
He:2014cra , with progress to certain subleading orders in both gravity and
gauge theories Campiglia2015 ; Campiglia2020 ; Campiglia:2021oqz ;
Strominger2014a ; Lysov:2014csa ; Donnelly:2016auv ; Speranza:2017gxd ;
Freidel:2020ayo ; Freidel:2020svx ; Freidel:2020xyx ; Freidel:2021dxw ;
Ciambelli:2021nmv ; Freidel:2021cjp ; Freidel:2021dfs ; Ciambelli:2021vnn ;
Geiller:2024bgf ; Geiller:2022vto .

In a brief Letter Nagy:2024dme , working in Yang-Mills (YM) theory, we
proposed a mechanism for extending the canonical phase space to accommodate
the construction of asymptotic symmetries and charges required for
establishing the soft theorems at arbitrary subleading orders333See also
Peraza:2023ivy  and Nagy:2022xxs , where this has been achieved in the
simplified set-ups of massless QED and self-dual theories in light-cone gauge,
respectively.. The proposal relies on a modification of the Stueckelberg
procedure Stueckelberg:1938hvi , originally introduced to restore broken gauge
symmetry. This extends our phase space with a new Goldstone-type field, which
transforms under the subleading asymptotic symmetries. We additionally put
forward a set of recursion relation for the e.o.m., valid at any order in the
radial expansion, which feed into recursion relations for the charges at all
orders.

In this longer article, we will expand on the proposal in Nagy:2024dme , and
present some new results that follow from our prescription. Firstly we will
give further details on the extended phase space construction, and prove that
it leads to appropriate charges, satisfying the expected charge algebra. We
consider a very broad set-up here, allowing for the presence of field-
dependent parameters with very general polyhomogeneous fall-offs in the radial
expansion. We also made no assumptions at this stage about the fall-off in the
coordinate dual to the energy ($u$ in Bondi coordinates), which means it can
be applied in principle to loop-level soft theorems as well (see e.g.
Sahoo:2018lxl ; Pasterski:2022djr ; Donnay:2022hkf ; Agrawal:2023zea ;
Choi:2024ygx ; Campiglia:2019wxe ; AtulBhatkar:2019vcb ).

We give explicit all-order expressions for the symmetry transformations of the
extended phase space Goldstone-like Stueckelberg field, which enter the
computation of the subn-charges. Interestingly, these turn out to be
controlled by the Bernoulli numbers, which arise from the perturbative
expansion of (an operator version of) the characteristic power series
generating the so-called Todd polynomial ToddPol .

The general construction is independent of gauge and coordinate choice, as
well as the choice of hypersurface on which the initial data is defined. We
choose to illustrate it here by working in Bondi coordinates $(u,r,z,\bar{z})$
around $\mathcal{I}^{+}$, and we look at two different gauge conditions,
namely light-cone and radial gauge. In these two contexts, we explicitly
derive recursion relations at all orders in the expansion in $r$, thus
formally justifying the choice of free data in the radiative phase space. We
additionally study the expansion in the $u$ coordinate, which, via a Fourier
transform, is related to the energy expansion relevant for subn-leading soft
theorems.

Our procedure allows us to give an elegant construction for the charges at
null infinity, valid at any subn-leading order. A standard renormalization
procedure can be carried out in order to avoid infrared divergences (see e.g.
Freidel:2019ohg ; Peraza:2023ivy ). We find that the charges can be neatly
organised in a radial expansion, which closely follows the expansion in the
new Goldstone field444See e.g. Freidel:2023gue ; Kampf:2023elx ; He:2024skc
for some other works where Goldstone modes arise at null infinity. introduced
via the Stueckelberg trick. We establish the existence of a closed sub-algebra
of charges at each order in the perturbative expansion, thus showing how a
robust truncation of the procedure can be established at any desired order for
practical calculations related to soft theorems. We also match with known
results from the literature at level $n=0$ (Strominger:2013lka ) and $n=1$
(Campiglia:2021oqz ). Moreover, the charges themselves turn out to satisfy
recursions that allow us to construct any order from previous orders (first
noted in Campiglia:2021oqz  for leading and subleading), and also
straightforwardly match the abelian hierarchy of subn-leading charges
Campiglia:2018dyi .

One of the more intriguing discoveries of the celestial holography program has
been the presence of certain infinite dimensional algebras in sectors of Yang-
Mills and gravity (where it is called the $w_{1+\infty}$ algebra, first
defined by Bakas:1989xu  and connected to gravity in Strominger:2021mtt , see
also Pope:1989ew ; Fairlie:1990wv ; Pope:1991ig ; Bu:2022iak ;
Bittleston:2023bzp ; Bittleston:2024rqe ; Taylor:2023ajd ; Kmec:2024nmu ;
Monteiro:2022lwm ; Himwich:2021dau ; Lipstein:2023pih ; Lipstein:2023tapp ;
Adamo:2021lrv  for further work and deformations). The quatization of these
algebras (known as $W_{1+\infty}$, Pope:1991zka ) is expected to be a quantum
symmetry of the S-matrix, and it is interesting that one can trace its origin
to classical symmetries and their charge algebra (see e.g. Strominger:2021mtt
for YM and Freidel:2021ytz ; Geiller:2024bgf  for gravity). We find that a
subset of our recursion relations encompasses the sectors displaying these
symmetries.

The article is organised as follows. In section 2 we give details of the
general phase space calculation. In section 3, working in Bondi coordinates
around null infinity we give the e.o.m. at all orders in $r$, and the proofs
of the recursion relations in both light-cone and radial gauge. The
construction of the necessary charges and their recursion relations at all
orders in given in section 4. In section 5 the charge algebra valid at any
subleading order is constructed and shown to close order by order. In section
6, we show the link between the recursive formulas in section 3 and the
charges constructed in section 4, as well as showing that our results for the
leading and subleading charges agree with previous proposals from the
literature Strominger:2013lka ; Campiglia:2021oqz  and the full abelian case
Campiglia:2018dyi ; Peraza:2023ivy . In section 7 we identify, via the
recursion relations, the subsectors of our theory that carry the infinite
symmetry algebras along the lines of the discussion in Freidel:2023gue . In
section 8 we provide an outlook of the results, and propose future
explorations. The appendices contain detailed computations, useful formulae
and discussions that compliment the main text. In Appendix A, we give details
of calculations pertaining to the general extended phase space. We collect
some useful formulae in Appendix B. In Appendix C, we provide proofs for the
derivation of the equations of motion and Bianchi identities at arbitrary
order in the radial expansion, which were used in finding the recursive
formulae of section 3. Finally, in Appendix D, we turn to the so-called flat
Bondi coordinates, which differ from the standard Bondi through the vanishing
of the $g_{uu}$ component of the metric, as well as the celestial sphere being
mapped into the complex plane $\mathbb{C}^{2}$ via the stereographic
projection. We see that our procedure continues to hold, and find simplified
recursion relations in these coordinates.

##  2 Extended phase space in arbitrary gauge and coordinates

We begin by reviewing the general procedure introduced in Nagy:2024dme . The
starting point is YM theory with the standard equation of motion

$\mathcal{E}_{\nu}\equiv D^{\mu}\mathcal{F}_{\mu\nu}=0$ (1)

where

$\mathcal{F}_{\mu\nu}=\partial_{\mu}\mathcal{A}_{\nu}-\partial_{\nu}\mathcal{A}_{\mu}-i[\mathcal{A}_{\mu},\mathcal{A}_{\nu}]$
(2)

The gauge fields transform in the usual way:

$\mathcal{A}^{\prime}_{\mu}=e^{i\Lambda}\mathcal{A}_{\mu}e^{-i\Lambda}+ie^{i\Lambda}\partial_{\mu}e^{-i\Lambda}.$
(3)

and we assume that $A_{\mu}$ satisfies some gauge condition

$\mathcal{G}(\mathcal{A}_{\mu})=0$ (4)

We will restrict to the residual gauge symmetries preserving the condition
above, as is standard when studying asymptotic symmetries. For now we do not
need to specify the coordinates. For explicit examples later on, we will work
in some appropriately chosen coordinates (these could be e.g. Bondi, flat
Bondi, or even light-cone coordinates). We find it convenient to adopt the
following notation for the coordinates:

$x^{\mu}=(\mathfrak{r},\vec{\mathbf{y}}),$ (5)

where $\mathfrak{r}$ is the expansion parameter, and $\vec{\mathbf{y}}$ are
the the coordinates of the hypersurface on which we define our free
data555Most commonly, in Bondi coordinates, we have $\mathfrak{r}=r$ and
$\vec{\mathbf{y}}=(u,z,\bar{z})$ describing $\mathcal{I}^{+}$. However, the
procedure applies in other coordinates as well..

We consider a very general expansion for our gauge field, in terms of
polyhomogeneous functions

$\mathcal{A}_{\mu}=\sum_{n,k}A_{\mu}^{(-n;k)}(\vec{\mathbf{y}})\frac{\text{log}^{k}\mathfrak{r}}{\mathfrak{r}^{n}}\
,$ (6)

where $n$ and $k$ are chosen such that
$\lim_{\mathfrak{r}\to\infty}\frac{\text{log}^{k}\mathfrak{r}}{\mathfrak{r}^{n}}=\mathcal{O}(1)$.
In practice, this means that we only allow terms with $n\geq 0$, and $k$ is
unconstrained, except for $n=0$, where we only allow a contribution from
$k=0$. We remark that upon summing all orders in the above, one could of
course use the Taylor series for the log, and ultimately have an expression of
the form
$\sum_{n=0}^{\infty}B_{\mu}^{(-n)}(\vec{\mathbf{y}})\frac{1}{\mathfrak{r}^{n}}$
. However in practice we will stop at a finite order in $n$ and $k$, in which
case the log terms become necessary (see e.g. Campiglia:2021oqz  in Lorenz
gauge). Certain gauge choices have the benefit of avoiding the log terms, as
we will see in subsection 2.2, where we work in axial gauge.

Then the phase space is described by

$\Gamma^{0}=\left\\{\mathfrak{A}^{0}|\text{with $\mathfrak{A}^{0}$ constrained
by \eqref{eom_gen} and \eqref{gf_gen} }\right\\}$ (7)

where $\mathfrak{A}^{0}$ comprises the necessary free data which determines
the radiative fields to all orders in $\mathfrak{r}$, via the equations of
motion and Bianchi identities.

In section 3, working in Bondi coordinates, we give expressions for the
equations of motion and Bianchi identities at all orders in $r$. Then, for
both radial and light-cone gauge, we give recursion relations valid at all
orders in the perturbation parameter, proving that the fields in the bulk can
be entirely reconstructed from the boundary data at null infinity.

The large gauge transformations responsible for the leading order soft
theorems are characterised by a parameter $\Lambda^{(0)}$, subject to a
constraint coming from the gauge condition (4):

$\mathcal{G}(D_{\mu}\Lambda^{(0)}(\vec{\mathbf{y}}))=0\ ,$ (8)

thus preserving $\Gamma^{0}$.

We now wish to construct an extended phase space, able to accommodate
additional large gauge transformations with a fall-off of the form 666Note
that small gauge transformations, i.e. those which vanish in the limit
$\mathfrak{r}\to\infty$, have been set to $0$ for simplicity, since they do
not contribute to the soft theorems.

$\Lambda_{+}(x)=\sum_{n,k}\mathfrak{r}^{n}\text{log}^{k}\mathfrak{r}\
\Lambda^{(n;k)}(\vec{\mathbf{y}})\ ,$ (9)

where $n$ and $k$ are chosen such that
$\mathfrak{r}^{n}\text{log}^{k}\mathfrak{r}$ diverges in the limit
$\mathfrak{r}\to\infty$. In Campiglia:2021oqz  this was achieved for at first
order in the radial expansion in YM in Lorenz gauge, working in Bondi
coordinates. In Nagy:2022xxs , a construction was given to all orders, for the
self-dual YM theory, working in light-cone gauge. Here we will show that the
construction extends to full YM theory, for $n=\infty$, with an arbitrary
gauge and coordinate choice.

First we note that in some cases one need to generalise (9) to a field-
dependent parameter777This might arise because we are imposing further
constraints on the transformation. For example, requiring that they preserve
Lorenz gauge can lead to field-dependent transformation parameters, see
Campiglia:2021oqz , and we consider here the most general situation:

$\displaystyle\breve{\Lambda}_{+}=\breve{\Lambda}_{+}(\mathcal{A}_{\mu}(x),\Lambda_{+}(x))=\sum_{n,k}\mathfrak{r}^{n}\text{log}^{k}\mathfrak{r}\
f^{(n;k)}(A_{\mu}^{(0;0)}(\vec{\mathbf{y}}),...,\Lambda_{+}^{(0;1)}(\vec{\mathbf{y}}),...)\
,$ (10)

where $n$ and $k$ are such that $\mathfrak{r}^{n}\text{log}^{k}\mathfrak{r}$
diverges as $\mathfrak{r}\to\infty$. Here $\mathcal{A}_{\mu}(x)$ is expanded
as in (6), and the $\Lambda_{+}(x)$ expansion is given in (9). We note that
certain gauges can simplify matters, for example in any axial gauge
$\mathcal{G}(A_{\mu})=a^{\mu}\mathcal{A}_{\mu}$, with $a^{\mu}$ a constant
vector, the gauge parameter is field independent $\breve{\Lambda}=\Lambda(x)$,
as we will see in subsection 2.2. However, for now we continue with the most
general solution described in (10).

The next step is to define an extended phase space capable of accomodating a
gauge transformation with a parameter given by (10). These tranformations
obviously violate the fall-off (6). We can interpret these violations as a
form of symmetry breaking, which leads us to using a modified Stueckelberg
trick in the construction of our phase space. The Stueckelberg
procedure888Initially introduced in the context of electromagnetism
Stueckelberg:1938hvi , also extended to (super)gravity Kuchar:1991xd ;
Nagy:2019ywi ; Bansal:2020krz . is usually employed when a gauge symmetry is
boken e.g. due to the presence of mass. It allows us to reinstate the
symmetry, by promoting the local transformation parameter to a new field,
which tranforms non-linearly. We will adapt it to our situation by defining
the object:

$\breve{\Psi}=\breve{\Psi}(\mathcal{A}_{\mu}(x),\Psi(x))=\sum_{n,k}\mathfrak{r}^{n}\text{log}^{k}\mathfrak{r}\
f^{(n;k)}(A^{(0;0)}_{\mu}(\vec{\mathbf{y}}),...,\Psi^{(0;1)}(\vec{\mathbf{y}}),...)$
(11)

with

$\Psi(x)=\sum_{n,k}\mathfrak{r}^{n}\text{log}^{k}\mathfrak{r}\
\Psi^{(n;k)}(\vec{\mathbf{y}}).$ (12)

and where $\breve{\Psi}$ has the same functional dependence on $A_{\mu}$ and
$\Psi$ and $\breve{\Lambda}_{+}$ does on $A_{\mu}$ and $\Lambda$.
Equivalently, $\breve{\Psi}$ is obtained from $\breve{\Lambda}_{+}$ by aplying
the replacement rule

$\Lambda_{+}(x)\quad\to\quad\Psi(x),$ (13)

We are now ready to define our extended phase space as:

$\Gamma_{\infty}^{\text{ext}}:=\Gamma^{0}\times\\{\Psi(x)\ |\ \Psi(x)\text{ as
defined in \eqref{gen_psi_no_fields}}\\}$ (14)

where $\Gamma^{0}$ is the original phase space defined in (7). We will see
that the object $\breve{\Psi}$ plays a crucial role in our construction. In
order to proceed with the construction of charges on our extended phase space,
we must first determine the transformation properties of our new field $\Psi$.
We generalise the procedure employed in Campiglia:2021oqz ; Nagy:2022xxs  and
start by defining the extended gauge field

$\tilde{\mathcal{A}}_{\mu}=e^{i\breve{\Psi}}\mathcal{A}_{\mu}e^{-i\breve{\Psi}}+ie^{i\breve{\Psi}}\partial_{\mu}e^{-i\breve{\Psi}}$
(15)

This mimics a gauge-transformed gauge field, with the parameter replaced by
our new field, which is a hallmark of the Stueckelberg procedure. We require
that is satisfies the following consistency condition:

$\delta_{\breve{\Lambda}}\tilde{\mathcal{A}}_{\mu}=\tilde{D}_{\mu}\breve{\Lambda}$
(16)

where

$\breve{\Lambda}=\Lambda^{(0)}+\breve{\Lambda}_{+}$ (17)

and $\tilde{D}_{\mu}$ is the gauge-covariant derivative w.r.t.
$\tilde{A}_{\mu}$. In the above, $\Lambda^{0}$ is the standard (leading order)
large gauge parameter. This is tantamount to saying that the extended field
transforms as a gauge field in the extended space. Note that when writing the
actual transformation, we are working only to linear order in the parameter,
as this will be sufficient for the construction of the charges. This is in
contrast with the Stueckelberg step (15), where all orders are necessary. The
consistency condition essentially states that the additional fields on the
boundary do not correspond to new fields in the bulk, but rather are coming
from pieces of the bulk field which were discarded when assuming the fall-off
in (6). Finally, as is standard in the Stueckelberg procedure, the
transformation of the original field $\mathcal{A}_{\mu}$ is unchanged.

Following a calculation similar to that in Nagy:2022xxs  (see Appendix A for
details), we rewrite the LHS of (16) as

$\delta_{\breve{\Lambda}}\tilde{\mathcal{A}}_{\mu}=e^{i\breve{\Psi}}\left\\{\delta_{\breve{\Lambda}}\mathcal{A}_{\mu}+D_{\mu}\left[e^{-i\breve{\Psi}}\mathcal{O}_{-i\breve{\Psi}}(\delta_{\breve{\Lambda}}\breve{\Psi})e^{i\breve{\Psi}}\right]\right\\}e^{-i\breve{\Psi}}$
(18)

where we have used that

$\delta e^{X}=e^{X}\mathcal{O}_{X}(\delta X)$ (19)

with

$\mathcal{O}_{X}:=\frac{1-e^{-ad_{X}}}{ad_{X}}\ .$ (20)

The above is defined via its series expansion

$\mathcal{O}_{X}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+1)!}(ad_{X})^{k}\ ,$
(21)

such that

$\displaystyle\mathcal{O}_{-i\breve{\Psi}}(\delta_{\breve{\Lambda}}\breve{\Psi})$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+1)!}(ad_{-i\breve{\Psi}})^{k}(\delta_{\breve{\Lambda}}\breve{\Psi})$
(22) $\displaystyle=$
$\displaystyle\delta_{\breve{\Lambda}}\breve{\Psi}+\frac{i}{2}[\breve{\Psi},\delta_{\breve{\Lambda}}\breve{\Psi}]-\frac{1}{6}[\breve{\Psi},[\breve{\Psi},\delta_{\breve{\Lambda}}\breve{\Psi}]]-\frac{i}{24}[\breve{\Psi},[\breve{\Psi},[\breve{\Psi},\delta_{\breve{\Lambda}}\breve{\Psi}]]]+\dots$

The RHS of (16) becomes

$\hat{D}_{\mu}{\breve{\Lambda}}=e^{i\breve{\Psi}}D_{\mu}(e^{-i\breve{\Psi}}{\breve{\Lambda}}e^{i\breve{\Psi}})e^{-i\breve{\Psi}}$
(23)

Then, plugging (18) and (23) into (16) we obtain:

$\delta_{\breve{\Lambda}}\mathcal{A}_{\mu}+D_{\mu}\left[e^{-i\breve{\Psi}}\mathcal{O}_{-i\breve{\Psi}}(\delta_{\breve{\Lambda}}\breve{\Psi})e^{i\breve{\Psi}}\right]=D_{\mu}\left(e^{-i\breve{\Psi}}{\breve{\Lambda}}e^{i\breve{\Psi}}\right)$
(24)

Let us now focus one one of the components of the equation above, call this
$w$. In any gauge, we can choose a $w$ such at $A_{w}^{(0,0)}\neq 0$. Then we
have

$\delta_{\breve{\Lambda}}A_{w}^{(-n;k)}:=\left(D_{w}{\breve{\Lambda}}\right)^{(-n;k)}=\delta_{n,0}\delta_{k,0}\partial_{w}\Lambda^{(0)}-i[A_{w}^{(-n;k)},\Lambda^{(0)}]\
,$ (25)

where we have imposed that only the leading order gauge parameter
$\Lambda^{0}$ contributes (the subleading gauge parameters in (9) will only
appear in the transformation of $\Psi$, and we recall that we are ignoring the
small gauge transformations). Making use of the above, we can rearrange (24)
into

$\mathcal{O}_{-i\breve{\Psi}}(\delta_{\breve{\Lambda}}\breve{\Psi})={\breve{\Lambda}}-e^{i\breve{\Psi}}\Lambda^{(0)}e^{-i\breve{\Psi}}$
(26)

This can be solved to any order in $r$ by inverting $\mathcal{O}_{-i\Psi}$:

$\delta_{\breve{\Lambda}}\breve{\Psi}=\mathcal{O}_{-i\breve{\Psi}}^{-1}({\breve{\Lambda}}-e^{i\breve{\Psi}}\Lambda^{(0)}e^{-i\breve{\Psi}})$
(27)

We will make use of the formal expansion

$\left(\frac{1-e^{-X}}{X}\right)^{-1}=\sum_{m=0}^{\infty}\frac{B^{+}_{m}X^{m}}{m!}$
(28)

where $B^{+}_{n}$ are the Bernoulli numbers

$B^{+}_{0}=1,\quad B^{+}_{1}=\frac{1}{2},\quad B^{+}_{2}=\frac{1}{6},\quad...$
(29)

Using this in (27) we find 999The proof is completely analogous to the one in
Appendix A3 of Nagy:2022xxs .:

$\delta_{\breve{\Lambda}}^{[m]}\breve{\Psi}=\frac{B_{m}^{+}}{m!}(ad_{-i\breve{\Psi}})^{m}\left[{\breve{\Lambda}}+(-1+2\delta_{m,1})\Lambda^{(0)}\right]$
(30)

where $\delta_{\breve{\Lambda}}^{[m]}\breve{\Psi}$ denotes the variation of
$\breve{\Psi}$ at order $m$ in $\breve{\Psi}$. The next step is to extract the
transformation of $\Psi$ from the above. This will of course depend on the
exact form of $\breve{\Psi}$ as a function of $\Psi$ and $\mathcal{A}_{\mu}$,
so it will be done case by case, depending on the gauge choice (4).
Nevertheless, we can proceed in the general case, as it is the $\breve{\Psi}$
that will appear in the construction of the charges, and so it is its
transformation that will be relevant for computing the charge algebra. Before
continuing with the charge algebra, we pause to make some remarks about the
transformation of the Stueckelberg fields. Firstly, we note that, at 0th order
in the field, it transforms via a shift

$\delta_{\breve{\Lambda}}^{[0]}\breve{\Psi}=\breve{\Lambda}-\Lambda^{(0)}=\breve{\Lambda}_{+}$
(31)

so they have an interpretation as Goldstone modes for the symmetry breaking in
the bulk.

Secondly, recall that

$B_{2k+1}=0,\quad\text{for}\quad k>0$ (32)

i.e. only even powers of the fields will contribute to the transformation (30)
beyond $m=1$. We leave the interpretation of this observation for future work.

Finally, there is a close relation between the Bernoulli numbers and the
Riemann $\zeta$-function

$\zeta(m)=\frac{(-1)^{\frac{m}{2}+1}}{2}\frac{B_{m}^{+}(2\pi)^{m}}{m!},$ (33)

for $m$ even and larger than $1$. Then we can alternatively write (30), for
$m>1$, as

$\delta_{\breve{\Lambda}}^{[m]}\breve{\Psi}=\frac{2(-1)^{\tfrac{m}{2}+1}\zeta(m)}{(2\pi)^{m}}(ad_{-i\breve{\Psi}})^{m}\left[{\breve{\Lambda}}+(-1+2\delta_{m,1})\Lambda^{(0)}\right].$
(34)

###  2.1 Charge algebra

The charge, computed via the covariant space space formalism101010More details
on the computation of charges are given in section 4., can be expressed as a
two-dimensional integral, over a hypersurface $\mathcal{B}^{2}$

$\tilde{Q}_{\breve{\Lambda}}=\int_{\mathcal{B}^{2}}\text{tr}({\breve{\Lambda}}\tilde{\mathcal{F}}^{\mu\nu})dS_{\mu\nu},$
(35)

where $\tilde{\mathcal{F}}^{\mu\nu}$ is constructed from
$\tilde{\mathcal{A}}_{\mu}$, and takes the form

$\tilde{\mathcal{F}}_{\mu\nu}\equiv\partial_{\mu}\tilde{\mathcal{A}}_{\nu}-\partial_{\nu}\tilde{\mathcal{A}}_{\mu}-i[\tilde{\mathcal{A}}_{\mu},\tilde{\mathcal{A}}_{\nu}]=e^{i\breve{\Psi}}\mathcal{F}_{\mu\nu}e^{-i\breve{\Psi}}$
(36)

Let $x_{B}$ and $y_{B}$ the the coordinates on $\mathcal{B}_{2}$, then we have

$dS_{\mu\nu}=dx_{B}dy_{B}\sqrt{g_{B}}m_{\mu}n_{\nu}$ (37)

where $g_{B}$ is the induced metric on $\mathcal{B}_{2}$ and
$n_{\mu}$,$m_{\mu}$ are unit vectors orthogonal to $\mathcal{B}_{2}$. We can
now define the charge density

$\tilde{q}_{\breve{\Lambda}}=\text{tr}\left(\sqrt{g_{B}}{\breve{\Lambda}}\tilde{\mathcal{F}}_{mn}\right)^{(0)}$
(38)

where we have denoted

$\tilde{\mathcal{F}}_{mn}=\tilde{\mathcal{F}}^{\mu\nu}m_{\mu}n_{\nu}$ (39)

In terms of the original field strength, the charge density is

$\displaystyle\tilde{q}_{\breve{\Lambda}}=$
$\displaystyle\text{tr}\left(\sqrt{g_{B}}{\breve{\Lambda}}e^{i\breve{\Psi}}\mathcal{F}_{mn}e^{-i\breve{\Psi}}\right)^{(0)}$
(40) $\displaystyle=$
$\displaystyle\text{tr}\left(\sqrt{g_{B}}e^{-i\breve{\Psi}}{\breve{\Lambda}}e^{i\breve{\Psi}}\mathcal{F}_{mn}\right)^{(0)}$

We now want to check the charge algebra

$\\{\tilde{q}_{{\breve{\Lambda}}_{1}},\tilde{q}_{{\breve{\Lambda}}_{2}}\\}_{*}=\frac{1}{2}\left[\delta_{{\breve{\Lambda}}_{1}}\tilde{q}_{{\breve{\Lambda}}_{2}}+\tilde{q}_{\delta_{{\breve{\Lambda}}_{1}}{\breve{\Lambda}}_{2}}-(1\leftrightarrow
2)\right]$ (41)

Note that we are using a modified bracket Barnich:2010eb ; Barnich:2010xq ;
Barnich:2013sxa ; Fiorucci:2021pha ; Campiglia:2021oqz , as $\breve{\Lambda}$
can be field dependent in general. Explicitly we then have:

$\displaystyle\\{\tilde{q}_{{\breve{\Lambda}}_{1}},\tilde{q}_{{\breve{\Lambda}}_{2}}\\}_{*}=$
$\displaystyle\frac{1}{2}\Bigg{[}\text{tr}\left(\sqrt{g_{B}}\left(\delta_{{\breve{\Lambda}}_{1}}e^{-i\breve{\Psi}}\right){\breve{\Lambda}}_{2}e^{i\breve{\Psi}}\mathcal{F}_{mn}\right)^{(0)}+2\
\text{tr}\left(\sqrt{g_{B}}e^{-i\breve{\Psi}}\left(\delta_{{\breve{\Lambda}}_{1}}{\breve{\Lambda}}_{2}\right)e^{i\breve{\Psi}}\mathcal{F}_{mn}\right)^{(0)}$
(42)
$\displaystyle+\text{tr}\left(\sqrt{g_{B}}e^{-i\breve{\Psi}}{\breve{\Lambda}}_{2}\left(\delta_{{\breve{\Lambda}}_{1}}e^{i\breve{\Psi}}\right)\mathcal{F}_{mn}\right)^{(0)}+\text{tr}\left(\sqrt{g_{B}}e^{-i\breve{\Psi}}{\breve{\Lambda}}_{2}e^{i\breve{\Psi}}\delta_{{\breve{\Lambda}}_{1}}\mathcal{F}_{mn}\right)^{(0)}\Bigg{]}$
$\displaystyle-(1\leftrightarrow 2)$

Using (19) and (20), we have

$\displaystyle\delta_{{\breve{\Lambda}}_{1}}e^{-i\breve{\Psi}}=$
$\displaystyle-
ie^{-i\breve{\Psi}}\mathcal{O}_{-i\breve{\Psi}}\left(\delta_{{\breve{\Lambda}}_{1}}\breve{\Psi}\right)$
(43) $\displaystyle=$ $\displaystyle-
ie^{-i\breve{\Psi}}\left({\breve{\Lambda}_{1}}-e^{i\breve{\Psi}}\Lambda_{1}^{(0)}e^{-i\breve{\Psi}}\right)$
$\displaystyle=$ $\displaystyle-
ie^{-i\breve{\Psi}}{\breve{\Lambda}_{1}}+i\Lambda_{1}^{(0)}e^{-i\breve{\Psi}}$

where to get to the second line we used (26). Similarly, using (264), we write

$\displaystyle\delta_{{\breve{\Lambda}}_{1}}e^{i\breve{\Psi}}=$ $\displaystyle
i\mathcal{O}_{-i\breve{\Psi}}\left(\delta_{{\breve{\Lambda}}_{1}}\breve{\Psi}\right)e^{i\breve{\Psi}}$
(44) $\displaystyle=$ $\displaystyle
i\left({\breve{\Lambda}_{1}}-e^{i\breve{\Psi}}\Lambda_{1}^{(0)}e^{-i\breve{\Psi}}\right)e^{i\breve{\Psi}}$
$\displaystyle=$ $\displaystyle
i\breve{\Lambda}_{1}e^{i\breve{\Psi}}-ie^{i\breve{\Psi}}\Lambda_{1}^{(0)}$

To compute the final term in (40) we use (25) to write

$\delta_{\breve{\Lambda}_{1}}\mathcal{F}_{mn}=-i[\mathcal{F}_{mn},\Lambda_{1}^{(0)}]$
(45)

remembering that we have set the small gauge transformations to vanish.
Finally, plugging (43), (44), and (45) into (42), we get

$\\{\tilde{q}_{{\breve{\Lambda}}_{1}},\tilde{q}_{{\breve{\Lambda}}_{2}}\\}_{*}=\tilde{q}_{[{\breve{\Lambda}}_{1},{\breve{\Lambda}}_{2}]_{*}}$
(46)

as required, where the deformed bracket is as expected for field dependent
parameters Barnich:2010eb ; Barnich:2010xq ; Barnich:2013sxa ;
Fiorucci:2021pha ; Campiglia:2021oqz :

$[{\breve{\Lambda}}_{1},{\breve{\Lambda}}_{2}]_{*}=-i[{\breve{\Lambda}}_{1},{\breve{\Lambda}}_{2}]+\delta_{{\breve{\Lambda}}_{1}}{\breve{\Lambda}}_{2}-\delta_{{\breve{\Lambda}}_{2}}{\breve{\Lambda}}_{1}$
(47)

###  2.2 Case study: Bondi coordinates in axial gauge

Let us make the previous section more concrete, by looking at a specific
example of coordinates, namely the Bondi coordinates $(u,r,z,\bar{z})$ around
$\mathcal{I}^{+}$, with line element

$ds^{2}=-du^{2}-2dudr+2r^{2}\gamma_{z\bar{z}}dzd\bar{z}$ (48)

where

$\gamma_{z\bar{z}}=\frac{2}{(1+z\bar{z})^{2}}\ .$ (49)

Let us choose an axial gauge, i.e.

$\mathcal{G}(\mathcal{A}_{\mu})=a^{\mu}\mathcal{A}_{\mu}\equiv 0\ ,$ (50)

where $a^{\mu}$ is some constant vector. Different choices of $a^{\mu}$ give
different gauges of interest for various applications. Later in the article we
will work in the light-cone, and radial gauges respectively. For now we
proceed with a general $a^{\mu}$. In this gauge, the expansion of the gauge
fields (6) simplifies to

$\mathcal{A}_{\mu}=\sum_{n=0}^{\infty}\frac{A_{\mu}^{(-n)}(u,z,\bar{z})}{r^{n}}\
,$ (51)

i.e. the log terms are no longer present. The radiative phase space will be as
in (7). The leading order large gauge transformations are constrained by the
gauge choice, and explicitly we have:

$a^{\mu}D_{\mu}\Lambda^{(0)}=a^{\mu}\partial_{\mu}\Lambda^{(0)}\equiv 0\ ,$
(52)

where in the first equality we made use of (50). We see that the only
constraint on the gauge parameter is that it does not depend on the linear
combination of coordinates $a_{\mu}x^{\mu}$. We are now looking to construct
an extended phase space, able to accommodate a gauge parameter

$\Lambda_{+}(x)=\sum_{n=1}^{\infty}r^{n}\Lambda^{(n)}(u,z,\bar{z})\ ,$ (53)

where we highlight the absence of the logarithmic terms, reflecting the
absence of these terms in the expansion of $\mathcal{A}_{\mu}$ above. At this
stage, one could choose to impose the gauge fixing condition on the new terms,
i.e.

$a^{\mu}\partial_{\mu}\Lambda_{+}(x)=0$ (54)

though we emphasize that this is not required by our procedure. In either
case, the upshot is that we do not need a field-dependent parameter in this
case. Thus (10) simplifies significantly, and we have

$\breve{\Lambda}_{+}(x)=\Lambda_{+}(x)$ (55)

We are now ready to construct our Stueckelberg field, which is simply

$\breve{\Psi}(x)=\Psi(x)=\sum_{k=1}^{\infty}r^{k}\Psi^{(k)}(u,z,\bar{z})$ (56)

The extended gauge field will be as in (15). Upon applying the consistency
condition (16), and following the steps in the previous section, we now
explicitly obtain the transformation rule for the Stuckelberg field

$\delta_{\Lambda}\Psi=\mathcal{O}_{-i\Psi}^{-1}({\Lambda}-e^{i\Psi}\Lambda^{(0)}e^{-i\Psi})$
(57)

with $\mathcal{O}$ as in (20). Working at order $m$ in $\Psi$, this can be
written in terms of the Bernoulli numbers (29) as

$\delta_{\Lambda}^{[m]}\Psi=\frac{B_{m}^{+}}{m!}(ad_{-i\Psi})^{m}\left[{\Lambda}+(-1+2\delta_{m,1})\Lambda^{(0)}\right]$
(58)

Let us write the first few orders explicitly. At $[m]=0$ we have

$\delta_{\Lambda}^{[0]}\Psi=\Lambda-\Lambda^{(0)}=\Lambda_{+}$ (59)

as expected for a Stuckelberg-type field. At linear order in $\Psi$ we have

$\delta_{\Lambda}^{[1]}\Psi=-\frac{i}{2}[\Psi,\Lambda+\Lambda^{(0)}]\ ,$ (60)

while at second order the transformation becomes

$\delta_{\Lambda}^{[2]}\Psi=-\frac{1}{12}[\Psi,[\Psi,\Lambda-\Lambda^{(0)}]]\
.$ (61)

As we mentioned in the previous section, all odd orders beyond $m=1$ will
vanish

$\delta_{\Lambda}^{[2k+1]}\Psi=0,\qquad k\geq 1$ (62)

The results above are exact in the coordinates, so in general we can expand
them to any order in $r$. We remark that we have

$\left(\delta_{\Lambda}^{[m]}\Psi\right)^{(n)}=0\qquad\text{for}\ n<m\ ,$ (63)

due to the fact that $\Psi$ starts at linear order in $r$.

For future use, let us also introduce the notation

$\overset{j}{\delta}:=\sum_{k=0}^{j}\delta^{[k]}$ (64)

which gives the variation up to order $j$ in the Stueckelberg field.

In Bondi coordinates, the charge will be defined on the celestial sphere at
$\mathcal{I}^{+}_{-}$, via

$\tilde{Q}_{\Lambda}=\int_{S^{2}}\text{tr}(\Lambda\tilde{\mathcal{F}}_{ru})r^{2}\gamma_{z\bar{z}}dzd\bar{z}\
,$ (65)

with

$\tilde{\mathcal{F}}_{ru}=e^{i\Psi}\mathcal{F}_{ru}e^{-i\Psi}$ (66)

We can now define the charge aspect as

$\displaystyle\tilde{q}_{\Lambda}$ $\displaystyle=\text{tr}(\Lambda
r^{2}\tilde{\mathcal{F}}_{ru})^{(0)}$ (67)
$\displaystyle=\text{tr}(e^{-i\Psi}\Lambda e^{i\Psi}\mathcal{F}_{ru})^{(-2)}$
$\displaystyle=\text{tr}\left[\left(e^{ad_{-i\Psi}}\Lambda\right)\mathcal{F}_{ru}\right]^{(-2)}$

The charge algebra now becomes

$\displaystyle\\{\tilde{q}_{{\Lambda}_{1}},\tilde{q}_{{\Lambda}_{2}}\\}_{*}=$
$\displaystyle\frac{1}{2}\left[\delta_{{\Lambda}_{1}}\tilde{q}_{{\Lambda}_{2}}-(1\leftrightarrow
2)\right]$ (68) $\displaystyle=$
$\displaystyle-i\tilde{q}_{[{\Lambda}_{1},{\Lambda}_{2}]}$

as needed. This is of course simpler than the expression in subsection 2.1,
because we no longer have field-dependent parameters. See section 4 for a more
detailed discussion of the charge construction and algebra at various orders.

The gauge choice will play an important role in our proof of the recursion
relations for the equations of motion and charges at all orders, in the
following sections. We will focus on two particular gauge choices. One is the
light-cone gauge

$a^{\mu}=(1,0,0,0)\quad\Rightarrow\quad\mathcal{A}_{u}=0$ (69)

together with a constraint to ensure radiative solutions

$A_{r}^{(0)}=A_{r}^{(-1)}=0\ .$ (70)

At this stage, one may choose to impose (54) , which results in111111See
Nagy:2022xxs  for an example of this in self-dual YM.

$\Lambda_{+}=\Lambda_{+}(r,z,\bar{z}),\quad\Rightarrow\Psi=\Psi(r,z,\bar{z})$
(71)

though as we mentioned above this is not needed for our procedure, and we can
simply work with a general

$\Lambda_{+}=\Lambda_{+}(u,r,z,\bar{z}),\quad\Rightarrow\Psi=\Psi(u,r,z,\bar{z})$
(72)

The other gauge choice is the radial gauge, defined by

$\mathcal{A}_{r}=0$ (73)

where we additionally impose the usual constraint at $\mathcal{I}^{+}$

$A_{u}^{(0)}=0\ .$ (74)

##  3 Equations of motion and recursion relations at all orders

In this section we study in detail the properties of the radiative phase
space, first providing the YM e.o.m. to all orders in the radial expansion in
Bondi coordinates. Working in two different gauge choices, namely light-cone
gauge and radial gauge, we give recursion relations that allow us to construct
the fields to all orders.

We work with the Bondi coordinates $(u,r,z,\bar{z})$ defined in (48) and (49).
We want to study Yang-Mills theory at $\mathcal{I}^{+}$, which in Bondi
coordinates is obtained by taking the limit $r\rightarrow\infty$ and it is
parametrized by $(u,z,\bar{z})$. We will use the radial coordinate as the
variable for the power expansions121212We use Mathematica to compute most of
the calculations regarding the expansions and manipulations with
coefficients..

Let $M:=\mathbb{R}^{1,3}$ be the smooth manifold considered with coordinates
defined (48) and (49). The components of a generic $(k,l)$-tensor
$\mathcal{T}$ on $M$ can be expanded in a $r-$power series near
$\mathcal{I}^{+}$ as:

${\mathcal{T}^{\mu_{1}\dots\mu_{k}}}_{\nu_{1}\dots\nu_{l}}(u,r,z,\bar{z})=\sum_{n\in\mathbb{Z}}\frac{{{T^{(-n)}}^{\mu_{1}\dots\mu_{k}}}_{\nu_{1}\dots\nu_{l}}(u,z,\bar{z})}{r^{n}},$
(75)

where each component
${\mathcal{T}^{\mu_{1}\dots\mu_{k}}}_{\nu_{1}\dots\nu_{l}}(u,r,z,\bar{z})$ is
a $C^{\infty}$ function on $M$ that can be written as a power series expansion
in $1/r$. We are interested in the subset $C^{\infty}_{r}(M)$ of functions
with finite limit when $r\rightarrow+\infty$, defined as follows131313Note
that in the sum in (76) we let $n$ assume negative values and we kill all the
coefficients $f^{(-n)}(u,z,\bar{z})$ with $n<0$. The reason why we do not
define $C^{\infty}_{r}(M)$ by summing only on $n\in\mathbb{N}$ is that we want
to construct a projector $\mathcal{P}_{-n}$ that acts on $C^{\infty}_{r}(M)$
with both positive and negative values of $n$ (see (273)), so that equations
like (274)-(276) are well defined. Thus, we want the set (76) to take into
account all $n\in\mathbb{Z}$.:

$\displaystyle C^{\infty}_{r}(M):=\\{f\in C^{\infty}(M)\ |\
f(u,r,z,\bar{z})=\sum_{n\in\mathbb{Z}}r^{-n}f^{(-n)}(u,z,\bar{z}),\
f^{(-n)}(u,z,\bar{z})=0\ \forall n<0\\}$ (76)

Therefore, given $f\in C^{\infty}_{r}(M)$, the limit when
$r\rightarrow+\infty$ is simply given by the smooth function $f^{(0)}$, which
can be thought of as taking values in $\mathcal{I}^{+}$. We also assume that
$f^{(n)}\in C^{\infty}(\mathcal{I}^{+})$ for each $n\in\mathbb{Z}$.

###  3.1 YM e.o.m. in Bondi coordinates

The YM equations of motion are given by the vanishing of the following tensor,

$\mathcal{E}_{\mu}:=D^{\nu}\mathcal{F}_{\nu\mu}=\nabla^{\nu}\mathcal{F}_{\nu\mu}-i[\mathcal{A}^{\nu},\mathcal{F}_{\nu\mu}].$
(77)

The vanishing of $\mathcal{E}_{\mu}$ implies in particular that
$\mathcal{E}_{\mu}\in C^{\infty}_{r}(M)$ in (76), when expressed in Bondi
coordinates $(u,r,z,\bar{z})$. Then, one can compute the components of
$\mathcal{E}_{\mu}$ at each order $1/r^{n}$, with $n\in\mathbb{N}$. Let

$\mathcal{E}_{\mu}=\sum_{n\in\mathbb{N}}r^{-n}E_{\mu}^{(-n)}(u,z,\bar{z})$
(78)

denote the $r$ power expansion. A complete derivation of the following
equations can be found in Appendix C. Here, we present the result for each
$E_{\mu}^{(-n)}$,

$\displaystyle E_{\mu}^{(-n)}$
$\displaystyle=-\partial_{u}F^{(-n)}_{r\mu}+\tilde{\delta}_{\mu}^{\rho}(n-1)(F^{(1-n)}_{u\rho}-F^{(1-n)}_{r\rho})+2\partial_{(z}(\gamma_{z\bar{z}}^{-1}F^{(2-n)}_{\bar{z})\mu})-2\bar{\delta}_{\mu}^{\rho}(\partial_{(z}\gamma_{z\bar{z}}^{-1})F^{(2-n)}_{\bar{z})\rho}$
(79)
$\displaystyle\quad+i\sum_{k=0}^{n}\left(2[A^{(k-n)}_{(u},F^{(-k)}_{r)\mu}]-[A^{(k-n)}_{r},F^{(-k)}_{r\mu}]-2\gamma_{z\bar{z}}^{-1}[A^{(k-n)}_{(z},F^{(2-k)}_{\bar{z})\mu}]\right),$

where,

$\displaystyle\bar{\delta}_{\mu}^{\nu}$
$\displaystyle:=\mathrm{diag}(1,1,0,0),$ (80)
$\displaystyle\tilde{\delta}_{\mu}^{\nu}(n)$
$\displaystyle:=n\delta_{\mu}^{\nu}-2\bar{\delta}_{\mu}^{\nu}.$ (81)

More explicitly, each of the components of $E_{\mu}^{(-n)}$ is given by

$\displaystyle E_{u}^{(-n)}$
$\displaystyle=\partial_{u}F^{(-n)}_{ur}+(n-3)F^{(1-n)}_{ur}-2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(2-n)}_{u|\bar{z})}$
(82)
$\displaystyle\quad+i\sum_{k=0}^{n}\left([A^{(k-n)}_{r}-A^{(k-n)}_{u},F^{(-k)}_{ur}]+2\gamma_{z\bar{z}}^{-1}[A^{(k-n)}_{(z},F^{(2-k)}_{u|\bar{z})}]\right),$
$\displaystyle E_{r}^{(-n)}$
$\displaystyle=(n-3)F^{(1-n)}_{ur}-2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(2-n)}_{r|\bar{z})}$
(83)
$\displaystyle\quad+i\sum_{k=0}^{n}\left([A^{(k-n)}_{r},F^{(-k)}_{ur}]+2\gamma_{z\bar{z}}^{-1}[A^{(k-n)}_{(z},F^{(2-k)}_{r|\bar{z})}]\right),$
$\displaystyle E_{z}^{(-n)}$
$\displaystyle=-\partial_{u}F^{(-n)}_{rz}+(n-1)(F^{(1-n)}_{uz}-F^{(1-n)}_{rz})-\partial_{z}(\gamma_{z\bar{z}}^{-1}F^{(2-n)}_{z\bar{z}})$
(84)
$\displaystyle\quad+i\sum_{k=0}^{n}\left(2[A^{(k-n)}_{(u},F^{(-k)}_{r)z}]-[A^{(k-n)}_{r},F^{(-k)}_{rz}]+\gamma_{z\bar{z}}^{-1}[A^{(k-n)}_{z},F^{(2-k)}_{z\bar{z}}]\right),$
$\displaystyle E_{\bar{z}}^{(-n)}$
$\displaystyle=-\partial_{u}F^{(-n)}_{r\bar{z}}+(n-1)(F^{(1-n)}_{u\bar{z}}-F^{(1-n)}_{r\bar{z}})+\partial_{\bar{z}}(\gamma_{z\bar{z}}^{-1}F^{(2-n)}_{z\bar{z}})$
(85)
$\displaystyle\quad+i\sum_{k=0}^{n}\left(2[A^{(k-n)}_{(u},F^{(-k)}_{r)\bar{z}}]-[A^{(k-n)}_{r},F^{(-k)}_{r\bar{z}}]-\gamma_{z\bar{z}}^{-1}[A^{(k-n)}_{\bar{z}},F^{(2-k)}_{z\bar{z}}]\right).$

In the following, we study these equations for two particular gauge choices,
light-cone and radial gauges. First, we have to define a subset of solutions,
given by certain fall-offs compatible with radiative fields. Then, we compute
the equations of motion ($E_{\mu}^{(-n)}=0$), and finally we deduce a
recursive formula that allows to compute the components of the field strength
up to functions on the sphere $S^{2}$ with coordinates $(z,\bar{z})$, i.e.,
the celestial sphere. Schematically, one has

$F_{ur}^{(-n)}=G[\partial_{u}^{-1},\gamma_{z\bar{z}},\partial_{z},\partial_{\bar{z}},A_{z}^{(0)},A_{\bar{z}}^{(0)},F_{ur}^{(-n+1)},...,F_{ur}^{(-2)}],$
(86)

where $\partial_{u}^{-1}:=\int_{-\infty}^{u}du$ and $G$ is a functional that
contains the differential operators
$\partial_{u}^{-1},\partial_{z},\partial_{\bar{z}}$ and depends on the field
strength via $F_{ur}^{(-n+1)},...,F_{ur}^{(-2)}$.

###  3.2 Light-cone gauge

Our first case study is the light-cone gauge. This gauge is characterised by
the vanishing of the $u$-component of the gauge vector,

$\mathcal{A}_{u}=0.$ (87)

As we mentioned above, we are interested in a subset of solutions that are
radiative. This can be achieved by taking the following conditions on the
$r$-component of $\mathcal{A}_{\mu}$,

$\qquad A_{r}^{(0)}=A_{r}^{(-1)}=0.$ (88)

From (87) and (88) we get the following conditions on $\mathcal{F}_{\mu\nu}$,

$F_{ur}^{(-n)}=F_{rz}^{(-n)}=F_{r\bar{z}}^{(-n)}=0\qquad\mathrm{for}\ n=0,1.$
(89)

Equations (82)-(85) simplify to the following set of identities,

$\displaystyle E_{u}^{(-n)}$
$\displaystyle=\partial_{u}F^{(-n)}_{ur}+(n-3)F^{(1-n)}_{ur}-2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(2-n)}_{u|\bar{z})}$
(90)
$\displaystyle\quad+i\sum_{k=0}^{n-2}\left([A^{(k-n)}_{r},F^{(-k)}_{ur}]+2\gamma_{z\bar{z}}^{-1}[A^{(2+k-n)}_{(z},F^{(-k)}_{u|\bar{z})}]\right),\qquad
n\geq 2$ $\displaystyle E_{r}^{(-n)}$
$\displaystyle=(n-3)F^{(1-n)}_{ur}-2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(2-n)}_{r|\bar{z})}$
(91)
$\displaystyle\quad+i\sum_{k=2}^{n-2}\left([A^{(k-n)}_{r},F^{(-k)}_{ur}]+2\gamma_{z\bar{z}}^{-1}[A^{(2+k-n)}_{(z},F^{(-k)}_{r|\bar{z})}]\right),\qquad
n\geq 4$ $\displaystyle E_{z}^{(-n)}$
$\displaystyle=-\partial_{u}F^{(-n)}_{rz}+(n-1)(F^{(1-n)}_{uz}-F^{(1-n)}_{rz})-\partial_{z}(\gamma_{z\bar{z}}^{-1}F^{(2-n)}_{z\bar{z}})$
(92)
$\displaystyle\quad+i\sum_{k=0}^{n-2}\left([A^{(k-n)}_{r},F^{(-k)}_{uz}-F^{(-k)}_{rz}]+\gamma_{z\bar{z}}^{-1}[A^{(2+k-n)}_{z},F^{(-k)}_{z\bar{z}}]\right),\qquad
n\geq 2$ $\displaystyle E_{\bar{z}}^{(-n)}$
$\displaystyle=-\partial_{u}F^{(-n)}_{r\bar{z}}+(n-1)(F^{(1-n)}_{u\bar{z}}-F^{(1-n)}_{r\bar{z}})+\partial_{\bar{z}}(\gamma_{z\bar{z}}^{-1}F^{(2-n)}_{z\bar{z}})$
(93)
$\displaystyle\quad+i\sum_{k=0}^{n-2}\left([A^{(k-n)}_{r},F^{(-k)}_{u\bar{z}}-F^{(-k)}_{r\bar{z}}]-\gamma_{z\bar{z}}^{-1}[A^{(2+k-n)}_{\bar{z}},F^{(-k)}_{z\bar{z}}]\right),\qquad
n\geq 2$

####  3.2.1 Recursive formula for the gauge vector

In what follows we prove that given the initial data

$A_{z}^{(0)}(u,z,\bar{z}),\quad A_{\bar{z}}^{(0)}(u,z,\bar{z})\ ,$ (94)

and families of functions on the sphere

$\\{A_{z}^{(-n,0)},A_{\bar{z}}^{(-n,0)},A_{r}^{(-n-2,0)}\in
C^{\infty}(S^{2})\\}_{n\geq 1}\ ,$ (95)

where $T^{(-n,k)}$ denotes the coefficient of $\frac{u^{k}}{r^{n}}$ in a
formal expansion of the tensor $T$ in $r$ and $u$, it is possible to find the
solution of the equation of motion at arbitrary order in the $1/r$-expansion.
In particular, we present a recursive formula that allows us to compute
$A_{\mu}^{(-n)}$ for all $n\in\mathbb{N}$, thus providing the explicit
expression of the solution of the e.o.m. at each order.

First of all, we write explicitly the definition of the field strength tensor
in light-cone gauge (87) and with fall-off (88) at a given order $n$:

$\displaystyle F_{ur}^{(-n)}$ $\displaystyle=\partial_{u}A_{r}^{(-n)},\qquad
n\geq 2$ (96) $\displaystyle F_{uz}^{(-n)}$
$\displaystyle=\partial_{u}A_{z}^{(-n)},\qquad n\geq 2$ (97) $\displaystyle
F_{rz}^{(-n)}$
$\displaystyle=(1-n)A_{z}^{(1-n)}-\partial_{z}A_{r}^{(-n)}-i\sum_{k=0}^{n-2}[A_{r}^{(k-n)},A_{z}^{(-k)}],\qquad
n\geq 2$ (98) $\displaystyle F_{z\bar{z}}^{(-n)}$
$\displaystyle=2\partial_{[z}A_{\bar{z}]}^{(-n)}-i\sum_{k=0}^{n}[A_{z}^{(k-n)},A_{\bar{z}}^{(-k)}],\qquad
n\geq 2\ ,$ (99)

where $F_{u\bar{z}}^{(-n)}$ and $F_{r\bar{z}}^{(-n)}$ are completely analogous
to (97) and (98), respectively. Recall that $F_{ur}^{(-n)}=F_{rz}^{(-n)}=0$
for $n=0,1$. From the above equations we notice that

$\displaystyle F_{u\mu}^{(-n)}$ $\displaystyle\quad\mathrm{depends}\
\mathrm{on}\quad A_{\mu}^{(-n)}\quad\text{for}\quad\mu=r,z,\bar{z},$ (100)
$\displaystyle F_{rz}^{(-n)}$ $\displaystyle\quad\mathrm{depends}\
\mathrm{on}\quad\\{A_{r}^{(-k)}\\}_{k\leq n},\\{A_{z}^{(-k)}\\}_{k<n},$ (101)
$\displaystyle F_{z\bar{z}}^{(-n)}$ $\displaystyle\quad\mathrm{depends}\
\mathrm{on}\quad\\{A_{z}^{(-k)},A_{\bar{z}}^{(-k)}\\}_{k\leq n}.$ (102)

Suppose we know
$\\{A_{z}^{(-k)}(u,z,\bar{z}),A_{\bar{z}}^{(-k)}(u,z,\bar{z}),A_{r}^{(-k-2)}(u,z,\bar{z})\\}_{k<n}$
for some arbitrary $n\in\mathbb{N}$; we want to find
$\\{A_{z}^{(-n)}(u,z,\bar{z}),A_{\bar{z}}^{(-n)}(u,z,\bar{z}),A_{r}^{(-n-2)}(u,z,\bar{z})\\}$.
Although it is useful to express the recursion step as above, we emphasize
that $A_{r}^{(-2)}$ is not part of the initial data, but can be obtained from
(94) and (95).

Indeed, $A_{r}^{(-2)}$ can be obtained from (96) and (90) with $n=2$:

$\displaystyle\partial_{u}F^{(-2)}_{ur}$
$\displaystyle=2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(0)}_{u|\bar{z})}-2i\gamma_{z\bar{z}}^{-1}[A^{(0)}_{(z},F^{(0)}_{u|\bar{z})}],$
(103) $\displaystyle\Longrightarrow A_{r}^{(-2)}$
$\displaystyle=2\gamma_{z\bar{z}}^{-1}\partial_{u}^{-1}\left(\partial_{(z}A_{\bar{z})}^{(0)}-i\partial_{u}^{-1}[A_{(z}^{(0)},\partial_{u}A_{\bar{z})}^{(0)}]\right),$
(104)

where in the second line we used (97) and the symbol $\partial_{u}^{-1}=\int
du$ defined previously. For $n\geq 3$ we use (96) and (91) to find:

$\displaystyle A_{r}^{(-n)}$
$\displaystyle=\frac{1}{n-2}\partial_{u}^{-1}\left[2\gamma_{z\bar{z}}^{-1}\partial_{(z}F_{r|\bar{z})}^{(1-n)}-i\sum_{k=2}^{n-1}\left([A_{r}^{(k-n-1)},F_{ur}^{(-k)}]+2\gamma_{z\bar{z}}^{-1}[A_{(z}^{(1+k-n)},F_{r|\bar{z})}^{(-k)}]\right)\right]$
(105)

By observing the equations above and recalling (100)-(101) we notice that, in
general:

$\displaystyle A_{r}^{(-n)}$ $\displaystyle\quad\mathrm{depends}\
\mathrm{on}\quad A_{r}^{(-n+1)},\\{A_{\mu}^{(-k)}\\}_{k\leq
n-2}\quad\text{and}\quad A_{r}^{(-n,0)}$ (106)

Therefore, we conclude that knowing $\\{A_{\mu}^{(-k)}\\}_{k<n}$ is sufficient
to determine $A_{r}^{(-n)}$ via equations (104) and (105). Each function
$\\{A_{r}^{(-n,0)}\\}_{n\geq 2}$ is the constant of integration of the
operator $\partial_{u}^{-1}$.

Now, we solve for $A_{z}^{(-n)}$. At leading order, that is, for $n=0$, the
$z$ and $\bar{z}$ components of the gauge field constitute the initial data,
so we consider $n\geq 1$. From (98) we have:

$\displaystyle A_{z}^{(-n)}$
$\displaystyle=-\frac{1}{n}\left(\partial_{z}A_{r}^{(-n-1)}-i\sum_{k=2}^{n+1}[A_{z}^{(k-n-1)},A_{r}^{(-k)}]+F_{rz}^{(-n-1)}\right).$
(107)

However, thanks to (101) we know that, in general, $F_{rz}^{(-n-1)}$ does
depend on $A_{z}^{(-n)}$. Hence, we need an explicit expression for it, which
is provided by the $z$ component (92) of the e.o.m.

$\displaystyle F^{(-n-1)}_{rz}$
$\displaystyle=nA_{z}^{(-n)}-\partial_{u}^{-1}\bigg{[}nF^{(-n)}_{rz}+\partial_{z}(\gamma_{z\bar{z}}^{-1}F^{(1-n)}_{z\bar{z}})$
(108)
$\displaystyle\quad-i\sum_{k=0}^{n-1}\left([A^{(k-n-1)}_{r},\partial_{u}A_{z}^{(-k)}-F^{(-k)}_{rz}]+\gamma_{z\bar{z}}^{-1}[A^{(1+k-n)}_{z},F^{(-k)}_{z\bar{z}}]\right)\bigg{]}.$

Putting (107) and (108) together we obtain (for $n\geq 1$):

$\displaystyle A_{z}^{(-n)}$
$\displaystyle=\frac{1}{2n}\bigg{\\{}-\partial_{z}A_{r}^{(-n-1)}+i\sum_{k=2}^{n+1}[A_{z}^{(k-n-1)},A_{r}^{(-k)}]$
(109)
$\displaystyle\quad+\partial_{u}^{-1}\bigg{[}nF^{(-n)}_{rz}+\partial_{z}(\gamma_{z\bar{z}}^{-1}F^{(1-n)}_{z\bar{z}})$
$\displaystyle\quad-i\sum_{k=0}^{n-1}\left([A^{(k-n-1)}_{r},\partial_{u}A_{z}^{(-k)}-F^{(-k)}_{rz}]+\gamma_{z\bar{z}}^{-1}[A^{(1+k-n)}_{z},F^{(-k)}_{z\bar{z}}]\right)\bigg{]}\bigg{\\}}.$

The above equation together with the (88), (101), (102) and (106) imply that:

$\displaystyle A_{z}^{(-n)}$ $\displaystyle\quad\mathrm{depends}\
\mathrm{on}\quad A_{r}^{(-n-1)},A_{r}^{(-n)},\\{A_{\mu}^{(-k)}\\}_{k<n}.$
(110)

For instance, $A_{z}^{(-1)}$ depends just on the initial data:

$\displaystyle A_{z}^{(-1)}$
$\displaystyle=-\frac{1}{2}\left(\partial_{z}A_{r}^{(-2)}-i[A_{z}^{(0)},A_{r}^{(-2)}]\right)$
(111)
$\displaystyle\quad+\frac{1}{2}\partial_{u}^{-1}\left(\partial_{z}(\gamma_{z\bar{z}}^{-1}F^{(0)}_{z\bar{z}})-i[A^{(-2)}_{r},\partial_{u}A^{(0)}_{z}]-i\gamma_{z\bar{z}}^{-1}[A^{(0)}_{z},F^{(0)}_{z\bar{z}}]\right).$

Observe that since we start with the fall-offs (88), the inductive step is
indeed satisfied. In other words, the solution for $A_{z}^{(-n)}$ is
completely determined by $A_{r}^{(-n-1)},A_{r}^{(-n)}$,
$\\{A_{\mu}^{(-k)}\\}_{k<n}$ and the functions on the sphere
$\\{A_{z}^{(-n,0)}(z,\bar{z})\\}$. Note that it is also possible to find
$A_{z}^{(-n)}$ by integrating $F_{uz}^{(-n)}$, which in turn can be obtained
through the Bianchi identity (293), achieving the same result.

In Figure 1, the relation between the indices is schematically depicted.

Figure 1: The solid arrows indicate which $A_{r}^{(-k)}$, $A_{z}^{(-k)}$ enter
the recursion formula for $A_{r}^{(-n)}$, while the dotted arrows indicate
which $A_{r}^{(-k)}$ and $A_{z}^{(-k)}$ enter the recursion formula for
$A_{z}^{(-n)}$. The component $A_{\bar{z}}^{(-n)}$ is not drawn, since the
same result as for $A_{z}^{(-n)}$ is valid.

To conclude, the above procedure allows us to compute the solution to the
e.o.m. $A_{\mu}^{(-n)}$ at an arbitrary order $n\in\mathbb{N}$ of the large
$r$-expansion, by recursively constructing $A_{\mu}^{(-k)}$ for $k=0,\dots,n$,
starting from the initial data $A_{z}^{(0)}$ and $A_{\bar{z}}^{(0)}$. Our
recursive relation can be written in a more compact way as follows:

$\displaystyle A_{r}^{(-2)}$
$\displaystyle=2\gamma_{z\bar{z}}^{-1}\partial_{u}^{-2}\left(D_{(z|}\partial_{u}\mathcal{A}_{|\bar{z})}\right)^{(0)}$
(112) $\displaystyle A_{r}^{(-n)}$
$\displaystyle=\frac{1}{n-2}\partial_{u}^{-1}\left[2\gamma_{z\bar{z}}^{-1}\left(D_{(z}\mathcal{F}_{r|\bar{z})}\right)^{(1-n)}-i\left([\mathcal{A}_{r},\partial_{u}\mathcal{A}_{r}]\right)^{(-n-1)}\right],\qquad
n\geq 3$ (113) $\displaystyle A_{z}^{(-n)}$
$\displaystyle=-\frac{1}{2n}\big{\\{}(D_{z}\mathcal{A}_{r})^{(-n-1)}-\partial_{u}^{-1}\big{(}nF_{rz}^{(-n)}+(D_{z}(\gamma_{z\bar{z}}^{-1}\mathcal{F}_{z\bar{z}}))^{(1-n)}$
(114)
$\displaystyle\quad-i([\mathcal{A}_{r},\partial_{u}\mathcal{A}_{z}-\mathcal{F}_{rz}])^{(-n-1)}\big{)}\big{\\}},\qquad
n\geq 1$

Recursive formula for $F_{ur}^{(-n)}$: As will be seen in section 4, the $ru$
component of the field strength plays a crucial role in the construction of
the charges. Thus it is useful to give a recursive formula for the different
coefficients of the $r$-expansion of $F_{ur}^{(-n)}$, with $n\geq 2$.

Consider $\\{F_{ur}^{(-n,0)}\in C^{\infty}(S^{2})\\}_{n\geq 0}$, a family of
functions on the sphere, then, by (90), we can obtain $F_{ur}^{-n}$ in terms
of $F_{ur}^{(-n,0)}$ and $\\{F_{ur}^{-k}\\}_{k<n}$, by using the
$\partial_{u}^{-1}$ operator.

It immediately follows from (90) that

$\displaystyle F_{ur}^{(-n)}$
$\displaystyle=\partial_{u}^{-1}\bigg{(}(3-n)F^{(-n+1)}_{ur}+2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(-n+2)}_{u|\bar{z})}$
(115)
$\displaystyle\quad-i\sum_{k=0}^{n-2}\left([A^{(k-n)}_{r},F^{(-k)}_{ur}]+2\gamma_{z\bar{z}}^{-1}[A^{(2+k-n)}_{(z},F^{(-k)}_{u|\bar{z})}]\right)\bigg{)}.$

The constant of integration for the operator $\partial_{u}^{-1}$ is given by
$F_{ur}^{(-n,0)}$, which we can introduce by fixing the lower limit in the
anti-derivative as $-\infty$,

$\displaystyle F_{ur}^{(-n)}(u,z,\bar{z})$
$\displaystyle=F_{ur}^{(-n,0)}(z,\bar{z})+\int_{-\infty}^{u}du\bigg{(}(3-n)F^{(-n+1)}_{ur}+2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(-n+2)}_{u|\bar{z})}$
(116)
$\displaystyle\quad-i\sum_{k=0}^{n-2}\left([A^{(k-n)}_{r},F^{(-k)}_{ur}]+2\gamma_{z\bar{z}}^{-1}[A^{(2+k-n)}_{(z},F^{(-k)}_{u|\bar{z})}]\right)\bigg{)},$

####  3.2.2 Decays on null infinity in the radiative phase space

As it is well known in the sub-$n$-theorems literature (see for example
Hamada:2018vrw ; Li:2018gnc ; Campiglia:2018dyi , the $u$-behaviour of the
fields translates as derivatives via a Fourier transform in $u$. Let us assume
that we are working at tree-level. This implies that the decay of the
$u$-derivative of the initial data ($\partial_{u}A_{z}^{(0)}$) is faster than
any polynomial in $u$.

Let $\mathfrak{F}(A^{(0)}_{z})(\omega,z,\bar{z})$ be the time-Fourier
transform of $A_{z}(u,z,\bar{z})$. Then, the $\omega\rightarrow 0$ expansion
of the tree-level amplitude (see Hamada:2018vrw ; Li:2018gnc ) implies the
following fall off in $\omega$ for the radiative free data for the gauge
vector,

$\mathfrak{F}(A^{(0)}_{z})(\omega,\hat{x})=\omega^{-1}\mathfrak{F}(A^{(0)}_{z})^{-1}(\hat{x})+\mathfrak{F}(A^{(0)}_{z})^{0}(\hat{x})+\omega\mathfrak{F}(A^{(0)}_{z})^{1}(\hat{x})+\ldots,$
(117)

where the upper index outside the bracket indicates the order in the
$\omega$-expansion. Undoing the Fourier transform, $A_{z}(u,z,\bar{z})$
behaves like

$A^{(0)}_{z}(u,\hat{x})=\theta(u)A_{z}^{(\theta)}(\hat{x})+\delta(u)A_{z}^{(\delta)}(\hat{x})+\delta^{\prime}(u)A_{z}^{\left(\delta^{\prime}\right)}(\hat{x})+\ldots,$
(118)

with $\theta$ a step function. Thus, the $u\rightarrow\pm\infty$ falloffs for
the null infinity free data are as follows

$A^{(0)}_{z}(u,\hat{x})\stackrel{{\scriptstyle
u\rightarrow\pm\infty}}{{=}}A^{(0,0)\pm}_{z}(\hat{x})+o(u^{-\infty}),$ (119)

where $o(u^{-\infty})$ is a remainder that goes to zero faster than any power
of $1/u$.

In what follows we compute the $u$-behaviour of the functions
$A_{r}^{(-n)},A_{z}^{(-n)}$ and $F_{ru}^{(-n)}$ as $u\rightarrow-\infty$. Let
us start with the the following boundary conditions on the field strength at
$\mathcal{I}^{+}_{+}$,

$\lim_{u\rightarrow+\infty}F^{(-2)}_{ru}(u,z,\bar{z})=0,\quad\lim_{u\rightarrow+\infty}F_{rz}^{(-2)}(u,z,\bar{z})=0,\quad\lim_{u\rightarrow+\infty}F_{r\bar{z}}^{(-2)}(u,z,\bar{z})=0,$
(120)

which arise due to the absence of massive colored fields. The standard
$u\rightarrow-\infty$ fall-off for the radiative data $A_{z}^{(0)}$ is the
following,

$A_{z}^{(0)}(u,z,\bar{z})=A_{z}^{(0,0)}(z,\bar{z})+o(u^{-\infty}).$ (121)

This translates to $F_{ru}^{(-2)}$ via (104) as follows

$F_{ru}^{(-2)}(u,z,\bar{z})=F_{ru}^{(-2,0)}(z,\bar{z})+o(u^{-\infty}).$ (122)

We can write

$\partial_{u}F_{ru}^{(-2)}=2\gamma_{z\bar{z}}^{-1}\partial_{(z}F_{\bar{z})u}^{(0)}+2\gamma_{z\bar{z}}^{-1}[A_{(z}^{(0)},F_{\bar{z})u}^{(0)}]=:2D_{(z}F_{\bar{z})u}^{(0)},$
(123)

where
$D_{z}:=\gamma_{z\bar{z}}^{-1}\left(\partial_{z}+[A_{z}^{(0)},\cdot]\right)$.
Using the results in the previous subsections, we will show here how the
integration from $A_{z}^{(0)}(u,z,\bar{z})$ can be carried out to all the
coefficients (with the corresponding constants of integration as functions on
the sphere).

First, observe that $A_{r}^{(-2)}=O(u)$, since
$\partial_{u}A_{r}^{(-2)}+F_{ru}^{(-2)}=0$. Then, if $A_{r}^{(-2,0)}$ is the
finite value of the limit

$\lim_{u\rightarrow-\infty}A_{r}^{(-2)}+uF_{ru}^{(-2,0)}=A_{r}^{(-2,0)}(z,\bar{z}),$
(124)

where the second term on the LHS above has the role of substracting the linear
dependence in $u$, then

$A_{r}^{(-2)}(u,z,\bar{z})=-uF_{ru}^{(-2,0)}(z,\bar{z})+A_{r}^{(-2,0)}(z,\bar{z})+o(u^{-\infty}).$
(125)

This is consistent with (104). Next, the equation for $A_{z}^{(-1)}$ is given
by (111),

$\displaystyle A_{z}^{(-1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}u\left(\partial_{z}F_{ru}^{(-2,0)}-i[A_{z}^{(0,0)},F_{ru}^{(-2,0)}]+\partial_{z}(\gamma_{z\bar{z}}^{-1}F^{(0,0)}_{z\bar{z}})-i\gamma_{z\bar{z}}^{-1}[A^{(0,0)}_{z},F^{(0,0)}_{z\bar{z}}]\right)$
$\displaystyle-\frac{1}{2}\left(\partial_{z}A_{r}^{(-2,0)}-i[A_{z}^{(0,0)},A_{r}^{(-2,0)}]\right)+A_{z}^{(-1,0)}+o(u^{-\infty}).$

Observe that $A_{z}^{(-1)}$ contains up to linear terms in $u$. The
coefficients $F_{zu}^{(0)}$, $F_{zu}^{(-1)}$ and $F_{zr}^{(-2)}$ are found
directly

$\displaystyle F_{zu}^{(0)}$
$\displaystyle=-\partial_{u}A_{z}^{(0)}=o(u^{-\infty}),\quad
F_{zu}^{(-1)}=-\partial_{u}A_{z}^{(-1)}=O(1),$ (127) $\displaystyle
F_{zr}^{(-2)}$
$\displaystyle=\partial_{z}A_{r}^{(-2)}-A_{z}^{(-1)}+[A_{z}^{(0)},A_{r}^{(-2)}]=O(u).$
(128)

The second equation is consistent with the definition of $A_{z}^{(-1)}$ via
(107),

$\partial_{u}F^{(-2)}_{rz}=\partial_{u}A_{z}^{(-1)}-F^{(-n)}_{rz}+\partial_{z}(\gamma_{z\bar{z}}^{-1}F^{(0)}_{z\bar{z}})-i\left([A^{(-2)}_{r},\partial_{u}A_{z}^{(0)}-F^{(0)}_{rz}]+\gamma_{z\bar{z}}^{-1}[A^{(0)}_{z},F^{(0)}_{z\bar{z}}]\right).$
(129)

From (116) we have

$\displaystyle F_{ur}^{(-3)}$ $\displaystyle=$
$\displaystyle\partial_{u}^{-1}\left(2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(-1)}_{u|\bar{z})}-2i\gamma_{z\bar{z}}^{-1}[A^{(-1)}_{(z},F^{(0)}_{u|\bar{z})}]\right.$
$\displaystyle\left.-2i\gamma_{z\bar{z}}^{-1}[A^{(0)}_{(z},F^{(-1)}_{u|\bar{z})}]\right).$

From equations (127) and (3.2.2), we deduce that $F_{ru}^{(-3)}$ is $O(u)$.
Indeed, the first and third terms within the brackets are $O(1)$ in $u$, while
the second term contains a product of a linear term and
$\partial_{u}A_{z}^{(0)}$, which is $o(u^{-\infty})$; therefore, after
integration, the second term is $O(1)$. Then, we can proceed in an analogous
way, since now we have $A_{r}^{(-3)}=O(u^{2})$. Therefore, inductively from
(105), (109) and (116) we have that each $F_{ru}^{(-n-2)}$ will be $O(u^{n})$.

###  3.3 Radial gauge

Next we study the radial gauge, characterised by,

$\mathcal{A}_{r}=0,$ (131)

and let us consider the fall-off,

$A_{u}^{(0)}=0,$ (132)

which further specifies the gauge. Thus, this choice implies the following
identities,

$F_{ur}^{(-n)}=F_{rz}^{(-n)}=F_{r\bar{z}}^{(-n)}=0\qquad\mathrm{for}\ n=0,1.$
(133)

Equations (82)-(85) result in,

$\displaystyle E_{u}^{(-n)}$
$\displaystyle=\partial_{u}F^{(-n)}_{ur}+(n-3)F^{(1-n)}_{ur}-2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(2-n)}_{u|\bar{z})}$
(134)
$\displaystyle\quad-i\sum_{k=0}^{n-1}[A^{(k-n)}_{u},F^{(-k)}_{ur}]+2i\gamma_{z\bar{z}}^{-1}\sum_{k=0}^{n-2}[A^{(2+k-n)}_{(z},F^{(-k)}_{u|\bar{z})}],\qquad
n\geq 2,$ $\displaystyle E_{r}^{(-n)}$
$\displaystyle=(n-3)F^{(1-n)}_{ur}-2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(2-n)}_{r|\bar{z})}+2i\gamma_{z\bar{z}}^{-1}\sum_{k=2}^{n-2}[A^{(2+k-n)}_{(z},F^{(-k)}_{r|\bar{z})}],\qquad
n\geq 4,$ (135) $\displaystyle E_{z}^{(-n)}$
$\displaystyle=-\partial_{u}F^{(-n)}_{rz}+(n-1)(F^{(1-n)}_{uz}-F^{(1-n)}_{rz})-\partial_{z}(\gamma_{z\bar{z}}^{-1}F^{(2-n)}_{z\bar{z}})$
(136)
$\displaystyle\quad+i\sum_{k=0}^{n-1}[A^{(k-n)}_{u},F^{(-k)}_{rz}]+i\gamma_{z\bar{z}}^{-1}\sum_{k=0}^{n-2}[A^{(2+k-n)}_{z},F^{(-k)}_{z\bar{z}}],\qquad
n\geq 2,$ $\displaystyle E_{\bar{z}}^{(-n)}$
$\displaystyle=-\partial_{u}F^{(-n)}_{r\bar{z}}+(n-1)(F^{(1-n)}_{u\bar{z}}-F^{(1-n)}_{r\bar{z}})+\partial_{\bar{z}}(\gamma_{z\bar{z}}^{-1}F^{(2-n)}_{z\bar{z}})$
(137)
$\displaystyle\quad+i\sum_{k=0}^{n-1}[A^{(k-n)}_{u},F^{(-k)}_{r\bar{z}}]-i\gamma_{z\bar{z}}^{-1}\sum_{k=0}^{n-2}[A^{(2+k-n)}_{\bar{z}},F^{(-k)}_{z\bar{z}}],\qquad
n\geq 2.$

Observe the almost symmetric behaviour of $r$ and $u$ with respect to the
previous case (eqn. (90)-(93)), although there are differences which we will
see below.

####  3.3.1 Recursive formula for the gauge vector

As in the previous section, we will prove that given the initial data
$A_{z}^{(0)}(u,z,\bar{z}),A_{\bar{z}}^{(0)}(u,z,\bar{z})$ and
$\\{A_{z}^{(-n,0)},A_{\bar{z}}^{(-n,0)},A_{u}^{(-n-1,0)}\\}_{n\geq 1}$, it is
possible to find the solution of the equation of motion at arbitrary order in
the $1/r$-expansion. We present recursive formulas (similar to (105),(109))
that allow us to compute $A_{\mu}^{(-n)}$ for all $n\geq 1$, thus providing
the explicit expression of the e.o.m. solution at each order.

First of all, it is useful to write explicitly the definition of the field
strength tensor in radial gauge. From (131) and (132), at a given order $n$ we
have

$\displaystyle F_{ur}^{(-n)}$ $\displaystyle=(n-1)A_{u}^{(1-n)},\qquad n\geq
2,$ (138) $\displaystyle F_{uz}^{(-n)}$
$\displaystyle=2\partial_{[u}A_{z]}^{(-n)}-i\sum_{k=0}^{n-1}[A_{u}^{(k-n)},A_{z}^{(-k)}]=\partial_{u}A_{z}^{(-n)}-(D_{z}A_{u})^{(-n)},$
(139) $\displaystyle F_{rz}^{(-n)}$ $\displaystyle=(1-n)A_{z}^{(1-n)},\qquad
n\geq 2,$ (140) $\displaystyle F_{z\bar{z}}^{(-n)}$
$\displaystyle=2\partial_{[z}A_{\bar{z}]}^{(-n)}-i\sum_{k=0}^{n}[A_{z}^{(k-n)},A_{\bar{z}}^{(-k)}].$
(141)

Recall that $F_{ur}^{(-n)}=F_{rz}^{(-n)}=0$ for $n=0,1$. From the above
equations we notice that:

$\displaystyle F_{r\mu}^{(-n)}$ $\displaystyle\quad\mathrm{depends}\
\mathrm{on}\quad A_{\mu}^{(1-n)},\qquad(n\geq 2),$ (142) $\displaystyle
F_{uz}^{(-n)}$ $\displaystyle\quad\mathrm{depends}\
\mathrm{on}\quad\\{A_{u}^{(-k)},A_{z}^{(-k)}\\}_{k\leq n},$ (143)
$\displaystyle F_{z\bar{z}}^{(-n)}$ $\displaystyle\quad\mathrm{depends}\
\mathrm{on}\quad\\{A_{z}^{(-k)},A_{\bar{z}}^{(-k)}\\}_{k\leq n}.$ (144)

Suppose we know
$\\{A_{z}^{(-k)}(u,z,\bar{z}),A_{\bar{z}}^{(-k)}(u,z,\bar{z}),A_{u}^{(-k-1)}(u,z,\bar{z})\\}_{k<n}$
for some arbitrary $n\in\mathbb{N}$; we want to find
$\\{A_{z}^{(-n)}(u,z,\bar{z}),A_{\bar{z}}^{(-n)}(u,z,\bar{z}),A_{r}^{(-n-2)}(u,z,\bar{z})\\}$.
First, we solve for the $u$ component of the gauge field. At leading order we
know that $A_{u}^{(0)}$ vanishes. $A_{u}^{(-1)}$ can be obtained from (134)
and (138) with $n=2$:

$\displaystyle\partial_{u}F^{(-2)}_{ur}$
$\displaystyle=2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(0)}_{u|\bar{z})}-2i\gamma_{z\bar{z}}^{-1}[A^{(0)}_{(z},F^{(0)}_{u|\bar{z})}],$
(145) $\displaystyle\Longrightarrow A_{u}^{(-1)}$
$\displaystyle=2\gamma_{z\bar{z}}^{-1}\left(\partial_{(z}A_{\bar{z})}^{(0)}-i\partial_{u}^{-1}[A_{(z}^{(0)},\partial_{u}A_{\bar{z})}^{(0)}]\right),$
(146)

where in the second line we used $F_{uz}^{(0)}=\partial_{u}A_{z}^{(0)}$. For
$n\geq 2$ we make use of (135), (138) and (140) to find

$\displaystyle A_{u}^{(-n)}$
$\displaystyle=-\frac{2\gamma_{z\bar{z}}^{-1}}{n}\left(\partial_{(z}A_{\bar{z})}^{(1-n)}+\sum_{k=1}^{n-1}\frac{ik}{1-n}[A^{(1+k-n)}_{(z},A_{\bar{z})}^{(-k)}]\right).$
(147)

We notice that, in general:

$\displaystyle A_{u}^{(-n)}$ $\displaystyle\quad\mathrm{depends}\
\mathrm{on}\quad\\{A_{z}^{(-k)},A_{\bar{z}}^{(-k)}\\}_{k<n}\quad\text{and}\quad
A_{u}^{(-n,0)}.$ (148)

Therefore, we conclude that knowing
$\\{A_{z}^{(-k)},A_{\bar{z}}^{(-k)}\\}_{k<n}$ and the functions on the sphere
$\\{A_{u}^{(-n,0)}\\}$ is sufficient to determine $A_{u}^{(-n)}$ via equations
(146) and (147).

Now, we look at the $A_{z}^{(-n)}$ coefficients. At leading order, the $z$ and
$\bar{z}$ components of the gauge field constitute the initial data. For
$n\geq 1$ we have $A_{z}^{(-n)}=-n^{-1}F_{rz}^{(-n-1)}$, from (140). However,
thanks to (143) we know that, in general, $F_{rz}^{(-n-1)}$ does depend on
$A_{z}^{(-n)}$. Hence, we need an explicit expression for it, which is
provided by the $z$ component (136) of the e.o.m.,

$\displaystyle F^{(-n-1)}_{rz}$
$\displaystyle=nA_{z}^{(-n)}-\partial_{u}^{-1}\bigg{(}n\partial_{z}A_{u}^{(-n)}+nF^{(-n)}_{rz}+\partial_{z}(\gamma_{z\bar{z}}^{-1}F^{(1-n)}_{z\bar{z}})$
(149) $\displaystyle\quad-
in\sum_{k=1}^{n}[A_{z}^{(k-n)},A_{u}^{(-k)}]-i\sum_{k=0}^{n}[A^{(k-n-1)}_{u},F^{(-k)}_{rz}]-i\gamma_{z\bar{z}}^{-1}\sum_{k=0}^{n-1}[A^{(1+k-n)}_{z},F^{(-k)}_{z\bar{z}}]\bigg{)}.$

After some algebra we obtain, for $n\geq 1$:

$\displaystyle A_{z}^{(-n)}$
$\displaystyle=\frac{1}{2}\partial_{u}^{-1}\bigg{(}(1-n)A_{z}^{(1-n)}+\partial_{z}\left(A_{u}^{(-n)}+\frac{1}{n}\gamma_{z\bar{z}}^{-1}F^{(1-n)}_{z\bar{z}}\right)$
(150)
$\displaystyle\quad-\frac{i}{n}\sum_{k=1}^{n}[A_{z}^{(k-n)},(2n-k)A_{u}^{(-k)}+\gamma_{z\bar{z}}^{-1}F^{(1-k)}_{z\bar{z}}]\bigg{)}.$

The above equation together with (144) and (148) imply that,

$A_{z}^{(-n)}\quad\mathrm{depends}\ \mathrm{on}\quad
A_{u}^{(-n)},\\{A_{z}^{(-k)},A_{\bar{z}}^{(-k)},A_{u}^{(-k)}\\}_{k<n}\quad\text{and}\quad
A_{z}^{(-n,0)}.$ (151)

As an example, $A_{z}^{(-1)}$ depends just on the initial data and on
$A_{u}^{(-1)}$:

$\displaystyle A_{z}^{(-1)}$
$\displaystyle=\frac{1}{2}\partial_{u}^{-1}\left(\partial_{z}A_{u}^{(-1)}+\partial_{z}(\gamma_{z\bar{z}}^{-1}F^{(0)}_{z\bar{z}})-i[A_{z}^{(0)},A_{u}^{(-1)}+\gamma_{z\bar{z}}^{-1}F^{(0)}_{z\bar{z}}]\right)$
(152)

In other words, the solution for $A_{z}^{(-n)}$ is completely determined by
our initial hypothesis and by the above solution for $A_{u}^{(-n)}$. As in the
light-cone gauge, it is also possible to compute $A_{z}^{(-n)}$ by integrating
$F_{uz}^{(-n)}$, which in turn can be obtained through the Bianchi identity
(297). This shows that the results are consistent with the algebraic structure
of the theory. Figure 2 shows the relation between the coefficients in the
radial expansion.

Figure 2: The solid arrows indicate which $A_{u}^{(-k)}$, $A_{z}^{(-k)}$ enter
the recursion formula for $A_{u}^{(-n)}$, while the dotted arrows indicate
which $A_{u}^{(-k)}$ and $A_{z}^{(-k)}$ enter the recursion formula for
$A_{z}^{(-n)}$. The component $A_{\bar{z}}^{(-n)}$ is not drawn, since the
same result as for $A_{z}^{(-n)}$ is valid.

Recursive formula for $F_{ur}^{(-n)}$: we can immediately write a recursive
formula for the $u-r$component of the field strength, by virtue of (138),

$\displaystyle F_{ur}^{(-2)}$
$\displaystyle=2\gamma_{z\bar{z}}^{-1}\partial_{u}^{-1}\left(D_{(z}\partial_{u}\mathcal{A}_{\bar{z})}\right)^{(0)}$
(153) $\displaystyle F_{ur}^{(-n)}$
$\displaystyle=\frac{2\gamma_{z\bar{z}}^{-1}}{n-2}\left(D_{(z|}\mathcal{F}_{r|\bar{z})}\right)^{(1-n)},\qquad
n\geq 3$ (154)

Explicitly:

$\displaystyle F_{ur}^{(-2)}$
$\displaystyle=2\gamma_{z\bar{z}}^{-1}\left(\partial_{(z}A_{\bar{z})}^{(0)}-i\partial_{u}^{-1}[A_{(z}^{(0)},\partial_{u}A_{\bar{z})}^{(0)}]\right)$
(155) $\displaystyle F_{ur}^{(-n)}$
$\displaystyle=-2\gamma_{z\bar{z}}^{-1}\left(\partial_{(z}A_{\bar{z})}^{(2-n)}+\sum_{k=1}^{n-2}\frac{ik}{2-n}[A^{(2+k-n)}_{(z},A_{\bar{z})}^{(-k)}]\right),\qquad
n\geq 3$ (156)

Formula for $F_{rz}^{(-n)}$ for $n\geq 2$:

$\displaystyle F_{rz}^{(-n)}$
$\displaystyle=\frac{1}{2}\partial_{u}^{-1}\bigg{(}F_{rz}^{(1-n)}-\partial_{z}(F_{ur}^{(-n)}+\gamma_{z\bar{z}}^{-1}F_{z\bar{z}}^{(2-n)})$
(157)
$\displaystyle\quad+i\sum_{k=2}^{n}[A_{z}^{(k-n)},\frac{2n-k-1}{k-1}F_{ur}^{(-k)}+\gamma_{z\bar{z}}^{-1}F_{z\bar{z}}^{(2-k)}]\bigg{)}$

####  3.3.2 Decays on null infinity in the radiative phase space

As in the light-cone gauge, the standard $u\rightarrow-\infty$ fall-offs for
the radiative data $A_{z}^{(0)}$,

$A_{z}^{(0)}(u,z,\bar{z})=A_{z}^{(0,0)}(z,\bar{z})+o(u^{-\infty}),$ (158)

translates to $F_{ru}^{(-2)}$ via (138) and (146) as follows

$F_{ru}^{(-2)}(u,z,\bar{z})=F_{ru}^{(-2,0)}(z,\bar{z})+o(u^{-\infty}).$ (159)

We can organise the three non-vanishing components of $A_{\mu}$ by their
$u$-decay,

$\displaystyle A_{z}^{(0)},A_{\bar{z}}^{(0)}$ $\displaystyle=$ $\displaystyle
O(1),\quad A_{u}^{(-1)}=O(1),$ (160) $\displaystyle
A_{z}^{(-1)},A_{\bar{z}}^{(-1)}$ $\displaystyle=$ $\displaystyle O(u),\quad
A_{u}^{(-2)}=O(u),$ (161) $\displaystyle A_{z}^{(-2)},A_{\bar{z}}^{(-2)}$
$\displaystyle=$ $\displaystyle O(u^{2}),\quad A_{u}^{(-3)}=O(u^{2}),\dots$

which also translates into the six components of $F_{\mu\nu}$ as is shown in
the following table,

$o(u^{-\infty})$ | $F_{uz}^{(0)}$ | $F_{u\bar{z}}^{(0)}$ |  |  |  |
---|---|---|---|---|---|---
$O(1)$ | $F_{uz}^{(-1)}$ | $F_{u\bar{z}}^{(-1)}$ | $F_{ru}^{(-2)}$ | $F_{z\bar{z}}^{(0)}$ |  |
$O(u)$ | $F_{uz}^{(-2)}$ | $F_{u\bar{z}}^{(-2)}$ | $F_{ru}^{(-3)}$ | $F_{z\bar{z}}^{(-1)}$ | $F_{rz}^{(-2)}$ | $F_{r\bar{z}}^{(-2)}$
$O(u^{2})$ | $F_{uz}^{(-3)}$ | $F_{u\bar{z}}^{(-3)}$ | $F_{ru}^{(-4)}$ | $F_{z\bar{z}}^{(-2)}$ | $F_{rz}^{(-3)}$ | $F_{r\bar{z}}^{(-3)}$
$\cdots$ |  |  |  |  |  |
$O(u^{n})$ | $F_{uz}^{(-n-1)}$ | $F_{u\bar{z}}^{(-n-1)}$ | $F_{ru}^{(-n-2)}$ | $F_{z\bar{z}}^{(-n)}$ | $F_{rz}^{(-n-1)}$ | $F_{r\bar{z}}^{(-n-1)}$
Table 1:

##  4 Charges

In this section we compute the charges that arise when renormalizing the
symplectic form, obtaining finite expressions at the corner
$\mathcal{I}^{+}_{-}$. In the extended phase space presented in (14), the
local gauge transformations are parametrized by a $\mathfrak{g}$-valued
function $\Lambda$, giving the usual infinitesimal variation, (16).

The bulk action,

$S[\tilde{\mathcal{A}}_{\mu}]=\int_{D}\text{tr}\left(\tilde{\mathcal{F}}_{\mu\nu}\tilde{\mathcal{F}}^{\mu\nu}\right)dvol,$
(162)

with $\tilde{\mathcal{F}}_{\mu\nu}$ as in (36), and where $D$ is some region
with its boundary $\partial D:=\Sigma_{t_{0}}\cup\Sigma_{t_{1}}$ two Cauchy
surfaces along Minkowski (see Figure 3).

We remark that the Lagrangian (162), constructed from
$\tilde{\mathcal{A}}_{\mu}$ of (15), takes the same form as the YM Lagrangian
constructed from $\mathcal{A}_{\mu}$. This can be seen immediately from the
fact that $\tilde{\mathcal{A}}_{\mu}$ has the functional form of a gauge
transformation, with the parameter replaced by our new field $\breve{\Psi}$,
as is characteristic of the Stueckelberg procedure. In order to compute the
relevant charges though, it is crucial to take the field
$\tilde{\mathcal{A}}_{\mu}$ as our fundamental field in the covariant phase
space formalism.

From (162), one can then compute the bulk symplectic form straightforwardly,

$\Omega^{\text{bulk}}[\delta_{1},\delta_{2}]=-\int_{\Sigma}\text{tr}\left(\delta_{1}\tilde{\mathcal{F}}^{\mu\nu}\wedge\delta_{2}\tilde{\mathcal{A}}_{\nu}\right)dS_{\mu}-(1\leftrightarrow
2),$ (163)

for some Cauchy surface $\Sigma$ in Minkowski space. The symplectic form can
be used to obtain the canonical charges associated with the infinitesimal
symmetries (16), by means of the relation

$\Omega^{\text{bulk}}[\delta_{\Lambda},\delta]=\delta\tilde{Q}_{\Lambda},$
(164)

giving the following result:

$\tilde{Q}_{\Lambda}=\int_{\Sigma}\partial_{\nu}\text{tr}(\Lambda\tilde{\mathcal{F}}^{\mu\nu})dS_{\mu}.$
(165)

We stress that this charge is computed within the bulk. As it stands, the
expression for the charge is a total derivative, which implies that

$\tilde{Q}_{\Lambda}=\int_{S^{2}}\text{tr}(\Lambda\tilde{\mathcal{F}}^{\mu\nu})dS_{\mu\nu},$
(166)

where we are using that $\partial\Sigma=i_{0}\simeq S^{2}$, with $i_{0}$
denoting spatial infinity. We can identify $\partial\Sigma$ with
$\mathcal{I}^{+}_{-}$ using Bondi coordinates. Indeed, when working in these
coordinates, the symplectic form at null infinity $\mathcal{I}^{+}$ is found
by taking $\Sigma$ to be the hypersurface $\\{t=r+u=constant\\}$, and taking
the limit $t\rightarrow+\infty$ at fixed $u$. In that way, $\mathcal{I}^{+}$
is parametrized by the null coordinate $u$. Therefore, by taking the limit
$r\rightarrow+\infty$, while maintaining $u$ fixed, we have
$\Sigma\rightarrow\mathcal{I}$ and
$\partial\Sigma\rightarrow\mathcal{I}^{+}_{-}$ (see Figure 3). The
conservation of the charges from $\mathcal{I}^{-}_{+}$ to
$\mathcal{I}^{+}_{-}$ is showed via the antipodal mapping at
$\mathcal{I}^{+}_{-}$ (cf. Strominger:2013lka ). This antipodal map can be
directly translated to $\Psi$. Indeed, if we defined $\Psi^{+}$ and $\Psi^{-}$
for the retarded and advanced coordinates respectively, then

$\Psi^{+}(\hat{x})=\Psi^{-}(-\hat{x}),$ (167)

where $\hat{x}$ are the stereographic projection from the complex plane to the
Riemann sphere. Thus, on the celestial sphere (i.e., $\mathcal{I}^{+}_{-}$),
the charge is the limit $r\rightarrow+\infty$ and $u\rightarrow-\infty$ in the
quantity

$\tilde{Q}_{\Lambda}=\int_{S^{2}}\text{tr}(\Lambda\tilde{\mathcal{F}}_{ru})r^{2}\gamma_{z\bar{z}}dzd\bar{z}.$
(168)

Of course, this expression can diverge, and in general, a renormalization
procedure has to be carried out to give meaning to the expression (see
Freidel:2019ohg ; Peraza:2023ivy ).

Figure 3: Cauchy Surfaces $\Sigma_{t_{0}},\Sigma_{t_{1}}$, with $t_{0}<t_{1}$,
and constant $u$ ray. The sets (topologically spheres) in the intersection of
the hypersurfaces with $\\{u=u_{0}\\}$ correspond to the celestial sphere in
the limit $t\rightarrow+\infty$ with $u$ fixed.

Assuming the renormalization procedure has been carried out (as is the case
for Maxwell, Peraza:2023ivy ), we will focus on the finite term in (168).
Abusing notation, we will call such a finite part charge and denote it also by
$\tilde{Q}_{\Lambda}$. We will have the expansion in $r$, and the expansion in
$\Psi$ and $\Lambda_{+}$. A careful counting of the orders of each field will
indeed show that the first few orders in our result are consistent with
previous works in the literature.

###  4.1 Charges in the light-cone gauge

Let us start by computing each order in $r$ of $r^{2}\tilde{\mathcal{F}}_{ru}$
141414Recall that the $r^{2}$ comes from the volume element in $dS^{2}$ in
standard Bondi coordinates.. Note that we can re-write
$\tilde{\mathcal{F}}_{\mu\nu}$ as

$\displaystyle\mathcal{F}_{\mu\nu}=$ $\displaystyle
e^{i\Psi}\mathcal{F}_{\mu\nu}e^{-i\Psi}$ (169) $\displaystyle=$ $\displaystyle
e^{ad_{i\Psi}}\mathcal{F}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\mathcal{F}_{\mu\nu}+\frac{e^{ad_{i\Psi}}-1}{ad_{i\Psi}}[i\Psi,\mathcal{F}_{\mu\nu}]$
$\displaystyle=$
$\displaystyle\mathcal{F}_{\mu\nu}+\frac{1-e^{ad_{i\Psi}}}{ad_{-i\Psi}}[i\Psi,\mathcal{F}_{\mu\nu}]$
$\displaystyle=$
$\displaystyle\mathcal{F}_{\mu\nu}+\mathcal{O}_{-i\Psi}[i\Psi,\mathcal{F}_{\mu\nu}]$

where to get to the last line we used (20). Then we have

$\left(r^{2}\tilde{\mathcal{F}}_{ru}\right)^{(-i)}=F_{ru}^{(-2-i)}+\left(\mathcal{O}_{-i\Psi}\left([i\Psi,\mathcal{F}_{ru}]\right)\right)^{(-2-i)}.$
(170)

Explicitly, the first three coefficients are

$\displaystyle\left(r^{2}\tilde{\mathcal{F}}_{ru}\right)^{(0)}$
$\displaystyle=$ $\displaystyle
F_{ru}^{(-2)}+[i\Psi^{(1)},F_{ru}^{(-3)}]+\frac{1}{2}[i\Psi^{(1)},[i\Psi^{(1)},F_{ru}^{(-4)}]]+[i\Psi^{(2)},F_{ru}^{(-4)}]+...,$
$\displaystyle\left(r^{2}\tilde{\mathcal{F}}_{ru}\right)^{(-1)}$
$\displaystyle=$ $\displaystyle
F_{ru}^{(-3)}+[i\Psi^{(1)},F_{ru}^{(-4)}]+\frac{1}{2}[i\Psi^{(1)},[i\Psi^{(1)},F_{ru}^{(-5)}]]+[i\Psi^{(2)},F_{ru}^{(-5)}]+...,$
$\displaystyle\left(r^{2}\tilde{\mathcal{F}}_{ru}\right)^{(-2)}$
$\displaystyle=$ $\displaystyle
F_{ru}^{(-4)}+[i\Psi^{(1)},F_{ru}^{(-5)}]+\frac{1}{2}[i\Psi^{(1)},[i\Psi^{(1)},F_{ru}^{(-6)}]]+[i\Psi^{(2)},F_{ru}^{(-6)}]+...,$

Since we take
$\Lambda(r,z,\bar{z})=\sum_{k=0}^{+\infty}r^{-k}\Lambda^{(k)}(z,\bar{z})$, the
finite term in the charge is given by,

$\tilde{Q}_{\Lambda}=\int_{S^{2}}\text{tr}\left(\sum_{l=0}\Lambda^{(l)}\left(r^{2}\tilde{\mathcal{F}}_{ru}\right)^{(-l)}\right)dS_{S^{2}},$
(171)

where $dS_{S^{2}}=\gamma_{z\bar{z}}dzd\bar{z}$. Then, in terms of expansion
(170), it is,

$\displaystyle\tilde{Q}_{\Lambda}$ $\displaystyle=$
$\displaystyle\int_{S^{2}}\text{tr}\left(\sum_{l=0}\Lambda^{(l)}\left(F_{ru}^{(-2-l)}+[i\Psi^{(1)},F_{ru}^{(-3-l)}]\right.\right.$
(172)
$\displaystyle\left.\left.+\frac{1}{2}[i\Psi^{(1)},[i\Psi^{(1)},F_{ru}^{(-4-l)}]]+[i\Psi^{(2)},F_{ru}^{(-4-l)}]+...\right)\right)dS_{S^{2}}.$

For each $l\geq 0$, consider the extended charge density associated to
$\Lambda^{(l)}$, defined as

$\displaystyle\tilde{q}_{\Lambda^{(l)}}$ $\displaystyle:=$
$\displaystyle\text{tr}\left(\Lambda^{(l)}\left(F_{ru}^{(-2-l)}+[i\Psi^{(1)},F_{ru}^{(-3-l)}]\right.\right.$
(173)
$\displaystyle+\left.\left.\frac{1}{2}[i\Psi^{(1)},[i\Psi^{(1)},F_{ru}^{(-4-l)}]]+[i\Psi^{(2)},F_{ru}^{(-4-l)}]+...\right)\right),$

and the corresponding charge is denoted by

$\tilde{Q}_{\Lambda^{(l)}}:=\int_{S^{2}}\tilde{q}_{\Lambda^{(l)}}dS_{S^{2}},$
(174)

while the total charge is given by

$\tilde{Q}_{\Lambda}=\int_{S^{2}}\sum_{l=0}^{+\infty}\tilde{q}_{\Lambda^{(l)}}dS^{2}.$
(175)

Observe that $\tilde{q}_{\Lambda^{(l)}}$ is linear in both $\Lambda^{(l)}$ and
the coefficients $F_{ru}^{(-2-i-l)}$, for $i\geq 0$. The approximation up to
$j$-th order in $\Psi$ to $\tilde{q}_{\Lambda^{(l)}}$ is defined as

$\overset{j}{q}_{\Lambda^{(l)}}:=\text{tr}\left(\Lambda^{(l)}\left(F_{ru}^{(-2-l)}+[i\Psi^{(1)},F_{ru}^{(-3-l)}]+...+\mathfrak{ord}^{(j)}_{i\Psi}[F_{ru}^{(-2-j-l)}]\right)\right),$
(176)

where we have introduced the operator

$\displaystyle\mathfrak{ord}^{(j)}_{i\Psi}=$
$\displaystyle\sum_{k=1}^{j}\frac{1}{k!}\sum_{i_{1}+...+i_{k}=j}ad_{i\Psi^{(i_{1})}}(...(ad_{i\Psi^{(i_{k})}})...)[\cdot]$
(177) $\displaystyle=$
$\displaystyle\left(e^{ad_{i\psi}}-1\right)^{(j)}[\cdot]\ ,$

i.e. $\mathfrak{ord}^{(j)}_{i\Psi}$ denotes nested commutators of $i\Psi$
which add up to $\mathcal{O}(r^{j})$.

We can re-write the charge $\overset{j}{q}_{\Lambda^{(l)}}$ by observing that
$\mathfrak{ord}^{(j)}_{\Psi}$ is symmetric in $i_{1},...,i_{k}$ and using
iteratively the identity

$\text{tr}(A[B,C])=\text{tr}([-B,A]C),$ (178)

as follows,

$\overset{j}{q}_{\Lambda^{(l)}}:=\text{Tr}\left(F_{ru}^{(-2-l)}\Lambda^{(l)}-[i\Psi^{(1)},\Lambda^{(l)}]F_{ru}^{(-3-l)}+...+\mathfrak{ord}^{(j)}_{-i\Psi}[\Lambda^{(l)}]F_{ru}^{(-2-j-l)}\right).$
(179)

Similarly, we define the $j$-th approximation to the charge
$\tilde{Q}_{\Lambda^{(l)}}$ as

$\overset{j}{Q}_{\Lambda^{(l)}}=\int_{S^{2}}\overset{j}{q}_{\Lambda^{(l)}}dS_{S^{2}}$
(180)

In what follows we will define a hierarchy of subn-charges, labeled by
$j+l\geq 0$ in the set of charge densities sequences
$\\{\\{\overset{j}{q}_{\Lambda^{(l)}}\\}_{l\geq 0}\\}_{j\geq 0}$, such that at
each cut-off in the expansion of $\Psi$ we have a closed charge algebra, that
approximates $\tilde{q}_{\Lambda^{(l)}}$. Each algebra will correspond to the
radiative, linear, quadratic, …, approximations.

###  4.2 Leading Charges from zeroth order in fields

We start with the zeroth order in the hierarchy, given by only one charge,
$\\{\overset{0}{q}_{\Lambda^{(0)}}\\}$. It corresponds to the usual charge
derived from the radiative data Strominger:2013lka ; Strominger:2017zoo . It
can be written simply as

$\overset{0}{q}_{\Lambda^{(0)}}=\text{tr}\left(\Lambda^{(0)}F_{ru}^{(-2)}\right)\
.$ (181)

This is an infinite dimensional space of charges, parametrized by
$\Lambda^{(0)}$151515Recall that $\Lambda^{(0)}$ is a $\mathfrak{g}$-valued
function on the sphere, for the Lie algebra $\mathfrak{g}$ of the Lie group
$G$.. The definition is geometrically consistent with imposing the vanishing
of terms containing $\Psi$ or any of the coefficients in $\Lambda_{+}$, eqn.
(53). In other words, this corresponds to a vanishing of the rest of the
charge aspect approximations,

$\overset{0}{q}_{\Lambda^{(l)}}=0\quad\forall l\geq 1.$ (182)

###  4.3 Sub-leading Charge from up to linear order in fields

In the first truncation of the hierarchy that contains the subleading charge,
we only consider linear terms in $\Psi$ or $\Lambda_{0}$ for the leading
charge and the linear in $\Lambda_{+}$ and $\Psi$-independent terms for the
subleading.

Again, this is consistent with imposing $\Lambda_{+}^{(l)}=0$ for all $l\geq
2$ and taking only linear variations of the field $\Psi$. Then,

$\displaystyle\overset{1}{q}_{\Lambda^{(0)}}$ $\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda^{(0)}\left(F_{ru}^{(-2)}+[i\Psi^{(1)},F_{ru}^{(-3)}]\right)\right),$
(183) $\displaystyle\overset{0}{q}_{\Lambda^{(1)}}$ $\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda^{(1)}F_{ru}^{(-3)}\right).$ (184)

We remark that this linear approximation in $\Psi$ and $\Lambda_{+}$ is
equivalent to the linearization Campiglia:2021oqz

$\tilde{\mathcal{A}_{\mu}}=\mathcal{A}_{\mu}+D_{\mu}\Psi.$ (185)

Observe also that the functional dependence of
$\overset{0}{q}_{\Lambda^{(1)}}$ on $\mathcal{F}_{ru}^{(-3)}$ is the same as
the leading charge $\overset{0}{q}_{\Lambda^{(0)}}$ on
$\mathcal{F}_{ru}^{(-2)}$. Thus, we can write the charge
$\overset{1}{q}_{\Lambda^{(0)}}$ in the following way,

$\overset{1}{q}_{\Lambda^{(0)}}=\overset{0}{q}_{\Lambda^{(0)}}-\overset{0}{q}_{[i\Psi,\Lambda^{(0)}]^{(1)}}.$
(186)

thanks to identity $\text{tr}(A[B,C])=-\text{tr}([B,A]C)$. In (186), we have
used the notation

$[i\Psi,\Lambda^{(m)}]^{(n)}\equiv[i\Psi^{(n-m)},\Lambda^{(m)}]$ (187)

The first term in (186) is in the zeroth order charge algebra, while the
second one is a subleading contribution from the first order charge algebra.
Notice that we can keep track of which algebra we are referring to with the
charge $\overset{j}{q}_{\Lambda^{(l)}}$ by computing $j+l$.

These results can be connected with those in Campiglia2021 , in view of the
recursion formulas found in section 3. The details are provided in section 6.

###  4.4 Sub-sub leading Charge from up to second order in fields

For the expressions in this subsection, we have up to quadratic terms in
$\Psi$, and $\Lambda_{+}^{(2)}$. The charge aspects are

$\displaystyle\overset{2}{q}_{\Lambda^{(0)}}$ $\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda^{(0)}\left(F_{ru}^{(-2)}+[i\Psi^{(1)},F_{ru}^{(-3)}]+\frac{1}{2}[i\Psi^{(1)},[i\Psi^{(1)},F_{ru}^{(-4)}]]+[i\Psi^{(2)},F_{ru}^{(-4)}]\right)\right),$
(188) $\displaystyle\overset{1}{q}_{\Lambda^{(1)}}$ $\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda^{(1)}\left(F_{ru}^{(-3)}+[i\Psi^{(1)},F_{ru}^{(-4)}]\right)\right),$
(189) $\displaystyle\overset{0}{q}_{\Lambda^{(2)}}$ $\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda^{(2)}F_{ru}^{(-4)}\right).$ (190)

In defining the above charge aspects, we took the same approach as in the
subleading case by considering a cut-off in the $r$-expansion of the fields,
i.e.,

$\Lambda^{(n)}=\Psi^{(n)}=0,\quad\forall n\geq 3,$ (191)

which gives our candidate for the sub2-leading charge. We can write a
recursive expression of the charges in terms of the previous ones,

$\displaystyle\overset{2}{q}_{\Lambda^{(0)}}$ $\displaystyle=$
$\displaystyle\overset{1}{q}_{\Lambda^{(0)}}-\overset{0}{q}_{[i\Psi,\Lambda^{(0)}]^{(2)}}+\overset{0}{q}_{\frac{1}{2}[i\Psi,[i\Psi,\Lambda^{(0)}]]^{(2)}},$
(192) $\displaystyle\overset{1}{q}_{\Lambda^{(1)}}$ $\displaystyle=$
$\displaystyle\overset{0}{q}_{\Lambda^{(1)}}-\overset{0}{q}_{[i\Psi,\Lambda^{(1)}]^{(2)}},$
(193)

where observe that, as before, we are using the convention that
$\overset{j}{q}_{\Lambda^{(l)}}$ is in the level $j+l$ of the hierarchy, for
some LGT $\Lambda$.

###  4.5 Subn-leading charges from the truncation up to $n-$th order in the
fields

In view of the previous examples, given $n>2$, the charge aspects are defined
recursively as follows,

$\displaystyle\overset{0}{q}_{\Lambda^{(n)}}$ $\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda^{(n)}F_{ru}^{(-2-n)}\right),$ (194)
$\displaystyle\overset{1}{q}_{\Lambda^{(n-1)}}$ $\displaystyle=$
$\displaystyle\overset{0}{q}_{\Lambda^{(n-1)}}+\overset{0}{q}_{[-i\Psi,\Lambda^{(n-1)}]^{(n)}},$
(195) $\displaystyle...$ (196) $\displaystyle\overset{n-1}{q}_{\Lambda^{(1)}}$
$\displaystyle=$
$\displaystyle\overset{n-2}{q}_{\Lambda^{(1)}}+\sum_{k=1}^{n-1}\overset{0}{q}_{\left(\frac{1}{k!}ad^{k}_{-i\Psi}(\Lambda^{(1)})\right)^{(n)}},$
(197) $\displaystyle\overset{n}{q}_{\Lambda^{(0)}}$ $\displaystyle=$
$\displaystyle\overset{n-1}{q}_{\Lambda^{(0)}}+\sum_{k=1}^{n}\overset{0}{q}_{\left(\frac{1}{k!}ad^{k}_{-i\Psi}(\Lambda^{(0)})\right)^{(n)}},$
(198)

where, in order to be able to display the charges separately, we are
identifying, by abuse of notation, that $\Lambda^{(n)}$ is the large gauge
transformation whose only coefficient is at $r^{n}$ and its value is
$\Lambda^{(n)}$. For an arbitrary large gauge transformation $\Lambda$, the
total charge is given by the sum, cf. (175).

This way of writing the charges comes from a dressing of the large gauge
symmetry parameter by the Stueckelberg field, which can be seen directly from
the charge expression (165) as

$\text{tr}\left(\Lambda
e^{i\Psi}\mathcal{F}_{ru}e^{-i\Psi}\right)=\text{tr}\left(e^{-i\Psi}\Lambda
e^{i\Psi}\mathcal{F}_{ru}\right)=\text{tr}\left(e^{ad_{-i\Psi}}\Lambda\mathcal{F}_{ru}\right).$
(199)

Now, trivially, we have that for any $0\leq j\leq n$,

$\overset{n-j}{q}_{\Lambda^{(j)}}=\overset{0}{q}_{\Lambda^{(j)}}+\overset{0}{q}_{[-i\Psi,\Lambda^{(j)}]^{(j+1)}}+...+\sum_{k=1}^{n-j}\overset{0}{q}_{\left(\frac{1}{k!}ad^{k}_{-i\Psi}(\Lambda^{(j)})\right)^{(n)}},$
(200)

Observe that we can rewrite the charges in terms of the operator
$\mathcal{O}$, simply by working up to its combined $n-$th order in
$(\Psi,\Lambda)$, for $1\leq j\leq n$,

$\displaystyle\overset{0}{q}_{\Lambda^{(n)}}$ $\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda^{(n)}F_{ru}^{(-2-n)}\right)$ (201)
$\displaystyle\overset{n-j}{q}_{\Lambda^{(j)}}$ $\displaystyle=$
$\displaystyle\overset{0}{q}_{[e^{ad_{-i\Psi}}(\Lambda^{(j)})]^{\\{n\\}}},$
(202)

where the superscript $\\{n\\}$ in
$[e^{ad_{-i\Psi}}(\Lambda^{(j)})]^{\\{n\\}}$ denotes that we are considering
the expression only up to the $n$-th order in combinations of $\Psi$ and
$\Lambda^{(j)}$ (recall that $\Lambda^{(j)}$ has order $j$, fixed). Of course,
this is consistent with (168), and is a compact expression that contains all
the information up to $n$-th order.

##  5 Charge Algebra at each level

In this section, we present the closure of each charge algebra in the
hierarchy presented in the previous section. At each level of the hierarchy,
we show that the charges act canonically, and the phase space has a natural
structure. It is a non-trivial check since it has to be deduced from the
definitions of each phase space.

We are assuming we are working in a field-independent frame for the large
gauge transformations (i.e., $\delta_{\Lambda_{1}}\Lambda_{2}=0$). Otherwise,
a simple generalization can be carried by exchanging the bracket by a modified
bracket (47).

Fixing $n>0$, the first question we have to address is how
$\overset{n-j}{Q}_{\Lambda^{(j)}}$ acts on the extended phase space, in
particular on $\Psi$, for some function on the sphere $\Lambda^{(j)}$ that
generates a Large Gauge Transformation of order $j$ in the $r$-expansion. Now,
by inspection of (202), we see that the expression involves the coefficients
$\Psi^{(1)},...,\Psi^{(n-j)}$ of the Stueckelberg field. Therefore, its action
on the phase space must involve only those fields. Moreover, for the action to
be canonical,

$\Omega^{\\{n\\}}[\\{\overset{n-j}{Q}_{\Lambda^{(j)}},\cdot\\},\delta]=\delta\overset{n-j}{Q}_{\Lambda^{(j)}},$
(203)

where $\Omega^{\\{n\\}}$ is the symplectic form at $\mathcal{I}^{+}$ (i.e.,
the renormalized version of (163)) up to $n-$th order in $\Psi$. Therefore,
since we are allowing for up to $(n-j)$-th order in $\Psi$, then the action of
the variation associated to $\\{\overset{n-j}{Q}_{\Lambda^{(j)}},\cdot\\}$ on
the coefficients of the Stueckelberg field $\Psi$ is given by the
$(n-j)$-approximation of the variation by $\Lambda^{(l)}$,

$\\{\overset{n-j}{Q}_{\Lambda^{(j)}},\cdot\\}[\Psi]=\overset{n-j}{\delta}_{\Lambda^{(j)}}[\Psi],$
(204)

where recall that $\overset{j}{\delta}:=\sum_{k=0}^{j}\delta^{[k]}$, as
defined in subsection 2.2. Observe that this is a minimal requirement for
equation (203) to hold consistently with (164).

In what follows, we consider only charge densities (the same results hold for
the integrated charge).

Figure 4: Diagram showing the organization of the subalgebras of charges.
Along the horizontal axis, we increase in the power of $r$ involved in $\Psi$,
while the vertical axis indicates the sub${}^{n}-$leading charges. In circles,
the first sub${}^{n}-$leading charge for each $n$. The curled brackets
indicate the recursion (202).

###  5.1 Zeroth order Charge algebra

Recall that in the radiative phase space, we have $\Psi\equiv 0$, and
therefore the action of a large gauge transformation $\Lambda$ is simply

$\delta_{\Lambda}A_{z}^{(0)}=D^{(0)}_{z}\Lambda^{(0)}.$ (205)

Due to the cut-off of the order in $\Psi$, observe that the action of the
charge on the symplectic space is given by the identity,

$\\{\overset{0}{q}_{\Lambda^{(0)}},\cdot\\}=\overset{0}{\delta}_{\Lambda^{(0)}}.$
(206)

Here, again, we are identifying the large gauge transformation $\Lambda$ with
only one coefficient at $r^{0}$ with its value, $\Lambda^{(0)}$, in order to
ease the notation. Then, it is straightforward to prove that the zeroth-order
charge algebra is closed,

$\\{\overset{0}{q}_{\Lambda_{1}^{(0)}},\overset{0}{q}_{\Lambda_{2}^{(0)}}\\}=\overset{0}{q}_{-i[\Lambda_{1},\Lambda_{2}]^{(0)}}.$
(207)

Let us denote this algebra as $\mathfrak{p}_{0}$, generated by all possible
$\overset{0}{q}_{\Lambda^{(0)}}$, with
$\Lambda^{(0)}:S^{2}\rightarrow\mathbb{R}$.

###  5.2 First order charge algebra

Recall that by abuse of notation we are denoting $\Lambda^{(n)}$ as the large
gauge transformation whose only coefficient is at $r^{n}$ and its value is
$\Lambda^{(n)}$.

We have the identities,

$\displaystyle\\{\overset{0}{q}_{\Lambda^{(1)}},\cdot\\}$ $\displaystyle=$
$\displaystyle\delta^{[0]}_{\Lambda^{(1)}},$ (208)
$\displaystyle\\{\overset{1}{q}_{\Lambda^{(0)}},\cdot\\}$ $\displaystyle=$
$\displaystyle\overset{1}{\delta}_{\Lambda^{(0)}}=\delta^{[0]}_{\Lambda^{(0)}}+\delta^{[1]}_{\Lambda^{(0)}}$
(209)

First, we compute the bracket for the radiative subalgebra,

$\displaystyle\\{\overset{1}{q}_{\Lambda_{1}^{(0)}},\overset{1}{q}_{\Lambda_{2}^{(0)}}\\}=$
$\displaystyle\overset{1}{\delta}_{\Lambda_{1}^{(0)}}\text{Tr}\left(\Lambda_{2}^{(0)}F_{ru}^{(-2)}-[i\Psi^{(1)},\Lambda_{2}^{(0)}]F_{ru}^{(-3)}\right)$
$\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda_{2}^{(0)}\delta^{[1]}_{\Lambda_{1}^{(0)}}F_{ru}^{(-2)}-[i\overset{1}{\delta}_{\Lambda_{1}^{(0)}}\Psi^{(1)},\Lambda_{2}^{(0)}]F_{ru}^{(-3)}-[i\Psi^{(1)},\Lambda_{2}^{(0)}]\overset{1}{\delta}_{\Lambda_{1}^{(0)}}F_{ru}^{(-3)}\right)$
$\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda_{2}^{(0)}[i\Lambda_{1}^{(0)},F_{ru}^{(-2)}]+i[i[\Psi^{(1)},\Lambda_{1}^{(0)}],\Lambda_{2}^{(0)}]F_{ru}^{(-3)}\right.$
$\displaystyle=$
$\displaystyle\left.-[i\Psi^{(1)},\Lambda_{2}^{(0)}][i\Lambda_{1}^{(0)},F_{ru}^{(-3)}]\right),$

where in the first line we used (183), and in the last line we used that
$\Lambda_{1}^{(0)}$ has only the zeroth-order coefficient in the
$r$-expansion, simplifying (30). Then, using identity (267), we have,

$\displaystyle\\{\overset{1}{q}_{\Lambda_{1}^{(0)}},\overset{1}{q}_{\Lambda_{2}^{(0)}}\\}$
$\displaystyle=$
$\displaystyle\text{Tr}\left(-i[\Lambda_{1}^{(0)},\Lambda_{2}^{(0)}]F_{ru}^{(-2)}+[i\Psi^{(1)},i[\Lambda^{(0)}_{1},\Lambda_{2}^{(0)}]]F_{ru}^{(-3)}\right),$
$\displaystyle=$
$\displaystyle\overset{1}{q}_{-i[\Lambda^{(0)}_{1},\Lambda_{2}^{(0)}]}.$

The previous step can be written in a clear form by observing that

$[\Lambda_{1}^{(0)},\Lambda_{2}^{(0)}]=[\Lambda_{1},\Lambda_{2}]^{(0)}.$ (210)

Next, we compute the mixed Poisson bracket between leading and sub-leading
charges,

$\displaystyle\\{\overset{0}{q}_{\Lambda_{1}^{(1)}},\overset{1}{q}_{\Lambda_{2}^{(0)}}\\}=$
$\displaystyle\overset{0}{\delta}_{\Lambda_{1}^{(1)}}\text{Tr}\left(\Lambda_{2}^{(0)}F_{ru}^{(-2)}-[i\Psi^{(1)},\Lambda_{2}^{(0)}]F_{ru}^{(-3)}\right)$
(211) $\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda_{2}^{(0)}\overset{0}{\delta}_{\Lambda_{1}^{(1)}}F_{ru}^{(-2)}-[i\overset{0}{\delta}_{\Lambda_{1}^{(1)}}\Psi^{(1)},\Lambda_{2}^{(0)}]F_{ru}^{(-3)}-[i\Psi^{(1)},\Lambda_{2}^{(0)}]\overset{0}{\delta}_{\Lambda_{1}^{(1)}}F_{ru}^{(-3)}\right)$
$\displaystyle=$
$\displaystyle\text{Tr}\left(-i[\Lambda_{1}^{(1)},\Lambda_{2}^{(0)}]F_{ru}^{(-3)}\right)$
$\displaystyle=$
$\displaystyle\overset{0}{q}_{-i[\Lambda_{1}^{(1)},\Lambda_{2}^{(0)}]}=\overset{0}{q}_{-i[\Lambda_{1},\Lambda_{2}]^{(1)}}.$

To get from the second to the third line we use that in general
$\delta_{\Lambda^{+}}F_{ru}^{(-2-k)}=0$ and the variation (59).

Similarly, we do a consistency check for the anti-symmetric bracket,

$\displaystyle\\{\overset{1}{q}_{\Lambda_{2}^{(0)}},\overset{0}{q}_{\Lambda_{1}^{(1)}}\\}$
$\displaystyle=$
$\displaystyle\overset{1}{\delta}_{\Lambda_{2}^{(0)}}\text{Tr}\left(\Lambda_{1}^{(1)}F_{ru}^{(-3)}\right),$
(212) $\displaystyle=$
$\displaystyle\text{Tr}\left(-i\Lambda_{1}^{(1)}[F_{ru}^{(-3)},\Lambda_{2}^{(0)}]\right),$
(213) $\displaystyle=$
$\displaystyle\overset{0}{q}_{-i[\Lambda_{2},\Lambda_{1}]^{(1)}}.$ (214)

Finally, the sub-leading/sub-leading components of the Poisson bracket vanish,

$\\{\overset{0}{q}_{\Lambda_{1}^{(1)}},\overset{0}{q}_{\Lambda_{2}^{(1)}}\\}={\delta}^{[0]}_{\Lambda_{1}^{(1)}}\text{Tr}\left(\Lambda_{2}^{(1)}F_{ru}^{(-3)}\right)=0,$
(215)

since $\overset{0}{\delta}_{\Lambda_{1}^{(1)}}$ acting on $\Lambda_{2}^{(1)}$
vanishes, as well as on $F_{ru}^{(-3)}$.

Summarizing,

$\displaystyle\\{\overset{1}{q}_{\Lambda_{1}^{(0)}},\overset{1}{q}_{\Lambda_{2}^{(0)}}\\}$
$\displaystyle=$
$\displaystyle\overset{1}{q}_{-i[\Lambda_{1}^{(0)},\Lambda_{2}^{(0)}]},$ (216)
$\displaystyle\\{\overset{0}{q}_{\Lambda_{1}^{(1)}},\overset{1}{q}_{\Lambda_{2}^{(0)}}\\}$
$\displaystyle=$
$\displaystyle-\\{\overset{1}{q}_{\Lambda_{2}^{(0)}},\overset{0}{q}_{\Lambda_{1}^{(1)}}\\}=\overset{0}{q}_{-i[\Lambda_{1},\Lambda_{2}]^{(1)}}$
(217)
$\displaystyle\\{\overset{0}{q}_{\Lambda_{1}^{(1)}},\overset{0}{q}_{\Lambda_{2}^{(1)}}\\}$
$\displaystyle=$ $\displaystyle 0.$ (218)

Therefore, we have a first order Poisson algebra, which we will denote by
$\mathfrak{p}_{1}$, generated by functions
$\Lambda^{(0)},\Lambda^{(1)}:S^{2}\rightarrow\mathbb{R}$. Observe that
$\mathfrak{p}_{0}\subset\mathfrak{p}_{1}$.

###  5.3 General case

In this subsection, we are using the following standard identities,

$\displaystyle[A,[B,C]]$ $\displaystyle=$ $\displaystyle[[A,B],C]+[B,[A,C]],$
(219)
$\displaystyle\text{tr}\left(\left[A,ad_{A_{1}}(...(ad_{A_{n}}(B))...)\right]\right)$
$\displaystyle=$
$\displaystyle\text{tr}\bigg{(}ad_{A_{1}}(...ad_{A_{n}}([A,B])...)$ (220)
$\displaystyle+\sum_{k=1}^{n}ad_{A_{1}}(...ad_{[A,A_{k}]}(...ad_{A_{n}}(B))...).\bigg{)}$

####  5.3.1 Brackets with leading charge

First, we compute the brackets with the leading charge
$\overset{n}{q}_{\Lambda^{(0)}}$. The action of $\delta_{\Lambda^{(0)}}$ on
$F_{ru}^{(l)}$ is

$\delta_{\Lambda^{(0)}}F_{ru}^{(l)}=i[\Lambda^{(0)},F_{ru}^{(l)}],$ (221)

while its action on $\Psi$ is (cf. (30)) ,

$\overset{n}{\delta}_{\Lambda^{(0)}}\Psi=i[\Lambda^{(0)},\Psi].$ (222)

Then,

$\displaystyle\overset{n}{\delta}_{\Lambda_{1}^{(0)}}\overset{n-j}{q}_{\Lambda_{2}^{(j)}}$
$\displaystyle=$
$\displaystyle\overset{n}{\delta}_{\Lambda_{1}^{(0)}}\overset{0}{q}_{[e^{ad_{-i\Psi}}(\Lambda_{2}^{(j)})]^{\\{n\\}}}$
(223) $\displaystyle=$
$\displaystyle\overset{n}{\delta}_{\Lambda_{1}^{(0)}}\left(\overset{0}{q}_{\Lambda_{2}^{(j)}}+\overset{0}{q}_{[-i\Psi,\Lambda_{2}^{(j)}]^{(j+1)}}+...+\sum_{k=1}^{n-j}\overset{0}{q}_{\left(\frac{1}{k!}ad^{k}_{-i\Psi}(\Lambda_{2}^{(j)})\right)^{(n)}}\right)$
(224) $\displaystyle=$
$\displaystyle\overset{0}{q}_{-i[\Lambda_{1}^{(0)},\Lambda_{2}^{(j)]}}+\overset{0}{q}_{[-i\Psi,-i[\Lambda_{1}^{(0)},\Lambda_{2}^{(j)]}]^{(j+1)}}+...+\sum_{k=1}^{n-j}\overset{0}{q}_{\left(\frac{1}{k!}ad^{k}_{-i\Psi}(-i[\Lambda_{1}^{(0)},\Lambda_{2}^{(j)}])\right)^{(n)}},$
(225)

where we use repeatedly (220).

Then,

$\overset{n}{\delta}_{\Lambda_{1}^{(0)}}\overset{n-j}{q}_{\Lambda_{2}^{(j)}}=\overset{n-j}{q}_{-i[\Lambda_{1}^{(0)},\Lambda_{2}^{(j)}]}.$
(226)

####  5.3.2 Bracket with sub-leading charge

Next, we compute the brackets with the sub-leading charge,
$\overset{n-1}{q}_{\Lambda^{(1)}}$. In this case, we have

$\displaystyle\overset{n-1}{\delta}_{\Lambda_{1}^{(1)}}\overset{n-j}{q}_{\Lambda_{2}^{(j)}}$
$\displaystyle=$
$\displaystyle\overset{n-1}{\delta}_{\Lambda_{1}^{(1)}}\left(\overset{0}{q}_{\Lambda_{2}^{(j)}}+\overset{0}{q}_{[-i\Psi,\Lambda_{2}^{(j)}]^{(j+1)}}+...+\sum_{k=1}^{n-j}\overset{0}{q}_{\left(\frac{1}{k!}ad^{k}_{-i\Psi}(\Lambda_{2}^{(j)})\right)^{(n)}}\right)$
(227)

Since $\delta_{\Lambda^{+}}F_{ru}=0$, the first term vanishes. In particular,

$\displaystyle\overset{n-1}{\delta}_{\Lambda_{1}^{(1)}}\overset{0}{q}_{\Lambda_{2}^{(n)}}=0,$
(228)

which is consistent with the cut-off at order $n$ in $\Psi$. For the second
term, we have,

$\displaystyle\overset{n-1}{\delta}_{\Lambda_{1}^{(1)}}\overset{0}{q}_{[-i\Psi,\Lambda_{2}^{(j)}]^{(j+1)}}$
$\displaystyle=$
$\displaystyle\overset{n-1}{\delta}_{\Lambda_{1}^{(1)}}\overset{0}{q}_{[-i\Psi^{(1)},\Lambda_{2}^{(j)}]}$
(229) $\displaystyle=$
$\displaystyle\overset{0}{q}_{-i[\Lambda_{1}^{(1)},\Lambda_{2}^{(j)}]}.$ (230)

As a intermediate step, we show the third term,

$\displaystyle\overset{n-1}{\delta}_{\Lambda_{1}^{(1)}}\overset{0}{q}_{[-i\Psi,\Lambda_{2}^{(j)}]^{(j+2)}+\frac{1}{2}[-i\Psi,[-i\Psi,\Lambda_{2}^{(j)}]]^{(j+2)}}$
$\displaystyle=$
$\displaystyle\overset{n-1}{\delta}_{\Lambda_{1}^{(1)}}\overset{0}{q}_{[-i\Psi^{(2)},\Lambda_{2}^{(j)}]+\frac{1}{2}[-i\Psi^{(1)},[-i\Psi^{(1)},\Lambda_{2}^{(j)}]]}$
$\displaystyle=$
$\displaystyle\overset{0}{q}_{[-\frac{i}{2}[-i\Psi^{(1)},\Lambda_{1}^{(1)}],\Lambda_{2}^{(j)}]+\frac{1}{2}[-i\Lambda_{1}^{(1)},[-i\Psi^{(1)},\Lambda_{2}^{(j)}]]}$
$\displaystyle+\overset{0}{q}_{\frac{1}{2}[-i\Psi^{(1)},[-i\Lambda_{1}^{(1)},\Lambda_{2}^{(j)}]]}$
$\displaystyle=$
$\displaystyle\overset{0}{q}_{[-i\Psi^{(1)},-i[\Lambda_{1}^{(1)},\Lambda_{2}^{(j)}]]}.$

In general, we have that each $\Psi^{(1)},...,\Psi^{(n)}$ transforms
accordingly to,

$\displaystyle\Psi^{(1)}$ $\displaystyle\mapsto$
$\displaystyle\Lambda_{1}^{(1)},$ $\displaystyle\Psi^{(2)}$
$\displaystyle\mapsto$
$\displaystyle-\frac{i}{2}[\Psi^{(1)},\Lambda_{1}^{(1)}],$
$\displaystyle\Psi^{(3)}$ $\displaystyle\mapsto$
$\displaystyle-\frac{i}{2}[\Psi^{(2)},\Lambda_{1}^{(1)}]-\frac{1}{12}[\Psi^{(1)},[\Psi^{(1)},\Lambda_{1}^{(1)}]],$
$\displaystyle...,$

and therefore each term in (227) moves on to the left (using (220) and Bianchi
identity). Then,

$\overset{n-1}{\delta}_{\Lambda_{1}^{(1)}}\overset{n-j}{q}_{\Lambda_{2}^{(j)}}=\left\\{\begin{array}[]{cc}\overset{n-j}{q}_{-i[\Lambda_{1}^{(1)},\Lambda_{2}^{(j)}]}&\text{if
}j<n,\\\ 0&\text{if }j=n.\end{array}\right.$

####  5.3.3 General expression

As it was already discussed above, in the general case the action of
$\overset{n-k}{q}_{\Lambda^{(k)}}$ on the phase space is

$\\{\overset{n-k}{q}_{\Lambda^{(k)}},\cdot\\}=\overset{n-k}{\delta}_{\Lambda^{(k)}}.$
(231)

Consider the bracket with $\overset{n-j}{q}_{\Lambda_{2}^{(j)}}$, for some
$0\leq j\leq n$,

$\displaystyle\overset{n-k}{\delta}_{\Lambda_{1}^{(k)}}\overset{n-j}{q}_{\Lambda_{2}^{(j)}}$
$\displaystyle=$
$\displaystyle\overset{n-k}{\delta}_{\Lambda_{1}^{(k)}}\left(\overset{0}{q}_{\Lambda_{2}^{(j)}}+\overset{0}{q}_{[-i\Psi,\Lambda_{2}^{(j)}]^{(j+1)}}+...+\sum_{k=1}^{n-j}\overset{0}{q}_{\left(\frac{1}{k!}ad^{k}_{-i\Psi}(\Lambda_{2}^{(j)})\right)^{(n)}}\right)$
(232)

By the identities in section 2, each $\Psi^{(1)},...,\Psi^{(n)}$ transforms
under $\overset{n-k}{\delta}_{\Lambda_{1}^{(k)}}$ accordingly to

$\displaystyle\Psi^{(1)}$ $\displaystyle\mapsto$ $\displaystyle 0,$ (233)
$\displaystyle\Psi^{(2)}$ $\displaystyle\mapsto$ $\displaystyle 0,$ (235)
$\displaystyle...$ $\displaystyle\Psi^{(k)}$ $\displaystyle\mapsto$
$\displaystyle\Lambda_{1}^{(k)},$ (236) $\displaystyle\Psi^{(k+1)}$
$\displaystyle\mapsto$
$\displaystyle-\frac{i}{2}[\Psi^{(1)},\Lambda_{1}^{(k)}],$ (237)
$\displaystyle\Psi^{(k+2)}$ $\displaystyle\mapsto$
$\displaystyle-\frac{i}{2}[\Psi^{(2)},\Lambda_{1}^{(k)}]-\frac{1}{12}[\Psi^{(1)},[\Psi^{(1)},\Lambda_{1}^{(k)}]],$
(239) $\displaystyle...,$

and therefore each term in (232) moves to the left $k$ steps. Then, we arrive
at the following result, for $0\leq j,k\leq n$,

$\\{\overset{n-k}{q}_{\Lambda_{1}^{(k)}},\overset{n-j}{q}_{\Lambda_{2}^{(j)}}\\}=\left\\{\begin{array}[]{lc}\overset{n-k-j}{q}_{-i[\Lambda^{(k)},\Lambda^{(j)}]}&\text{if
}j+k\leq n,\\\ 0&\text{otherwise.}\end{array}\right.$ (240)

With this result, we have a closed Poisson algebra, $\mathfrak{p}_{n}$ such
that the action of the charges is canonical and we have the following chain

$\mathfrak{p}_{0}\subset...\subset\mathfrak{p}_{n-1}\subset\mathfrak{p}_{n}.$
(241)

##  6 Charges as corner terms - comparison with previous results

In this brief section, we are connecting the recursive formulas in section 3
with the charges constructed in section 4. We show that subsets of our charges
match with previous results. In particular, we can match our $n=1$ level with
the one in the literature for subleading charges in Yang-Mills Casali:2014xpa
; Campiglia:2021oqz , and the complete hierarchy of charges for the abelian
case Campiglia:2018dyi ; Hamada:2018vrw ; Li:2018gnc ; Peraza:2023ivy .

We first provide the usual corner interpretation for the leading charge. By
the fall-offs in subsubsection 3.2.2,
$F_{ru}^{(-2)}(u,z,\bar{z})=F_{ru}^{(-2)}+0(u^{-\infty})$, we have that,

$\displaystyle\overset{0}{Q}_{\Lambda^{(0)}}=\int_{S^{2}}\overset{0}{q}_{\Lambda^{(0)}}dS_{S^{2}}$
$\displaystyle=$
$\displaystyle\int_{S^{2}}\text{tr}\left(\Lambda^{(0)}F_{ru}^{(-2)}\right)dS_{S^{2}}$
(242) $\displaystyle=$
$\displaystyle\int_{S^{2}}\text{tr}\left(\Lambda^{(0)}F_{ru}^{(-2,0)}\right)dS_{S^{2}}.$
(243)

This is the standard leading charge (cf. Strominger:2013lka ).

Next, we compare the results for the subleading charge with the ones obtained
in Campiglia:2021oqz . We rewrite equations (183) as follow,

$\displaystyle\overset{0}{q}_{\Lambda^{(1)}}$ $\displaystyle=$
$\displaystyle\text{Tr}\left(\Lambda^{(1)}F_{ru}^{(-3)}\right),$ (244)
$\displaystyle\overset{1}{q}_{\Lambda^{(0)}}$ $\displaystyle=$
$\displaystyle\overset{0}{q}_{\Lambda^{(0)}}+\overset{0}{q}_{[-i\Psi^{(1)},\Lambda^{(0)}]}.$
(245)

In the prescription given in Campiglia:2021oqz , $\Psi^{(1)}$ acts as the
generator of the linear extension for the radiative phase space,

$\Gamma^{ext}_{1}:=\\{\tilde{\mathcal{A}}_{\mu}=\mathcal{A}_{\mu}+\mathcal{D}_{\mu}\Psi^{(1)},\quad\mathcal{A}_{\mu}\in\Gamma^{rad},\quad\Psi^{(1)}\in
C^{\infty}(S^{2})\\},$ (246)

where $\Gamma^{rad}$ is the radiative phase space. Observe that the leading
charge, (245), has the same expression as in equation (4.21) in
Campiglia:2021oqz , by restoring the coefficients $i$ in the reference. For
the subleading charge, we have to use the recursive relations for
$F_{ru}^{(-3)}$.

Let denote $D^{-}_{z}:=\partial_{z}-i[A_{z}^{(0,0)},\cdot]$, which is the
covariant derivative with respect to the $u\rightarrow-\infty$ limit in
$A_{z}^{(0)}$. Then, $A_{z}^{(-1)}$ can be written as (cf. (3.2.2)),

$A_{z}^{(-1)}=A_{z}^{(-1,0)}-\frac{1}{2}D^{-}_{z}A_{r}^{(-2,0)}+\frac{1}{2}uD^{-}_{z}(F_{ru}^{(-2,0)}+\gamma_{z\bar{z}}^{-1}F_{z\bar{z}}^{(0,0)})+o(u^{-\infty}).$
(247)

From here, a direct computation gives,

$F_{uz}^{(-1)}=\frac{1}{2}D^{-}_{z}(F_{ru}^{(-2,0)}+\gamma_{z\bar{z}}^{-1}F_{z\bar{z}}^{(0,0)})+o(u^{-\infty}),$
(248)

and analogous for $F_{u\bar{z}}^{(-1)}$. From (3.2.2), we have,

$\partial_{u}F_{ru}^{(-3)}=\gamma_{z\bar{z}}^{-1}D^{-}_{(z}D^{-}_{\bar{z})}(F_{ru}^{(-2,0)}+\gamma_{z\bar{z}}^{-1}F_{z\bar{z}}^{(0,0)})+o(u^{-\infty}).$
(249)

This is exactly the integrated (modulus the weight u) quantity in equation
(4.17) in Campiglia:2021oqz , where the authors use the integrated version of
the charges, along $\mathcal{I}^{+}$. To relate the integrated charges with
the ones here, one should impose the boundary conditions

$F_{ru}^{(-2)}(u,z,\bar{z}),F_{z\bar{z}}^{(0)}(u,z,\bar{z})\sim
o(u^{-\infty}),\quad\text{as}\quad u\rightarrow+\infty,$ (250)

and integrate by parts using
$\partial_{u}^{-1}(u\partial_{u}X)=uX-\partial_{u}^{-1}X$. Therefore, we
arrive at the same result for the subleading charge.

As an interesting corollary of our construction, observe that for the abelian
case we can immediately see the coefficients of the field $\Psi$ as the
Goldstone modes associated to the subleading charges, by constructing a
symplectic form which satisfies (203) and such that the modes
$\\{\Psi^{(i)}\\}_{i\geq 1}$ live on the boundary $\mathcal{I}^{+}_{-}$. In
accordance with Peraza:2023ivy , for a fixed $n\geq 0$ we have to evaluate the
abelian version of (163) at null infinity and after a renormalization in both
$r$ and $u$,

$\Omega^{n}(\delta_{1},\delta_{2})=\Omega^{rad}(\delta_{1},\delta_{2})+\sum_{i=1}^{n}\int_{S^{2}}\delta_{1}F_{ru}^{(-2-i,0)}\wedge\delta_{2}\Psi^{(i)}\gamma_{z\bar{z}}dzd\bar{z},$
(251)

where
$\Omega^{rad}(\delta_{1},\delta_{2})=\int_{\mathcal{I}^{+}}\delta_{1}\partial_{u}A^{(z}\wedge\delta_{2}A^{\bar{z})}dudzd\bar{z}$
is the usual radiative phase space symplectic form. Of course, the analogue of
this symplectic form in Yang-Mills theory also implies (203), but
$F_{ru}^{(-2-i,0)}$ and $\Psi^{(i)}$ are no longer canonical conjugates (due
to the action of the variations on $\Psi$).

##  7 Relations with infinite-dimensional algebras

In Freidel:2023gue , a subset of the charge aspects of YM theory were shown,
up to quadratic order in the creation and annihilation operators, to satisfy
the infinite dimensional YM version of the $w_{1+\infty}$ algebra
Strominger:2021mtt

$\left[S_{m}^{p,a}(\bar{z}),S_{n}^{q,b}(\bar{z})\right]=-i\ f^{ab}_{\ \
c}S_{m+n}^{p+q-1,c}(\bar{z})$ (252)

where $1-p\leq m\leq p-1$, $p,q$ are half-integers with $p,q>1$, and
$m+p\in\mathbb{Z}$ (similarly for $(q,n)$).

Crucial to find (252) were the recursion relations for the charge aspects.
Working in flat Bondi coordinates for simplicity (see Appendix D), and in
radial gauge with

$\mathcal{A}_{r}=0,\quad\text{and \quad}A_{u}^{(0)}=0,$ (253)

these relations can be written as161616The difference in sign is due to
converting to our conventions for the commutator.:

$\partial_{u}\mathcal{R}_{s}=\partial_{z}\mathcal{R}_{s-1}-i[A_{z}^{(0)},\mathcal{R}_{s-1}]=D_{z}^{(0)}\mathcal{R}_{s-1}$
(254)

where $\mathcal{R}_{s}$ are spin charge aspects, the first few given
explicitly by:

$\mathcal{R}_{-1}=F_{\bar{z}u}^{(0)},\quad\mathcal{R}_{0}=\tfrac{1}{2}\left(F_{ru}^{(-2)}+F_{\bar{z}z}^{(0)}\right),\quad\mathcal{R}_{1}=F_{rz}^{(-2)}$
(255)

The generators in (252) are constructed, after a renormalisation procedure, by
smearing the charge aspects with a Lie algebra valued function on the
sphere171717See Freidel:2023gue  for details..

First, observe that we can consider $F_{uz}^{(0)}$ as a lowest weight field on
the sphere, since its limit when $u\rightarrow-\infty$ vanishes, see Table 1.
The next weight corresponds to $O(1)$, and this is indeed the $u$ fall-off of
$F_{ru}^{(-2)}$ and $F_{\bar{z}z}^{(0)}$. Finally $F_{rz}^{(-2)}$ will be
quadratic in $u$, as detailed in Table 1181818The fall-off in $u$ is the same
in both standard Bondi and flat Bondi coordinates..

We note that the recursions above arise as a special case of the recursion
relations we give in Appendix D. At $s=0$, we have, from (254) and (255):

$\partial_{u}F_{ru}^{(-2)}+\partial_{u}F_{\bar{z}z}^{(0)}=2\left(\partial_{z}F_{\bar{z}u}^{(0)}-i[A_{z}^{(0)},F_{\bar{z}u}^{(0)}]\right)$
(256)

This follows from our relation (336), can be rearranged in the form of (256)
as below:

$\displaystyle\partial_{u}F_{ur}^{(-2)}$
$\displaystyle=2\left(\partial_{(z}\partial_{u}A_{\bar{z})}^{(0)}-i[A_{(z}^{(0)},\partial_{u}A_{\bar{z})}^{(0)}]\right)$
(257)
$\displaystyle=2\left(\partial_{z}F_{u\bar{z}}^{(0)}-i[A_{z}^{(0)},F_{u\bar{z}}^{(0)}]\right)-\partial_{u}F_{z\bar{z}}^{(0)}$

At $s=1$ we have

$\partial_{u}F_{rz}^{(-2)}=\tfrac{1}{2}\left(\partial_{z}F_{ru}^{(-2)}+\partial_{z}F_{\bar{z}z}^{(0)}-i[A_{z}^{(0)},F_{ru}^{(-2)}+F_{\bar{z}z}^{(0)}]\right)$
(258)

This immediately follows from our equation (337) with $n=2$

$\displaystyle 2\partial_{u}F_{rz}^{(-2)}$
$\displaystyle=-\partial_{z}(F_{ur}^{(-2)}+F_{z\bar{z}}^{(0)})+i[A_{z}^{(0)},F_{ur}^{(-2)}+F_{z\bar{z}}^{(0)}]\
.$ (259)

Higher orders are then reached via repeatedly acting with teh operator
$\partial_{u}^{-1}\cdot D_{z}^{(0)}$, see (254). We remark that this tower of
charges, which are linear in the anti-holomorphic component
$A_{\bar{z}}^{(0)}$ and of arbitrary order in the holomorphic components
$A_{z}^{(0)}$ are closely related to a self-dual subsector of the theory. One
could of course also construct the anti-holomorphc tower, by exchanging $z$
and $\bar{z}$

$\bar{\mathcal{R}}_{-1}=F_{zu}^{(0)},\quad\bar{\mathcal{R}}_{0}=\tfrac{1}{2}\left(F_{ru}^{(-2)}+F_{z\bar{z}}^{(0)}\right),\quad\bar{\mathcal{R}}_{1}=F_{r\bar{z}}^{(-2)}$
(260)

and postulating the recursion relation

$\partial_{u}\bar{\mathcal{R}}_{s}=\partial_{\bar{z}}\bar{\mathcal{R}}_{s-1}-i[A_{\bar{z}}^{(0)},\bar{\mathcal{R}}_{s-1}]=D_{\bar{z}}^{(0)}\bar{\mathcal{R}}_{s-1}$
(261)

Again, the first two steps will follow from our (336) and the conjugate of
(337), with all higher orders following from the action of $D_{\bar{z}}$. Of
course this (anti-self-dual) subsector will also possess a copy of the
infinite-dimensional symmetry algebra. If would be interesting to see whether
the commutation of charges between the towers could belong to some deformation
of these infinite algebras. We note that similar considerations apply for the
gravity theory, see Geiller:2024bgf .

##  8 Conclusions

Expanding on our proposal in Nagy:2024dme , we have shown how the phase space
at null infinity can be enlarged in order to accommodate subn-leading charges,
associated to the soft theorems for Yang-Mills theory. The procedure is
independent of gauge and coordinate choices, and we have allowed for a very
general fall-off in the expansion coordinate, thus making our procedure
potentially applicable to both tree- and loop-level effects.

We also provide expressions for the equations of motion for Yang-Mills theory
in the $r$-expansion, order by order, in standard Bondi coordinates. As an
example, we provide the explicit equations in both radial and light-cone
gauges using the usual radiative fall-offs compatible with tree-level. In both
cases, we show the existence of a recursive relation for the sub-leading
coefficients to all orders in the $r$-expansion for the field strength and the
gauge vector in terms of the leading component, which establishes the usual
convention of taking $A_{z}^{(0)}(u,z,\bar{z})$ as free initial data. Of
course, it is not clear whether it is possible to obtain such recursive
formulas for other gauge choices. We leave this question for future work.

The Stueckelberg procedure191919See e.g. Stueckelberg:1938hvi ;
Henneaux:1989zc ; Nagy:2019ywi ; Bansal:2020krz . is generalized from our
previous construction in the context of self-dual Yang-Mills Nagy:2022xxs  to
full Yang-Mills. The Stueckelberg trick has previously been employed in
seemingly very different contexts, but the unifying principle is the presence
of a local broken symmetry, though in our case this is subtly related to the
radial expansion at null infinity. In particular, the Goldstone-like modes
that constitute the expansion of the Stueckelberg field naturally serve as the
indicators of the power in the radial expansion, since they serve as the
target of the action of over-leading large gauge transformations acting on the
whole phase space. This, in turn, allows us to construct an extended phase
space that contains a clear hierarchy of subspaces. The tangent subspaces
(where variations live) are determined by the successive terms in the
perturbative expansion of the characteristic power series generating the so-
called Todd polynomial. An interesting insight here is that the canonical
action of the charges within the hierarchy (204) is controlled by the
Bernoulli numbers $B_{n}$. Since $B_{2k+1}$ are vanishing for $k>0$, it
appears that the canonical action in higher levels is controlled by the even
subn-leading charges. We will explore this further in future work.

The charges are related to the $r$-expansion of the field strength (we assume
renormalization, cf. (Peraza:2023ivy ), and can be computed directly. The
charge algebra is closed for each phase space. Using the recursion formulas
derived in section 3, we can compare the formulas of the sub-leading charge
with those in the literature Campiglia:2021oqz ; Casali:2014xpa . The charges
can also be constructed via an algebraically simple recursive relation.

Several interesting future lines of work emerge from our results.

On the one hand, the relation between over-leading large gauge transformations
and the symplectic structure of the phase space could shed light on the
symmetries of the theories, and therefore provide a deep understanding of the
classical starting point for a possible quantization of the theory. A natural
question is whether this extends to gravity, where work at the first few
subleading orders already exists in some contexts (e.g., Cachazo:2014fwa ;
Campiglia:2021oqz ; Freidel:2021dfs ; Fuentealba:2022xsz ), see also results
in the Newman-Penrose formalism Geiller:2024bgf . Our construction is
particularly well-suited for gravity, via the finite action of a
“Stueckelberg” diffeomorphism $\xi$ acting on the metric $g$,

$\breve{g}=e^{\mathcal{L}_{\xi}}g.$ (262)

We expect this extension to be more involved than the Yang-Mills case, due to
the non-linearities in both $\xi$ and $g$ in the definitions of the
gravitational charges.

In Nagy:2022xxs , the self-dual sector of gravity was considered at all
subleading orders, and a simple double copy dictionary was constructed from
Yang-Mills to gravity, for the extended radiative phase space, and the tower
of symmetries. A natural question prompted by our results here is whether this
can be extended to the full YM and gravitational theories202020For some
previous work on double copy relations at null infinity, see also
Campiglia:2021srh ; Adamo:2021dfg ; Ferrero:2024eva ; Mao:2021kxq ;
Godazgar:2021iae .

On the other hand, understanding the symmetries that govern the properties of
the S-matrix is crucial for establishing the conjectured holographic principle
for asymptotically flat spacetimes (e.g., Pasterski:2021raf ; Raclariu:2021zjz
and references therein). Recently, studies in some sub-sectors of gravity
(e.g. Freidel:2021ytz ) and in the celestial holography program (e.g.
Guevara:2021abz ; Strominger:2021mtt ) have shown the presence of infinite
dimensional algebras, known as $w_{1+\infty}$ algebras. It would be
interesting to see how these fit in the extended gravity case via our
proposal. In this direction, from a geometric point of view and on more
speculative grounds, it would be interesting to explore the connection between
the action of large diffeomorphisms and the universal corner algebra
$\mathfrak{diff}(S^{2})\times\mathfrak{gl}(2,\mathbb{R})\times\mathbb{R}^{2}$
Ciambelli:2022cfr . In particular, a possible extended graded algebraic
structure with the subalgebra
$\mathfrak{diff}(S^{2})\times\mathfrak{gl}(2,\mathbb{R})$ as base level could
determine the emergence of the $w_{1+\infty}$ algebra in a clear geometric
picture.

Finally, in the context of scattering amplitudes, the next step is to directly
apply the ideas in this article to the calculation of subn-leading soft
theorems at loop level Bianchi:2014gla ; Sahoo:2018lxl ; Pasterski:2022djr ;
Donnay:2022hkf ; Agrawal:2023zea ; Choi:2024ygx ; Campiglia:2019wxe ;
AtulBhatkar:2019vcb , via the Ward identities 212121For recent results at the
first subleading level see Choi:2024ygx . A simplified set-up which holds
promise as a starting point for going to arbitrary orders is the self-dual
sector This also has the benefit of being one-loop exact, for both Yang-Mills
and gravity Bern:1993qk ; Mahlon:1993si ; Bern:1996ja ; Bern:1998xc ;
Monteiro:2022nqt , and it preserves the infinite dimensional algebras above at
loop-level Ball:2021tmb ; Mago:2021wje ; Banerjee:2023jne ; Monteiro:2022xwq .

###### Acknowledgements.

We thank Miguel Campiglia and Marc Geiller for useful discussions. G.P. is
funded by STFC Doctoral Studentship 2023. S.N. is supported in part by STFC
consolidated grant T000708. J.P. was partially funded by Fondo Clemente
Estable Project FCE_1_2023_1_175902 and by CSIC Group 883174. Currently funded
by a Postdoctoral Fellowship at Concordia University, Montreal.

##  Appendix A Extended phase space calculations

Equation (18):

$\displaystyle\delta_{\Lambda}\hat{\mathcal{A}}_{I}$ $\displaystyle=$
$\displaystyle\delta_{\Lambda}(e^{i\Psi}\mathcal{A}_{I}e^{-i\Psi}+ie^{i\Psi}\partial_{I}e^{-i\Psi})$
(263) $\displaystyle=$
$\displaystyle\delta_{\Lambda}(e^{-X}\mathcal{A}_{I}e^{X}+ie^{-X}\partial_{I}e^{X})$
$\displaystyle=$ $\displaystyle-\mathcal{O}_{X}(\delta
X)e^{-X}\mathcal{A}_{I}e^{X}+e^{-X}(\delta_{\Lambda}\mathcal{A}I)e^{X}+e^{-X}\mathcal{A}_{I}e^{X}\mathcal{O}_{X}(\delta
X)+$ $\displaystyle-i\mathcal{O}_{X}(\delta
X)e^{-X}\partial_{I}e^{X}+ie^{-X}\partial_{I}\left[e^{X}\mathcal{O}_{X}(\delta
X)\right]$ $\displaystyle=$ $\displaystyle
e^{-X}\big{\\{}\delta_{\Lambda}\mathcal{A}_{I}+\mathcal{A}_{I}e^{X}\mathcal{O}_{X}(\delta
X)e^{-X}-e^{X}\mathcal{O}_{X}(\delta X)e^{-X}\mathcal{A}_{I}+$
$\displaystyle\quad\quad+ie^{X}\mathcal{O}_{X}(\delta
X)\partial_{I}e^{-X}+i\partial_{I}\left[e^{X}\mathcal{O}_{X}(\delta
X)\right]e^{-X}\big{\\}}e^{X}$ $\displaystyle=$ $\displaystyle
e^{i\Psi}\left\\{\delta_{\Lambda}\mathcal{A}_{I}+D_{I}\left[e^{-i\Psi}\mathcal{O}_{-i\Psi}(\delta\Psi)e^{i\Psi}\right]\right\\}e^{-i\Psi}$

where we defined $X:=-i\Psi$ and we used:

$\displaystyle\delta e^{-X}$ $\displaystyle=$
$\displaystyle-\mathcal{O}_{X}(\delta X)e^{-X}$ (264) $\displaystyle
e^{-X}\partial_{I}e^{X}$ $\displaystyle=$
$\displaystyle-(\partial_{I}e^{-X})e^{X}$ (265)

Equation (23):

$\displaystyle\hat{D}_{I}\Lambda$ $\displaystyle=$
$\displaystyle\partial_{I}\Lambda-i[\hat{\mathcal{A}}_{I},\Lambda]$ (266)
$\displaystyle=$
$\displaystyle\partial_{I}\Lambda-i[e^{i\Psi}\mathcal{A}_{I}e^{-i\Psi}+ie^{i\Psi}\partial_{I}e^{-i\Psi},\Lambda]$
$\displaystyle=$ $\displaystyle e^{i\Psi}\partial_{I}(e^{-i\Psi}\Lambda
e^{i\Psi})e^{-i\Psi}-e^{i\Psi}[\mathcal{A}_{I},e^{-i\Psi}\Lambda
e^{i\Psi}]e^{-i\Psi}$ $\displaystyle=$ $\displaystyle
e^{i\Psi}D_{I}(e^{-i\Psi}\Lambda e^{i\Psi})e^{-i\Psi}$

##  Appendix B Useful formulae

In this appendix, we provide some of the formulae which are used throughout
the text. The proof of each identity is straightforward.

First, the usual trace identities,

$\displaystyle\text{tr}\left(A[B,C]\right)$ $\displaystyle=$
$\displaystyle\text{tr}\left([A,B]C\right),$ (267)
$\displaystyle\text{tr}\left(Aad_{B}^{k}(C)\right)$ $\displaystyle=$
$\displaystyle(-1)^{k}\text{tr}\left(ad_{B}^{k}(A)C\right).$ (268)

Next, we provide a few representations for the operator $\mathcal{G}$ when
evaluated at $\mathcal{A}_{\mu}$, i.e.,
$\mathcal{G}_{\Psi}(\mathcal{A}_{\mu})$.

$\displaystyle\mathcal{G}_{\Psi}(\mathcal{A}_{\mu})$ $\displaystyle=$
$\displaystyle
e^{i\Psi}\mathcal{A}_{\mu}e^{-i\Psi}+ie^{i\Psi}\partial_{\mu}e^{-i\Psi},$
(269) $\displaystyle=$
$\displaystyle\mathcal{A}_{\mu}-D_{\mu}i\Psi+\frac{1}{2}[D_{\mu}i\Psi,i\Psi]-\frac{1}{3!}[[D_{\mu}i\Psi,i\Psi],i\Psi]+...,$
(270) $\displaystyle=$
$\displaystyle\mathcal{A}_{\mu}+\sum_{k=1}\frac{1}{k!}ad^{k-1}_{i\Psi}(D_{\mu}i\Psi),$
(271) $\displaystyle=$
$\displaystyle\mathcal{A}_{\mu}+\mathcal{O}_{-i\Psi}(D_{\mu}i\Psi).$ (272)

##  Appendix C YM e.o.m. in radial expansion

In order to study $\mathcal{T}$ at a generic order $n$ in the $1/r$-expansion,
it is useful to introduce the following map:

$\displaystyle\mathcal{P}_{-n}\colon C_{r}^{\infty}(M)$
$\displaystyle\longrightarrow C_{r}^{\infty}(M)$ (273) $\displaystyle f$
$\displaystyle\longmapsto r^{-n}f^{(-n)}$

It is easy to see that
$\mathcal{P}_{-m}\circ\mathcal{P}_{-n}(f)=\delta_{mn}\mathcal{P}_{-n}(f)$, for
all $m,n\in\mathbb{Z}$ and for all $f\in C^{\infty}_{r}(M)$, meaning that the
map $\mathcal{P}_{-n}$ is a projector. Moreover, as a consequence of the
definition of the set $C^{\infty}_{r}(M)$, we have that
$\mathcal{P}_{-n}(f)=0$ for all $n<0$, which is consistent with equation (75).

######  Lemma 1.

$\forall m,n\in\mathbb{N}$ and $\forall f,g\in C^{\infty}_{r}(M)$ the
following identities hold:

$\displaystyle\mathcal{P}_{-n}(r^{-m}f)$
$\displaystyle=r^{-m}\mathcal{P}_{m-n}(f)$ (274)
$\displaystyle\mathcal{P}_{-n}(\partial_{r}f)$
$\displaystyle=\partial_{r}\mathcal{P}_{1-n}(f)$ (275)
$\displaystyle\mathcal{P}_{-n}(fg)$
$\displaystyle=\sum_{k\in\mathbb{N}}\mathcal{P}_{k-n}(f)\mathcal{P}_{-k}(g)$
(276)

###### Proof.

$\displaystyle\mathcal{P}_{-n}(r^{-m}f)$
$\displaystyle=\mathcal{P}_{-n}\sum_{k\in\mathbb{Z}}r^{-m-k}f^{(-k)}=r^{-n}f^{(m-n)}=r^{-m}\mathcal{P}_{m-n}(f)$
(277) $\displaystyle\mathcal{P}_{-n}(\partial_{r}f)$
$\displaystyle=\mathcal{P}_{-n}\sum_{k\in\mathbb{Z}}(-k)r^{-k-1}f^{(-k)}=(1-n)r^{-n}f^{(1-n)}=\partial_{r}\mathcal{P}_{1-n}(f)$
(278) $\displaystyle\mathcal{P}_{-n}(fg)$
$\displaystyle=\mathcal{P}_{-n}\sum_{k,l\in\mathbb{Z}}r^{-k-l}f^{(-k)}g^{(-l)}=r^{-n}\sum_{k\in\mathbb{N}}f^{(k-n)}g^{(-k)}=\sum_{k\in\mathbb{N}}\mathcal{P}_{k-n}(f)\mathcal{P}_{-k}(g)$
(279)

∎

Using this lemma, one can prove equation (79):

###### Proof.

From (77):

$\displaystyle\mathcal{E}_{\mu}$
$\displaystyle=g^{\nu\rho}D_{\rho}\mathcal{F}_{\nu\mu}$ (280)
$\displaystyle=-2D_{(u}\mathcal{F}_{r)\mu}+D_{r}\mathcal{F}_{r\mu}+2r^{-2}\gamma_{z\bar{z}}^{-1}D_{(z}\mathcal{F}_{\bar{z})\mu}$
(281)
$\displaystyle=2\left(-\partial_{(u}\mathcal{F}_{r)\mu}+\Gamma^{\rho}_{ur}\mathcal{F}_{\rho\mu}+\Gamma^{\rho}_{\mu(u}\mathcal{F}_{r)\rho}+i[\mathcal{A}_{(u},\mathcal{F}_{r)\mu}]\right)$
(282)
$\displaystyle\quad+\partial_{r}\mathcal{F}_{r\mu}-\Gamma^{\rho}_{rr}\mathcal{F}_{\rho\mu}-\Gamma^{\rho}_{r\mu}\mathcal{F}_{r\rho}-i[\mathcal{A}_{r},\mathcal{F}_{r\mu}]$
$\displaystyle\quad+2r^{-2}\gamma_{z\bar{z}}^{-1}\left(\partial_{(z}\mathcal{F}_{\bar{z})\mu}-\Gamma^{\rho}_{z\bar{z}}\mathcal{F}_{\rho\mu}-\Gamma^{\rho}_{\mu(z}\mathcal{F}_{\bar{z})\rho}-i[\mathcal{A}_{(z},\mathcal{F}_{\bar{z})\mu}]\right)$
$\displaystyle=-2\left(\partial_{(u}\mathcal{F}_{r)\mu}-i[\mathcal{A}_{(u},\mathcal{F}_{r)\mu}]\right)+r^{-1}(\mathcal{F}_{uz}\delta_{z\mu}+\mathcal{F}_{u\bar{z}}\delta_{\bar{z}\mu})$
(283)
$\displaystyle\quad+\partial_{r}\mathcal{F}_{r\mu}-r^{-1}(\mathcal{F}_{rz}\delta_{z\mu}+\mathcal{F}_{r\bar{z}}\delta_{\bar{z}\mu})-i[\mathcal{A}_{r},\mathcal{F}_{r\mu}]$
$\displaystyle\quad+2r^{-2}\gamma_{z\bar{z}}^{-1}\\{\partial_{(z}\mathcal{F}_{\bar{z})\mu}-r\gamma_{z\bar{z}}(\mathcal{F}_{u\mu}-\mathcal{F}_{r\mu})-r^{-1}\mathcal{F}_{(z\bar{z})}\delta_{r\mu}-i[\mathcal{A}_{(z},\mathcal{F}_{\bar{z})\mu}]\\}$
$\displaystyle\quad+r^{-1}(\mathcal{F}_{uz}-\mathcal{F}_{rz})\delta_{z\mu}+r^{-1}(\mathcal{F}_{u\bar{z}}-\mathcal{F}_{r\bar{z}})\delta_{\bar{z}\mu}+(r\gamma_{z\bar{z}})^{-2}(\partial_{z}\gamma_{z\bar{z}}\delta_{z\mu}-\partial_{\bar{z}}\gamma_{z\bar{z}}\delta_{\bar{z}\mu})\mathcal{F}_{z\bar{z}}$
$\displaystyle=-2\left(\partial_{(u}\mathcal{F}_{r)\mu}-i[\mathcal{A}_{(u},\mathcal{F}_{r)\mu}]\right)+\partial_{r}\mathcal{F}_{r\mu}-i[\mathcal{A}_{r},\mathcal{F}_{r\mu}]-2r^{-1}\mathcal{F}_{ur}(\delta_{u\mu}+\delta_{r\mu})$
(284)
$\displaystyle\quad+2r^{-2}\gamma_{z\bar{z}}^{-1}\left(\partial_{(z}\mathcal{F}_{\bar{z})\mu}-i[\mathcal{A}_{(z},\mathcal{F}_{\bar{z})\mu}]\right)+(r\gamma_{z\bar{z}})^{-2}(\partial_{z}\gamma_{z\bar{z}}\delta_{z\mu}-\partial_{\bar{z}}\gamma_{z\bar{z}}\delta_{\bar{z}\mu})\mathcal{F}_{z\bar{z}}$

Thus, $\forall n\in\mathbb{N}$:

$\displaystyle E_{\mu}^{(-n)}$
$\displaystyle=r^{n}\mathcal{P}_{-n}(\mathcal{E}_{\mu})$ (286)
$\displaystyle=r^{n}\left\\{-\partial_{u}\mathcal{P}_{-n}(\mathcal{F}_{r\mu})-\partial_{r}\mathcal{P}_{1-n}(\mathcal{F}_{u\mu}-\mathcal{F}_{r\mu})+i\mathcal{P}_{-n}(2[\mathcal{A}_{(u},\mathcal{F}_{r)\mu}]-[\mathcal{A}_{r},\mathcal{F}_{r\mu}])\right\\}$
(287)
$\displaystyle\quad-2r^{n-1}\mathcal{P}_{1-n}(\mathcal{F}_{ur})(\delta_{u\mu}+\delta_{r\mu})+2r^{n-2}\gamma_{z\bar{z}}^{-1}\left\\{\partial_{(z}\mathcal{P}_{2-n}(\mathcal{F}_{\bar{z})\mu})-i\mathcal{P}_{2-n}([\mathcal{A}_{(z},\mathcal{F}_{\bar{z})\mu}])\right\\}$
$\displaystyle\quad+r^{n-2}\gamma_{z\bar{z}}^{-2}(\partial_{z}\gamma_{z\bar{z}}\delta_{z\mu}-\partial_{\bar{z}}\gamma_{z\bar{z}}\delta_{\bar{z}\mu})\mathcal{P}_{2-n}(\mathcal{F}_{z\bar{z}})$
$\displaystyle=-\partial_{u}F^{(-n)}_{r\mu}+(n-1)(F^{(1-n)}_{u\mu}-F^{(1-n)}_{r\mu})-2F^{(1-n)}_{ur}(\delta_{u\mu}+\delta_{r\mu})$
(288)
$\displaystyle\quad+2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(2-n)}_{\bar{z})\mu}+\gamma_{z\bar{z}}^{-2}(\partial_{z}\gamma_{z\bar{z}}\delta_{z\mu}-\partial_{\bar{z}}\gamma_{z\bar{z}}\delta_{\bar{z}\mu})F^{(2-n)}_{z\bar{z}}$
$\displaystyle\quad+i\sum_{k=0}^{n}\left(2[A^{(k-n)}_{(u},F^{(-k)}_{r)\mu}]-[A^{(k-n)}_{r},F^{(-k)}_{r\mu}]\right)-2i\gamma_{z\bar{z}}^{-1}\sum_{k\in\mathbb{N}}[A^{(2+k-n)}_{(z},F^{(-k)}_{\bar{z})\mu}]$
(289)
$\displaystyle=-\partial_{u}F^{(-n)}_{r\mu}+\tilde{\delta}_{\mu}^{\rho}(n)F^{(1-n)}_{u\rho}-\tilde{\delta}_{\mu}^{\rho}(n)F^{(1-n)}_{r\rho}$
(290)
$\displaystyle\quad+2\gamma_{z\bar{z}}^{-1}\partial_{(z}F^{(2-n)}_{\bar{z})\mu}+\gamma_{z\bar{z}}^{-2}(\partial_{z}\gamma_{z\bar{z}}\delta_{z\mu}-\partial_{\bar{z}}\gamma_{z\bar{z}}\delta_{\bar{z}\mu})F^{(2-n)}_{z\bar{z}}$
$\displaystyle\quad+i\sum_{k=0}^{n}\left(2[A^{(k-n)}_{(u},F^{(-k)}_{r)\mu}]-[A^{(k-n)}_{r},F^{(-k)}_{r\mu}]-2\gamma_{z\bar{z}}^{-1}[A^{(k-n)}_{(z},F^{(2-k)}_{\bar{z})\mu}]\right)$

with $\tilde{\delta}_{\mu}^{\nu}(n)$ as in (81). ∎

###  C.1 Bianchi identities

For the recursive formulas it is useful to have explicitly written Bianchi
identities for certain gauge and fall-offs conditions (for example, (87) and
(88). We can think of Bianchi identities as the vanishing of a $(0,3)-$tensor,

$\displaystyle\mathcal{B}_{\mu\nu\rho}:=D_{\mu}\mathcal{F}_{\nu\rho}+\circlearrowleft=0.$
(291)

Then, by defining the $r-$expansion of $\mathcal{B}_{\mu\nu\rho}$ as

$\mathcal{B}_{\mu\nu\rho}=\sum_{n\in\mathbb{N}}r^{-n}B^{(-n)}_{\mu\nu\rho},$
(292)

we can directly compute Bianchi identities in light-cone gauge at a given
order $n\in\mathbb{N}$:

$\displaystyle B_{urz}^{(-n)}$
$\displaystyle=\partial_{u}F_{rz}^{(-n)}+(n-1)F_{uz}^{(1-n)}+\partial_{z}F_{ur}^{(-n)}$
(293)
$\displaystyle\quad+i\sum_{k=0}^{n-2}[A_{r}^{(k-n)},F_{uz}^{(-k)}]-i\sum_{k=2}^{n}[A_{z}^{(k-n)},F_{ur}^{(-k)}],\qquad
n\geq 2$ $\displaystyle B_{ur\bar{z}}^{(-n)}$
$\displaystyle=\partial_{u}F_{r\bar{z}}^{(-n)}+(n-1)F_{u\bar{z}}^{(1-n)}+\partial_{\bar{z}}F_{ur}^{(-n)}$
(294)
$\displaystyle\quad+i\sum_{k=0}^{n-2}[A_{r}^{(k-n)},F_{u\bar{z}}^{(-k)}]-i\sum_{k=2}^{n}[A_{\bar{z}}^{(k-n)},F_{ur}^{(-k)}],\qquad
n\geq 2$ $\displaystyle B_{uz\bar{z}}^{(-n)}$
$\displaystyle=\partial_{u}F_{z\bar{z}}^{(-n)}-\partial_{z}F_{u\bar{z}}^{(-n)}+\partial_{\bar{z}}F_{uz}^{(-n)}+2i\sum_{k=0}^{n}[A_{[z}^{(k-n)},F_{u|\bar{z}]}^{(-k)}]$
(295) $\displaystyle B_{rz\bar{z}}^{(-n)}$
$\displaystyle=(1-n)F_{z\bar{z}}^{(1-n)}-\partial_{z}F_{r\bar{z}}^{(-n)}+\partial_{\bar{z}}F_{rz}^{(-n)}$
(296)
$\displaystyle\quad-i\sum_{k=0}^{n-2}[A_{r}^{(k-n)},F_{z\bar{z}}^{(-k)}]+2i\sum_{k=2}^{n}[A^{(k-n)}_{[z},F_{r|\bar{z}]}^{(-k)}],\qquad
n\geq 2$

Bianchi identities in radial gauge at a given order $n\in\mathbb{N}$:

$\displaystyle B_{urz}^{(-n)}$
$\displaystyle=\partial_{u}F_{rz}^{(-n)}+(n-1)F_{uz}^{(1-n)}+\partial_{z}F_{ur}^{(-n)}$
(297)
$\displaystyle-i\sum_{k=0}^{n-1}[A_{u}^{(k-n)},F_{rz}^{(-k)}]-i\sum_{k=2}^{n}[A_{z}^{(k-n)},F_{ur}^{(-k)}],\qquad
n\geq 2$ $\displaystyle B_{ur\bar{z}}^{(-n)}$
$\displaystyle=\partial_{u}F_{r\bar{z}}^{(-n)}+(n-1)F_{u\bar{z}}^{(1-n)}+\partial_{\bar{z}}F_{ur}^{(-n)}$
(298)
$\displaystyle-i\sum_{k=0}^{n-1}[A_{u}^{(k-n)},F_{r\bar{z}}^{(-k)}]-i\sum_{k=2}^{n}[A_{\bar{z}}^{(k-n)},F_{ur}^{(-k)}],\qquad
n\geq 2$ $\displaystyle B_{uz\bar{z}}^{(-n)}$
$\displaystyle=\partial_{u}F_{z\bar{z}}^{(-n)}-\partial_{z}F_{u\bar{z}}^{(-n)}+\partial_{\bar{z}}F_{uz}^{(-n)}$
(299)