Celestial $Lw_{1+\infty}$ charges from a twistor action

Adam Kmec111<EMAIL_ADDRESS>Lionel Mason222
<EMAIL_ADDRESS>Romain Ruzziconi333
<EMAIL_ADDRESS>Akshay Yelleshpur Srikant444
<EMAIL_ADDRESS>

Mathematical Institute, University of Oxford,
Andrew Wiles Building, Radcliffe Observatory Quarter,
Woodstock Road, Oxford, OX2 6GG, UK

The celestial $Lw_{1+\infty}$ symmetries in asymptotically flat spacetimes
have a natural geometric interpretation on twistor space in terms of Poisson
diffeomorphisms. Using this framework, we provide a first-principle derivation
of the canonical generators associated with these symmetries starting from the
Poisson BF twistor action for self-dual gravity. We express these charges as
surface integrals over the celestial sphere in terms of spacetime data at null
infinity. The connection between twistor space and spacetime expressions at
$\mathscr{I}$ is achieved via an integral formula for the asymptotic Bianchi
identities due to Bramson and Tod. Finally, we clarify how $Lw_{1+\infty}$
transformations are symmetries of gravity from a phase space perspective by
showing the invariance of the asymptotic Bianchi identities.

###### Contents

  1. 1 Introduction
  2. 2 Twistor space action
  3. 3 Asymptotic twistor space and the Bramson-Tod integral formula
  4. 4 $Lw_{1+\infty}$ symmetry algebra
  5. 5 Charge algebra
  6. 6 Discussion
  7. A Recursion relation for $f_{s}$
  8. B Symmetry algebra computation
  9. C Čech version
  10. D BMS Transformations
  11. E Vectors and Forms on $\mathbb{CP}^{n}$

##  1 Introduction

Finding the symmetries of gravity in asymptotically flat spacetimes has been a
long quest with several important turning points. The seminal work of Bondi,
van der Burg, Metzner, and Sachs [1, 2, 3] showed that the so-called BMS
group, which is an infinite-dimensional enhancement of the Poincaré group with
supertranslations, plays a crucial role in the description of radiative
spacetimes at null infinity. It was later suggested that the BMS group could
itself be enhanced into larger symmetry groups, such as extended [4, 5, 6],
generalized [7, 8, 9], or Weyl [10] BMS groups (see also [11, 12, 13, 14, 15,
16] for the corresponding discussions at spacelike infinity). These symmetries
have been shown to be of major importance to understand the infrared structure
of the gravitational scattering, and the interplay between soft theorems and
memory effects [17]. More recently, in the context of celestial holography, it
has been shown that a larger group of symmetries with Lie algebra
$Lw_{1+\infty}$ emerges from the collinear limit of scattering amplitudes [18,
19]. The action of these symmetries for flat spacetime is not geometric at
null infinity, $\mathscr{I}$, but can be traced back to Penrose’s non-linear
graviton construction of self-dual spacetimes from twistor space data [20, 21]
where they act by local structure preserving diffeomorphisms. The connection
between this geometric construction and the collinear limit was presented in
[22]. Nevertheless, patterns of the $Lw_{1+\infty}$ symmetries have been
identified in the gravitational phase space at null infinity in a series of
beautiful works [23, 24, 25], including in the expansion of the metric in
terms of multipole moments [26].

An essential ingredient in the identification of the symmetries is the
construction of the associated surface charges. Various Noetherian procedures
have been developed over the years to tackle this subtle problem and provide a
first-principle derivation of these canonical generators [27, 28, 29, 30, 31].
The covariant phase space methods were successfully applied to obtain the BMS
charges [32, 33, 6, 34, 9, 12, 35, 10]. However, the charges of [24, 25]
associated with the $Lw_{1+\infty}$ symmetries were not derived from first
principles, but rather via some recursion relations and symmetries arguments.
Truncation of the Bianchi identities was assumed at a certain order to
preserve the form of these recursion relations at all orders in the expansion.
The obstruction to obtain the celestial $Lw_{1+\infty}$ charges from first
principles essentially relies on the non-local action of these symmetries on
spacetime data [36, 37], which makes the direct application of the standard
covariant phase space methods obscure.

In this work, we propose a first-principle derivation of the surface charges
associated with the $Lw_{1+\infty}$ symmetries. To do so, we start from the
action describing self-dual gravity on twistor space [38], where the action of
the celestial symmetries becomes local and has a natural geometric
interpretation as Poisson diffeomorphisms [39, 22]. At null infinity, this
self-dual phase space agrees with that of conventional gravity. In this
framework, the charge algebra for these symmetries is easily derived, by
contrast with the spacetime computations which are quite cumbersome and can
only be achieved at linear order [24, 25, 40]. Our approach involves several
salient features of the covariant phase space methods, such as a treatment of
non-integrability and the use of a modified bracket [6, 9, 41, 42, 43, 44, 45,
46]. Using the Bramsom-Tod integral formula [47, 48], we relate our twistor
space expressions for the symplectic structure and the charges in terms of
spacetime data. The recursion relations of [24, 25] derive naturally from the
integral formula and the on-shell conditions for self-duality implied by the
twistor action. In particular, this demonstrates that the truncation of the
Bianchi identities postulated in [24, 25, 40] precisely corresponds to the
self-dual sector of gravity; essentially they correspond the asymtptotic
expansion of Newman’s $\mathcal{H}$-space constructed from data at null
infinity [49].

The self-duality equations are well-known to be an integrable system [50, 51,
52] and the recursion relations of [24, 25, 40] can be understood to be those
arising from the associated recursion operators for the self-duality equations
as described in [53] for gravity and [51] for Yang-Mills. These recursion
relations extend the self-duality equations to an integrable hierarchy, and
here its definition is adapted to data at null infinity. In particular, we see
that $Lw_{1+\infty}$ arises by application of these recursion operators to the
BMS group.

We also check the consistency of the action of the symmetries on twistor space
with the one expected on spacetime and show the invariance of the recursion
relations. This clarifies in which sense $Lw_{1+\infty}$ is a symmetry of the
gravitational solution space. Finally, we compare our results with those of
[24, 25] and almost find perfect agreement. The differences occur in the
expression of the surface charges for superrotation and higher spin charges.
In particular, our derivation offers a new and unambiguous proposal for the
expression of the angular momentum at $\mathscr{I}$ in the presence of
radiation.

The paper is organized as follows. In Section 2, we review the self-dual
gravity action on twistor space, which is the starting point of our analysis.
We also introduce notations and conventions for the rest of the paper. In
Section 3, we discuss the asymptotic twistor space and present the Bramson-Tod
integral formula which relates twistor space to data at $\mathscr{I}$. We show
that the spacetime recursion relations automatically follow from this integral
transform and the equations of motion on twistor space. In Section 4, we show
that, for a natural gauge fixing adapted to the asymptotic twistor space, the
residual gauge transformations on twistor space form the $Lw_{1+\infty}$
algebra. We compute the action of these symmetries on spacetime data and show
the invariance of the recursion relations. In Section 5, we apply the
covariant phase space methods on the twistor action to derive the charge
algebra associated with $Lw_{1+\infty}$. We translate these expressions on
spacetime using the Bramson-Tod formula. In Section, 6, we conclude with some
remarks and perspectives. The paper is also completed by several Appendices.
Appendix A provides expressions valid at all spin $s$, Appendix B provides
details on the algebra computation, Appendix C presents the Bramson-Tod
integral formula in the original Čech picture, instead of the Dolbeault
picture adopted in the core of the text, Appendix D derives the full BMS
transformations from the Bramson-Tod integral formula, and finally, Appendix E
provides useful formulae for projective forms used throughout the paper.

##  2 Twistor space action

In this section, we introduce the twistor space action describing self-dual
gravity [38]. We also review some basics of twistor space as well as notations
and conventions used in the rest of the paper.

### Notation and conventions.

Non-projective twistor space is denoted $\mathbb{T}=\mathbb{C}^{4}$ with
coordinates

$Z^{A}=(\mu^{\dot{\alpha}},\lambda_{\alpha}).$ (2.1)

Here $\alpha=0,1$ and $\dot{\alpha}=\dot{0},\dot{1}$ are standard two-
component spinor indices for the anti-self-dual (ASD) and self-dual (SD)
spinors respectively. We use the conventions
$\omega^{\alpha}=\epsilon^{\alpha\beta}\omega_{\beta}$,
$\omega_{\alpha}=\omega^{\beta}\epsilon_{\beta\alpha}$ to raise/lower spinor
indices, where $\epsilon_{\alpha\beta}$ is the Levi-Civita symbol
($\epsilon_{\alpha\beta}=-\epsilon_{\beta\alpha}$,
$\epsilon_{01}=1=\epsilon_{\dot{0}\dot{1}}$). We will use spinor helicity
notation for the spinor inner products

$\langle\omega\lambda\rangle=\omega^{\alpha}\lambda_{\alpha},\qquad[\mu\pi]=\mu^{\dot{\alpha}}\pi_{\dot{\alpha}}.$

In Lorentz signature, the complex conjugation on space-time induces a map from
non-projective twistor space $\mathbb{T}$ to its dual $\mathbb{T}^{*}$ leading
to a hermitian inner product given by

$\bar{Z}_{A}=(\bar{\lambda}_{\dot{\alpha}},\bar{\mu}^{\alpha}),\qquad
Z^{A}\bar{Z}_{A}=\mu^{\dot{\alpha}}\bar{\lambda}_{\dot{\alpha}}+\lambda_{\alpha}\bar{\mu}^{\alpha}.$

This is preserved by $SU(2,2)$, the spin group of the conformal group.

We will also take $Z^{A}$ to be homogeneous coordinates on the corresponding
projective space

$\mathbb{CP}^{3}=\\{Z^{A}|Z^{A}\sim tZ^{A},t\in\mathbb{C}^{*}\\}\,.$ (2.2)

In flat space a point $x$ in Minkowski space $\mathbb{M}$ is represented in
twistor space as a complex projective line or Riemann sphere,
$X\simeq\mathbb{CP}^{1}$ via the incidence relation

$\mu^{\dot{\alpha}}=ix^{\alpha\dot{\alpha}}\lambda_{\alpha}\,,$ (2.3)

this being two linear equations on the four twistor coordinates, it defines
the complex projective line $X$ which has homogeneous coordinates
$\lambda_{\alpha}$.

We take flat projective twistor space to be the open subset of the complex
three-dimensional projective space given by

$\mathbb{PT}=\mathbb{CP}^{3}-\mathbb{CP}^{1}\,,$ (2.4)

where
$\mathbb{CP}^{1}=\\{(\mu^{\dot{\alpha}},\lambda_{\alpha})\in\mathbb{CP}^{3}|\lambda_{{\alpha}}=0\\}$
is the line corresponding to the vertex of the light cone at infinity
$\mathscr{I}$. The quotient from $\mathbb{C}^{4}$ to $\mathbb{CP}^{3}$ follows
by quotienting along $\Upsilon=Z^{A}\partial_{A}$ and
$\bar{\Upsilon}=\bar{Z}^{A}\bar{\partial}_{A}$, the holomorphic and anti-
holomorphic Euler vector fields, respectively. Define a $(p,q)$-form
$\omega\in\Omega^{p,q}(m,n)$ on $\mathbb{PT}$ of homogeneity degree $(m,n)$ on
the projective space to be a section of $\mathcal{O}(m,n)$ on twistor space by

$\Upsilon\lrcorner\,\,\omega=0=\bar{\Upsilon}\lrcorner\,\,\omega\quad,\quad\mathcal{L}_{\Upsilon}\omega=m\omega\quad,\quad\mathcal{L}_{\bar{\Upsilon}}\omega=n\omega\,.$
(2.5)

In particular, we abuse notation to define the line bundles
$\mathcal{O}(n)=\mathcal{O}(n,0)$–here we will be considering mostly smooth
rather than holomorphic sections of the line bundle $\mathcal{O}(n)$ in the
context of the Dolbeault picture, by contrast with the Čech picture where this
notation would denote the sheaf of holomorphic sections.

With these notations, the homogeneous measure on the Riemann sphere reads as

$D\lambda=\langle\lambda d\lambda\rangle,\qquad
D\bar{\lambda}=[\bar{\lambda}d\bar{\lambda}]\,,$ (2.6)

and these will be used to define homogeneous 1-forms on the lines $X$
corresponding to points of space-time $\mathbb{M}$. Moreover, the holomorphic
volume form on $\mathbb{PT}$ is provided by

$D^{3}Z=\frac{1}{4!}\epsilon_{ABCD}Z^{A}\mathrm{d}Z^{B}\wedge\mathrm{d}Z^{C}\wedge\mathrm{d}Z^{D}\,,$
(2.7)

where $\epsilon_{ABCD}$ is the Levi-Civita symbol in 4 indices. The Poincaré
group is the subgroup of the conformal group that preserves the infinity
twistor

$I^{AB}=\begin{pmatrix}\epsilon^{\dot{\alpha}\dot{\beta}}&0\\\
0&0\end{pmatrix},\qquad I_{AB}=\begin{pmatrix}0&0\\\
0&\epsilon^{\alpha\beta}\end{pmatrix}\,.$ (2.8)

With this, we can define a Poisson bracket on twistor space through

$\\{f,g\\}=I^{AB}\partial_{B}f\partial_{A}g=\epsilon^{\dot{\alpha}\dot{\beta}}\frac{\partial
f}{\partial\mu^{\dot{\beta}}}\frac{\partial g}{\partial\mu^{\dot{\alpha}}}\,,$
(2.9)

which closes when $f,g$ are sections of $\mathcal{O}(2)$. This bracket can be
naturally extended to act on forms of various homogeneities via the Lie
derivative as follows:

$\\{\omega,\sigma\\}=I^{AB}\mathcal{L}_{\partial_{B}}\omega\wedge\mathcal{L}_{\partial_{A}}\sigma=\epsilon^{\dot{\alpha}\dot{\beta}}\mathcal{L}_{\partial_{\dot{\beta}}}\omega\wedge\mathcal{L}_{\partial_{\dot{\alpha}}}\sigma\,.$
(2.10)

### Deforming the complex structure on twistor space.

The complex structure of flat twistor space $\mathbb{PT}$ corresponds to the
splitting of the exterior derivative into its holomorphic and anti-holomorphic
parts

$\partial=dZ^{A}\frac{\partial}{\partial
Z^{A}},\qquad\bar{\partial}=d\bar{Z}^{A}\frac{\partial}{\partial{\bar{Z}}^{A}}\,.$
(2.11)

A deformation of the complex structure can be expressed as

$\bar{\partial}\quad\longrightarrow\quad\bar{\nabla}=\bar{\partial}+V\ ,\qquad
V=V^{B}_{A}d\bar{Z}^{A}\otimes\frac{\partial}{\partial Z^{B}}\,,$ (2.12)

so that $V\in\Omega^{0,1}(\mathbb{PT},T^{1,0}\mathbb{PT})$, i.e., an anti-
holomorphic $(0,1)$-form on $\mathbb{PT}$, with values on the holomorphic
tangent bundle $T^{1,0}\mathbb{PT}$. In general, such a $\bar{\nabla}$ defines
an _almost-complex structure_ on twistor space. To be a complex structure it
must be integrable, $\bar{\nabla}^{2}=0$, which gives the Maurer-Cartan
equation

$\bar{\partial}V+\frac{1}{2}[V,V]=0\,.$ (2.13)

Here $[\cdot,\cdot]$ is the Schouten–Nijenhuis bracket defined as the
commutator between the vector fields combined with the wedge product on the
$(0,1)$-forms. As we shall explain in Section 3, asymptotic twistor space
[54], and more generally, the nonlinear graviton [21], derives this
deformation from a Hamiltonian so as to preserve the Poisson structure (2.10)
arising from the infinity twistor (although the latter does so in a Čech
framework). Thus $V=\\{h,\cdot\\}$ and the deformed complex structure reads as

$\bar{\nabla}=\bar{\partial}+\\{h,\cdot\\}\,,$ (2.14)

where $h\in\Omega^{0,1}(\mathbb{PT},\mathcal{O}(2))$ is the Hamiltonian for
the deformation. The integrability condition for $\bar{\nabla}$ (2.13) is then
essentially equivalent to

$\bar{\partial}h+\frac{1}{2}\\{h,h\\}=0\,.$ (2.15)

### Self-dual gravity.

We now introduce the twistor space action for gravity with only self-dual
interactions. The Poisson BF theory for self-dual gravity is given by

$\boxed{S[g,h]=\frac{1}{2\pi
i}\int_{\mathbb{PT}}g\wedge\left(\bar{\partial}h+\frac{1}{2}\\{h,h\\}\right)\wedge
D^{3}Z\,,}$ (2.16)

where $D^{3}Z\in\Omega^{3,0}(\mathbb{PT},\mathcal{O}(4))$ is the holomorphic
top form defined in (2.7), and $g\in\Omega^{0,1}(\mathbb{PT},\mathcal{O}(-6))$
and $h\in\Omega^{0,1}(\mathbb{PT},\mathcal{O}(2))$ are the off-shell twistor
data for respectively the ASD and SD parts of the dynamical gravitational
field. The action is invariant under gauge transformations

$\begin{split}\delta_{(\xi,\phi)}h&=\bar{\nabla}\xi\,,\\\
\delta_{(\xi,\phi)}g&=\\{g,\xi\\}+\bar{\nabla}\phi\,,\end{split}$ (2.17)

where $(\xi,\phi)\in\mathcal{O}(2)\oplus\mathcal{O}(-6)$ are smooth and
arbitrary gauge parameters. In these transformations, we are using the bracket
(2.10) as well as the deformed complex structure introduced in (2.14). The
equations of motion of the action (2.16) are given by

$\begin{split}\bar{\nabla}g=\bar{\partial}g+\\{h,g\\}&=0\,,\\\
\bar{\partial}h+\frac{1}{2}\\{h,h\\}&=0\,.\end{split}$ (2.18)

One can check explicitly that the equations of motion are invariant under the
gauge transformations (2.17) although it also follows from the corresponding
invariance of the Lagrangian. On-shell, it follows from (2.18) that the field
$g$ is $\bar{\nabla}$-closed and $h$ satisfies the integrability condition
(2.15). In particular, we note that the linearized version of this second
equation is $\bar{\nabla}\delta h=0$. Moreover, from (2.17), the fields are
defined up to $\bar{\nabla}$-exact terms. In other words, the on-shell field
$g$ and linearized perturbation $\delta h$ around $h$ modulo gauge
transformations define elements of the cohomology groups for the deformed
complex structure denoted by
$H^{0,1}_{\bar{\nabla}}(\mathbb{PT},\mathcal{O}(-6))$ and
$H^{0,1}_{\bar{\nabla}}(\mathbb{PT},\mathcal{O}(2))$.

We can connect these fields to conventional descriptions of self-dual gravity
via integral formulae in linear theory as follows. The Weyl tensor has a
spinorial decomposition into an ASD and SD part

$C_{\mu\nu\rho\sigma}=\psi_{\alpha\beta\gamma\delta}\epsilon_{\dot{\alpha}\dot{\beta}}\epsilon_{\dot{\gamma}\dot{\delta}}+\tilde{\psi}_{\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta}}\epsilon_{\alpha\beta}\epsilon_{\gamma\delta}\,,$
(2.19)

where
$\psi_{\alpha\beta\gamma\delta},\tilde{\psi}_{\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta}}$
are the ASD and SD fully symmetric Weyl spinors respectively. In linear
theory, the Penrose transform relates these to twistor data by

$\begin{split}g\in
H^{0,1}_{\bar{\nabla}}(\mathbb{PT},\mathcal{O}(-6))&\cong\\{\psi_{\alpha\beta\gamma\delta}\,\,\text{on}\,\,\mathbb{M}|\nabla^{\alpha\dot{\alpha}}\psi_{\alpha\beta\gamma\sigma}=0\\}\,,\\\
\delta h\in
H^{0,1}_{\bar{\nabla}}(\mathbb{PT},\mathcal{O}(2))&\cong\\{\tilde{\psi}_{\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta}}\,\,\text{on}\,\,\mathbb{M}|\nabla^{\alpha\dot{\alpha}}\tilde{\psi}_{\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta}}=0\\}\,.\end{split}$
(2.20)

The correspondence above is given explicitly by the integral formulae over the
Riemann sphere $X\subset\mathbb{PT}$ given by (2.3)

$\displaystyle\psi_{\alpha\beta\gamma\delta}$ $\displaystyle=\frac{1}{2\pi
i}\int_{\mathbb{CP}^{1}}\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}\lambda_{\delta}\,\,g|_{X}\wedge\mathrm{D}\lambda\,,\qquad
g\in H^{0,1}_{\bar{\nabla}}(\mathbb{PT},\mathcal{O}(-6))\,,$
$\displaystyle\psi_{\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta}}$
$\displaystyle=\frac{1}{2\pi
i}\int_{\mathbb{CP}^{1}}\mathrm{D}\lambda\wedge\frac{\partial^{4}\delta
h}{\partial\mu^{\dot{\alpha}}\partial\mu^{\dot{\beta}}\partial\mu^{\dot{\gamma}}\partial\mu^{\dot{\delta}}}|_{X}\,,\qquad\delta
h\in H^{0,1}_{\bar{\nabla}}(\mathbb{PT},\mathcal{O}(2))\,.$ (2.21)

To work on a curved self-dual background, rather than in linear theory, we
must perform the nonlinear graviton construction. Briefly, the lines $X$
corresponding to points $x\in\mathbb{M}$ become deformed by (2.14) but
nevertheless survive as a 4-complex-dimensional family that forms a generic
self-dual space-time $\mathcal{M}$ with points $x\in\mathcal{M}$ corresponding
to holomorphic curves $X=\mathbb{CP}^{1}$ of degree one, but now for the
deformed complex structure on curved twistor space, $\mathbb{P}\mathcal{T}$.

On such a curved background, the first of (2.21) still makes good sense to
give linearized anti-self-dual fields. As such backgrounds admit covariantly
constant undotted spinors, there are no Buchdahl conditions, and
$\psi_{\alpha\beta\gamma\delta}$ will again satisfy the vacuum Bianchi
identities

$\nabla^{\alpha\dot{\alpha}}\psi_{\alpha\beta\gamma\delta}=0\,,$ (2.22)

where now $\nabla_{\alpha\dot{\alpha}}$ is the covariant derivative on the SD
background. For the elements of
$H^{0,1}_{\bar{\nabla}}(\mathbb{PT},\mathcal{O}(2))$ only a potential modulo
gauge description survives from which one can find the SD metric perturbation
[55, 56] although later it will be sufficient for us to simply use the
correspondence with $\mathscr{I}$.

##  3 Asymptotic twistor space and the Bramson-Tod integral formula

The Penrose transform (2.21) allows us to relate fields on twistor space to
fields on space-time. However, those formulae are adapted to representing the
fields in the bulk space-time, whereas here in this section, we will use
formulae adapted to null infinity originally introduced by Bramson and Tod
[47, 48] for $g$ and by Sparling [57, 58] for $h$. In doing so, we introduce
the notion of asymptotic twistor space and homogeneous coordinates at
$\mathscr{I}$. Asymptotic twistor space encodes the gravitational data at
$\mathscr{I}$ into a deformed complex structure on twistor space and makes
contact between Newman’s $\mathcal{H}$-space construction and Penrose’s
nonlinear graviton construction [54].

### Asymptotic twistor space and homogeneous coordinates on $\mathscr{I}$

To obtain the asymptotic twistor space, we complexify the usual Bondi
coordinate $u_{B}$ at null infinity to obtain
$\mathscr{I}_{\mathbb{C}}=\mathbb{C}\times S^{2}$ (the real $\mathscr{I}$ can
be recovered by just imposing a reality condition on $u$ and making sure that
the functions on $\mathscr{I}_{\mathbb{C}}$ are holomorphic in $u$). Twistor
space then fibers over $\mathbb{PT}\rightarrow\mathscr{I}_{\mathbb{C}}$. The
use of homogeneous coordinates on $\mathbb{PT}$ then suggests homogeneous
coordinates on $\mathscr{I}$ which have the advantage of manifesting full
Lorentz invariance, rather than the usual Bondi coordinates
$(u_{B},z,\bar{z})$ which exhibit $SO(3)$ invariance only [57].

Thus, we introduce homogeneous coordinates $\lambda_{\alpha}$ on the celestial
sphere with the usual $z=\lambda_{1}/\lambda_{0}$. The Lorentz group acts in
the usual way on the spinor index $\alpha=0,1$, and the reduction from
$SL(2,\mathbb{C})$ to $SU(2)$ follows from the introduction of a vector
$T_{\alpha\dot{\alpha}}$ satisfying $T^{2}=2$. This allows us to define

$\hat{\lambda}_{\alpha}=T_{\alpha\dot{\alpha}}\bar{\lambda}^{\dot{\alpha}}\,,$
(3.1)

so that the round sphere metric is written as

$ds^{2}_{S^{2}}=\frac{D\lambda
D\hat{\lambda}}{\langle\lambda\hat{\lambda}\rangle^{2}}=\frac{dzd\bar{z}}{(1+|z|^{2})^{2}}\,.$
(3.2)

We lift the Bondi coordinate to a homogeneous coordinate $u$ of weight $(1,1)$
by

$u=\langle\lambda\hat{\lambda}\rangle u_{B}\,.$ (3.3)

This definition follows from Penrose’s strong conformal structure [59, 60]
that fixes the ratio $du^{2}:ds^{2}_{S^{2}}$ and a conformal rescaling of the
sphere metric to remove the $\langle\lambda\hat{\lambda}\rangle^{2}$ factor.
It can now be checked from standard transformation laws that $u$ is invariant
under Lorentz boosts and that the Lorentz group acts in the normal linear way
on $\lambda_{\alpha}$.

With this, the projection $\mathbb{PT}\rightarrow\mathscr{I}_{\mathbb{C}}$ is
defined through

$p:\mathbb{PT}\to\mathscr{I}_{\mathbb{C}},\quad(\mu^{\dot{\alpha}},\lambda_{\alpha})\to(u=[\mu\bar{\lambda}],\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}})\,,$
(3.4)

where $u$ is a null complexifed time, and
$(\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}})$ the homogeneous coordinates
on the celestial sphere. Homogeneous functions on $\mathscr{I}$ satisfy

$f(t\bar{t}u,t\lambda,\bar{t}\bar{\lambda})=t^{m}\bar{t}^{n}f(u,\lambda,\bar{\lambda})\,,$
(3.5)

and naturally descend from sections of $\mathcal{O}(m,n)$ on twistor space
(2.5) using the projection (3.4). The homogeneity weights $m,n$ are related to
the usual spin and boost weights [61] through

$s=\frac{m-n}{2},\qquad w=\frac{m+n}{2}.$ (3.6)

The dual vector fields on the Riemann sphere to the forms (2.6) define the
Edth operator and its complex conjugate

$\eth=\frac{\hat{\lambda}_{\alpha}}{\langle\hat{\lambda}\lambda\rangle}\frac{\partial}{\partial\lambda_{\alpha}}\quad,\quad\bar{\eth}=\frac{\lambda_{\alpha}}{\langle\lambda\hat{\lambda}\rangle}\frac{\partial}{\partial\hat{\lambda}_{\alpha}}\,,$
(3.7)

which can act on sections of $\mathcal{O}(m,n)$. The action of the Edth
operators changes the weights

$\begin{split}\eth&:\mathcal{O}(m,n)\rightarrow\mathcal{O}(m-2,n)\,,\\\
\bar{\eth}&:\mathcal{O}(m,n)\rightarrow\mathcal{O}(m,n-2)\,.\end{split}$ (3.8)

Under the projection map (3.4), we decompose the twistor coordinate
$\mu^{\dot{\alpha}}$ into the image and kernel of $p$

$\mu^{\dot{\alpha}}=\frac{u\hat{\bar{\lambda}}^{\dot{\alpha}}}{\langle\hat{\lambda}\lambda\rangle}+q\bar{\lambda}^{\dot{\alpha}}\quad,\quad
q=\frac{[\mu\hat{\bar{\lambda}}]}{\langle\lambda\hat{\lambda}\rangle}\,,$
(3.9)

where $q\in\mathcal{O}(1,-1)$ is the fiber coordinate over the projection to
$\mathscr{I}$. We make an important distinction between the Edth operator on
twistor space and the one on $\mathscr{I}$

$\begin{split}\partial_{0}&=\left.\frac{\hat{\lambda}_{\alpha}}{\langle\hat{\lambda}\lambda\rangle}\frac{\partial}{\partial\lambda_{\alpha}}\right|_{\mu}=\eth-\bar{q}\partial_{\bar{u}}+\frac{u}{\langle\hat{\lambda}\lambda\rangle^{2}}\partial_{q}\,,\\\
\bar{\partial}_{0}&=\left.\frac{\lambda_{\alpha}}{\langle\lambda\hat{\lambda}\rangle}\frac{\partial}{\partial\hat{\lambda}_{\alpha}}\right|_{\mu}=\bar{\eth}-q\partial_{u}+\frac{\bar{u}}{\langle\hat{\lambda}\lambda\rangle^{2}}\partial_{\bar{q}}\,.\end{split}$
(3.10)

In particular, one can check that $\bar{\eth}u=0$, but
$\bar{\partial}_{0}u\neq 0$.

In this set-up, the deformation of the complex structure on $\mathbb{PT}$
introduced in (2.14) can be constructed from the asymptotic shear
$\bar{\sigma}(u,\lambda,\bar{\lambda})$, which is a section of
$\mathcal{O}(1,-3)$, as

$\boxed{h(\mu,\lambda,\bar{\lambda})=\int^{u}\bar{\sigma}(u^{\prime},\lambda,\bar{\lambda})du^{\prime}D\bar{\lambda}\,.}$
(3.11)

We will also use the notation $h=h_{0}D\bar{\lambda}$. Here
$u=[\mu\bar{\lambda}]$ and $h$ generates the shear according to

$\frac{\partial
h}{\partial\mu^{\dot{\alpha}}}=\bar{\lambda}_{\dot{\alpha}}\bar{\sigma}([\mu\bar{\lambda}],\lambda,\bar{\lambda})D\bar{\lambda}.$
(3.12)

One can check that this choice ensures that $h$ satisfies its equation of
motion (2.18).

### Peeling and recursion relations.

See chapter 9 of [60] for full details of the following. In asymptotically
flat spacetimes, the large $r$ behaviour of the Weyl tensor in a neighbourhood
of $\mathscr{I}$ is dictated by the peeling theorem. We denote the components
of the Weyl tensor (2.19) by

$\begin{split}\Psi_{0}&=C_{\mu\nu\rho\sigma}l^{\mu}m^{\nu}l^{\rho}m^{\sigma}=\psi_{\alpha\beta\gamma\delta}\iota^{\alpha}\iota^{\beta}\iota^{\gamma}\iota^{\delta},\\\
\Psi_{1}&=C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}l^{\rho}m^{\sigma}=\psi_{\alpha\beta\gamma\delta}o^{\alpha}\iota^{\beta}\iota^{\gamma}\iota^{\delta},\\\
\Psi_{2}&=\frac{1}{2}(C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}l^{\rho}n^{\sigma}-C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}m^{\rho}\bar{m}^{\sigma})=\psi_{\alpha\beta\gamma\delta}o^{\alpha}o^{\beta}\iota^{\gamma}\iota^{\delta},\\\
\Psi_{3}&=C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}n^{\rho}\bar{m}^{\sigma}=\psi_{\alpha\beta\gamma\delta}o^{\alpha}o^{\beta}o^{\gamma}\iota^{\delta},\\\
\Psi_{4}&=C_{\mu\nu\rho\sigma}n^{\mu}\bar{m}^{\nu}n^{\rho}\bar{m}^{\sigma}=\psi_{\alpha\beta\gamma\delta}o^{\alpha}o^{\beta}o^{\gamma}o^{\delta}\,,\end{split}$
(3.13)

where $(l,n,m,\bar{m})$ is the Newman-Unti co-tetrad [62] and $(\iota,o)$ is
the associated spinor dyad [63, 64]; thus $\langle\iota o\rangle=1$ and
$l^{\alpha\dot{\alpha}}=o^{\alpha}\bar{o}^{\dot{\alpha}}$ and
$l^{a}\partial_{a}=\partial/\partial r$, and
$n^{\alpha\dot{\alpha}}=\iota^{\alpha}\bar{\iota}^{\dot{\alpha}}$ with
$n^{a}\partial_{a}=\partial_{u}$ at $\mathscr{I}$ and the spinors are parallel
propagated along $\partial_{r}$. The peeling theorem gives

$\Psi_{n}=\frac{\Psi_{n}^{0}}{r^{5-n}}+\mathcal{O}(r^{n-6})\,,\qquad
n=0,1,2,3,4\,.$ (3.14)

We have the following relation with the asymptotic shear

$\Psi^{0}_{4}=\ddot{\sigma},\qquad\Psi^{0}_{3}=\bar{\eth}\dot{\sigma},$ (3.15)

and asymptotic Bianchi identities imply the following time evolution equations

$\partial_{u}\Psi_{n}^{0}=\bar{\eth}\Psi_{n+1}^{0}+(3-n)\bar{\sigma}\Psi_{n+2}^{0},\qquad
n=0,1,2,3\,.$ (3.16)

Using the notations of [24, 25], we denote

$\mathcal{Q}_{s}=\Psi^{0}_{2-s},\qquad s=-1,0,1,2\,,$ (3.17)

where $s$ denotes the spin-weight of these Newman-Penrose scalars under
Lorentz transformations on the celestial sphere; this suggests a continuation
of the series in $s$ which can be motivated as follows.

One can show that the subleading orders in $r$ of $\Psi_{1}$, $\Psi_{2}$,
$\Psi_{3}$, $\Psi_{4}$ are completely fixed by the Bianchi identities once the
leading orders $\Psi_{1}^{0}$, $\Psi_{2}^{0}$, $\Psi_{3}^{0}$, $\Psi_{4}^{0}$
are given. By contrast there is no Bianchi identity fixing subleading orders
in $\Psi_{0}$ in $r$ in terms of $\Psi^{0}_{0}$ and so its expansion in $r$
involves an infinite tower of independent data. Following [25], we write the
expansion of $\Psi_{0}$ as

$\Psi_{0}=\frac{\Psi^{0}_{0}}{r^{5}}+\sum_{s=3}^{\infty}\frac{1}{r^{s+3}}\frac{(-1)^{s}}{(s-2)!}\left(\eth^{s-2}\mathcal{Q}_{s}+\Phi_{s-2}\right)\,,$
(3.18)

where this splitting of the sub-leading terms is chosen so that the evolution
equations in $u$ can be cast into the form

$\partial_{u}\mathcal{Q}_{s}=\bar{\eth}\mathcal{Q}_{s-1}+(s+1)\bar{\sigma}\mathcal{Q}_{s-2}\,,$
(3.19)

and the infinitesimal extended BMS transformations [65, 66] are

$\delta_{(T,Y,\bar{Y})}\mathcal{Q}_{s}=\left(f\partial_{u}+Y\eth+\bar{Y}\bar{\eth}+\frac{3-s}{2}\eth
Y+\frac{3+s}{2}\bar{\eth}\bar{Y}\right)\mathcal{Q}_{s}+(s+2)\mathcal{Q}_{s-1}\bar{\eth}f\,,$
(3.20)

where $f=T+\frac{u}{2}(\eth Y+\bar{\eth}\bar{Y})$, with
$T=T(\lambda,\bar{\lambda})\in\mathcal{O}(1,1)$ the supertranslations and
$Y(\lambda)\in\mathcal{O}(2,0)$, $\bar{Y}(\bar{\lambda})\in\mathcal{O}(0,2)$
the superrotations. The terms $\Phi_{s-2}$ in (3.18) can be shown to be
expressible as non-local expressions of the $\mathcal{Q}_{s}$’s appearing in
lower order [25]. Neglecting these terms would yield a truncation of the
Bianchi identities for $s\geq 4$ in the full theory of gravity [24].

We see that the recursion relations (3.19) are valid for any spin
$s=-1,0,1,2,3\ldots$ and naturally extend (3.16), as do the transformation
laws (3.20) under BMS. As we shall explain, both the recursion relations of
the $\mathcal{Q}_{s}$ and the transformation laws will arise naturally from
twistor space via a natural extension of the Bramson-Tod integral formula.

### Bramson-Tod integral formula and its extension.

The original Bramson-Tod integral formula [48] relates twistor space
cohomology classes in $H^{1}(\mathbb{PT},\mathcal{O}(-6))$ in the Čech
representation to solutions to the asymptotic Bianchi identities (3.16). Here,
we reformulate this integral in the Dolbeault framework to connect more
directly to the ingredient $g$ in the twistor action (2.16). In particular, it
extends directly to give the $\mathcal{Q}_{s}$ providing their origin in
Noether’s theory. The two pictures are completely equivalent on shell, see for
example [67, 68]; see Appendix C for the Čech version.

The integral formula is given by

$\boxed{\mathcal{Q}_{s}=\int q^{s+2}\mathrm{d}q\wedge
g|_{X_{(u,\lambda,\bar{\lambda})}}=\int
q^{s+2}g_{1}\mathrm{d}q\mathrm{d}\bar{q}\,,}$ (3.21)

where $X_{(u,\lambda,\bar{\lambda})}$ is the fiber over $\mathscr{I}$ for the
projection (3.4), and where
$g|_{X_{(u,\lambda,\bar{\lambda})}}=g_{1}\mathrm{d}\bar{q}$, with
$g_{1}\in\mathcal{O}(-5,-1)$, denotes the component of the form $g$ along the
fiber. Here, the integration takes place on the fiber with coordinates
$(q,\bar{q})$ as introduced in (3.9). From now on, we will also assume the
following holomorphicity condition with respect to $u$:

$\hat{\lambda}^{\alpha}\frac{\partial}{\partial\bar{\mu}^{\alpha}}\lrcorner\,\,g=0\,.$
(3.22)

As discussed at the beginning of this section, this ensures that, on shell, we
can analytically continue our twistor data from $\mathscr{I}$ to
$\mathscr{I}_{\mathbb{C}}$. In the next section, we will re-interpret this
condition as a gauge-fixing condition on twistor space.

Using the Bramson-Tod formula (3.21), we now show that the closure condition
$\bar{\nabla}g=0$, which corresponds to the equation of motion for $g$ given
in (2.18), is equivalent to the recursion relations

$\boxed{\partial_{u}\mathcal{Q}_{s}=\bar{\eth}\mathcal{Q}_{s-1}+(s+1)\bar{\sigma}\mathcal{Q}_{s-2},\qquad
s=-1,0,1,2\ldots}$ (3.23)

discussed in the previous paragraph. This justifies the identification of the
left-hand side of (3.21) with the spin $s$ momenta discussed in the previous
paragraph. Using the Bramson-Tod integral formula, the recursion relations
impose a constraint equation on $g_{1}$

$\begin{split}\partial_{u}\mathcal{Q}_{s}-\bar{\eth}\mathcal{Q}_{s-1}-(s+1)\bar{\sigma}\mathcal{Q}_{s-2}&=\int\left[q^{s+2}\partial_{u}g_{1}-q^{s+1}\bar{\eth}g_{1}-(s+1)\bar{\sigma}q^{s}g_{1}\right]\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\int
q^{s+1}[q\partial_{u}g_{1}-\bar{\eth}g_{1}+\bar{\sigma}\partial_{q}g_{1}]\mathrm{d}q\mathrm{d}\bar{q}\,,\end{split}$
(3.24)

where we have integrated by parts and dropped the boundary terms as the fibers
are taken to have no boundary. The LHS is zero if the integrand is a total
derivative, hence the condition on $g_{1}$ is

$q\partial_{u}g_{1}-\bar{\eth}g_{1}+\bar{\sigma}\partial_{q}g_{1}-\frac{1}{\langle\hat{\lambda}\lambda\rangle^{2}}\partial_{\bar{q}}g_{0}=0\,,$
(3.25)

where $g_{0}\in\mathcal{O}(-4)$ is an arbitrary function which does no depend
on $\bar{u}$, and the $\langle\hat{\lambda}\lambda\rangle$ factor was inserted
for future convenience. This constraint equation arises naturally from the on-
shell condition $\bar{\nabla}g=0$ for $g$ in an appropriate basis of
$(0,1)$-forms on twistor space. This is adapted to the projection over
$\mathscr{I}$ by first introducing the variables

$\bar{\mathfrak{q}}=\frac{\langle\bar{\mu}\hat{\lambda}\rangle}{\langle\lambda\hat{\lambda}\rangle^{2}}=\frac{\bar{q}}{\langle\lambda\hat{\lambda}\rangle}\quad,\quad\bar{u}_{B}=\frac{\bar{u}}{\langle\hat{\lambda}\lambda\rangle}.$
(3.26)

These have no anti-holomorphic homogeneity weight (and holomorphic weights
$-2$ and $0$ respectively) and so we have

$\begin{split}\bar{\partial}\bar{\mathfrak{q}}&=\frac{\langle\mathrm{d}\bar{\mu}\hat{\lambda}\rangle}{\langle\lambda\hat{\lambda}\rangle^{2}}-\bar{\mathfrak{q}}\frac{\langle\mathrm{d}\hat{\lambda}\lambda\rangle}{\langle\hat{\lambda}\lambda\rangle}+\bar{u}_{B}\bar{e}^{0}\,,\\\
\bar{\partial}\bar{u}_{B}&=\frac{\langle\mathrm{d}\bar{\mu}\lambda\rangle}{\langle\hat{\lambda}\lambda\rangle}-\bar{u}_{B}\frac{\langle\mathrm{d}\hat{\lambda}\lambda\rangle}{\langle\hat{\lambda}\lambda\rangle}\,,\\\
\bar{e}^{0}&=\frac{\mathrm{D}\bar{\lambda}}{\langle\lambda\hat{\lambda}\rangle^{2}},\end{split}$
(3.27)

where these have vanishing contraction $\bar{\Upsilon}\lrcorner\,\,\cdot=0$;
they are all necessarily $\bar{\partial}$-closed. We expand our generic
(0,1)-form $g$ in this basis

$g=g_{0}\bar{e}^{0}+g_{\bar{\mathfrak{q}}}\bar{\partial}\bar{\mathfrak{q}}+g_{\bar{u}_{B}}\bar{\partial}\bar{u}_{B}\,.$
(3.28)

Now we impose the axial gauge condition (3.22)

$\hat{\lambda}^{\alpha}\frac{\partial}{\partial\bar{\mu}^{\alpha}}\lrcorner\,\,g=0\quad\Longrightarrow\quad
g_{\bar{u}_{B}}=0\,,$ (3.29)

which will result in holomorphicity in $u$ of the other components when $g$ is
on-shell. In this gauge, we find

$\bar{\partial}g=\bar{\partial}g_{0}\wedge\bar{e}^{0}+\bar{\partial}g_{\bar{\mathfrak{q}}}\wedge\bar{\partial}\bar{\mathfrak{q}}\,.$
(3.30)

For a general section $F$ of $\mathcal{O}(n)$, $\bar{\partial}F$ can be
expanded in the basis (3.27) as

$\begin{split}\bar{\partial}F&=\frac{\partial
F}{\partial\bar{\lambda}_{\dot{\alpha}}}\mathrm{d}\bar{\lambda}_{\dot{\alpha}}+\frac{\partial
F}{\partial\bar{\mu}^{\alpha}}\mathrm{d}\bar{\mu}^{\alpha}\\\
&=-\bar{\partial}_{0}F\,\mathrm{D}\bar{\lambda}+\bar{\lambda}_{\dot{\alpha}}\frac{\partial
F}{\partial\bar{\lambda}_{\dot{\alpha}}}\frac{\langle\mathrm{d}\hat{\lambda}\lambda\rangle}{\langle\hat{\lambda}\lambda\rangle}+\partial_{\bar{u}_{B}}F\frac{\langle\mathrm{d}\bar{\mu}\lambda\rangle}{\langle\hat{\lambda}\lambda\rangle}+\partial_{\bar{\mathfrak{q}}}F\frac{\langle\mathrm{d}\bar{\mu}\hat{\lambda}\rangle}{\langle\lambda\hat{\lambda}\rangle^{2}}\\\
&=\left(\frac{\bar{u}}{\langle\hat{\lambda}\lambda\rangle^{2}}\partial_{\bar{q}}F-\bar{\partial}_{0}F\right)\mathrm{D}\bar{\lambda}+\bar{\Upsilon}F\frac{\langle\mathrm{d}\hat{\lambda}\lambda\rangle}{\langle\hat{\lambda}\lambda\rangle}+\partial_{\bar{\mathfrak{q}}}F\bar{\partial}\bar{\mathfrak{q}}+\partial_{\bar{u}_{B}}F\bar{\partial}\bar{u}_{B}\\\
&=\left(q\partial_{u}F-\bar{\eth}F\right)\mathrm{D}\bar{\lambda}+\partial_{\bar{\mathfrak{q}}}F\bar{\partial}\bar{\mathfrak{q}}+\partial_{\bar{u}_{B}}F\bar{\partial}\bar{u}_{B}\,,\end{split}$
(3.31)

where we have used $\bar{\Upsilon}F=0$ and the push-forward of the Edth
operator from twistor space to $\mathscr{I}$ in (3.10). The non-linear term in
the complex structure is given by

$\begin{split}\\{h,g\\}&=\epsilon^{\dot{\beta}\dot{\alpha}}\mathcal{L}_{\partial_{\dot{\alpha}}}h\wedge\mathcal{L}_{\partial_{\dot{\beta}}}g\\\
&=\bar{\sigma}^{0}\bar{\lambda}^{\dot{\alpha}}\frac{\partial
g_{\bar{\mathfrak{q}}}}{\partial\mu^{\dot{\alpha}}}\mathrm{D}\bar{\lambda}\wedge\bar{\partial}\bar{\mathfrak{q}}\\\
&=\bar{\sigma}^{0}\partial_{q}g_{\bar{\mathfrak{q}}}\mathrm{D}\bar{\lambda}\wedge\bar{\partial}\bar{\mathfrak{q}}\,.\end{split}$
(3.32)

Therefore, the $\bar{\nabla}$-closure condition in the basis (3.27) is

$\bar{\nabla}g=\left[q\partial_{u}g_{\bar{\mathfrak{q}}}-\bar{\eth}g_{\bar{\mathfrak{q}}}+\bar{\sigma}^{0}\partial_{q}g_{\bar{\mathfrak{q}}}+\frac{1}{\langle\hat{\lambda}\lambda\rangle}\partial_{\bar{q}}g_{0}\right]\mathrm{D}\bar{\lambda}\wedge\bar{\partial}\bar{\mathfrak{q}}+\partial_{\bar{u}_{B}}g_{\bar{\mathfrak{q}}}\,\bar{\partial}\bar{u}_{B}\wedge\bar{\partial}\bar{\mathfrak{q}}+\partial_{\bar{u}_{B}}g_{0}\,\bar{\partial}\bar{u}_{B}\wedge\bar{e}^{0}.$
(3.33)

The on-shell condition for $g$ then gives the conditions that the components
satisfy

$\displaystyle\partial_{\bar{u}}g_{1}=0=\partial_{\bar{u}}g_{0}\,,$ (3.34)
$\displaystyle q\partial_{u}g_{1}-\bar{\eth}g_{1}$
$\displaystyle+\bar{\sigma}\partial_{q}g_{1}-\frac{1}{\langle\hat{\lambda}\lambda\rangle^{2}}\partial_{\bar{q}}g_{0}=0\,,$
(3.35)

where we have made the redefinition
$g_{\bar{\mathfrak{q}}}=g_{1}\langle\lambda\hat{\lambda}\rangle$ to obtain the
corrected weighted function in the integral formula. The first conditions give
holomorphicity in the $u$-coordinate and are a direct consequence of the axial
gauge choice (3.22). The second gives the condition that if $g$ is
$\bar{\nabla}$-closed, the Bramson-Tod integral formula gives solutions to
(3.23) as desired.

##  4 $Lw_{1+\infty}$ symmetry algebra

In [22], the $Lw_{1+\infty}$ symmetry algebra was geometrically realized as
the algebra of Hamiltonians for local holomorphic Poisson diffeomorphisms on
the region of twistor space that excludes $\lambda_{0}=0$ and $\lambda_{1}=0$.
These are the local symmetries of twistor space in the Čech description of
Penrose’s original nonlinear graviton construction [21]. In that discussion,
the Hamiltonians for the diffeomorphisms play a role also as Čech
representatives of $H^{1}(\mathbb{PT},\mathcal{O}(2))$ which also define
infinitesimal deformations and hence generic self-dual gravitons via the
Penrose transform.

Here we take a different approach, deriving the symmetries from those of the
self-dual twistor action. Being a Dolbeault description, these are smooth
rather than holomorphic functions, essentially being functions of the six real
coordinates of twistor space, rather than the three complex variables as in
the previous paragraph. So we perform various gauge fixings to bring the gauge
freedom down to functions of three rather than six variables. In this way, we
will arrive at $Lw_{1+\infty}$.

Thus, in this section, we perform a residual gauge symmetry analysis on
twistor space (see e.g. [69] for a review). We start from the solution space
associated with (2.18) and impose a natural gauge fixing. We then derive the
subset of gauge transformations (2.17) that preserve this gauge fixing and
show that they form the $Lw_{1+\infty}$ algebra after using an appropriate
bracket. We also compute the transformation of the solution space under the
residual gauge transformations and deduce the action of $Lw_{1+\infty}$
symmetries on spacetime using the Bramson-Tod integral formula (3.21). As in
the Čech description of [22], we will see that these gauge transformations
correspond to an element $\delta h\in
H_{\bar{\nabla}}^{0,1}(\mathbb{PT},\mathcal{O}(2))$ defining a self-dual
graviton, now in a Dolbeault representation with the gauge transformation
being obtained via an integral formula.

### Residual gauge symmetries.

In the first instance, we will choose the following gauge fixing conditions on
the forms $(g,h)$ on asymptotic twistor space:

$\displaystyle\hat{\lambda}^{\alpha}\frac{\partial}{\partial\bar{\mu}^{\alpha}}\lrcorner\,\,g$
$\displaystyle=0\,,$ (4.1)
$\displaystyle\frac{\partial}{\partial\bar{\mu}^{\alpha}}\lrcorner\,\,h$
$\displaystyle=0\,.$ (4.2)

These choices are analogous to those in [70] adapted to a fibration of twistor
space over Euclidean signature space-time. Here we use the same general idea,
but for the fibration of twistor space over $\mathscr{I}_{\mathbb{C}}$. As
discussed above after (3.22), the first condition gives the on-shell
components of $g$ independence of $\bar{u}$ that follows from the analytic
extension of the fields to complex $u$ in $\mathscr{I}_{\mathbb{C}}$. For
homogeneity $\geq-1$ the cohomology up the $\mathbb{CP}^{1}$ fibers vanishes
and so a gauge can be found so that $h$ is trivial up the fibers in the
$\partial_{\bar{q}}$ direction. However,
$H^{1}(\mathbb{CP}^{1},\mathcal{O}(n))\neq 0$ for $n<-1$ so we cannot ask this
of $g$. The condition on $h$ restricts us to the solution space in which we
can hope to interpret the $\mathrm{D}\bar{\lambda}$ component as coming from
the shear as in (3.11) although we will see that it is not quite sufficient.
The gauge conditions restrict our gauge transformations in the following way

$\displaystyle\hat{\lambda}^{\alpha}\frac{\partial}{\partial\bar{\mu}^{\alpha}}\lrcorner\,\,\delta_{(\xi,\phi)}g$
$\displaystyle=0\quad\Longrightarrow\quad\partial_{\bar{u}}\phi=0\,,$ (4.3)
$\displaystyle\frac{\partial}{\partial\bar{\mu}^{\alpha}}\lrcorner\,\,\delta_{(\xi,\phi)}h$
$\displaystyle=0\quad\Longrightarrow\quad\frac{\partial\xi}{\partial\bar{\mu}^{\alpha}}=0\,,$
(4.4)

where we used (2.17). From this we see that the $\xi$ are independent of
$\bar{\mu}^{\alpha}$ while being arbitrary in
$\mu^{\dot{\alpha}},\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}}$ so, taking
homogeneities into account, we are now reduced to gauge freedom depending on
four variables rather than the three we are aiming for.

In the following, we will therefore also impose the further gauge condition

$\mathcal{L}_{\partial_{q}}h=0\,.$ (4.5)

Together with (4.2), this ensures the compatibility with $\eqref{h and sigma}$
where the shear is seen as a function defined on $\mathscr{I}_{\mathbb{C}}$
uplifted to twistor space, and therefore does not depend on $q$; the residual
gauge freedoms at this point will be the supertranslations and this gauge
choice is tantamount to performing the Penrose transform to a potential modulo
gauge for the self-dual field. Using (2.17), this additional gauge condition
(4.5) implies

$\partial_{q}(q\partial_{u}-\bar{\eth}+\bar{\sigma}\partial_{q})\xi=0\,.$
(4.6)

We can solve this formally by expanding $\xi$ as a finite polynomial in $q$

$\xi_{s}=\sum_{n=0}^{s+1}\xi_{s,n}q^{n}\,,$ (4.7)

where $\xi_{s,n}=\xi_{s,n}(u,\lambda,\bar{\lambda})$ and the gauge condition
(4.6) gives the following recursion relations

$\partial_{u}\xi_{s,n-1}-\bar{\eth}\xi_{s,n}+(n+1)\bar{\sigma}\xi_{s,n+1}=0\,,\quad\quad\xi_{s,s+1}:=\tau_{s}(\lambda,\bar{\lambda})\,.$
(4.8)

These recursion relations can be solved by an integral formula similar to
(3.21)

$\xi_{s,n}=\int q^{-n-1}\mathrm{d}q\wedge\delta
h_{s}|_{X_{(u,\lambda,\bar{\lambda})}}\,,$ (4.9)

where $\delta h_{s}\in H^{0,1}_{\bar{\nabla}}(\mathbb{PT},\mathcal{O}(2))$ is
an infinitesimal deformation around the background $h$. In this way, the local
Hamiltonian vector field generated by $\xi_{s}$ under the Poisson bracket
(2.10) corresponds to the infinitesimal deformation $\delta h_{s}$ around the
curved background $h$. The point here is that, like the holomorphic Poisson
diffeomorphisms of [22], $\xi_{s}$ is generically singular on twistor space
whereas $\delta h_{s}$, which is gauge equivalent to $\bar{\nabla}\xi_{s}$,
will need to be global and smooth to represent a Dolbeault class. However, the
integral formula is somewhat implicit and it will be instructive to solve the
recursion relations (4.8) more directly.

To proceed, we consider a polynomial of order $s+1$ in $\mu^{\dot{\alpha}}$

$\tilde{\xi}_{s}=\frac{1}{(s+1)!}\frac{\partial^{s+1}\tau_{s}}{\partial\bar{\lambda}^{\dot{\alpha}_{1}}\cdots{\partial\bar{\lambda}^{\dot{\alpha}_{s+1}}}}\mu^{\dot{\alpha}_{1}}\cdots\mu^{\dot{\alpha}_{s+1}}=\sum_{n=-1}^{s}\frac{u^{s-n}q^{n+1}}{(s-n)!}\bar{\eth}^{s-n}\tau_{s}\,,$
(4.10)

where $\tau_{s}=\tau_{s}(\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}})$ and
is valued in $\mathcal{O}(1-s,1+s)$ and $s=\\{-1,0,1,\dots\\}$. When
$\bar{\eth}^{s+2}\tau_{s}=0$ these coincide with the twistor space
$Lw_{1+\infty}$ generators given in [22] and this ansatz will be consistent
with the wedge condition, but we will work more generally here following [24].
This ansatz preserves the two first gauge conditions (4.1) and (4.2), i.e. it
satisfies (4.4). Furthermore, $\tilde{\xi}_{s}$ solves (4.6) in the absence of
shear. Acting with this parameter on the curved background given by $h$, gives

$\delta_{\tilde{\xi}_{s}}h=\bar{\nabla}\tilde{\xi}_{s}=\left[\bar{\sigma}\sum_{n=0}^{s}\frac{n+1}{(s-n)!}u^{s-n}q^{n}\bar{\eth}^{s-n}\tau_{s}-\frac{u^{s+1}}{(s+1)!}\bar{\eth}^{s+2}\tau_{s}\right]\mathrm{D}\bar{\lambda}\,,$
(4.11)

where we used $\mathcal{L}_{\partial_{u}}h=\bar{\sigma}D\bar{\lambda}$ (3.11)
to make the shear appear. As we can see this gives a polynomial in $q$ which
is not compatible with the gauge condition (4.5). Another way of saying this
is that $\tilde{\xi}_{s}$ does not solve (4.6). The first few examples are

$\begin{split}\bar{\nabla}\tilde{\xi}_{-1}&=-\bar{\eth}\tau_{-1}\mathrm{D}\bar{\lambda}\,,\\\
\bar{\nabla}\tilde{\xi}_{0}&=\left[\tau_{0}\bar{\sigma}-u\bar{\eth}^{2}\tau_{0}\right]\mathrm{D}\bar{\lambda}\,,\\\
\bar{\nabla}\tilde{\xi}_{1}&=\left[(u\bar{\eth}\tau_{1}+2q\tau_{1})\bar{\sigma}-\frac{u^{2}}{2}\bar{\eth}^{3}\tau_{1}\right]\mathrm{D}\bar{\lambda}\,,\\\
\bar{\nabla}\tilde{\xi}_{2}&=\left[\left(\frac{u^{2}}{2}\bar{\eth}^{2}\tau_{2}+2uq\bar{\eth}\tau_{2}+3q^{2}\tau_{2}\right)\bar{\sigma}-\frac{u^{3}}{6}\bar{\eth}^{4}\tau_{2}\right]\mathrm{D}\bar{\lambda}\,,\\\
\bar{\nabla}\tilde{\xi}_{3}&=\left[\left(\frac{u^{3}}{6}\bar{\eth}^{3}\tau_{3}+u^{2}q\bar{\eth}^{2}\tau_{3}+3uq^{2}\bar{\eth}\tau_{3}+4q^{3}\tau_{3}\right)\bar{\sigma}-\frac{u^{4}}{24}\bar{\eth}^{5}\tau_{3}\right]\mathrm{D}\bar{\lambda}\,,\end{split}$
(4.12)

and the issue arises for $s\geq 1$. To remedy this, we use the linear
dependence of $q$ in $\bar{\nabla}$ to cancel the highest order in $q$ terms
by adding an $\mathcal{O}(2)$ function $f_{s}$ to $\tilde{\xi}_{s}$ of a lower
order in $q$:

$\xi_{s}=\tilde{\xi}_{s}+f_{s}\,.$ (4.13)

We must now do this order by order. For $s=-1,0,1,2,3$, this gives the field-
depended parameters

$\begin{split}f_{-1}&=0\,,\\\ f_{0}&=0\,,\\\
f_{1}&=-2\tau_{1}\partial_{u}^{-1}\bar{\sigma}\,,\\\
f_{2}&=-3q\tau_{2}\partial_{u}^{-1}\bar{\sigma}-3\bar{\eth}(\tau_{2}\partial_{u}^{-2}\bar{\sigma})-2\bar{\eth}\tau_{2}\partial_{u}^{-1}(u\bar{\sigma})\,,\\\
f_{3}&=-4q^{2}\tau_{3}\partial_{u}^{-1}\bar{\sigma}-3q\bar{\eth}\tau_{3}\partial_{u}^{-1}(u\bar{\sigma})-4q\bar{\eth}(\tau_{3}\partial_{u}^{-2}\bar{\sigma})-\bar{\eth}^{2}\tau_{3}\partial_{u}^{-1}(u^{2}\bar{\sigma})\\\
&\qquad\qquad\qquad\qquad+8\tau_{3}\partial_{u}^{-1}(\bar{\sigma}\partial_{u}^{-1}\bar{\sigma})-3\bar{\eth}(\bar{\eth}\tau_{3}\partial_{u}^{-2}(u\bar{\sigma}))-4\bar{\eth}^{2}(\tau_{3}\partial_{u}^{-3}\bar{\sigma})\,.\end{split}$
(4.14)

We refer to Appendix A for expressions valid for all spin $s$.

### Asymptotic symmetry algebra.

We now compute the asymptotic symmetry algebra formed by the residual gauge
transformations on twistor space. We focus on the
$\xi\in\Omega^{0,0}(\mathbb{PT},\mathcal{O}(2))$, as the parameters
$\phi\in\Omega^{0,0}(\mathbb{PT},\mathcal{O}(-6))$ will be shown to generate
trivial gauge transformations in Section 5. The requirement (4.5) is
technically demanding as it yields field dependence in the residual gauge
transformations, see (4.14). However, this feature often arises in the gauge-
fixing approach to asymptotic symmetries [5, 71, 72, 73, 74], and tools have
been developed over the years to deal with these subtleties. In particular,
the residual gauge transformations form an algebra, provided one uses a
modified Lie bracket [5] which takes into account the field dependence of the
parameters. In our case, we define the bracket

$\\{\xi_{s},\xi_{s^{\prime}}\\}_{*}=\\{\xi_{s},\xi_{s^{\prime}}\\}+\delta_{\xi_{s}}\xi_{s^{\prime}}-\delta_{\xi_{s^{\prime}}}\xi_{s}\,,$
(4.15)

where the first term in the right-hand side is the Poisson bracket on twistor
space (2.10). The two additional terms take into account the field dependence
in $\xi$ and ensure that the bracket closes. Denoting
$\xi_{s}\equiv\xi(\tau_{s})$ the expression (4.13) with (4.10)-(4.14), one can
show that

$\boxed{\\{\xi(\tau_{s}),\xi(\tau_{s^{\prime}})\\}_{*}=\xi((s^{\prime}+1)\tau_{s^{\prime}}\bar{\eth}\tau_{s}-(s+1)\tau_{s}\bar{\eth}\tau_{s^{\prime}})\,.}$
(4.16)

The gauge parameter on the right-hand side is again of the form (4.13) with
(4.10)-(4.14). We checked explicitly that the algebra closes for all $s$ at
leading order in the fields. Furthermore, we also showed that the algebra
closes up to $s=5$ to all orders in the fields. We refer to Appendix B for
more details on these computations and explicit expressions for all $s$. We
conclude that the symmetry algebra of the residual gauge diffeomorphisms $\xi$
is given by the $Lw_{1+\infty}$ algebra. Notice that, in this approach, there
is no wedge condition on the parameters, which is in agreement with the phase
space discussions on spacetime in [24, 25].

### Action of $Lw_{1+\infty}$.

Let us now compute the action of the above residual gauge transformations on
the solution space, and deduce the action of $Lw_{1+\infty}$ on spacetime.
Here we focus on the residual gauge transformations associated with
$\xi\in\Omega^{0,0}(\mathbb{PT},\mathcal{O}(2))$ symmetries. The starting
point for the transformations of the fields is (2.17), together with the
explicit form of the gauge parameters (4.13) discussed in the previous
section. Denoting $\delta_{\xi_{s}}\equiv\delta_{s}$, the transformation of
$h$ reads as

$\begin{split}\delta_{-1}h&=-\bar{\eth}\tau_{-1}\mathrm{D}\bar{\lambda}\,,\\\
\delta_{0}h&=(\tau_{0}\bar{\sigma}-u\bar{\eth}^{2}\tau_{0})\mathrm{D}\bar{\lambda}\,,\\\
\delta_{1}h&=\left(2\bar{\eth}(\tau_{1}\partial_{u}^{-1}\bar{\sigma})+u\bar{\sigma}\bar{\eth}\tau_{1}-\frac{u^{2}}{2}\bar{\eth}^{3}\tau_{1}\right)\mathrm{D}\bar{\lambda}\,,\\\
\delta_{2}h&=\left(\frac{u^{2}}{2}\bar{\sigma}\bar{\eth}^{2}\tau_{2}+2\bar{\eth}(\bar{\eth}\tau_{2}\partial_{u}^{-1}(u\bar{\sigma}))+3\bar{\eth}^{2}(\tau_{2}\partial_{u}^{-2}\bar{\sigma})-3\tau_{2}\bar{\sigma}\partial_{u}^{-1}\bar{\sigma}-\frac{u^{3}}{6}\bar{\eth}^{4}\tau_{2}\right)\mathrm{D}\bar{\lambda}\,,\\\
\delta_{3}h&=\left(\frac{u^{3}}{6}\bar{\sigma}\bar{\eth}^{3}\tau_{3}+\bar{\eth}(\bar{\eth}^{2}\tau_{3}\partial_{u}^{-1}(u^{2}\bar{\sigma}))+3\bar{\eth}^{2}(\bar{\eth}\tau_{3}\partial_{u}^{-2}(u\bar{\sigma}))-3\bar{\sigma}\bar{\eth}\tau_{3}\partial_{u}^{-1}(u\bar{\sigma})\right.\\\
&\quad\quad\left.+4\bar{\eth}^{3}(\tau_{3}\partial_{u}^{-3}\bar{\sigma})-8\bar{\eth}(\tau_{3}\partial_{u}^{-1}(\bar{\sigma}\partial_{u}^{-1}\bar{\sigma}))-4\bar{\sigma}\bar{\eth}(\tau_{3}\partial_{u}^{-2}\bar{\sigma})-\frac{u^{4}}{24}\bar{\eth}^{5}\tau_{3}\right)\mathrm{D}\bar{\lambda}\,,\end{split}$
(4.17)

and so on. Notice, in particular, that the spin $s=-1$ symmetry is acting non-
trivially on $h$. Using (3.11), we can easily deduce the action on the
asymptotic shear as $\delta_{s}\bar{\sigma}=\partial_{u}\delta_{s}h_{0}$. We
display them explicitly for completeness:

$\displaystyle\delta_{-1}\bar{\sigma}$ $\displaystyle=0\,,$
$\displaystyle\delta_{0}\bar{\sigma}$
$\displaystyle=\tau_{0}\partial_{u}\bar{\sigma}-\bar{\eth}^{2}\tau_{0}\,,$
$\displaystyle\delta_{1}\bar{\sigma}$
$\displaystyle=\left(u\bar{\eth}\tau_{1}\partial_{u}+2\tau_{1}\bar{\eth}+3\bar{\eth}\tau_{1}\right)\bar{\sigma}-u\bar{\eth}^{3}\tau_{1}\,,$
(4.18) $\displaystyle\delta_{2}\bar{\sigma}$
$\displaystyle=\frac{u^{2}}{2}(\bar{\eth}^{2}\tau_{2}\partial_{u}\bar{\sigma}-\bar{\eth}^{4}\tau_{2})+u(3\bar{\eth}^{2}\tau_{2}\bar{\sigma}+2\bar{\eth}\tau_{2}\bar{\eth}\bar{\sigma})+3(\bar{\eth}^{2}-\partial_{u}\bar{\sigma})(\tau_{2}\partial_{u}^{-1}\bar{\sigma})-3\tau_{2}\bar{\sigma}^{2}\,,$
$\displaystyle\delta_{3}\bar{\sigma}$
$\displaystyle=\frac{u^{3}}{6}(\bar{\eth}^{3}\tau_{3}\partial_{u}\bar{\sigma}-\bar{\eth}^{5}\tau_{3})+\frac{u^{2}}{2}(3\bar{\eth}^{3}\tau_{3}\bar{\sigma}+2\bar{\eth}^{2}\tau_{3}\bar{\eth}\bar{\sigma})+3u\left((\bar{\eth}^{2}-\partial_{u}\bar{\sigma})(\bar{\eth}\tau_{3}\partial_{u}^{-1}\bar{\sigma})-\tau_{3}\bar{\sigma}^{2}\right)$
$\displaystyle+4\bar{\eth}^{3}(\tau_{3}\partial_{u}^{-2}\bar{\sigma})-3\bar{\eth}^{2}(\bar{\eth}\tau_{3}\partial_{u}^{-2}\bar{\sigma})-\bar{\eth}\tau_{3}\partial_{u}\bar{\sigma}\partial_{u}^{-2}\bar{\sigma}-4\tau_{3}\partial_{u}\bar{\sigma}\partial_{u}^{-2}\bar{\eth}\bar{\sigma}-12\bar{\sigma}\bar{\eth}(\tau_{3}\partial_{u}^{-1}\bar{\sigma})-8\tau_{3}\bar{\eth}\sigma\partial_{u}^{-1}\bar{\sigma}\,,$

and so on. These transformations reproduce correctly those of [24, 25]. In
particular, the terms linear in the shear agree with the transformations
deduced from twistor space in [37]. The spin $s=-1$ symmetry does not act on
the shear, which explains why it was invisible in the phase space approach on
spacetime [24, 25]. The $s=0,1$ transformations reproduce the expected
transformations under supertranslations $T$ and superrotations $\bar{Y}$ by
identifying $T=\tau_{0}$ and $\bar{Y}=2\tau_{1}$ [65, 66].

The transformations of the spin $s$ momenta $\mathcal{Q}_{s}$ introduced above
can be computed similarly, using the Bramson-Tod integral formula (3.21).
Starting from

$\delta_{s^{\prime}}\mathcal{Q}_{s}=\int
q^{s+2}\mathrm{d}q\wedge\\{g,\xi_{s^{\prime}}\\}|_{X_{(u,\lambda,\bar{\lambda})}}\,,$
(4.19)

and using again (2.17), we find explicitly, on-shell,

$\displaystyle\delta_{-1}\mathcal{Q}_{s}$ $\displaystyle=0\,,$
$\displaystyle\delta_{0}\mathcal{Q}_{s}$
$\displaystyle=\tau_{0}\partial_{u}\mathcal{Q}_{s}+(s+2)\mathcal{Q}_{s-1}\bar{\eth}\tau_{0}\,,$
$\displaystyle\delta_{1}\mathcal{Q}_{s}$
$\displaystyle=\left(u\bar{\eth}\tau_{1}\partial_{u}+2\tau_{1}\bar{\eth}+(s+3)\bar{\eth}\tau_{1}\right)\mathcal{Q}_{s}+u(s+2)\mathcal{Q}_{s-1}\bar{\eth}^{2}\tau_{1}\,,$
(4.20) $\displaystyle\delta_{2}\mathcal{Q}_{s}$
$\displaystyle=\frac{u^{2}}{2}\left(\bar{\eth}^{2}\tau_{2}\partial_{u}\mathcal{Q}_{s}+(s+2)\mathcal{Q}_{s-1}\bar{\eth}^{3}\tau_{2}\right)+u\left(2\bar{\eth}\tau_{s}\bar{\eth}\mathcal{Q}_{s}+(s+3)\mathcal{Q}_{s}\bar{\eth}^{2}\tau_{2}\right)$
$\displaystyle\qquad+3\tau_{2}\bar{\eth}\mathcal{Q}_{s+1}+(s+4)\bar{\eth}\tau_{2}\mathcal{Q}_{s+1}-3\left(\partial_{u}\mathcal{Q}_{s}+(s+2)\mathcal{Q}_{s-1}\bar{\eth}\right)\left(\tau_{2}\partial_{u}^{-1}\bar{\sigma}\right)\,,$
$\displaystyle\delta_{3}\mathcal{Q}_{s}$
$\displaystyle=\frac{u^{3}}{6}\left(\bar{\eth}^{3}\tau_{3}\partial_{u}\mathcal{Q}_{s}+(s+2)\mathcal{Q}_{s-1}\bar{\eth}^{4}\tau_{3}\right)+\frac{u^{2}}{2}\left(2\bar{\eth}^{2}\tau_{3}\bar{\eth}\mathcal{Q}_{s}+(s+3)\bar{\eth}^{3}\tau_{3}\mathcal{Q}_{s}\right)$
$\displaystyle\qquad+u\left(3\bar{\eth}\tau_{3}\bar{\eth}\mathcal{Q}_{s+1}+(s+4)\bar{\eth}^{2}\tau_{3}\mathcal{Q}_{s+1}-3\left(\partial_{u}\mathcal{Q}_{s}+(s+2)\mathcal{Q}_{s-1}\bar{\eth}\right)(\bar{\eth}\tau_{3}\partial_{u}^{-1}\bar{\sigma})\right)$
$\displaystyle\qquad+4\tau_{3}\bar{\eth}\mathcal{Q}_{s+2}+(s+5)\bar{\eth}\tau_{3}\mathcal{Q}_{s+2}-4\left(2\bar{\eth}\mathcal{Q}_{s}+(s+3)\mathcal{Q}_{s}\bar{\eth}\right)(\tau_{3}\partial_{u}^{-1}\bar{\sigma})$
$\displaystyle\qquad-\left(\partial_{u}\mathcal{Q}_{s}+(s+2)\mathcal{Q}_{s-1}\bar{\eth}\right)\left(\bar{\eth}\tau_{3}\partial_{u}^{-2}\bar{\sigma}+4\tau_{3}\partial_{u}^{-2}\bar{\eth}\bar{\sigma}\right)\,.$

We note for both the transformation of $\bar{\sigma}$ and $\mathcal{Q}_{s}$
there is a recursive structure. In particular, the expressions with a non-zero
power of $u$ are built from a lower spin transformation of the same object.
The recursive nature is made manifest by expressing a general spin $s$
variations in terms of the $\beta_{n,p}$ found in Appendix A

$\begin{split}\delta_{s}\bar{\sigma}&=\partial_{u}\left(\bar{\sigma}\sum_{p=0}^{\lfloor\frac{s}{2}\rfloor}\beta_{1,p}-\sum_{p=0}^{\lfloor\frac{s+1}{s}\rfloor}\bar{\eth}\beta_{0,p}\right)\,,\\\
\delta_{s^{\prime}}\mathcal{Q}_{s}&=\sum_{p=0}^{\lfloor\frac{s^{\prime}}{2}\rfloor}\left[\beta_{1,p}\partial_{u}\mathcal{Q}_{s}+(s+2)\bar{\eth}\beta_{1,p}\mathcal{Q}_{s-1}\right]\\\
&\qquad\qquad+\sum_{n=1}^{s^{\prime}}\sum_{p=0}^{\lfloor\frac{s^{\prime}-n}{2}\rfloor}\left[(n+1)\beta_{n+1,p}\bar{\eth}\mathcal{Q}_{s+n-1}+(s+2+n)\bar{\eth}\beta_{n+1,p}\mathcal{Q}_{s+n-1}\right]\,.\end{split}$
(4.21)

Again, the symmetry $s=-1$ does not act on these objects, and the cases
$s=0,1$ reproduce the part of the BMS symmetries associated with $T$ and
$\bar{Y}$ (the transformation under $Y$ is deduced in Appendix D). To our
knowledge, this is the first time the transformations (4.20) are deduced
systematically for all $s$. This illustrates the power of the formula (3.21).
Finally, let us mention that the action of the residual gauge transformations
on the solution space forms a representation of the $Lw_{1+\infty}$
symmetries:

$[\delta_{\xi},\delta_{\xi^{\prime}}]=\delta_{\\{\xi,\xi^{\prime}\\}_{*}}\,.$
(4.22)

This can be checked explicitly on all the fields using the following relations
on twistor space:

$[\delta_{\xi},\delta_{\xi^{\prime}}]g=\\{g,\\{\xi,\xi^{\prime}\\}_{*}\\}\,,\qquad[\delta_{\xi},\delta_{\xi^{\prime}}]h=\bar{\nabla}\\{\xi,\xi^{\prime}\\}_{*}\,,$
(4.23)

where the Jacobi identity on the twistor space bracket (2.10) was used. These
relations also ensure that the modified bracket (4.15) satisfies the Jacobi
identity.

### Invariance of the recursion relations under $Lw_{1+\infty}$.

We now make the following important observation: the recursion relations
(3.23) are invariant under the action of the $Lw_{1+\infty}$. This is a direct
consequence of the fact that the equation of motion for $g$ in (2.18) is
invariant under gauge transformations (2.17), combined with the fact that the
recursion relations (3.23) are deduced from the closure condition of $g$
through the Bramson-Tod integral formula (3.21). As a consistency check, one
can also verify explicitly the invariance of (3.23) using the spacetime
transformations found in (4.18) and (4.20). More explicitly, for each $s$, one
can show the recursion relations are invariant under the higher spin
transformations by lifting to twistor space. We first re-write the recursion
relations using the Bramson-Tod integral formula

$\begin{split}\delta_{s^{\prime}}\left(\partial_{u}\mathcal{Q}_{s}-\bar{\eth}\mathcal{Q}_{s-1}-(s+1)\bar{\sigma}\mathcal{Q}_{s-2}\right)&=\delta_{s^{\prime}}\int
q^{s+1}\left[q\partial_{u}g_{1}-\bar{\eth}g_{1}+\bar{\sigma}\partial_{q}g_{1}\right]\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\frac{1}{\langle\lambda\hat{\lambda}\rangle}\delta_{s^{\prime}}\int
q^{s+1}\left[\bar{\nabla}g_{\bar{\mathfrak{q}}}\right]_{\mathrm{D}\bar{\lambda}}\mathrm{d}q\mathrm{d}\bar{q}\,,\end{split}$
(4.24)

where
$\left[\bar{\nabla}g_{\bar{\mathfrak{q}}}\right]_{\mathrm{D}\bar{\lambda}}$
denotes the $\mathrm{D}\bar{\lambda}$ component of the form. The variation of
the form is given by

$\begin{split}\delta_{s^{\prime}}\bar{\nabla}g_{\bar{\mathfrak{q}}}&=\bar{\partial}\delta_{s^{\prime}}g_{\bar{\mathfrak{q}}}+\\{\delta_{s^{\prime}}h,g_{\bar{\mathfrak{q}}}\\}+\\{h,\delta_{s^{\prime}}g_{\bar{\mathfrak{q}}}\\}\\\
&=\bar{\partial}\\{g_{\bar{\mathfrak{q}}},\xi_{s^{\prime}}\\}+\\{\bar{\nabla}\xi_{s^{\prime}},g_{\bar{\mathfrak{q}}}\\}+\\{h,\\{g_{\bar{\mathfrak{q}}},\xi_{s^{\prime}}\\}\\}\\\
&=\\{\bar{\partial}g_{\bar{\mathfrak{q}}},\xi_{s^{\prime}}\\}+\\{\\{h,\xi_{s^{\prime}}\\},g_{\bar{\mathfrak{q}}}\\}+\\{\\{\xi_{s^{\prime}},g_{\bar{\mathfrak{q}}}\\},h\\}\\\
&=\frac{1}{\langle\lambda\hat{\lambda}\rangle}\\{\partial_{\bar{q}}g_{0},\xi_{s^{\prime}}\\}\mathrm{D}\bar{\lambda}+\\{\partial_{\bar{\mathfrak{q}}}g_{\bar{\mathfrak{q}}},\xi_{s^{\prime}}\\}\bar{\partial}\bar{\mathfrak{q}}\,,\end{split}$
(4.25)

where we have used the equations of motion for $g_{\bar{\mathfrak{q}}}$ and
the Jacobi identity for (2.10). Finally using the restriction on the gauge
parameters (4.4) due to the gauge conditions, we find

$\delta_{s^{\prime}}\left(\partial_{u}\mathcal{Q}_{s}-\bar{\eth}\mathcal{Q}_{s-1}+(s+1)\bar{\sigma}\mathcal{Q}_{s-2}\right)=\frac{1}{\langle\hat{\lambda}\lambda\rangle^{2}}\int\partial_{\bar{q}}\left(q^{s+1}\\{g_{0},\xi_{s^{\prime}}\\}\right)\mathrm{d}q\mathrm{d}\bar{q}=0\,,$
(4.26)

a total boundary term that vanishes in the integral. Therefore, the higher
spin variations preserve the recursion relations

$\partial_{u}\delta_{s^{\prime}}\mathcal{Q}_{s}=\bar{\eth}\delta_{s^{\prime}}\mathcal{Q}_{s-1}+(s+1)\delta_{s^{\prime}}\bar{\sigma}\mathcal{Q}_{s-2}+(s+1)\bar{\sigma}\delta_{s^{\prime}}\mathcal{Q}_{s-2}\,,$
(4.27)

as desired. This clarifies in which sense $Lw_{1+\infty}$ are symmetries from
a spacetime perspective.

### Wedge condition

At this stage, as mentioned below (4.16), the wedge condition does not seem to
appear in our analysis. However, if one requires the radiative phase space to
obey the Schwartzian falloffs as in [24, 25, 75],

$\lim_{u\to\pm\infty}\bar{\sigma}=\mathcal{O}(u^{-s-\epsilon})\,,$ (4.28)

the $Lw_{1+\infty}$ algebra reduces to its wedge subalgebra,
$Lw_{1+\infty}^{\wedge}$, defined by the commutation relations

$[\tau_{s},\tau_{s^{\prime}}]=\epsilon^{\dot{\beta}\dot{\alpha}}\frac{\partial\tau_{s}}{\partial\bar{\lambda}^{\dot{\alpha}}}\frac{\partial\tau_{s^{\prime}}}{\partial\bar{\lambda}^{\dot{\beta}}}=(s^{\prime}+1)\tau_{s^{\prime}}\bar{\eth}\tau_{s}-(s+1)\tau_{s}\bar{\eth}\tau_{s^{\prime}}\,,$
(4.29)

together with the wedge condition on the parameters
$\tau_{s}(\lambda,\bar{\lambda})$

$\bar{\eth}^{s+2}\tau_{s}=0\,.$ (4.30)

Indeed, preserving (4.28) under the action of $Lw_{1+\infty}$ implies
$\delta_{s}\bar{\sigma}|_{\mathscr{I}_{\pm}}=0$, which requires setting all
the non-homogenous terms in the transformation written in (4.18) to $0$,
leading to (4.30). Notice that the field-dependent contributions in (4.13)
vanish at $\mathscr{I}^{+}_{\pm}$ if we impose the falloffs (4.28). Hence, we
have
$\xi_{s}|_{\mathscr{I}^{+}_{\pm}}=\tilde{\xi}_{s}|_{\mathscr{I}^{+}_{\pm}}$.
Interestingly, using the bracket (2.10), the parameters $\tilde{\xi}_{s}$
defined in (4.10) with the wedge condition (4.30) form the
$Lw_{1+\infty}^{\wedge}$ algebra

$\\{\tilde{\xi}(\tau_{s}),\tilde{\xi}(\tau_{s^{\prime}})\\}=\tilde{\xi}((s^{\prime}+1)\tau_{s^{\prime}}\bar{\eth}\tau_{s}-(s+1)\tau_{s}\bar{\eth}\tau_{s^{\prime}})|_{\wedge}\,.$
(4.31)

The explicit computations concerning the algebra can be found in Appendix B.
To connect with the geometric interpretation of $Lw_{1+\infty}^{\wedge}$
symmetries discussed in [22, 76], we can expand $\tilde{\xi}_{s}$ in the wedge
as

$\tilde{\xi}_{s}(\mu^{\dot{\alpha}},\lambda_{\alpha})=\sum_{|m|\leq
p-1}\sum_{a\in\mathbb{Z}}\tau_{m,a}^{p}w^{p}_{m,a}\,,\qquad
w^{p}_{m,a}:=\frac{(\mu^{\dot{0}})^{p+m-1}(\mu^{\dot{1}})^{p-m-1}}{2\lambda_{0}^{p-a-2}\lambda_{1}^{p+a-2}}\,,$
(4.32)

where we define $s=2p-3$ and $\tau_{m,a}^{p}$ are constants. Using these
definitions, the algebra (4.31) gives the well-known structure constants

$\\{w^{p}_{m,a},w^{q}_{n,b}\\}=(m(q-1)-n(p-1))w^{p+q-2}_{m+n,a+b}\,.$ (4.33)

As defined, the $\tilde{\xi}$ can be viewed as Lie algebra homomorphisms, and
one can show that the $\tau_{s}$ have a following expansion

$\tau_{s}(\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}})=\sum_{|m|\leq
p-1}\sum_{a\in\mathbb{Z}}\tau_{m,a}^{p}\tilde{w}^{p}_{m,a}\,,\qquad\tilde{w}^{p}_{m,a}:=\frac{(\bar{\lambda}^{\dot{0}})^{p+m-1}(\bar{\lambda}^{\dot{1}})^{p-m-1}}{2\lambda_{0}^{p-a-2}\lambda_{1}^{p+a-2}}\,,$
(4.34)

where the $\tilde{w}^{p}_{m,a}$ have the same structure constants under the
bracket (4.29). In the following, we will proceed without imposing the wedge
condition restriction (4.30), and keep the full $Lw_{1+\infty}$ symmetries on
the phase space.

##  5 Charge algebra

In this section, we construct the symplectic structure on twistor space, from
which we deduce the charges associated with the $Lw_{1+\infty}$ symmetries
discussed in the previous section. We show that these charges form a
representation of these symmetries, provided one uses an appropriate bracket.
We then translate these results to spacetime expressions and compare them with
previous expressions discussed in the literature.

### Charges on twistor space.

To find the symplectic structure we use the covariant phase space formalism
[28, 29, 30, 31]. The variation of the twistor action (2.16) yields

$\begin{split}\delta
S=&\int_{\mathbb{PT}}\mathrm{D}^{3}Z\wedge\left[\left(\bar{\partial}h+\frac{1}{2}\\{h,h\\}\right)\wedge\delta
g+\bar{\nabla}g\wedge\delta h\right]\\\
&+\int_{\mathbb{PT}}\mathrm{D}^{3}Z\wedge\frac{\partial}{\partial\mu^{\dot{\alpha}}}\left(g\wedge\epsilon^{\dot{\alpha}\dot{\beta}}\frac{\partial
h}{\partial\mu^{\dot{\beta}}}\wedge\delta
h\right)+\int_{\mathbb{PT}}\mathrm{D}^{3}Z\wedge\bar{\partial}(g\wedge\delta
h)\,.\end{split}$ (5.1)

The first term on the right-hand side corresponds to the equations of motion
(2.18). The second term vanishes due to the gauge fixing condition (4.2). The
last term is a total derivative term corresponding to the presymplectic
potential. Taking one more variation of the latter, and integrating over the
co-dimension $1$-surface
$\Sigma=\\{Z\in\mathbb{PT}|u_{B}-\bar{u}_{B}=\text{constant}\\},$555The space
of null twistors is the hypersurface
$PN=\\{Z\in\mathbb{PT}|u_{B}-\bar{u}_{B}=0\\}$. we obtain the presymplectic
form

$\boxed{\Omega(\delta g,\delta h)=\int_{\Sigma}\delta h\wedge\delta g\wedge
D^{3}Z\,.}$ (5.2)

The charges are then obtained by contracting the presymplectic form with the
residual gauge transformations obtained in the previous section. The charge
associated with the $\phi$ is given by

$\delta_{\phi}\lrcorner\,\,\Omega+\delta H_{\phi}=0\quad,\quad
H_{\phi}=\int_{\partial\Sigma}D^{3}Z\wedge h\phi\,.$ (5.3)

However, since $\partial\Sigma$ is the hypersurface in twistor space keeping
$u_{B}$ constant and real, in the gauge we are working, $H_{\phi}=0$. This is
because when pulled back to $\partial\Sigma$ the holomorphic top form $D^{3}Z$
expressed in the fibration coordinates contains a $\mathrm{D}\bar{\lambda}$,
see Appendix E for more details. In this case, the symplectic form is
degenerate on the phase space because $\delta_{\phi}$ is a non-zero vector in
general. Moreover,

$\delta_{\xi}\lrcorner\,\,\Omega+\cancel{\delta}H_{\xi}=0\,,$ (5.4)

where we have dropped the boundary terms associated with total $\partial_{q}$
terms because we assume the fibers have no boundary throughout. We also
discarded a total $\partial_{u}$ term on the right-hand side, which reads
explicitly as

$\int_{\Sigma}\mathrm{D}^{3}Z\wedge\partial_{u}\left(\partial_{q}\xi_{s}g\wedge\delta
h\right)=\\\ \left[\sum_{n=0}^{s}(n+1)\int_{\mathbb{CP}^{1}}\delta
h_{0}\mathcal{Q}_{n-2}\left(\frac{u^{s-n}}{(s-n)!}\bar{\eth}^{s-n}\tau_{s}+\alpha_{n+1}[\bar{\sigma},\tau_{s}]\right)\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\right]_{\iota^{0}}^{\iota^{+}}\,.$
(5.5)

This corner term vanishes for $s=0,1$ if we take the same falloffs in $u$ as
in [77, 78]. Furthermore, this term vanishes for all $s$ if we restrict
ourselves to the Schwartzian falloffs for the shear (4.28), as considered in
[24, 25]. We proceed by neglecting this boundary term, but without assuming
the wedge condition. We see from (5.4) that the charges associated with
$\xi\in\Omega^{0,0}(\mathbb{PT},\mathcal{O}(2))$ are non-vanishing, hence
confirming that the $Lw_{1+\infty}$ are indeed large gauge transformations on
twistor space. In particular, the charge associated with $s=-1$ does not
vanish, which means that the corresponding symmetry has a non-trivial action
on phase space. The charges are non-integrable (i.e. non $\delta$-exact on the
solution space) and can be split as

$\cancel{\delta}H_{\xi}=\delta H_{\xi}-\Xi_{\xi}[\delta]\,,$ (5.6)

where the integrable and non-integrable pieces are given by

$\boxed{H_{\xi}=\int_{\partial\Sigma}\xi g\wedge
D^{3}Z,}\qquad\Xi_{\xi}[\delta]=\int_{\partial\Sigma}\delta\xi g\wedge
D^{3}Z\,.$ (5.7)

The non-integrability of the charges, which appears for $s\geq 1$, can be
traced back to the field dependence of the parameter $\xi_{s}$ due to the
gauge condition (4.5). This might have been expected, as it is well known that
the charges associated to the BMS symmetries in asymptotically flat spacetimes
are already non-integrable [32, 33, 6, 34, 9, 12, 35, 10]. Several tools have
been developed to deal with this non-integrability, including a modified
bracket [6, 9, 41, 42, 43, 44, 45, 46] defined as

$\\{H_{\xi},H_{\xi^{\prime}}\\}_{\star}=\delta_{\xi}H_{\xi^{\prime}}-\Xi_{\xi}[\delta_{\xi^{\prime}}]\,.$
(5.8)

Using this bracket, one can easily show that

$\boxed{\\{H_{\xi},H_{\xi^{\prime}}\\}_{\star}=H_{\\{\xi,\xi^{\prime}\\}_{*}}\,,}$
(5.9)

where the bracket for vector fields was given in (4.15). Hence, the charges
form a representation of the $Lw_{1+\infty}$ symmetries on twistor space. The
consistency of this bracket with Jacobi identities follows directly from the
consistency of the bracket (4.15). Notice that a different split between
integrable and non-integrable pieces in (5.6)-(5.7),

$H_{\xi}\to
H^{\prime}_{\xi}=H_{\xi}+N_{\xi},\qquad\Xi_{\xi}\to\Xi^{\prime}_{\xi}=\Xi_{\xi}+\delta
N_{\xi}\,,$ (5.10)

would not affect $\cancel{\delta}H_{\xi}$, but it would produce a (trivial)
field-dependent 2-cocycle in the right-hand side of (5.9):

$\\{H^{\prime}_{\xi},H^{\prime}_{\xi^{\prime}}\\}_{\star}=H^{\prime}_{\\{\xi,\xi^{\prime}\\}_{*}}+K^{\prime}_{\xi,\xi^{\prime}},\qquad
K^{\prime}_{\xi,\xi^{\prime}}=\delta_{\xi}N_{\xi^{\prime}}-\delta_{\xi^{\prime}}N_{\xi}-N_{\\{\xi,\xi^{\prime}\\}_{*}}\,.$
(5.11)

This justifies a posteriori why the natural split chosen in (5.6)-(5.7) is the
most appropriate.

### Charges on spacetime.

We now translate the symplectic structure and charges obtained from first
principles on twistor space to spacetime expressions using the dictionary
discussed in section 3. Starting from the symplectic structure (5.2) and using
the Bramson-Tod integral formula (3.21), we find

$\begin{split}\Omega&=\int_{\Sigma}\delta h_{0}\delta
g\wedge\mathrm{d}q\wedge\mathrm{d}u\wedge\mathrm{D}\lambda\wedge\mathrm{D}\bar{\lambda}=-\int_{\mathscr{I}}\delta
h_{0}\delta\Psi_{4}^{0}\mathrm{d}u\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,.\end{split}$
(5.12)

Hence, on the radiative phase space coming from twistor space, the natural
symplectic pairing is between $h_{0}$ and $\Psi^{0}_{4}$. The standard
Ashtekar-Streubel symplectic structure on the radiative phase space [32] can
be obtained by integrating by parts with respect to $u$, using (3.15) and
(3.11), and assuming strong fall-offs on the news to get rid of the boundary
term:

$\Omega=\int_{\mathscr{I}}\delta\bar{\sigma}\delta\dot{\sigma}\mathrm{d}u\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,.$
(5.13)

In this case, the symplectic pairing is between the shear and the news. The
virtue of choosing the symplectic pairing $(h_{0},\Psi^{0}_{4})$ instead of
$(\bar{\sigma},\dot{\sigma})$ is that it allows us to see the non-trivial
action of the $s=-1$ symmetry on phase space (compare (4.17) with (4.18)).

Let us now focus on the integrable part of the charges (5.7) associated with
the $Lw_{1+\infty}$ symmetries. Again, using (3.21), as well as the explicit
expressions for the spin $s$ parameters (4.13) together with (4.10)-(4.14), we
find explicit expressions for the surface charges at a cut
$\mathcal{S}=\\{(u,z,\bar{z})\in\mathscr{I}|u=\text{constant}\\}$ of null
infinity:

$\boxed{\begin{split}H_{-1}&=\int_{\mathcal{S}}\tau_{-1}\mathcal{Q}_{-1}\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,\\\
H_{0}&=\int_{\mathcal{S}}\tau_{0}(\mathcal{Q}_{0}-u\bar{\eth}\mathcal{Q}_{-1})\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,\\\
H_{1}&=\int_{\mathcal{S}}\tau_{1}\Big{(}\mathcal{Q}_{1}-u\bar{\eth}\mathcal{Q}_{0}+\frac{u^{2}}{2}\bar{\eth}^{2}\mathcal{Q}_{-1}-2\mathcal{Q}_{-1}\partial_{u}^{-1}\bar{\sigma}\Big{)}\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,\\\
H_{2}&=\int_{\mathcal{S}}\tau_{2}\Big{(}\mathcal{Q}_{2}-u\bar{\eth}\mathcal{Q}_{1}+\frac{u^{2}}{2}\bar{\eth}\mathcal{Q}_{0}-\frac{u^{3}}{6}\bar{\eth}^{3}\mathcal{Q}_{-1}\\\
&\qquad\qquad\quad-3\mathcal{Q}_{0}\partial_{u}^{-1}\bar{\sigma}+3\bar{\eth}\mathcal{Q}_{-1}\partial_{u}^{-2}\bar{\sigma}+2\bar{\eth}\left(\mathcal{Q}_{-1}\partial_{u}^{-1}(u\bar{\sigma})\right)\Big{)}\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,\\\
H_{3}&=\int_{\mathcal{S}}\tau_{3}\Big{(}\mathcal{Q}_{3}-u\bar{\eth}\mathcal{Q}_{2}+\frac{u^{2}}{2}\bar{\eth}^{2}\mathcal{Q}_{1}-\frac{u^{3}}{6}\bar{\eth}^{3}\mathcal{Q}_{0}+\frac{u^{4}}{24}\bar{\eth}^{4}\mathcal{Q}_{-1}-4\mathcal{Q}_{1}\partial_{u}^{-1}\bar{\sigma}\\\
&\qquad\qquad\quad+3\bar{\eth}\left(\mathcal{Q}_{0}\partial_{u}^{-1}(u\bar{\sigma})\right)+4\bar{\eth}\mathcal{Q}_{0}\partial_{u}^{-2}\bar{\sigma}+8\mathcal{Q}_{-1}\partial_{u}^{-1}(\bar{\sigma}\partial_{u}^{-1}\bar{\sigma})\\\
&\qquad\qquad\quad-3\bar{\eth}\left(\bar{\eth}\mathcal{Q}_{-1}\partial_{u}^{-2}(u\bar{\sigma})\right)-\bar{\eth}^{2}\left(\mathcal{Q}_{-1}\partial_{u}^{-1}(u^{2}\bar{\sigma})\right)-4\bar{\eth}\mathcal{Q}_{-1}\partial_{u}^{-3}\bar{\sigma}\Big{)}\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,\,.\end{split}}$
(5.14)

Using the recursion relations presented in Appendix A, a general expression
for the charges valid for any spin $s$ can be presented:

$\begin{split}H_{s}&=\sum_{n=-1}^{s}\int_{\mathcal{S}}\left(\tau_{s}\frac{(-u)^{s-n}}{(s-n)!}\bar{\eth}^{s-n}\mathcal{Q}_{n}+\sum_{p=1}^{\lfloor\frac{s-n}{2}\rfloor}\beta_{n+1,p}[\tau_{s},\bar{\sigma}]\mathcal{Q}_{n}\right)\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\\\
&=\sum_{n=-1}^{s}\sum_{p=0}^{\lfloor\frac{s-n}{2}\rfloor}\int_{\mathcal{S}}\beta_{n+1,p}[\tau_{s},\bar{\sigma}]\mathcal{Q}_{n}\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,\end{split}$
(5.15)

where the $\beta_{n,p}$ are depended on the coordinates of $\mathcal{S}$, the
transformation parameter $\tau_{s}$, and the shear $\bar{\sigma}$. The
$\beta_{n,p}$ are defined recursively in (A.13).

Let us make a couple of comments concerning these surface charge expressions.
First, we observe that these charges are very similar to those constructed
heuristically in [24, 25]. By contrast with these works, we find a non-
vanishing spin $s=-1$ charge on spacetime, involving the momentum
$\mathcal{Q}_{-1}=\Psi^{0}_{3}$. This is consistent with the fact that the
spin $s=-1$ acts non-trivially on the phase space variable $h$. Notice that
this charge would disappear if the wedge condition (4.30) was imposed, see
(3.15) (in that case, the symmetry $s=-1$ would be a central term). We find a
similar contribution $\frac{(-u)^{s+1}}{(s+1)!}\mathcal{Q}_{-1}$, for all spin
$s$ charge, which differs from [24, 25] and which would disappear on the
wedge. Moreover, we find some terms $\sim\partial_{u}^{-1}\bar{\sigma}$ of
difference compared to [24, 25]. Despite these differences, one can verify
that our charges (5.14) satisfy the two following criteria:

$(i)$ The charges are conserved in time in the absence of radiation, i.e.

$\Psi^{0}_{4}=0\quad\Longrightarrow\quad\frac{d}{du}H_{s}=0\,.$ (5.16)

Notice that the non-radiative condition we are considering here is weaker than
that in [24, 25] where the additional condition $\Psi^{0}_{3}=0$ is imposed,
but it matches exactly the one proposed earlier in [79]. This makes a lot of
sense regarding our construction: the canonical variable $\Psi^{0}_{4}$
controls the radiation which spoils the conservation of the $Lw_{1+\infty}$
charges constructed out of $\mathcal{Q}_{s}$, with $s=-1,0,1,2,\ldots$ The
explicit expressions of the local fluxes $\mathcal{F}_{s}=\partial_{u}H_{s}$
read as

$\displaystyle\mathcal{F}_{-1}$
$\displaystyle=\int_{\mathcal{S}}\tau_{-1}\bar{\eth}\Psi_{4}^{0}\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,$
$\displaystyle\mathcal{F}_{0}$
$\displaystyle=\int_{\mathcal{S}}\tau_{0}\left(\bar{\sigma}\Psi_{4}^{0}-u\bar{\eth}^{2}\Psi_{4}^{0}\right)\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,$
$\displaystyle\mathcal{F}_{1}$
$\displaystyle=\int_{\mathcal{S}}\tau_{1}\left(\frac{u^{2}}{2}\bar{\eth}^{3}\Psi_{4}^{0}-u\bar{\eth}(\bar{\sigma}\Psi_{4}^{0})-2\bar{\eth}\Psi_{4}^{0}\partial_{u}^{-1}\bar{\sigma}\right)\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,$
(5.17) $\displaystyle\mathcal{F}_{2}$
$\displaystyle=\int_{\mathcal{S}}\tau_{2}\left(\frac{u^{2}}{2}\bar{\eth}^{2}(\bar{\sigma}\Psi_{4}^{0})-\frac{u^{3}}{6}\bar{\eth}^{4}\Psi_{4}^{0}-3\Psi_{4}^{0}\bar{\sigma}\partial_{u}^{-1}\bar{\sigma}+2\bar{\eth}\left(\bar{\eth}\Psi_{4}^{0}\partial_{u}^{-1}(u\bar{\sigma})\right)\right)\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,$
$\displaystyle\mathcal{F}_{3}$
$\displaystyle=\int_{\mathcal{S}}\tau_{3}\left(\frac{u^{4}}{24}\bar{\eth}^{5}\Psi_{4}^{0}-\frac{u^{3}}{6}\bar{\eth}^{3}(\bar{\sigma}\Psi_{4}^{0})-\bar{\eth}^{2}\left(\bar{\eth}\Psi_{4}^{0}\partial_{u}^{-1}(u^{2}\bar{\sigma})\right)-3\bar{\eth}\left(\bar{\eth}^{2}\Psi_{4}^{0}\partial_{u}^{-2}(u\bar{\sigma})\right)\right.$
$\displaystyle\qquad\qquad+\left.3\bar{\eth}\left(\Psi_{4}^{0}\bar{\sigma}\partial_{u}^{-1}(u\bar{\sigma})\right)+4\bar{\eth}(\bar{\sigma}\Psi_{4}^{0})\partial_{u}^{-2}\bar{\sigma}+8\bar{\eth}\Psi_{4}^{0}\partial_{u}^{-1}(\bar{\sigma}\partial_{u}^{-1}\bar{\sigma})-4\bar{\eth}^{3}\Psi_{4}^{0}\partial_{u}^{-3}\bar{\sigma}\right)\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,.$

Using the recursion relations found in Appendix A, one can further show that
the fluxes have a compact expression in terms of the $\beta_{n,p}$

$\mathcal{F}_{s}=\int_{\mathcal{S}}\Psi_{4}^{0}\left(\bar{\sigma}\sum_{p=0}^{\lfloor\frac{s}{2}\rfloor}\beta_{1,p}[\tau_{s},\bar{\sigma}]-\sum_{p=0}^{\lfloor\frac{s+1}{2}\rfloor}\bar{\eth}\beta_{0,p}[\tau_{s},\bar{\sigma}]\right)\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,$
(5.18)

showing that the charges $H_{s}$ satisfy (5.16).

$(ii)$ One can easily show that the integrated fluxes
$\int_{-\infty}^{+\infty}du\mathcal{F}_{s}$ are exactly the same as those
considered in [24, 25], provided one assumes the same falloffs in $u$ on the
shear given in (4.28) allowing for integration by parts. In particular, these
fluxes are consistent with the action of the $Lw_{1+\infty}$ symmetries on the
radiative phase space.

Let us emphasize that the phase space analysis on twistor space also provides
new proposals for the more standard BMS charges. Introducing the
supertranslation $T=\tau_{0}$ and superrotation $\bar{Y}=2\tau_{1}$
parameters, and using the correspondence (3.17), we find the BMS surface
charges:666In our conventions, $4\pi G=1$.

$\begin{split}H_{0}&=\int_{\mathcal{S}}T(\Psi^{0}_{2}-u\bar{\eth}\Psi^{0}_{3})\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,,\\\
H_{1}&=\frac{1}{2}\int_{\mathcal{S}}\bar{Y}\left(\Psi^{0}_{1}-u\bar{\eth}\Psi^{0}_{2}+\frac{u^{2}}{2}\bar{\eth}^{2}\Psi^{0}_{3}-2\Psi^{0}_{3}\partial_{u}^{-1}\bar{\sigma}\right)\mathrm{D}\lambda\mathrm{D}\bar{\lambda}\,.\end{split}$
(5.19)

Notably, we find new expressions for the mass and angular momentum in general
relativity, involving the field $h$ which is natural from a twistor space
perspective. These charges have the virtue of being fully compatible with the
whole tower of $Lw_{1+\infty}$ symmetries in asymptotically flat spacetime.

##  6 Discussion

In this work, we have shown that twistor space provides the natural framework
to understand the $Lw_{1+\infty}$ symmetries and their associated charges. The
crucial advantage of twistor space compared to space-time is that the action
of the symmetries becomes local and geometric, emerging from the gauge
symmetries of the twistor space action. This allows for the application of
standard covariant phase space methods. This analysis allowed us to understand
better the organization of the solutions of gravity in asymptotically flat
spacetimes.

Several questions remain to be understood. First, in this work, we essentially
focused on the self-dual sector of gravity for which the twistor actions take
a very simple form. Although at $\mathscr{I}$, the asymptotic solution space
of self-dual gravity is identical to that of gravity, it will start to differ
in the subleading orders of the expansion. It would be interesting to
understand where this difference exactly occurs and see to what extent the
$Lw_{1+\infty}$ symmetries survive beyond the self-dual sector. We remark that
a twistor action for full gravity was found by Sharma in [80], but being based
on reducing the twistor action for conformal gravity of [81], it does not sit
so well with the self-dual twistor action of [38] used here, see also [55, 82]
for related discussion.

Second, the gauge fixing conditions adopted here are quite strong and allow
for a direct relation between twistor space and $\mathscr{I}$ via the Bramson-
Tod formula. The drawback is that the expression of the residual gauge
parameters becomes cumbersome involving field dependence, which requires the
use of very refined tools, such as a modified Lie bracket. Another route
requiring less gauge fixing and using the Penrose transform to relate twistor
space data to the bulk space-time should also exist. We plan to return to this
alternative approach in future work.

Third, we have not been fully explicit about the recursion operator and
bihamiltonian structures implicitly used in this work, see chapter 6 of [51]
for an introduction and more complete references. These are common features of
completely integrable partial differential equations, such as the Korteweg de
Vries or nonlinear Schrödinger equations, in which there is an infinite
hierarchy of conserved quantities leading to the complete integrability in the
Liouville sense. Each conserved quantity generates its own flow or higher
symmetry, leading to a hierarchy of PDEs compatible with each other and the
original PDE, and these can be understood to be related to each other by a
recursion operator. This recursion operator is particularly simple on twistor
space where it is given in [51, 53] by multiplication of a Čech representative
by the spectral parameter or ratio of twistor coordinates
$\lambda_{1}/\lambda_{0}$. Its action on perturbations of the fields is then
simply obtainable from the Penrose transform integral formula using the Lax
pair. In this work, it is easily seen from our extension of the Bramson-Tod
formula that in adapting the construction to $\mathscr{I}$, we are instead
realizing the action of the recursion operator by multiplying Čech, or
Dolbeault representatives by a different twistor coordinate $q$. The action of
this recursion is then easily transferred to fields on $\mathscr{I}$ using the
Lax pair adapted to $\mathscr{I}$ as we have seen from our argument for the
various recursion relations such as (3.23). In the first instance, then, these
are the expected symmetries of the self-dual vacuum equations as follows from
their complete Liouville integrability as by now described by many authors,
see [50, 51, 53] and references therein.

Such symmetries are not expected to extend to the full Einstein vacuum
equations in the bulk which are not believed to be integrable. There is
therefore no direct evidence that these should be symmetries of the S-matrix
for gravity save that they act naturally at the boundary. However, the
integrability of the self-dual sector implies much more that symmetry of the
S-matrix, rather instead the vanishing of the scattering of all positive
helicity and the single negative and rest positive helicity amplitudes [55].
So, the lack of integrability does not preclude some natural action on the
amplitude, although it is expected to be more complicated than the Ward
identities arising from the extended BMS symmetries at low $s$.

Finally, in the current paper, we focus on the celestial symmetries of gravity
in asymptotically flat spacetimes. Of course, there are several possibilities
for extensions. One obvious direction would be to extend this analysis to
Yang-Mills theory and the $S$-algebra. Another would be to understand the
effect of a cosmological constant on the whole construction and see if we can
recover the recently-found deformations of $Lw_{1+\infty}$ in the presence of
a cosmological constant [83, 84]. We plan to address these questions in an
upcoming paper.

### Acknowledgements

We would like to thank Laurent Freidel and Marc Geiller for insightful
discussions on their work. We would also like to thank Tim Adamo, Giuseppe
Bogna, and Simon Heuveline for helpful discussions. R.R. is also grateful to
Glenn Barnich for related discussions during his Master’s thesis project in
2016. A.K. is supported by the STFC. R.R. is supported by the Titchmarsh
Research Fellowship at the Mathematical Institute and by the Walker Early
Career Fellowship in Mathematical Physics at Balliol College. A.Y.S. is
supported by the STFC grant ST/X000761/1. This work is supported by the Simons
Collaboration on Celestial Holography.

##  Appendix A Recursion relation for $f_{s}$

Here we derive general formulae for the $\xi_{s}$ which are defined in terms
of a field-independent part $\tilde{\xi}_{s}$ and a field-dependent part
$f_{s}$

$\xi_{s}=\tilde{\xi}_{s}+f_{s}[\tau_{s},\bar{\sigma}]\,.$ (A.1)

First, consider the deformed complex structure acting on the field-independent
part of $\xi_{s}$

$\displaystyle[\bar{\nabla}\tilde{\xi}_{s}]_{\mathrm{D}\bar{\lambda}}=\bar{\sigma}\frac{u^{s}}{s!}\bar{\eth}^{s}\tau_{s}-\frac{u^{s+1}}{(s+1)!}\bar{\eth}^{s+2}\tau_{s}+(s+1)q^{s}\tau_{s}+\bar{\sigma}\sum_{n=1}^{s-1}\frac{n+1}{(s-n)!}u^{s-n}q^{n}\bar{\eth}^{s-n}\tau_{s}\,.$
(A.2)

We wish to add $f_{s}$ to this such that it cancels all the $q$ dependence. To
this end, we make the Ansatz

$\displaystyle
f_{s}=\sum_{n=0}^{s}\alpha_{n}\left(\lambda,\bar{\lambda},\bar{\sigma},u\right)q^{n}$
(A.3)
$\displaystyle\qquad\qquad\implies[\bar{\nabla}f_{s}]_{\mathrm{D}\bar{\lambda}}=-\bar{\eth}\alpha_{0}+\alpha_{1}\bar{\sigma}+q^{s+1}\partial_{u}\alpha_{s}+q^{s}\left(\partial_{u}\alpha_{s-1}-\bar{\eth}\alpha_{s}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{n=1}^{s-1}q^{n}\left(\partial_{u}\alpha_{n-1}-\bar{\eth}\alpha_{n}+\alpha_{n+1}\left(n+1\right)\bar{\sigma}\right)\,.$
(A.4)

We will leave the dependence of the $\alpha_{n}$ implicit from here on. The
absence of $q^{s+1}$ and $q^{s}$ implies

$\displaystyle\partial_{u}\alpha_{s}=0,\qquad\partial_{u}\alpha_{s-1}-\bar{\eth}\alpha_{s}+\bar{\sigma}\left(s+1\right)\tau_{s}=0\implies\alpha_{s-1}=-\left(s+1\right)\tau_{s}\partial_{u}^{-1}\bar{\sigma}\,.$
(A.5)

We have set $\alpha_{s}=0$ in writing the last equation. The rest of the
$\alpha_{n}$ must satisfy the following recursion relation $(n=s-1,\dots 1)$

$\displaystyle\partial_{u}\alpha_{n-1}-\bar{\eth}\alpha_{n}+\left(n+1\right)\bar{\sigma}\alpha_{n+1}+\frac{\left(n+1\right)}{\left(s-n\right)!}u^{s-n}\bar{\sigma}\bar{\eth}^{s-n}\tau_{s}=0\,.$
(A.6)

We can immediately solve for

$\displaystyle\alpha_{s-2}$
$\displaystyle=\partial_{u}^{-1}\bar{\eth}\alpha_{s-1}-\frac{s}{(s-n)!}\partial_{u}^{-1}\left(u\bar{\sigma}\bar{\eth}\tau_{s}\right)$
(A.7)
$\displaystyle=-(s+1)\left(\partial_{u}^{-1}\bar{\eth}\right)\left(\tau_{s}\partial_{u}^{-1}\bar{\sigma}\right)-\frac{s}{(s-n)!}\partial_{u}^{-1}\left(u\bar{\sigma}\bar{\eth}\tau_{s}\right).$

These expressions are exact and these two are only _linear_ in $\bar{\sigma}$.
In general, we have the following expansion

$\displaystyle\alpha_{n}=\sum_{p=1}^{\lfloor\frac{s-n+1}{2}\rfloor}\beta_{n,p},\qquad\text{where
}\beta_{n,p}\text{ is of degree p in }\bar{\sigma},$ (A.8)

and $\lfloor n\rfloor$ is the floor function. The upper limit on the sum is
perhaps not obvious at this stage. However, we will derive this after solving
the recursion. We can now split (A.6) into different sets of recursion
relations, each of which is of a fixed degree in $\bar{\sigma}$. We display
the $p=1$ and $p>1$ cases separately for convenience.

$\displaystyle\partial_{u}\beta_{n-1,1}-\bar{\eth}\beta_{n,1}+\frac{\left(n+1\right)}{\left(s-n\right)!}u^{s-n}\bar{\sigma}\bar{\eth}^{s-n}\tau_{s}=0,\qquad$
$\displaystyle p=1,$ (A.9)
$\displaystyle\partial_{u}\beta_{n-1,p}-\bar{\eth}\beta_{n,p}+\bar{\sigma}(n+1)\beta_{n+1,p-1}=0,\qquad$
$\displaystyle p>1\,.$ (A.10)

The solution to the $p=1$ recursion can be found by inspection to be

$\displaystyle\beta_{n,1}$
$\displaystyle=-\sum_{r=0}^{s-n-1}\frac{(s-r+1)}{r!}\left(\partial_{u}^{-1}\bar{\eth}\right)^{s-n-1-r}\left(\bar{\eth}^{r}\tau_{s}\partial_{u}^{-1}\left(u^{r}\bar{\sigma}\right)\right),$
(A.11)
$\displaystyle=-\sum_{r=0}^{s-n-1}\frac{(s-r+1)}{r!}\mathcal{D}_{s-n,r-2}\left(\bar{\eth}^{r}\tau_{s}u^{r}\right).$

In the second equality above, for increased readability, we have introduced
the operator $\mathcal{D}_{n,r}$ which has the following action:

$\displaystyle\mathcal{D}_{n,r}f=\left[\left(\partial_{u}^{-1}\right)^{n-2-r}\bar{\eth}^{n-3-r}\right]\left(\bar{\sigma}f\right).$
(A.12)

Having obtained the $\beta_{n,1}$, we can use these as seeds in (A.10) and use
it to derive the following recursion relation in $p$:

$\displaystyle\beta_{n,p}=-\sum_{r=-2}^{s-n-3}(s-r-1)\mathcal{D}_{s-n,r}\beta_{s-r-1,p-1}\quad,\quad\beta_{n+1,0}:=\frac{u^{s-n}}{(s-n)!}\bar{\eth}^{s-n}\tau_{s}\,.$
(A.13)

The solution to this recursion can be compactly expressed using the operator
(A.12) as

$\displaystyle\beta_{n,p}$
$\displaystyle=(-1)^{p-1}\sum_{r_{1}=0}^{s-n-3}\sum_{r_{2}=0}^{r_{1}-2}\dots\sum_{r_{p-1}=0}^{r_{p-2}-2}\prod_{i=1}^{p-1}(s-r_{i}-1)\mathcal{D}_{s-n,r_{1}}\prod_{k=1}^{p-2}\mathcal{D}_{r_{k}+1,r_{k+1}}\left[\beta_{s-r_{p-1}-1,1}\right]$
$\displaystyle=(-1)^{p}\sum_{r_{1}=0}^{s-n-3}\sum_{r_{2}=0}^{r_{1}-2}\dots\sum_{r_{p}=0}^{s-n-1}\frac{1}{r_{p}!}\prod_{i=1}^{p}(s-r_{i}-1)\mathcal{D}_{s-n,r_{1}}\left[\prod_{k=1}^{p-2}\mathcal{D}_{r_{k}+1,r_{k+1}}\right]\mathcal{D}_{s-n,r_{p}-2}\left(\bar{\eth}^{r_{p}}\tau_{s}u^{r_{p}}\right)\,.$

It is now manifest that $\beta_{n,p}$ is of degree $p$ in $\bar{\sigma}$ since
it has $p$ insertions of the operator $\mathcal{D}_{n,r}$. We can also infer
the following limit on the value of $p$ for fixed $n,s$ from this formula.
Note that the ranges on the sums imply $0\leq r_{k}\leq s-n-(2k+1)$. In
particular, we have $0\leq r_{p-1}\leq s-n-2p+1$. For this to be non-empty, we
must have $p\leq\lfloor\frac{1}{2}\left(s-n+1\right)\rfloor$. Thus, for a
fixed spin, the gauge parameter must be

$\xi_{s}=\tilde{\xi}_{s}+\sum_{n=0}^{s}\sum_{p=1}^{\lfloor\frac{s-n+1}{2}\rfloor}\beta_{n,p}q^{n}\,.$
(A.14)

It can be checked that this formula is consistent with (4.14). With these
explicit forms of the gauge parameters in hand, one can check that they
satisfy the algebra (4.16). While we have not shown this analytically, we have
checked explicitly that the algebra is satisfied (including all the
corrections that are non-linear in $\bar{\sigma}$) for all pairs of spins up
to

$\left\\{\tilde{\xi}_{4},\tilde{\xi}_{2}\right\\}_{*}=\tilde{\xi}_{5}.$ (A.15)

##  Appendix B Symmetry algebra computation

Here we show that the field-independent part $\tilde{\xi}_{s}$ of the
generators $\xi_{s}$ closes under the bracket (2.10) when restricted to the
wedge sub-algebra. We begin with

$\begin{split}\tilde{\xi}_{s+s^{\prime}-1}&=\sum_{n=0}^{s+s^{\prime}}\frac{u^{s+s^{\prime}-n}q^{n}}{(s+s^{\prime}-n)!}\bar{\eth}^{s+s^{\prime}-n}((s^{\prime}+1)\tau_{s^{\prime}}\bar{\eth}\tau_{s}-(s+1)\tau_{s}\bar{\eth}\tau_{s^{\prime}})\\\
&=\sum_{n=0}^{s+s^{\prime}}(s^{\prime}+1)\frac{u^{n}q^{s+s^{\prime}-n}}{n!}\bar{\eth}^{n}(\tau_{s^{\prime}}\bar{\eth}\tau_{s})-(s\leftrightarrow
s^{\prime})\\\
&=\sum_{n=0}^{s+s^{\prime}}\sum_{m=0}^{n}(s^{\prime}+1)\frac{u^{n}q^{s+s^{\prime}-n}}{m!(n-m)!}\bar{\eth}^{m+1}\tau_{s}\bar{\eth}^{n-m}\tau_{s^{\prime}}\,,\end{split}$
(B.1)

now use the double sum identity

$\sum_{k=l}^{m}\sum_{j=l}^{k}a_{k,j}=\sum_{j=l}^{m}\sum_{k=j}^{m}a_{k,j}\,.$
(B.2)

We then obtain

$\begin{split}\tilde{\xi}_{s+s^{\prime}-1}&=\sum_{m=0}^{s+s^{\prime}}\sum_{n=m}^{s+s^{\prime}}(s^{\prime}+1)\frac{u^{n}q^{s+s^{\prime}-n}}{m!(n-m)!}\bar{\eth}^{m+1}\tau_{s}\bar{\eth}^{n-m}\tau_{s^{\prime}}-(s\leftrightarrow
s^{\prime})\\\
&=\sum_{m=0}^{s}\sum_{n=0}^{s+s^{\prime}-m}(s^{\prime}+1)\frac{u^{n+m}q^{s+s^{\prime}-n-m}}{m!n!}\bar{\eth}^{m+1}\tau_{s}\bar{\eth}^{n}\tau_{s^{\prime}}-(s\leftrightarrow
s^{\prime})\,,\end{split}$ (B.3)

where we have used the wedge condition $\bar{\eth}^{s+2}\tau_{s}=0$ and re-
indexed. We now impose the wedge condition on the $\tau_{s^{\prime}}$ and find

$\begin{split}\tilde{\xi}_{s+s^{\prime}-1}|_{\wedge}&=\sum_{m=0}^{s}\sum_{n=0}^{s^{\prime}}\frac{(s^{\prime}+1)u^{n+m}q^{s+s^{\prime}-n-m}}{m!n!}\bar{\eth}^{m+1}\tau_{s}\bar{\eth}^{n}\tau_{s^{\prime}}\\\
&\quad\quad\quad+\sum_{m=1}^{s}\frac{u^{s^{\prime}+m}q^{s-m}}{(m-1)!s^{\prime}!}\bar{\eth}^{m}\tau_{s}\bar{\eth}^{s^{\prime}+1}\tau_{s^{\prime}}-(s\leftrightarrow
s^{\prime})\,.\end{split}$ (B.4)

We now turn to the bracket between two $\tilde{\xi}_{s}$

$\begin{split}\\{\tilde{\xi}_{s},\tilde{\xi}_{s^{\prime}}\\}&=\partial_{u}\tilde{\xi}_{s}\partial_{q}\tilde{\xi}_{s^{\prime}}-\partial_{u}\tilde{\xi}_{s^{\prime}}\partial_{q}\tilde{\xi}_{s}\\\
&=\sum_{n=0}^{s}\sum_{m=0}^{s^{\prime}}\frac{(m+1)u^{s+s^{\prime}-n-m}q^{n+m}}{(s-n)!(s^{\prime}-m)!}\bar{\eth}^{s+1-n}\tau_{s}\bar{\eth}^{s^{\prime}-m}\tau_{s^{\prime}}-(s\leftrightarrow
s^{\prime})\\\
&=\sum_{n=0}^{s}\sum_{m=0}^{s^{\prime}}(s^{\prime}+1-m)\frac{u^{n+m}q^{s+s^{\prime}-n-m}}{n!m!}\bar{\eth}^{n+1}\tau_{s}\bar{\eth}^{m}\tau_{s^{\prime}}-(s\leftrightarrow
s^{\prime})\\\
&=\sum_{m=0}^{s}\sum_{n=0}^{s^{\prime}}\frac{(s^{\prime}+1)u^{n+m}q^{s+s^{\prime}-n-m}}{m!n!}\bar{\eth}^{m+1}\tau_{s}\bar{\eth}^{n}\tau_{s^{\prime}}-(s\leftrightarrow
s^{\prime})\\\
&\quad\quad\quad-\sum_{n=1}^{s+1}\sum_{m=1}^{s^{\prime}}\frac{u^{n+m-1}q^{s+s^{\prime}+1-n-m}}{(n-1)!(m-1)!}\bar{\eth}^{n}\tau_{s}\bar{\eth}^{m}\tau_{s^{\prime}}+(s\leftrightarrow
s^{\prime})\\\
&=\sum_{m=0}^{s}\sum_{n=0}^{s^{\prime}}\frac{(s^{\prime}+1)u^{n+m}q^{s+s^{\prime}-n-m}}{m!n!}\bar{\eth}^{m+1}\tau_{s}\bar{\eth}^{n}\tau_{s^{\prime}}-(s\leftrightarrow
s^{\prime})\\\
&\quad\quad\quad+\sum_{m=1}^{s}\frac{u^{s^{\prime}+m}q^{s-m}}{(m-1)!s^{\prime}!}\bar{\eth}^{m}\tau_{s}\bar{\eth}^{s^{\prime}+1}\tau_{s^{\prime}}-(s\leftrightarrow
s^{\prime})\,,\end{split}$ (B.5)

Therefore we have that in the wedge sub-algebra

$\\{\tilde{\xi}_{s},\tilde{\xi}_{s^{\prime}}\\}=\tilde{\xi}_{s+s^{\prime}-1}((s^{\prime}+1)\tau_{s^{\prime}}\bar{\eth}\tau_{s}-(s+1)\tau_{s}\bar{\eth}\tau_{s^{\prime}})|_{\wedge}\,.$
(B.6)

Outside the wedge, the RHS is given by

$\begin{split}\tilde{\xi}_{s+s^{\prime}-1}-\tilde{\xi}_{s+s^{\prime}-1}|_{\wedge}&=\sum_{m=0}^{s-2}\sum_{n=s^{\prime}+2}^{s^{\prime}+s-m}(s^{\prime}+1)\frac{u^{n+m}q^{s+s^{\prime}-n-m}}{m!n!}\bar{\eth}^{m+1}\tau_{s}\bar{\eth}^{n}\tau_{s^{\prime}}\\\
&+\sum_{m=s+1}^{s+s^{\prime}}\sum_{n=0}^{s+s^{\prime}-m}(s^{\prime}+1)\frac{u^{n+m}q^{s+s^{\prime}-n-m}}{m!n!}\bar{\eth}^{m+1}\tau_{s}\bar{\eth}^{n}\tau_{s^{\prime}}-(s\leftrightarrow
s^{\prime})\\\
&=\sum_{m=0}^{s^{\prime}-1}\sum_{n=m}^{s^{\prime}-1}(s^{\prime}+1)\frac{u^{s+1+n}q^{s^{\prime}-1-n}}{(n-m)!(s+1+m)!}\bar{\eth}^{s+2+m}\tau_{s}\bar{\eth}^{n-m}\tau_{s^{\prime}}\\\
&+\sum_{m=0}^{s-2}\sum_{n=m}^{s-2}(s^{\prime}+1)\frac{u^{s^{\prime}+2+n}q^{s-2-n}}{m!(s^{\prime}+2+n-m)!}\bar{\eth}^{m+1}\tau_{s}\bar{\eth}^{s^{\prime}+2+n-m}\tau_{s^{\prime}}-(s\leftrightarrow
s^{\prime})\\\
&=\sum_{m=0}^{s^{\prime}-1}\sum_{n=m}^{s^{\prime}-1}(s^{\prime}+1)\frac{u^{s+1+n}q^{s^{\prime}-1-n}}{(n-m)!(s+1+m)!}\bar{\eth}^{s+2+m}\tau_{s}\bar{\eth}^{n-m}\tau_{s^{\prime}}\\\
&-\sum_{m=0}^{s^{\prime}-2}\sum_{n=m}^{s^{\prime}-2}(s+1)\frac{u^{s+2+n}q^{s^{\prime}-2-n}}{m!(s+2+n-m)!}\bar{\eth}^{m+1}\tau_{s^{\prime}}\bar{\eth}^{s+2+n-m}\tau_{s}-(s\leftrightarrow
s^{\prime})\\\
&=\sum_{m=0}^{s^{\prime}-1}\sum_{n=m}^{s^{\prime}-1}\frac{(s^{\prime}+1)(s+2+n-m)-(s+1)m}{m!(s+2+n-m)!}u^{s+1+n}q^{s^{\prime}-1-n}\bar{\eth}^{m}\tau_{s^{\prime}}\bar{\eth}^{s+2+n-m}\tau_{s}\\\
&\hskip 60.0pt-(s\leftrightarrow s^{\prime})\,.\end{split}$ (B.7)

The field-independent part of the variation of the generators is given by

$\begin{split}\delta_{s}\xi_{s^{\prime}}-\delta_{s^{\prime}}\xi_{s}&=\sum_{m=0}^{s^{\prime}-1}\sum_{n=m}^{s^{\prime}-1}q^{s^{\prime}-1-n}u^{s+1+n}\frac{(s^{\prime}+1-m)(m+s)!}{s!m!(s+1+n)!}\bar{\eth}^{n-m}(\bar{\eth}^{m}\tau_{s^{\prime}}\bar{\eth}^{s+2}\tau_{s})-(s\leftrightarrow
s^{\prime})\\\
&=\sum_{m=0}^{s^{\prime}-1}\sum_{n=m}^{s^{\prime}-1}\sum_{k=m}^{n}(s^{\prime}+1-m)\frac{(m+s)!(n-m)!q^{s^{\prime}-1-n}u^{s+1+n}}{s!m!(n-k)!(k-m)!(s+1+n)!}\bar{\eth}^{k}\tau_{s^{\prime}}\bar{\eth}^{s+2+n-k}\tau_{s}\\\
&\hskip 60.0pt-(s\leftrightarrow s^{\prime})\\\
&=\sum_{m=0}^{s^{\prime}-1}\sum_{n=m}^{s^{\prime}-1}\frac{(s^{\prime}+1)(s+2+n-m)-(s+1)m}{m!(s+2+n-m)!}q^{s^{\prime}-1-n}u^{s+1+n}\bar{\eth}^{m}\tau_{s^{\prime}}\bar{\eth}^{s+2+n-m}\tau_{s}\\\
&\hskip 60.0pt-(s\leftrightarrow s^{\prime})\,.\end{split}$ (B.8)

A useful sum was used

$\begin{split}\sum_{k=0}^{m}(s^{\prime}+1-k)&\frac{(s+m)!(n-m)!}{m!(k-m)!}\\\
&=((s^{\prime}+1)(s+2+n-m)-m(s+1))\frac{s!(s+1+n)!(n-m)!}{m!(s+2+n-m)!}\,,\end{split}$
(B.9)

which relies on the use of the Chu-Vandermonde identity

${}_{2}F_{1}(-n,b,c;1)=\frac{(c-b)_{n}}{(c)_{n}}\quad,\quad n=0,1,2,\dots\,,$
(B.10)

where $(a)_{n}=\Gamma(a+n)/\Gamma(a)$ is the Pochhammer’s symbol. Therefore,
the algebra is shown to be satisfied to leading order in the fields

$\\{\xi_{s},\xi_{s^{\prime}}\\}_{*}=\tilde{\xi}_{s+s^{\prime}-1}+\mathcal{O}(\bar{\sigma})\,.$
(B.11)

##  Appendix C Čech version

The homogenous formalism has the advantage that the $u$-coordinate, under
Lorentz transformations, does not transform. As consequence $u$ is a weighted-
scalar valued in $\mathcal{O}(1,1)$. One can recover the transformation of
$u_{B}$

$u^{\prime}_{B}=\frac{u}{\langle\lambda^{\prime}\hat{\lambda}^{\prime}\rangle}=\frac{1+z\bar{z}}{|az+b|^{2}+|cz+d|^{2}}u_{B}\,,$
(C.1)

where the homogeneous coordinates on the sphere have global transformations

$\lambda^{\prime}_{\alpha}=L_{\alpha}^{\,\,\,\,\beta}\lambda_{\beta}\quad,\quad
L_{\alpha}^{\,\,\,\,\beta}=\begin{pmatrix}d&c\\\
b&a\end{pmatrix}\in\text{SL}(2,\mathbb{C})\,.$ (C.2)

Under these transformation laws, the Edth operator is not left invariant [85]:

$\eth^{\prime}=\eth+\frac{[\bar{\lambda}|\bar{L}TL|\hat{\lambda}\rangle}{\langle\hat{\lambda}^{\prime}\lambda^{\prime}\rangle\langle\hat{\lambda}\lambda\rangle}\lambda_{\alpha}\frac{\partial}{\partial\lambda_{\alpha}}\,,$
(C.3)

where

$[\bar{\lambda}|\bar{L}TL|\hat{\lambda}\rangle=\bar{\lambda}_{\dot{\beta}}\bar{L}_{\dot{\alpha}}^{\,\,\,\,\dot{\beta}}T^{\alpha\dot{\alpha}}L_{\alpha}^{\,\,\,\,\beta}\hat{\lambda}_{\beta}\,.$
(C.4)

When $L\in SU(2)$ corresponds to rotations on the sphere, this term in the
transformation of $\eth$ vanishes. Because of this fact, as written in (3.16),
the Bianchi identities are not manifestly Lorentz invariant. However, as a
curiosity, the recursion relations can be uplifted to twistor space [47] as

$\frac{\partial\psi_{n}}{\partial\mu^{\dot{\alpha}}}=\frac{\partial\psi_{n+1}}{\partial\bar{\lambda}^{\dot{\alpha}}}+(3-n)\bar{\lambda}_{\dot{\alpha}}\bar{\sigma}\psi_{n+2}\,.$
(C.5)

Here
$\psi_{n}((\lambda_{\alpha},\mu^{\dot{\alpha}}),\bar{\lambda}_{\dot{\alpha}})$
has weight $(-n-1,n-5)$ in $(\lambda_{\alpha},\mu^{\dot{\alpha}})$ and
$\bar{\lambda}_{\dot{\alpha}}$ respectively. This formula expresses full
Lorentz invariance. Then contraction of (C.5) with
$\bar{\lambda}_{\dot{\alpha}}$ gives

$\frac{\partial\psi_{n}}{\partial q}=(n-4)\psi_{n+1}\,,$ (C.6)

which can be solved in the form

$\psi_{n}=\sum_{m=0}^{4-n}(-1)^{m}\binom{4-n}{m}\Psi^{0}_{n+m}q^{m}\,.$ (C.7)

The Bramson-Tod integral formula in its original form solved (C.5) via a Čech
cohomology integral transform

$\psi_{n}=\oint\tilde{q}^{4-n}g(\mu^{\dot{\alpha}}+\tilde{q}\bar{\lambda}^{\dot{\alpha}},\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}})d\tilde{q}\,,$
(C.8)

where $g$ is a $\mathcal{O}(-6)$ valued holomorphic function with respect to
the deformed complex structure

$\frac{\partial
g}{\partial\bar{\lambda}^{\dot{\alpha}}}-\bar{\lambda}_{\dot{\alpha}}\bar{\sigma}\bar{\lambda}^{\dot{\beta}}\frac{\partial
g}{\partial\mu^{\dot{\beta}}}=0\quad,\quad\frac{\partial
g}{\partial\bar{\mu}^{\alpha}}=0\,.$ (C.9)

To see that this integral formula solves the Bianchi identities we introduce
for convenience
$\omega^{\dot{\alpha}}=\mu^{\dot{\alpha}}+\tilde{q}\bar{\lambda}^{\dot{\alpha}}$
and apply the necessary derivatives

$\begin{split}\frac{\partial\psi_{n}}{\partial\mu^{\dot{\alpha}}}&=\oint\tilde{q}^{4-n}\frac{\partial
g}{\partial\omega^{\dot{\alpha}}}d\tilde{q},\\\
\frac{\partial\psi_{n+1}}{\partial\bar{\lambda}^{\dot{\alpha}}}&=\oint\tilde{q}^{3-n}\left.\frac{\partial
g}{\partial\bar{\lambda}^{\dot{\alpha}}}\right|_{\mu}d\tilde{q}\\\
&=\oint\tilde{q}^{3-n}\left(\left.\frac{\partial
g}{\partial\bar{\lambda}^{\dot{\alpha}}}\right|_{\omega}+\tilde{q}\frac{\partial
g}{\partial\omega^{\dot{\alpha}}}\right)d\tilde{q}.\\\ \end{split}$ (C.10)

Therefore, combining these results and using the holomorphicity of $f$ in
(C.9)

$\begin{split}\frac{\partial\psi_{n}}{\partial\mu^{\dot{\alpha}}}-\frac{\partial\psi_{n+1}}{\partial\bar{\lambda}^{\dot{\alpha}}}&=-\sigma^{0}\bar{\lambda}_{\dot{\alpha}}\oint\tilde{q}^{3-n}\bar{\lambda}^{\dot{\beta}}\frac{\partial
g}{\partial\omega^{\dot{\beta}}}d\tilde{q}\\\
&=-\sigma^{0}\bar{\lambda}_{\dot{\alpha}}\oint\tilde{q}^{3-n}\frac{\partial
g}{\partial\tilde{q}}d\tilde{q}\\\
&=(3-n)\sigma^{0}\bar{\lambda}_{\dot{\alpha}}\psi_{n+2},\end{split}$ (C.11)

where in the last line integration parts was used. We can also understand
(C.7) by inserting the explicit expansion of $\mu^{\dot{\alpha}}$ into the
integral formula

$\psi_{n}=\oint(\tilde{q}-q)^{4-n}g(\mu_{0}^{\dot{\alpha}}+\tilde{q}\bar{\lambda}^{\dot{\alpha}},\lambda_{\alpha},\lambda_{\dot{\alpha}})d\tilde{q},$
(C.12)

where we introduce
$\mu_{0}^{\dot{\alpha}}=\frac{u\hat{\bar{\lambda}}^{\dot{\alpha}}}{\langle\hat{\lambda}\lambda\rangle}$
and we can define the quantities living on $\mathscr{I}$

$\Psi_{n}^{0}=\oint
q^{4-n}g(\mu_{0}^{\dot{\alpha}}+q\bar{\lambda}^{\dot{\alpha}},\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}})dq,$
(C.13)

which solve (3.16).

##  Appendix D BMS Transformations

Here we demonstrate that we can recover the BMS transformation of the
components of the Weyl spinor using only the integral transform and the
weights naturally present in the construction. Here we take the perspective of
doing infinitesimal transformations of the coordinates on twistor space and
therefore under the projection to the coordinates on $\mathscr{I}$. The
supertranslations are

$\delta_{\text{ST}}\mu^{\dot{\alpha}}=\frac{\partial
T}{\partial\bar{\lambda}_{\dot{\alpha}}}\quad,\quad\delta_{\text{ST}}\lambda^{\alpha}=0\,,$
(D.1)

where $T$ is valued in $\mathcal{O}(1,1)$. For superrotations we take

$\delta_{\text{SR}}\mu^{\dot{\alpha}}=\frac{1}{2}\frac{\partial^{2}\bar{Y}}{\partial\bar{\lambda}_{\dot{\alpha}}\partial\bar{\lambda}^{\dot{\beta}}}\mu^{\dot{\beta}}\quad,\quad\delta_{\text{SR}}\lambda^{\alpha}=\frac{1}{2}\frac{\partial^{2}Y}{\partial\lambda_{\alpha}\partial\lambda^{\beta}}\lambda^{\beta}\,,$
(D.2)

where $Y,\bar{Y}$ are valued in $\mathcal{O}(2,0)$ and $\mathcal{O}(0,2)$
respectively. All the variations can also be extended to
$\bar{\lambda}_{\dot{\alpha}}$ and $\bar{\mu}^{\alpha}$ by complex
conjugation. Furthermore, the BMS generators $\\{T,Y,\bar{Y}\\}$ are only
functions of $\lambda_{\alpha}$ and $\bar{\lambda}_{\dot{\alpha}}$. All
variations are taken on-shell and therefore $\partial_{\bar{u}}g_{1}=0$. We
begin by using the Bramson-Tod integral formula and inserting the
infinitesimal coordinate transformations. Starting with supertranslations

$\begin{split}\delta_{\text{ST}}\mathcal{Q}_{s}&=\int
q^{s+2}\left(\frac{\partial
g_{1}}{\partial\mu^{\dot{\alpha}}}\delta_{\text{ST}}\mu^{\dot{\alpha}}+\frac{\partial
g_{1}}{\partial\bar{\mu}^{\alpha}}\delta_{\text{ST}}\bar{\mu}^{\alpha}\right)\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\int q^{s+2}\left(\frac{\partial
g_{1}}{\partial\mu^{\dot{\alpha}}}\frac{\partial
T}{\partial\bar{\lambda}_{\dot{\alpha}}}+\partial_{\bar{q}}g_{1}\eth\bar{T}\right)\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\int
q^{s+2}\left(T\partial_{u}g_{1}-\partial_{q}g_{1}\bar{\eth}T\right)\mathrm{d}q\mathrm{d}\bar{q}\\\
&=T\partial_{u}\mathcal{Q}_{s}+(4-n)\mathcal{Q}_{s-1}\bar{\eth}T\,,\end{split}$
(D.3)

where integration by parts and the homogeneity of $T$ has been used. Now
moving on to superrotations

$\delta_{\text{SR}}\mathcal{Q}_{s}=\int q^{s+2}\left(\frac{\partial
g_{1}}{\partial\mu^{\dot{\alpha}}}\delta_{\text{SR}}\mu^{\dot{\alpha}}+\frac{\partial
g_{1}}{\partial\bar{\lambda}^{\dot{\alpha}}}\delta_{\text{SR}}\bar{\lambda}^{\dot{\alpha}}+\frac{\partial
g_{1}}{\partial\bar{\mu}^{\alpha}}\delta_{\text{SR}}\bar{\mu}^{\alpha}+\frac{\partial
g_{1}}{\partial\lambda^{\alpha}}\delta_{\text{SR}}\lambda^{\alpha}\right)\mathrm{d}q\mathrm{d}\bar{q}\,.$
(D.4)

Taking this term by term starting with the dotted spin bundle terms

$\begin{split}\int q^{s+2}\frac{\partial
g_{1}}{\partial\mu^{\dot{\alpha}}}\delta_{\text{SR}}\mu^{\dot{\alpha}}\mathrm{d}q\mathrm{d}\bar{q}&=\frac{1}{2}\int
q^{s+2}\epsilon^{\dot{\alpha}\dot{\beta}}\frac{\partial
g_{1}}{\partial\mu^{\dot{\beta}}}\left(u\bar{\eth}\left(\frac{\partial\bar{Y}}{\partial\bar{\lambda}^{\dot{\alpha}}}\right)+q\frac{\partial\bar{Y}}{\partial\bar{\lambda}^{\dot{\alpha}}}\right)\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\frac{1}{2}\int
q^{s+2}\left[\partial_{u}g_{1}\left(u\bar{\eth}\bar{Y}+2q\bar{Y}\right)-\partial_{q}g_{1}\left(u\bar{\eth}^{2}\bar{Y}+q\bar{\eth}\bar{Y}\right)\right]\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\frac{u}{2}\bar{\eth}\bar{Y}\partial_{u}\mathcal{Q}_{s}+\bar{Y}\partial_{u}\mathcal{Q}_{s+1}+\frac{s+3}{2}\bar{\eth}\bar{Y}\mathcal{Q}_{s}+\frac{u}{2}(s+2)\mathcal{Q}_{s-1}\bar{\eth}^{2}\bar{Y}\,,\end{split}$
(D.5)

where we only use the homogeneity of $\bar{Y}$ and integration by parts. The
next term gives

$\begin{split}\int q^{s+2}\frac{\partial
g_{1}}{\partial\bar{\lambda}^{\dot{\alpha}}}\delta_{\text{SR}}\bar{\lambda}^{\dot{\alpha}}\mathrm{d}q\mathrm{d}\bar{q}&=\frac{1}{2}\int
q^{s+2}\epsilon^{\dot{\alpha}\dot{\beta}}\frac{\partial
g_{1}}{\partial\bar{\lambda}^{\dot{\beta}}}\frac{\partial\bar{Y}}{\partial\bar{\lambda}^{\dot{\alpha}}}\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\frac{1}{2}\int
q^{s+2}\left[2\bar{Y}\bar{\partial}_{0}g_{1}-\bar{\lambda}^{\dot{\alpha}}\frac{\partial
g_{1}}{\partial\bar{\lambda}^{\dot{\alpha}}}\bar{\eth}\bar{Y}\right]\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\frac{1}{2}\int
q^{s+2}\left[2\bar{Y}\bar{\partial}_{0}g_{1}+(g_{1}+\bar{u}\partial_{\bar{u}}g_{1}+\bar{q}\partial_{\bar{q}}g_{1})\bar{\eth}\bar{Y}\right]\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\bar{Y}\int
q^{s+2}\left[\bar{\eth}-q\partial_{u}+\frac{\bar{u}}{\langle\hat{\lambda}\lambda\rangle^{2}}\partial_{\bar{q}}\right]g_{1}\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\bar{Y}\bar{\eth}\mathcal{Q}_{s}-\bar{Y}\partial_{u}\mathcal{Q}_{s+1}\,,\end{split}$
(D.6)

where we have used the fact that $g_{1}$ is a section of $\mathcal{O}(-5,-1)$
and on-shell $\partial_{\bar{u}}g_{1}=0$. Now looking at the un-dotted
transformations

$\begin{split}\int q^{s+2}\frac{\partial
g_{1}}{\partial\bar{\mu}^{\alpha}}\delta_{\text{SR}}\bar{\mu}^{\alpha}\mathrm{d}q\mathrm{d}\bar{q}&=-\frac{1}{2}\int
q^{s+2}\partial_{\bar{q}}g_{1}\left(\bar{u}\eth^{2}Y+\bar{q}\eth
Y\right)\mathrm{d}q\mathrm{d}\bar{q}=\frac{1}{2}\eth
Y\mathcal{Q}_{s}\,,\end{split}$ (D.7)

where we have used the homogeneity of $Y$ and integration by parts. The final
term gives us

$\begin{split}\int q^{s+2}\frac{\partial
g_{1}}{\partial\lambda^{\alpha}}\delta_{\text{SR}}\lambda^{\alpha}\mathrm{d}q\mathrm{d}\bar{q}&=\frac{1}{2}\int
q^{s+2}\left[2Y\partial_{0}g_{1}-\lambda^{\alpha}\frac{\partial
g_{1}}{\partial\lambda^{\alpha}}\eth Y\right]\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\frac{1}{2}\int q^{s+2}\left[\eth Y(u\partial_{u}g_{1}+(2-s)f)+2Y\left(\eth
g_{1}+\frac{u}{\langle\hat{\lambda}\lambda\rangle^{2}}\partial_{q}f\right)\right]\mathrm{d}q\mathrm{d}\bar{q}\\\
&=\frac{u}{2}\eth Y\partial_{u}\mathcal{Q}_{s}+\frac{2-s}{2}\eth
Y\mathcal{Q}_{s}+Y\eth\mathcal{Q}_{s}-(s+2)\frac{u}{\langle\hat{\lambda}\lambda\rangle^{2}}Y\mathcal{Q}_{s-1}\\\
&=\frac{u}{2}\eth Y\partial_{u}\mathcal{Q}_{s}+\frac{2-s}{2}\eth
Y\mathcal{Q}_{s}+Y\eth\mathcal{Q}_{s}+\frac{u}{2}(s+2)\mathcal{Q}_{s-1}[\bar{\eth},\eth]Y\,,\end{split}$
(D.8)

where we have used the homogeneity of $g_{1}$ and $\partial_{\bar{u}}g_{1}=0$.
A useful commutator between the $\eth$ and $\bar{\eth}$ operators was used

$[\eth,\bar{\eth}]=\frac{1}{\langle\hat{\lambda}\lambda\rangle^{2}}\left[\lambda_{\alpha}\frac{\partial}{\partial\lambda_{\alpha}}-\bar{\lambda}_{\dot{\alpha}}\frac{\partial}{\partial\bar{\lambda}_{\dot{\alpha}}}\right]\,.$
(D.9)

Putting all of this together, along with the assumption that $\bar{\eth}Y=0$,
the full infinitesimal BMS transformations give

$\delta_{(T,Y,\bar{Y})}\mathcal{Q}_{s}=\left(f\partial_{u}+Y\eth+\bar{Y}\bar{\eth}+\frac{3-s}{2}\eth
Y+\frac{3+s}{2}\bar{\eth}\bar{Y}\right)\mathcal{Q}_{s}+(s+2)\mathcal{Q}_{s-1}\bar{\eth}f\,,$
(D.10)

where $f=T+\frac{u}{2}\left(\eth Y+\bar{\eth}\bar{Y}\right)$.

##  Appendix E Vectors and Forms on $\mathbb{CP}^{n}$

Due to the projective nature of twistor space, it is worth discussing the
tangent and cotangent space on $\mathbb{CP}^{n}$ which is defined as

$\mathbb{CP}^{n}=\\{(Z^{0},\cdots,Z^{n})\in\mathbb{C}^{n+1}|(Z^{0},\cdots,Z^{n})\sim(tZ^{0},\cdots,tZ^{n}):t\in\mathbb{C}^{*}\\},$
(E.1)

where the $Z_{A}$ are called homogeneous coordinates. We wish to use these
coordinates rather than choosing a patch $U_{i}$ where $Z_{i}\neq 0$ and the
affine coordinates are $z_{j}=Z_{j}/Z_{i}$ such that $i\neq j$ to keep our
formula agnostic to the choice of the patch. To construct the tangent space,
we consider the space of all derivations on functions

$f:\mathbb{CP}^{n}\rightarrow\mathbb{C},$ (E.2)

which in homogenous coordinates can be viewed as

$f:\mathbb{C}^{n+1}\rightarrow\mathbb{C}\quad,\quad f(tZ)=f(Z),$ (E.3)

a function with zero homogeneity. Therefore the homogeneity vector field
$\Upsilon=Z^{A}\partial_{A}$ which generates rescaling will not appear, put
another way

$T^{1,0}_{Z}\mathbb{CP}^{n}\otimes\mathcal{O}(m,n)\cong
T^{1,0}_{Z}\mathbb{C}^{n+1}/\langle\Upsilon\rangle,$ (E.4)

where $\langle\Upsilon\rangle\subset T^{1,0}_{Z}\mathbb{C}^{n+1}$ and the
twisting of the line bundle $\mathcal{O}(m,n)$ appears as our vector fields
may have a non zero homogeneity. Therefore to get sections of
$T^{1,0}_{Z}\mathbb{CP}^{n}$ we must multiply our vector fields by sections of
$\mathcal{O}(-m,-n)$. The cotangent bundle is realized by
$\omega\in\Omega^{1,0}(\mathbb{C}^{n+1})$ such that
$\Upsilon\lrcorner\,\,\omega=0$. More generally sections of
$\Omega^{p,q}(\mathbb{CP}^{n})\otimes\mathcal{O}(m,n)$ are given by the
subspace of $\Omega^{p,q}(\mathbb{C}^{n+1})$ such that (2.5) are satisfied.
The next task is to define the exterior derivative on $\mathbb{CP}^{n}$ from
the one on $\mathbb{C}^{n+1}$, as these two objects do not coincide due to the
presence of the line bundle. By definition of a connection on a vector bundle,
we expand the exterior derivative acting on forms $\omega$ valued in
$\mathcal{O}(m,n)$ on $\mathbb{CP}^{n}$ as

$d_{\mathbb{CP}^{n}}\omega=\mathrm{d}\omega-m\theta\wedge\omega-n\bar{\theta}\wedge\omega,$
(E.5)

where $d=\mathrm{d}Z^{A}\frac{\partial}{\partial
Z^{A}}+\mathrm{d}\bar{Z}^{\bar{A}}\frac{\partial}{\partial\bar{Z}^{\bar{A}}}$.
By requiring that $d_{\mathbb{CP}^{n}}\omega$ is a form under the conditions
(2.5), we find that

$\Upsilon\lrcorner\,\,\theta=1\quad,\quad\bar{\Upsilon}\lrcorner\,\,\bar{\theta}=1,$
(E.6)

and all other contractions are zero. This means that the connection $1$-forms
must lie in the image of the homogeneity vector fields, and therefore can be
built in the process of building the cotangent space. As an example, let us
take the canonical bundle of $\mathbb{CP}^{n}$ which is defined as
$K_{\mathbb{CP}^{n}}=\bigwedge^{n}T^{*1,0}\mathbb{CP}^{n}$. Taking an n-form
on $\mathbb{C}^{n+1}$

$\omega=\sum_{A=0}^{n}f_{A}(Z)\mathrm{d}Z^{0}\wedge\cdots\wedge\hat{\mathrm{d}Z^{A}}\wedge\cdots\wedge\mathrm{d}Z^{n},$
(E.7)

where $\hat{dZ^{A}}$ corresponds to the omission. Now taking the contraction
with the homogeneity vector field

$\Upsilon\lrcorner\,\,\omega=\sum_{A,B=0}^{n}(-1)^{B}f_{A}(Z)Z_{B}\mathrm{d}Z^{0}\wedge\cdots\wedge\hat{\mathrm{d}Z^{A}}\wedge\cdots\wedge\hat{\mathrm{d}Z^{B}}\wedge\cdots\wedge\mathrm{d}Z^{n},$
(E.8)

which vanishes if $f_{A}(Z)=(-1)^{A}Z_{A}g(Z)$ by antisymmetry. Therefore we
can compactly write the forms in $K_{\mathbb{CP}^{n}}$ as

$\omega=g(Z)\epsilon_{A_{0}\cdots
A_{n}}Z^{A_{0}}\mathrm{d}Z^{A_{1}}\wedge\cdots\wedge\mathrm{d}Z^{A_{n}},$
(E.9)

with $g(Z)$ being a section of $\mathcal{O}(-n-1)$ and therefore
$K_{\mathbb{CP}^{n}}\cong\mathcal{O}(-n-1)$. Another example is provided by
the simplest case of $\mathbb{CP}^{1}$, which sees much use in twistor theory.
The unique sections of $T^{1,0}\mathbb{CP}^{1}\otimes\mathcal{O}(-2)$ and
$T^{*1,0}\mathbb{CP}^{1}\otimes\mathcal{O}(2)$ are given by

$[\eth]=\left[\frac{a^{\alpha}}{\langle
a\lambda\rangle}\frac{\partial}{\partial\lambda^{\alpha}}\right]\quad\quad
e^{0}=\mathrm{D}\lambda=\langle\lambda\mathrm{d}\lambda\rangle,$ (E.10)

where $[a]\in\mathbb{CP}^{1}$ and $[\lambda]\neq[a]$ is a reference spinor.
The definition of $[\eth]$ is independent of the spinor $a$ due to the
equivalence class and can be defined globally. The exterior derivative acting
on forms valued in $\mathcal{O}(m,n)$ is given by

$d_{\mathbb{CP}^{1}}=d-m\frac{\langle a\mathrm{d}\lambda\rangle}{\langle
a\lambda\rangle}\wedge-n\frac{[\bar{a}\mathrm{d}\bar{\lambda}]}{[\bar{a}\bar{\lambda}]}\wedge.$
(E.11)

On twistor space, we choose a decomposition that is adapted to $\mathscr{I}$,
in the notation introduced in (3.27)

$\Omega^{1}(\mathbb{PT})=\text{Span}\left\\{\mathrm{D}\lambda,\partial\mathfrak{q},\partial
u_{B},\bar{e}^{0},\bar{\partial}\bar{\mathfrak{q}},\bar{\partial}\bar{u}_{B}\right\\}\quad,\quad\theta=\frac{\langle\hat{\lambda}\mathrm{d}\lambda\rangle}{\langle\hat{\lambda}\lambda\rangle}\quad,\quad\bar{\theta}=\frac{\langle\mathrm{d}\hat{\lambda}\lambda\rangle}{\langle\hat{\lambda}\lambda\rangle}\,.$
(E.12)

Note that the connections $\theta,\bar{\theta}$ pull back to the same
expression on $\mathscr{I}$ and is a particularly natural choice. The
holomorphic top form on twistor space can be decomposed into the coordinate of
the fibration over $\mathscr{I}$

$\begin{split}\mathrm{D}^{3}Z&=\frac{1}{4!}\epsilon_{ABCD}Z^{A}\mathrm{d}Z^{B}\wedge\mathrm{d}Z^{C}\wedge\mathrm{d}Z^{D}\\\
&=\frac{1}{2}\mathrm{D}\lambda\wedge\mathrm{d}\mu^{\dot{\alpha}}\wedge\mathrm{d}\mu_{\dot{\alpha}}+\frac{1}{2}[\mu\mathrm{d}\mu]\wedge\mathrm{d}\lambda^{\alpha}\wedge\mathrm{d}\lambda_{\alpha}\\\
&=\mathrm{d}_{\mathbb{CP}^{1}}u\wedge\mathrm{d}_{\mathbb{CP}^{1}}q\wedge\mathrm{D}\lambda+q\mathrm{d}_{\mathbb{CP}^{1}}q\wedge\mathrm{D}\bar{\lambda}\wedge\mathrm{D}\lambda\\\
&=\langle\lambda\hat{\lambda}\rangle\mathrm{d}u_{B}\wedge\mathrm{d}_{\mathbb{CP}^{1}}q\wedge\mathrm{D}\lambda+q\mathrm{d}_{\mathbb{CP}^{1}}q\wedge\mathrm{D}\bar{\lambda}\wedge\mathrm{D}\lambda\,.\end{split}$
(E.13)

Once we restrict to a constant $u_{B}$ the first term vanishes. Note that
since the decomposition of $\mu^{\dot{\alpha}}$ does not preserve the complex
structure, the presence of a $\mathrm{D}\bar{\lambda}$ form is to be expected.
We also note that
$\mathrm{d}_{\mathbb{CP}^{1}}\bar{q}=\langle\lambda\hat{\lambda}\rangle\bar{\partial}\bar{\mathfrak{q}}$
and therefore the restriction of the form in the Bramson-Tod formula can be
equally written as

$g|_{X_{(u,\lambda,\bar{\lambda})}}=g_{1}\mathrm{d}_{\mathbb{CP}^{1}}\bar{q}=g_{\bar{\mathfrak{q}}}\bar{\partial}\bar{\mathfrak{q}}\,,$
(E.14)

in the notation of Section 3. We drop the $\mathbb{CP}^{1}$ subscript for
notational convenience in the rest of the text.

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