# Coadjoint representation of the BMS group on celestial Riemann surfaces

Glenn Barnich, Romain Ruzziconi

Coadjoint representation of the BMS group on celestial Riemann surfaces

Glenn Barnich

Physique Théorique et Mathématique

Université libre de Bruxelles and International Solvay Institutes

Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium.

E-mail<EMAIL_ADDRESS>
Romain Ruzziconi

Institute for Theoretical Physics, TU Wien

Wiedner Hauptstrasse 8, A-1040 Vienna, Austria

E-mail<EMAIL_ADDRESS>
Abstract. The coadjoint representation of the BMS group in four dimensions is
constructed in a formulation that covers both the sphere and the punctured
plane. The structure constants are worked out for different choices of bases.
The conserved current algebra of non-radiative asymptotically flat spacetimes
is explicitly interpreted in these terms.

###### Contents

  1. 1 Introduction
  2. 2 Poincaré and BMS algebras on the celestial sphere
  3. 3 Coadjoint representations of semi-direct product groups and algebras
  4. 4 General structure of the coadjoint representation of BMS4
    1. 4.1 Background structure
    2. 4.2 Coadjoint representation of the algebra
    3. 4.3 Coadjoint representation of the group
    4. 4.4 Weyl invariance
  5. 5 Realization on the sphere
    1. 5.1 Generalities
    2. 5.2 Adjoint and coadjoint representations of the group
    3. 5.3 Expansions
    4. 5.4 Structure constants
    5. 5.5 Coadjoint representation of the algebra
  6. 6 Realization on the punctured complex plane
    1. 6.1 Generalities
    2. 6.2 Adjoint and coadjoint representations of the group
    3. 6.3 Expansions
    4. 6.4 Structure constants
    5. 6.5 Coadjoint representation of the algebra
  7. 7 Comments on the cylinder
  8. 8 Identification in non-radiative asymptotically flat spacetimes
  9. 9 Discussion and perspectives
  10. Acknowledgments
  11. A Spin-weighted spherical harmonics

##  1 Introduction

The BMS group [1, 2, 3, 4, 5] is the symmetry group of four-dimensional
asymptotically flat spacetimes at null infinity. The precise form of this
group changes when one replaces the celestial sphere by a different Riemann
surface [6, 7]. Whereas unitary irreducible representations of the BMS group
are directly relevant for the quantum theory [8, 9, 10], the coadjoint
representation is intimately connected to classical solution space through the
momentum map. Unitary irreducible representations come later, after the
classification of coadjoint orbits, via geometric quantization.

In this paper, we provide the detailed construction of the coadjoint
representation of the BMS group and of the algebra on the celestial sphere and
the punctured plane. In the case of the sphere, we explicitly identify the
coadjoint representation in the gravitational data of non-radiative
spacetimes.

In order to set the stage, we start in section 2 by providing the (corrected)
commutation relations of the $\mathfrak{bms}_{4}$ algebra on the celestial
sphere. Since the structure of the BMS group is the same as that of the
Poincaré group in the sense that both are semi-direct product groups with an
abelian ideal, we recall in section 3 the structure of the coadjoint
representation of such groups and algebras [11].

In a next step in section 4, we provide a description of the coadjoint
representation of the BMS group in four dimensions in terms of suitably
weighted functions on a two-dimensional surface by focussing on local aspects.
For the presentation, after some generic considerations based on [12, 13], we
will use a Weyl covariant derivative instead of the standard “eth” operator
[5, 14, 15]. Note that our conventions for these derivatives differ somewhat
from those originally introduced in [16, 17, 18] for related reasons. The
description applies both to the “global” and “local” versions of the algebra
[19, 20, 21, 22], which are studied explicitly in sections 5 and 6,
respectively. In the former section, we use extensively results of sections
4.14 and 4.15 of [17]. Besides the standard choice of rotation and boost
generators as used originally in [3], we also provide explicit commutation
relations adapted to the
$\mathfrak{sl}(2,\mathbb{R})\times\mathfrak{sl}(2,\mathbb{R})$ decomposition
of the Lorentz algebra. In the latter section, our expansions follow the
standard conventions used in the context of two-dimensional conformal field
theories.

In section 7, we briefly comment the case of the cylinder. Finally, in the
case of the sphere and for non-radiative asymptotically flat spacetimes, we
explicitly construct the equivariant map from the free gravitational data at
${\mycal I}^{+}$ to the coadjoint representation in section 8.

The coadjoint representation of the BMS group in three dimensions [23] (see
also [24, 25]) has been investigated in [26]. In that case, the abelian factor
can be identified with the Lie algebra of the non abelian factor acted upon by
the adjoint representation, which simplifies the classification of coadjoint
orbits considerably. Furthermore, central extensions are the familiar ones
directly related to the Virasoro group and algebra. Neither of these
simplifications occur in four dimensions. As a consequence, we will not
discuss the classification of coadjoint orbits in this paper. Also, central
extensions that are relevant in the gravitational context are of a different
nature [27], and will not be considered here.

##  2 Poincaré and BMS algebras on the celestial sphere

The structure constants of the $\mathfrak{bms}_{4}$ algebra on the celestial
sphere have been worked out in [3] (see also [28] for corrections, and [29],
[30] for reviews). More details on the geometric interpretation can be found
in [5, 15] (see also [31, 17, 18, 32, 33]). In this section, we start by
providing the standard commutation relations of the $\mathfrak{bms}_{4}$
algebra in terms of rotation and boost generators.

Let $x^{a}$, $a=0,\dots 3$, be Cartesian coordinates on Minkowski spacetime
where $\eta_{ab}$ = $\text{diag}(1,-1,-1,-1)$ and its inverse $\eta^{ab}$ are
used to lower and raise indices. The starting point is the Poincaré algebra
with generators

$L^{ab}=L^{[ab]}=-(x^{a}\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x_{b}}$}\kern
1.0pt-x^{b}\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x_{a}}$}\kern 1.0pt),\quad P^{a}=\frac{\partial}{\partial x_{a}},$ (2.1)

satisfying

$\begin{split}[L^{ab},L^{cd}]&=-(\eta^{bc}L^{ad}-\eta^{ac}L^{bd}-\eta^{bd}L^{ac}+\eta^{ad}L^{bc}),\\\
[P^{a},L^{bc}]&=-(\eta^{ab}P^{c}-\eta^{ac}P^{b}).\end{split}$ (2.2)

When splitting into suitable combinations of rotation and boost generators and
of translation generators,

$\begin{split}L_{z}&=L^{12},\quad L^{\pm}=\pm iL^{23}+L^{13},\quad
K_{z}=L^{30},\quad K^{\pm}=\mp iL^{20}-L^{10},\\\ H&=P^{0},\quad
P_{z}=-\frac{1}{2}P^{3},\quad P^{\pm}=\frac{1}{2}(iP^{2}\pm
P^{1}),\end{split}$ (2.3)

the non-vanishing commutation relations of the Poincaré algebra become

$\begin{split}[L^{+},L^{-}]&=2iL_{z},\quad[L_{z},L^{\pm}]=\pm
iL^{\pm},\quad[K^{+},K^{-}]=-2iL_{z},\\\
[K_{z},K^{\pm}]&=L^{\pm},\quad[L^{+},K^{-}]=2K_{z},\quad[L^{-},K^{+}]=2K_{z},\\\
[L_{z},K^{\pm}]&=\pm iK^{\pm},\quad[L^{\pm},K_{z}]=-K^{\pm},\\\
[K_{z},H]&=2P_{z},\quad[K^{\pm},H]=\pm 2P^{\pm},\quad[L^{\pm},P_{z}]=\mp
P^{\pm},\quad[K_{z},P_{z}]=\frac{1}{2}H,\\\ [L_{z},P^{\pm}]&=\pm iP^{\pm},\
[L^{+},P^{-}]=-2P_{z}=-[L^{-},P^{+}],\\\
[K^{+},P^{-}]&=-H=-[K^{-},P^{+}].\end{split}$ (2.4)

In terms of spherical coordinates and a retarded time coordinate,

$r=\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}},\quad u=x^{0}-r,\quad
r\cos\theta=x^{3},\quad r\sin\theta e^{i\phi}=x^{1}+ix^{2},$ (2.5)

the Poincaré generators read

$\begin{split}L_{z}&=\partial_{\phi},\\\ L^{\pm}&=-e^{\pm
i\phi}\big{[}\partial_{\theta}\pm i\cot\theta\partial_{\phi}\big{]},\\\
K_{z}&=-(1+\frac{u}{r})\cos\theta(r\partial_{r})+\cos\theta(u\partial_{u})+(1+\frac{u}{r})\sin\theta\partial_{\theta},\\\
K^{\pm}&=e^{\pm
i\phi}\big{[}(1+\frac{u}{r})\sin\theta(r\partial_{r})-\sin\theta(u\partial_{u})+(1+\frac{u}{r})\cos\theta\partial_{\theta}\pm(1+\frac{u}{r})\frac{i}{\sin\theta}\partial_{\phi}\big{]},\\\
H&=\partial_{u},\\\
-2P_{z}&=\cos\theta(-\partial_{r}+\partial_{u})+\frac{1}{r}\sin\theta\partial_{\theta},\\\
\pm 2P^{\pm}&=e^{\pm
i\phi}\big{[}\sin\theta(\partial_{r}-\partial_{u})+\frac{1}{r}\cos\theta\partial_{\theta}\pm\frac{1}{r\sin\theta}\partial_{\phi}\big{]}.\end{split}$
(2.6)

As may be shown on general grounds or explicitly checked, the Poincaré algebra
in the form (2.4) may also be represented in terms of these generators
restricted to the surface $r=cte\to\infty$,

$\begin{split}L_{z}&=\partial_{\phi},\quad L^{\pm}=-e^{\pm
i\phi}\big{[}\partial_{\theta}\pm i\cot\theta\partial_{\phi}\big{]},\\\
K_{z}&=\cos\theta(u\partial_{u})+\sin\theta\partial_{\theta},\quad
K^{\pm}=e^{\pm
i\phi}\big{[}-\sin\theta(u\partial_{u})+\cos\theta\partial_{\theta}\pm\frac{i}{\sin\theta}\partial_{\phi}\big{]},\\\
H&=\partial_{u},\quad-2P_{z}=\cos\theta\partial_{u},\quad\pm 2P^{\pm}=-e^{\pm
i\phi}\sin\theta\partial_{u}.\end{split}$ (2.7)

The next step is to represent the Poincaré algebra at $u=0$. This can be done
by simply restricting the Lorentz generators to that surface, and by
representing the translation generators by suitable functions on that surface,

$\begin{split}L_{z}&=\partial_{\phi},\quad L^{\pm}=-e^{\pm
i\phi}\big{[}\partial_{\theta}\pm i\cot\theta\partial_{\phi}\big{]},\\\
K_{z}&=\sin\theta\partial_{\theta},\quad K^{\pm}=e^{\pm
i\phi}\big{[}\cos\theta\partial_{\theta}\pm\frac{i}{\sin\theta}\partial_{\phi}\big{]},\\\
H&=1={}_{0}Z_{0,0},\quad P_{z}=-\frac{1}{2}\cos\theta={}_{0}Z_{1,0},\quad
P^{\pm}=\mp\frac{1}{2}e^{\pm i\phi}\sin\theta={}_{0}Z_{1,\pm 1},\end{split}$
(2.8)

while in addition, defining for any function $f$ on the sphere,

$\begin{split}[L_{z},f]&=L_{z}(f),\quad[L^{\pm},f]=L^{\pm}(f),\\\
[K_{z},f]&=K_{z}(f)-\cos\theta f,\quad[K^{\pm},f]=K^{\pm}(f)+e^{\pm
i\phi}\sin\theta f.\end{split}$ (2.9)

When applied to the four functions in the last line of (2.8), this reproduces
the commutation relations of the Lorentz with the translation generators in
the Poincaré algebra, i.e., the last two lines of (2.4). The general
expression for the unnormalized spherical harmonics ${}_{s}Z_{j,m}$ are
explicitly given in Appendix A.

How the Poincaré algebra is enhanced to the $\mathfrak{bms}_{4}$ algebra in
the context of asymptotically flat spacetimes at null infinity is discussed in
the references at the beginning of this section. Besides the original
reference [3], we also refer to the re-derivation in [20] for more details.

In the $\mathfrak{bms}_{4}$ algebra, the commutation relations for the Lorentz
sub-algebra are unchanged and given in the first three lines of (2.4). The
commutation relations for the $\mathfrak{bms}_{4}$ algebra are then completed
by choosing a basis for functions on the sphere. Whatever basis is chosen, the
supertranslation generators $\mathcal{T}^{G}_{A}$ commute,

$[\mathcal{T}^{G}_{A},\mathcal{T}^{G}_{A^{\prime}}]=0.$ (2.10)

To get explicit structure constants for the commutators involving Lorentz and
supertranslation generators, one may start with an unnormalized basis
involving associated Legendre functions,

$\mathcal{T}^{U}_{j,m}=P_{j}^{m}(\cos\theta)e^{im\phi},\quad
P^{m}_{j}(x)=(-1)^{m}(1-x^{2})^{\frac{m}{2}}\frac{d^{m}}{dx^{m}}P_{j}(x).$
(2.11)

The action of the Lorentz generators on the $\mathcal{T}^{U}_{j,m}$ is then
worked out according to (2.9) by using suitable properties of associated
Legendre functions (see e.g. 8.733 and 8.735 of [34]),

$\displaystyle[L_{z},\mathcal{T}^{U}_{j,m}]=im\mathcal{T}^{U}_{j,m},$ (2.12)
$\displaystyle[L^{+},\mathcal{T}^{U}_{j,m}]=-\mathcal{T}^{U}_{j,m+1},$ (2.13)
$\displaystyle[L^{-},\mathcal{T}^{U}_{j,m}]=(j-m+1)(j+m)\mathcal{T}^{U}_{j,m-1},$
(2.14)
$\displaystyle[K_{z},\mathcal{T}^{U}_{j,m}]=\frac{(j-1)(j-m+1)}{2j+1}\mathcal{T}^{U}_{j+1,m}-\frac{{(j+2)(j+m)}}{2j+1}\mathcal{T}^{U}_{j-1,m},$
(2.15)
$\displaystyle[K^{+},\mathcal{T}^{U}_{j,m}]=\frac{j-1}{2j+1}\mathcal{T}^{U}_{j+1,m+1}+\frac{j+2}{2j+1}\mathcal{T}^{U}_{j-1,m+1},$
(2.16)
$[K^{-},\mathcal{T}^{U}_{j,m}]=-\frac{(j-1)(j-m+1)(j-m+2)}{2j+1}\mathcal{T}^{U}_{j+1,m-1}\\\
-\frac{(j+2)(j+m)(j+m-1)}{2j+1}\mathcal{T}^{U}_{j-1,m-1}.$ (2.17)

For a normalized basis in terms of standard spherical harmonics,

$\mathcal{T}^{S}_{j,m}={}_{0}Y_{j,m}=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)!}}P_{j}^{m}(\cos\theta)e^{im\phi},$
(2.18)

one finds instead

$\displaystyle[L_{z},\mathcal{T}^{S}_{j,m}]=im\mathcal{T}^{S}_{j,m},$ (2.19)
$\displaystyle[L^{\pm},\mathcal{T}^{S}_{j,m}]=\mp\sqrt{(j\mp m)(j\pm
m+1)}\mathcal{T}^{S}_{j,m\pm 1},$ (2.20)
$[K_{z},\mathcal{T}^{S}_{j,m}]=(j-1)\sqrt{\frac{(j+m+1)(j-m+1)}{(2j+1)(2j+3)}}\mathcal{T}^{S}_{j+1,m}\\\
-(j+2)\sqrt{\frac{(j+m)(j-m)}{(2j-1)(2j+1)}}\mathcal{T}^{S}_{j-1,m},$ (2.21)
$[K^{\pm},\mathcal{T}^{S}_{j,m}]=\pm(j-1)\sqrt{\frac{(j\pm m+2)(j\pm
m+1)}{(2j+1)(2j+3)}}\mathcal{T}^{S}_{j+1,m\pm 1}\\\ \pm(j+2)\sqrt{\frac{(j\mp
m)(j\mp m-1)}{(2j-1)(2j+1)}}\mathcal{T}^{S}_{j-1,m\pm 1}.$ (2.22)

This is the form under which the commutation relations between Lorentz and
supertranslation generators usually appear in the literature (with due care
devoted to various conventions and correction of misprints).

##  3 Coadjoint representations of semi-direct product groups and algebras

For a semi-direct product group of the form $G\ltimes_{\sigma}A$, with $G$ a
Lie group and $A$ an abelian Lie group seen as a vector space with the
addition, the group law is given by

$(f,\alpha)\cdot(g,\beta)=(f\cdot g,\alpha+\sigma_{f}(\beta)),$ (3.1)

while $\sigma$ is a representation of $G$ on $A$. The associated Lie algebra
is $\mathfrak{g}\oright_{\Sigma}A$, the Lie algebra of $A$ being identified
with $A$ itself, and

$[(X,\alpha),(Y,\beta)]=([X,Y],\Sigma_{X}\beta-\Sigma_{Y}\alpha),$ (3.2)

where $\Sigma$ is the differential of $\sigma$. The adjoint actions of the
group and the algebra are then given by

${\rm Ad}_{(f,\alpha)}(X,\beta)=({\rm Ad}_{f}X,\sigma_{f}\beta-\Sigma_{{\rm
Ad}_{f}X}\alpha),$ (3.3) ${\rm
ad}_{(X,\alpha)}(Y,\beta)=([X,Y],\Sigma_{X}\beta-\Sigma_{Y}\alpha).$ (3.4)

The dual space to the Lie algebra is given by $\mathfrak{g}^{*}\oplus A^{*}$,
with non-degenerate pairing denoted by

$\langle(j,p),(X,\alpha)\rangle=\langle j,X\rangle+\langle p,\alpha\rangle,$
(3.5)

and coadjoint actions

$\displaystyle\langle{\rm Ad}^{*}_{(f,\alpha)}(j,p),(Y,\beta)\rangle$
$\displaystyle=\langle(j,p),{\rm Ad}_{(f,\alpha)^{-1}}(Y,\beta)\rangle,$ (3.6)
$\displaystyle\langle{\rm ad}^{*}_{(X,\alpha)}(j,p),(Y,\beta)\rangle$
$\displaystyle=\langle(j,p),-{\rm ad}_{(X,\alpha)}(Y,\beta)\rangle.$ (3.7)

Defining $\times:A\oplus A^{*}\to\mathfrak{g}^{*}$ by

$\langle\alpha\times p,X\rangle=\langle p,\Sigma_{X}\alpha\rangle,$ (3.8)

and $\sigma^{*}$ to be the dual representation associated with $\sigma$,
$\sigma^{*}:G\times A^{*}\to A^{*}$,

$\langle\sigma^{*}_{f}p,\alpha\rangle=\langle p,\sigma_{f^{-}1}\alpha\rangle,$
(3.9)

the coadjoint representations are given by

${\rm Ad}^{*}_{(f,\alpha)}(j,p)=({\rm
Ad}^{*}_{f}j+\alpha\times\sigma^{*}_{f}p,\sigma^{*}_{f}p),$ (3.10) ${\rm
ad}^{*}_{(X,\alpha)}(j,p)=({\rm ad}^{*}_{X}j+\alpha\times p,\Sigma^{*}_{X}p).$
(3.11)

In terms of generators, $(e_{A},e_{\alpha})$ of
$\mathfrak{g}\oright_{\Sigma}A$, with $(e_{*}^{A},e_{*}^{\alpha})$ the
associated dual basis of $\mathfrak{g}^{*}\oplus A^{*}$,

$[e_{A},e_{B}]=f_{AB}^{C}e_{C},\quad[e_{A},e_{\alpha}]=f^{\beta}_{A\alpha}e_{\beta},\quad[e_{\alpha},e_{\beta}]=0,$
(3.12)

the coadjoint representation of the algebra (3.11) becomes

${\rm ad}^{*}_{e_{A}}e_{*}^{B}=-f^{B}_{AC}e_{*}^{C},\quad{\rm
ad}^{*}_{e_{\alpha}}e_{*}^{B}=0,\quad{\rm
ad}^{*}_{e_{A}}e_{*}^{\beta}=-f^{\beta}_{A\gamma}e_{*}^{\gamma},\quad{\rm
ad}^{*}_{e_{\alpha}}e_{*}^{\beta}=-f^{\beta}_{\alpha C}e_{*}^{C}.$ (3.13)

##  4 General structure of the coadjoint representation of BMS4

###  4.1 Background structure

####  4.1.1 Extended conformal transformations

Consider an $n$-dimensional Riemannian manifold with coordinates $x^{\alpha}$
and metric $g_{\alpha\beta}(x)$ which transforms under invertible coordinate
transformations $x^{\prime\alpha}=x^{\prime\alpha}(x)$ as

$g^{\prime}_{\gamma\delta}(x^{\prime})=g_{\alpha\beta}(x)\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial
x^{\alpha}}{\partial x^{\prime\gamma}}$}\kern
1.0pt\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial
x^{\beta}}{\partial x^{\prime\delta}}$}\kern 1.0pt.$ (4.1)

Conformal coordinate transformations are such invertible coordinate
transformations for which

$g^{\prime}_{\gamma\delta}(x^{\prime})=g_{\gamma\delta}(x^{\prime})\Omega^{2}(x^{\prime}).$
(4.2)

Consider then a two-dimensional surface $\mathcal{S}$ with coordinates
$x^{\alpha}=(\xi,\bar{\xi})$ and a conformally flat metric

$ds^{2}=g_{\alpha\beta}dx^{\alpha}dx^{\beta}=-2(P\bar{P})^{-1}d\xi
d\bar{\xi},$ (4.3)

for some nowhere vanishing $P(x)$. In this case, coordinate transformations of
the form

$\xi^{\prime}=\xi^{\prime}(\xi),\quad\bar{\xi}^{\prime}=\bar{\xi}^{\prime}(\bar{\xi})$
(4.4)

are conformal coordinate transformations with

$\Omega(x^{\prime})=\left[\frac{(P\bar{P})(x^{\prime})}{(P\bar{P})(x)}J\right]^{\frac{1}{2}},\quad
J=\frac{\partial\xi}{\partial\xi^{\prime}}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}.$
(4.5)

For such conformal transformations, the transformation law

$P^{\prime}(x^{\prime})=P(x)\frac{\partial\xi^{\prime}}{\partial\xi},\quad\bar{P}^{\prime}(x^{\prime})=\bar{P}(x)\frac{\partial\bar{\xi}^{\prime}}{\partial\bar{\xi}},$
(4.6)

induces the transformation (4.2) of the metric components. The more general
transformation law

$P^{\prime}(x^{\prime})=P(x)\frac{\partial\xi^{\prime}}{\partial\xi}e^{-E(x^{\prime})},\quad\bar{P}^{\prime}(x^{\prime})=\bar{P}(x)\frac{\partial\bar{\xi}^{\prime}}{\partial\bar{\xi}}e^{-\bar{E}(x^{\prime})},$
(4.7)

with $E$ a complex scalar field can be understood as follows. Writing the
metric with suitable zweibeins as,

$ds^{2}={e^{A}}_{\alpha}dx^{\alpha}\eta_{AB}{e^{B}}_{\beta}dx^{\beta},\quad\eta_{AB}=\begin{pmatrix}0&-1\\\
-1&0\end{pmatrix},\quad{e_{1}}^{\alpha}\partial_{\alpha}=P\partial,\quad{e_{2}}^{\alpha}\partial_{\alpha}=\bar{P}\bar{\partial},$
(4.8)

they correspond to the transformations of the zweibeins under conformal
coordinate transformations. At the same time, the imaginary part $iE_{I}$
produces a local rotation of the zweibeins while the real part of $E_{R}$
generates the Weyl rescaling of the metric. The associated conformal factor is

$\Omega(x^{\prime})=\left[\frac{(P\bar{P})(x^{\prime})}{(P\bar{P})(x)}J\right]^{\frac{1}{2}}e^{E_{R}(x^{\prime})}.$
(4.9)

Three relevant subclasses of the extended transformations (4.7) are

(i) Conformal coordinate transformation: taking $E=0=\bar{E}$ in (4.7) leads
back to (4.6), and thus, when applied to the metric components, to the same
transformations (4.2) as those coming from the conformal coordinate
transformation applied to the metric tensor.

(ii) Complex Weyl rescaling: taking $\xi^{\prime}=\xi$,
$\bar{\xi}^{\prime}=\bar{\xi}$ in (4.7) gives

$P^{\prime}(x)=P(x)e^{-E(x)},\quad\bar{P}^{\prime}(x)=\bar{P}(x)e^{-\bar{E}(x)}.$
(4.10)

This induces a real local Weyl rescaling on the metric,

$g^{\prime}_{\alpha\beta}(x)=e^{2E_{R}}g_{\alpha\beta}(x).$ (4.11)

When the real part $E_{R}=0$, the metric is unchanged.

(iii) Fixed conformal factor: fixing as in [5, 15] the conformal factor to be
a prescribed function of its arguments, $P(x)=P_{F}(x)$,
$\bar{P}(x)=\bar{P}_{F}(x)$ and $P^{\prime}(x^{\prime})=P_{F}(x^{\prime})$,
$\bar{P}^{\prime}(x^{\prime})=\bar{P}_{F}(x^{\prime})$ in (4.7), implies that
complex Weyl rescalings are frozen to

$e^{E(x^{\prime})}=\frac{P_{F}(x)}{P_{F}(x^{\prime})}\frac{\partial\xi^{\prime}}{\partial\xi}\Longleftrightarrow\left\\{\begin{array}[]{l}e^{E_{R}(x^{\prime})}=J^{-\frac{1}{2}}\left[\frac{(P_{F}\bar{P}_{F})(x)}{(P_{F}\bar{P}_{F})(x^{\prime})}\right]^{\frac{1}{2}},\\\
e^{iE_{I}(x^{\prime})}=\left[\frac{(P_{F}/\bar{P}_{F})(x)}{(P_{F}/\bar{P}_{F})(x^{\prime})}\left(\frac{\partial\xi^{\prime}/\partial\xi}{\partial\bar{\xi}^{\prime}/\partial\bar{\xi}}\right)\right]^{\frac{1}{2}},\end{array}\right.$
(4.12)

where $E=E_{R}+iE_{I}$. We will mostly be interested in 2 particular cases
below. The first is when $\mathcal{S}$ is a 2-sphere of radius $R$ with metric
$ds^{2}=-R^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})$ and
$\xi=\bar{\zeta}=\cot{\frac{\theta}{2}}e^{-i\phi}$, so that

$P_{S}=\frac{1+\xi\bar{\xi}}{R\sqrt{2}}=\bar{P}_{S},$ (4.13)

see [17] section 4.15 for details.

The second is the punctured complex plane, the complex plane with the origin
removed $\mathbb{C}_{0}=\mathbb{C}-\\{0\\}$, with standard metric
${ds}^{2}=-2dzd\bar{z}$ so that

$P=1={\bar{P}}.$ (4.14)

####  4.1.2 Conformal fields and weighted scalars

Under conformal coordinate transformations and complex Weyl rescalings, fields
$\phi^{\lambda,\bar{\lambda}}_{h,\bar{h}}$ of conformal dimensions
$(h,\bar{h})$ and Weyl weights $(\lambda,\bar{\lambda})$ transform as

${\phi^{\prime}}^{\lambda,\bar{\lambda}}_{h,\bar{h}}(x^{\prime})=e^{\lambda
E(x^{\prime})}e^{\bar{\lambda}\bar{E}(x^{\prime})}\Big{(}\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial\xi}{\partial\xi^{\prime}}$}\kern
1.0pt\Big{)}^{h}\Big{(}\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}$}\kern
1.0pt\Big{)}^{\bar{h}}\phi_{h,\bar{h}}^{\lambda,\bar{\lambda}}(x).$ (4.15)

It follows from (4.7) that the conformal dimensions and Weyl weights of $P$
are both $(-1,0)$ whereas those of $\bar{P}$ are both $(0,-1)$. These
quantities can be used to map the fields
$\phi^{\lambda,\bar{\lambda}}_{h,\bar{h}}$ into scalars $\eta^{s,w}$

$\eta^{s,w}=P^{h}\bar{P}^{\bar{h}}\phi_{h,\bar{h}}^{\lambda,\bar{\lambda}},$
(4.16)

of spin and conformal weights $[s,w]$,

$s=(h-\bar{h})-(\lambda-\bar{\lambda}),\quad
w=-(h+\bar{h})+(\lambda+\bar{\lambda}),$ (4.17)

which transform under conformal coordinate transformations and complex Weyl
rescalings as

$\eta^{\prime
s,w}(x^{\prime})=e^{wE_{R}(x^{\prime})}e^{-isE_{I}(x^{\prime})}\eta^{s,w}(x).$
(4.18)

####  4.1.3 Derivative operators

The only non-vanishing components of the Levi-Civita connection associated
with (4.3) are

$\Gamma^{\xi}_{\xi\xi}=-\partial\ln(P\bar{P}),\quad\Gamma^{\bar{\xi}}_{\bar{\xi}\bar{\xi}}=-\bar{\partial}\ln(P\bar{P}),$
(4.19)

where $\partial=\partial_{\xi},\bar{\partial}=\partial_{\bar{\xi}}$. Equation
(4.7) induces their transformation law under conformal coordinate
transformations combined with Weyl rescalings,

$\Gamma^{\prime\xi^{\prime}}_{\xi^{\prime}\xi^{\prime}}(x^{\prime})=\Gamma^{\xi}_{\xi\xi}(x)\frac{\partial\xi}{\partial\xi^{\prime}}+\frac{\partial\xi^{\prime}}{\partial\xi}\frac{\partial^{2}\xi}{\partial\xi^{\prime}\partial\xi^{\prime}}+2\partial^{\prime}E_{R}(x^{\prime}),$
(4.20)

with a similar transformation law for
$\Gamma^{\bar{\xi}}_{\bar{\xi}\bar{\xi}}$. In addition to the conformally flat
metric (4.3), one supposes that $\mathcal{S}$ is endowed with a Weyl
connection $({W},\bar{{W}})$ (see e.g. [12]) that transforms as

${W}^{\prime}(x^{\prime})=\Big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\Big{)}{W}(x)+2\partial^{\prime}E_{R}(x^{\prime}),\quad\bar{{W}}^{\prime}(x^{\prime})=\Big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\Big{)}\bar{{W}}(x)+2\bar{\partial}^{\prime}E_{R}(x^{\prime}).$
(4.21)

Using $P$, $\bar{P}$, ${W}$, $\bar{{W}}$, one can define

$\begin{split}&K=\frac{1}{2}(\partial\ln\mu-\Gamma^{\xi}_{\xi\xi})+{W}=\partial\ln\bar{P}+{W},\quad
O=\frac{1}{2}(\Gamma^{\xi}_{\xi\xi}-\partial\ln\mu)=-\partial\ln\bar{P}\quad\\\
&\bar{K}=\frac{1}{2}(\bar{\partial}\ln\bar{\mu}-\Gamma^{\bar{\xi}}_{\bar{\xi}\bar{\xi}})+\bar{{W}}=\bar{\partial}\ln
P+\bar{{W}},\quad\bar{O}=\frac{1}{2}(\Gamma^{\bar{\xi}}_{\bar{\xi}\bar{\xi}}-\bar{\partial}\ln\bar{\mu})=-\bar{\partial}\ln
P,\end{split}$ (4.22)

where $\mu=\frac{\bar{P}}{P}$ is a Beltrami differential. These objects
transform as

$\begin{split}K^{\prime}(x^{\prime})&=\Big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\Big{)}K(x)+\partial^{\prime}E(x^{\prime}),\quad
O^{\prime}(x^{\prime})=\Big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\Big{)}O(x)+\partial^{\prime}\bar{E}(x^{\prime}),\\\
\bar{K}^{\prime}(x^{\prime})&=\Big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\Big{)}\bar{K}(x)+\bar{\partial}^{\prime}\bar{E}(x^{\prime}),\quad\bar{O}^{\prime}(x^{\prime})=\Big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\Big{)}\bar{O}(x)+\bar{\partial}^{\prime}E(x^{\prime}).\end{split}$
(4.23)

The Weyl covariant derivative can then be defined as

$\begin{split}D\phi_{h,\bar{h}}^{\lambda,\bar{\lambda}}&=[\nabla+(h-\lambda)K+(h-\bar{\lambda})O]\phi^{\lambda,\bar{\lambda}}_{h,\bar{h}},\\\
\bar{D}\phi_{h,\bar{h}}^{\lambda,\bar{\lambda}}&=[\bar{\nabla}+(\bar{h}-\bar{\lambda})\bar{K}+(\bar{h}-\lambda)\bar{O}]\phi^{\lambda,\bar{\lambda}}_{h,\bar{h}},\end{split}$
(4.24)

where $\nabla\equiv\nabla_{\xi}$ and $\bar{\nabla}\equiv\nabla_{\bar{\xi}}$
are the components of the covariant derivative associated to the Levi-Civita
connection (4.19). Notice that the field $P$ ($\bar{P}$) of Weyl weights
$(\lambda,\bar{\lambda})=(-1,0)$ (resp. $(\lambda,\bar{\lambda})=(0,-1)$) and
conformal dimensions $(h,\bar{h})=(-1,0)$ (resp. $(h,\bar{h})=(0,-1)$) are
holomorphic (resp. anti-holomorphic) with respect to the Weyl covariant
derivative, namely $\bar{D}P=0$ (resp. $D\bar{P}=0$). Under conformal
coordinate transformations and complex Weyl rescalings, we have

$\begin{split}(D\phi_{h,\bar{h}}^{\lambda,\bar{\lambda}})^{\prime}(x^{\prime})&=e^{\lambda
E(x^{\prime})}e^{\bar{\lambda}\bar{E}(x^{\prime})}\Big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\Big{)}^{h+1}\Big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\Big{)}^{\bar{h}}(D\phi^{\lambda,\bar{\lambda}}_{h,\bar{h}})(x),\\\
(D\phi^{\lambda,\bar{\lambda}}_{h,\bar{h}})^{\prime}(x^{\prime})&=e^{\lambda
E(x^{\prime})}e^{\bar{\lambda}\bar{E}(x^{\prime})}\Big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\Big{)}^{h}\Big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\Big{)}^{\bar{h}+1}(\bar{D}\phi^{\lambda,\bar{\lambda}}_{h,\bar{h}})(x).\end{split}$
(4.25)

Therefore, the operator $D$ ($\bar{D}$) acts on fields of Weyl weights
$(\lambda,\bar{\lambda})$ and conformal dimensions $(h,\bar{h})$ to produce
fields of Weyl weights $(\lambda,\bar{\lambda})$ and conformal dimensions
$(h+1,\bar{h})$ (resp. $(h,\bar{h}+1)$).

In the following we will assume that the only fields carrying non-vanishing
Weyl weights are $P,\bar{P}$. All other fields are thus of the form
$\phi_{h,\bar{h}}^{0,0}$ with associated scalars
$\eta^{s,w}=P^{h}\bar{P}^{\bar{h}}\phi_{h,\bar{h}}^{0,0}$ so that

$s=h-\bar{h},\quad w=-(h+\bar{h}),\quad
h=\frac{s-w}{2},\quad\bar{h}=-\frac{s+w}{2}.$ (4.26)

If

$\eth\eta^{s,w}=P^{h+1}\bar{P}^{\bar{h}}(\nabla\phi_{h,\bar{h}}^{0,0}),\quad\bar{\eth}\eta^{s,w}=P^{h}\bar{P}^{\bar{h}+1}(\bar{\nabla}\phi_{h,\bar{h}}^{0,0}),$
(4.27)

then

$\begin{split}\eth\eta^{s,w}=P\bar{P}^{-s}\partial(\bar{P}^{s}\eta^{s,w})=P(\partial-
sO)\eta^{s,w},\\\
\bar{\eth}\eta^{s,w}=\bar{P}P^{s}\bar{\partial}(P^{-s}\eta^{s,w})=\bar{P}(\bar{\partial}+s\bar{O})\eta^{s,w}.\end{split}$
(4.28)

in agreement with expressions (4.14.34) and (4.14.33) of [17]. Under conformal
coordinate transformations and complex Weyl rescalings,

$\begin{split}(\eth\eta^{s,w})^{\prime}(x^{\prime})=e^{(w-1)E_{R}(x^{\prime})}e^{-i(s+1)E_{I}(x^{\prime})}\big{[}\eth+(w-s)P\partial
E_{R}(x^{\prime}(x))\big{]}\eta^{s,w}(x),\\\
(\bar{\eth}\eta^{s,w})^{\prime}(x^{\prime})=e^{(w-1)E_{R}(x^{\prime})}e^{-i(s-1)E_{I}(x^{\prime})}\big{[}\bar{\eth}+(w+s)\bar{P}\bar{\partial}E_{R}(x^{\prime}(x))\big{]}\eta^{s,w}(x).\end{split}$
(4.29)

Hence, the scalars $\eth\eta^{s,w}$ and $\bar{\eth}\eta^{s,w}$ transform as
scalars of weights $[s+1,w-1]$ respectively $[s-1,w-1]$ only if $w=s\iff h=0$
respectively $w=-s\iff\bar{h}=0$. Alternatively, one may limit oneself to
complex Weyl rescalings with $E_{R}=0$ so that only rotations of the zweibeins
are allowed, with no Weyl rescaling of the metric. In this case, only spin
weight $s$ is relevant.

When using the Weyl covariant derivative $D$ instead of the covariant
derivative $\nabla$ associated to the Christoffel connection, this issue does
not arise. Denoting $\mathcal{D}\eta^{s,w}$ and $\bar{\mathcal{D}}\eta^{s,w}$
the images under the mapping (4.16) of
$D\phi_{h,\bar{h}}^{\lambda,\bar{\lambda}}$ and of
$\bar{D}\phi_{h,\bar{h}}^{\lambda,\bar{\lambda}}$, respectively, we have

$\begin{split}\mathcal{D}\eta^{s,w}&=\left[\eth+\left(\frac{s-w}{2}\right)(\mathcal{O}+\mathcal{K})\right]\eta^{s,w}=P\left[\partial-\frac{s}{2}(O-K)-\frac{w}{2}(O+K)\right]\eta^{s,w},\\\
\bar{\mathcal{D}}\eta^{s,w}&=\left[\bar{\eth}-\left(\frac{w+s}{2}\right)(\bar{\mathcal{O}}+\bar{\mathcal{K}})\right]\eta^{s,w}=\bar{P}\left[\bar{\partial}+\frac{s}{2}(\bar{O}-\bar{K})-\frac{w}{2}(\bar{O}+\bar{K})\right]\eta^{s,w},\end{split}$
(4.30)

where $\mathcal{O}=PO$, $\mathcal{K}=PK$, $\bar{\mathcal{O}}=\bar{P}\bar{O}$,
$\bar{\mathcal{K}}=\bar{P}\bar{K}$. Under conformal coordinate transformations
and complex Weyl rescalings, we now have

$\begin{split}(\mathcal{D}\eta^{s,w})^{\prime}(x^{\prime})&=e^{(w-1)E_{R}(x^{\prime})}e^{-i(s+1)E_{I}(x^{\prime})}(\mathcal{D}\eta^{s,w})(x),\\\
(\bar{\mathcal{D}}\eta^{s,w})^{\prime}(x^{\prime})&=e^{(w-1)E_{R}(x^{\prime})}e^{-i(s-1)E_{I}(x^{\prime})}(\bar{\mathcal{D}}\eta^{s,w})(x).\end{split}$
(4.31)

Therefore, the operator $\mathcal{D}$ ($\bar{\mathcal{D}}$) acts on weighted
scalars $[s,w]$ to produce weighted scalars $[s+1,w-1]$ (resp. $[s-1,w-1]$).
The following property holds:

$[\mathcal{D},\bar{\mathcal{D}}]\eta^{s,w}=-P\bar{P}\big{(}s\partial\bar{\partial}\ln(P\bar{P})+\frac{s-w}{2}\bar{\partial}W+\frac{s+w}{2}\partial\bar{W}\big{)}\eta^{s,w}.$
(4.32)

Note that $R=-2P\bar{P}\partial\bar{\partial}\ln(P\bar{P})$ is the scalar
curvature of $\mathcal{S}$.

####  4.1.4 Ingredients

In the considerations below, all fields except for $P,\bar{P}$ have Weyl
weights $(0,0)$. We need the following ingredients:

(i) Supertranslation field: a real conformal field $\tilde{\mathcal{T}}$ of
dimensions $(-\frac{1}{2},-\frac{1}{2})$ and its assocated weighted scalar
$\mathcal{T}$ under the map (4.16) of weights $[0,1]$.

(ii) Superrotation field: a conformal field $\tilde{\mathcal{Y}}$ of
dimensions $(-1,0)$, its associated weighted scalar $\mathcal{Y}$ of weights
$[-1,1]$, and the complex conjugates $\bar{\tilde{\mathcal{Y}}}$ and
$\bar{\mathcal{Y}}$. These fields satisfy the conformal Killing equation which
becomes

$\bar{\mathcal{D}}\mathcal{Y}=0\iff\bar{D}\tilde{\mathcal{Y}}=0,$ (4.33)

together with the complex conjugate relations. Locally, the solutions are
simply $\tilde{\mathcal{Y}}=\tilde{\mathcal{Y}}(\xi)$ and
$\mathcal{Y}=P^{-1}\tilde{\mathcal{Y}}(\xi)$, with $\tilde{\mathcal{Y}}(\xi)$
arbitrary. This will not be the case when taking global restrictions into
account. Note also that, because $s=-w$ for $\mathcal{Y}$, it follows from the
second of (4.30) that the first of (4.33) can also be written using
$\bar{\eth}$ instead of $\bar{\mathcal{D}}$.

(iii) Supermomentum: a real conformal field $\tilde{\mathcal{P}}$ of
dimensions $(\frac{3}{2},\frac{3}{2})$ and its associated weighted scalar
$\mathcal{P}$ of weights $[0,-3]$.

(iv) Super angular momentum: a conformal field $\tilde{\mathcal{J}}$ of
dimensions $(1,2)$ and its associated weighted scalar $\mathcal{J}$ of weights
$[-1,-3]$, together with the complex conjugates $\bar{\tilde{\mathcal{J}}}$
and $\bar{\mathcal{J}}$. We consider equivalence classes $[\mathcal{J}]$ such
that $\mathcal{J}\sim\mathcal{J}+\mathcal{D}\mathcal{L}$ with $\mathcal{L}$
characterized by the weights $[-2,-2]$ and their complex conjugates. In this
case, it follows from the first of (4.30) that, since $s=w$ for $\mathcal{L}$,
one may also write $\eth\mathcal{L}$ in the equivalence relation. Similarly,
we consider equivalence classes $[\tilde{\mathcal{J}}]$ such that
$\tilde{\mathcal{J}}\sim\tilde{\mathcal{J}}+D\tilde{\mathcal{L}}$ with
$\tilde{\mathcal{L}}$ characterized by the conformal dimensions $(0,2)$ and
their complex conjugates). These equivalence classes may be called super
angular momenta.

The conformal dimensions, the Weyl weights, and the spin and conformal weights
of the different ingredients are summarized in the tables 1 below. The objects
$\tilde{d\mu}$, $d\mu$ represent the integration measure (see below).

$\phi_{h,\bar{h}}$ | $\tilde{\mathcal{T}}$ | $\tilde{\mathcal{Y}}$ | $\tilde{\mathcal{P}}$ | $\tilde{\mathcal{J}}$ | $\tilde{d\mu}$ | $P$ | $D$
---|---|---|---|---|---|---|---
$h$ | $-\frac{1}{2}$ | $-1$ | $\frac{3}{2}$ | $1$ | $-1$ | $-1$ | $1$
$\bar{h}$ | $-\frac{1}{2}$ | $0$ | $\frac{3}{2}$ | $2$ | $-1$ | $0$ | $0$
$\lambda$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-1$ | $0$
$\bar{\lambda}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$

$\eta^{s,w}$ $\mathcal{T}$ $\mathcal{Y}$ $\mathcal{P}$ $\mathcal{J}$ $d\mu$
$\mathcal{D}$ $s$ $0$ $-1$ $0$ $-1$ $0$ $1$ $w$ $1$ $1$ $-3$ $-3$ $2$ $-1$

Table 1: Dimensions and weights

Under complex conjugation, $\overline{(h,\bar{h})}=(\bar{h},h)$,
$\overline{(\lambda,\bar{\lambda})}=(\bar{\lambda},\lambda)$,
$\overline{[s,w]}=[-s,w]$.

###  4.2 Coadjoint representation of the algebra

####  4.2.1 Weighted scalars

In terms of above ingredients, the $\mathfrak{bms}_{4}$ algebra may be defined
by triplets $s=(\mathcal{Y},\bar{\mathcal{Y}},\mathcal{T})$ with the
commutation relations

$[(\mathcal{Y}_{1},\bar{\mathcal{Y}}_{1},\mathcal{T}_{1}),(\mathcal{Y}_{2},\bar{\mathcal{Y}}_{2},\mathcal{T}_{2})]=(\hat{\mathcal{Y}},\hat{\bar{\mathcal{Y}}},\hat{\mathcal{T}}),$
(4.34)

where

$\left\\{\begin{aligned}
\hat{\mathcal{Y}}&=\mathcal{Y}_{1}\mathcal{D}\mathcal{Y}_{2}-\mathcal{Y}_{2}\mathcal{D}\mathcal{Y}_{1}\,,\\\
\hat{\mathcal{T}}&=\mathcal{Y}_{1}\mathcal{D}\mathcal{T}_{2}-\frac{1}{2}\mathcal{D}\mathcal{Y}_{1}\mathcal{T}_{2}-(1\leftrightarrow
2)+{\rm c.c.}\,.\\\ \end{aligned}\right.$ (4.35)

Elements of the type $(\mathcal{Y},\bar{\mathcal{Y}},0)$ form a sub-algebra
$\mathfrak{g}$. As usual, we identify the individual entries of the
triplets/doublets with the triplets/doublets where all other entries are zero.
For weighted scalar $\eta^{s,w}$, a representation of $\mathfrak{g}$ is
defined by

$\mathcal{Y}\cdot\eta^{s,w}=\mathcal{Y}\mathcal{D}\eta^{s,w}+\frac{s-w}{2}\mathcal{D}\mathcal{Y}\eta^{s,w},\quad\bar{\mathcal{Y}}\cdot\eta^{s,w}=\bar{\mathcal{Y}}\bar{\mathcal{D}}\eta^{s,w}-\frac{s+w}{2}\bar{\mathcal{D}}\bar{\mathcal{Y}}\eta^{s,w}.$
(4.36)

At this stage, we note that this representation, and also the
$\mathfrak{bms}_{4}$ algebra above and the coadjoint representation below, may
also be written with $\eth,\bar{\eth}$ instead of
$\mathcal{D},\bar{\mathcal{D}}$ because the additional terms cancel.

In the notation of section 3, we thus have
$X=(\mathcal{Y},\bar{\mathcal{Y}})$, $\alpha=\mathcal{T}$, and

$\Sigma_{X}\alpha=(\mathcal{Y},\bar{\mathcal{Y}})\cdot\mathcal{T}=\mathcal{Y}\mathcal{D}\mathcal{T}-\frac{1}{2}\mathcal{D}\mathcal{Y}\mathcal{T}+{\rm
c.c.}\,.$ (4.37)

Elements of $\mathfrak{bms}^{*}_{4}$ are denoted by triplets
$([\mathcal{J}],[\bar{\mathcal{J}}],\mathcal{P})$ where the pairing is given
by

$\langle([\mathcal{J}],[\bar{\mathcal{J}}],\mathcal{P}),(\mathcal{Y},\bar{\mathcal{Y}},\mathcal{T})\rangle=\int_{\mathcal{S}}d\mu\,[\bar{\mathcal{J}}\mathcal{Y}+\mathcal{J}\bar{\mathcal{Y}}+\mathcal{P}\mathcal{T}].$
(4.38)

The measure

$d\mu(\xi,\bar{\xi})=\frac{iC}{P\bar{P}}d\xi\wedge d\bar{\xi},$ (4.39)

for some normalization constant $C$, has dimensions $(0,0)$ and weights
$[0,2]$. At this stage, we assume that the integral annihilates total
$\mathcal{D}$ and $\bar{\mathcal{D}}$ derivatives. Furthermore, we require the
pairing to be non-degenerate, which can only be the case when taking quotients
with respect to the equivalence relations
${\mathcal{J}}\sim{\mathcal{J}}+\mathcal{D}{\mathcal{L}}$ and
${\bar{\mathcal{J}}}\sim{\bar{\mathcal{J}}}+\bar{\mathcal{D}}{\bar{\mathcal{L}}}$.
Concrete realizations where these assumptions hold will be discussed below.

From the definition of the coadjoint representation (3.7), it then follows
that

$\begin{split}{\rm
ad}^{*}_{(\mathcal{Y},\bar{\mathcal{Y}},\mathcal{T})}\mathcal{J}&=\bar{\mathcal{Y}}\bar{\mathcal{D}}\mathcal{J}+2\bar{\mathcal{D}}\bar{\mathcal{Y}}\mathcal{J}+\mathcal{D}(\mathcal{Y}\mathcal{J})+\frac{1}{2}\mathcal{T}\bar{\mathcal{D}}\mathcal{P}+\frac{3}{2}\bar{\mathcal{D}}\mathcal{T}\mathcal{P},\\\
{\rm
ad}^{*}_{(\mathcal{Y},\bar{\mathcal{Y}},\mathcal{T})}\mathcal{P}&=\mathcal{Y}\mathcal{D}\mathcal{P}+\frac{3}{2}\mathcal{D}\mathcal{Y}\mathcal{P}+{\rm
c.c.}\,,\end{split}$ (4.40)

where the third term in the first of the above equations does not appear but
can be added because it is equivalent to zero. This is useful in order to have
a transformation law consistent with the conformal dimensions of
$\mathcal{J}$.

Remarks:

(i) The definition makes sense on the level of equivalence classes,

${\rm
ad}^{*}_{(\mathcal{Y},\bar{\mathcal{Y}},\mathcal{T})}([0],[0],0)=([0],[0],0),$
(4.41)

because

$\bar{\mathcal{Y}}\bar{\mathcal{D}}\mathcal{D}\mathcal{L}+2\bar{\mathcal{D}}\bar{\mathcal{Y}}\mathcal{D}\mathcal{L}=\mathcal{D}(\bar{\mathcal{Y}}\bar{\mathcal{D}}\mathcal{L}+2\bar{\mathcal{D}}\bar{\mathcal{Y}}\mathcal{L}).$
(4.42)

(ii) Since the inner product involves complex conjugation, we have

${\rm
ad}^{*}_{\mathcal{Y}}\mathcal{J}=\bar{\mathcal{Y}}\bar{\mathcal{D}}\mathcal{J}+2\bar{\mathcal{D}}\bar{\mathcal{Y}}\mathcal{J},\quad{\rm
ad}^{*}_{\bar{\mathcal{Y}}}\mathcal{J}=\mathcal{D}(\mathcal{Y}\mathcal{J})\sim
0,\quad{\rm
ad}^{*}_{\mathcal{Y}}\mathcal{P}=\bar{\mathcal{Y}}\bar{\mathcal{D}}\mathcal{P}+\frac{3}{2}\bar{\mathcal{D}}\bar{\mathcal{Y}}\mathcal{P},$
(4.43)

together with the complex conjugates of these relations.

(iii) In the notation of section 3, $j=([\mathcal{J}],[\bar{\mathcal{J}}])$,
$p=\mathcal{P}$ and

$\begin{split}\Sigma^{*}_{X}p&=\mathcal{Y}\mathcal{D}\mathcal{P}+\frac{3}{2}\mathcal{D}\mathcal{Y}\mathcal{P}+{\rm
c.c.}\,,\\\ \alpha\times
p&=([\frac{1}{2}\mathcal{T}\bar{\mathcal{D}}\mathcal{P}+\frac{3}{2}\bar{\mathcal{D}}\mathcal{T}\mathcal{P}],[\frac{1}{2}\mathcal{T}\mathcal{D}\mathcal{P}+\frac{3}{2}\mathcal{D}\mathcal{T}\mathcal{P}])\,.\end{split}$
(4.44)

This relation encodes the change of super angular momentum under an
infinitesimal supertranslation, which depends linearly on supermomentum. Note
also that when using a vector $\mathcal{T}$ which is not real, the
contribution of $\alpha\times p$ to $\mathcal{J}$ is
$\frac{1}{2}\bar{\mathcal{T}}\bar{\mathcal{D}}\mathcal{P}+\frac{3}{2}\bar{\mathcal{D}}\bar{\mathcal{T}}\mathcal{P}$.

(iv) On the level of integrands, if we define

$\mathcal{J}^{u}_{s}=\bar{\mathcal{J}}\mathcal{Y}+\mathcal{J}\bar{\mathcal{Y}}+\mathcal{P}\mathcal{T},$
(4.45)

equation (3.7) reads

${\rm
ad}^{*}_{s_{1}}\mathcal{J}^{u}_{s_{2}}=-\mathcal{J}^{u}_{[s_{1},s_{2}]}+\mathcal{D}\mathcal{L}_{s_{1},s_{2}}+\bar{\mathcal{D}}\bar{\mathcal{L}}_{s_{1},s_{2}},$
(4.46)

where

$\mathcal{L}_{s_{1},s_{2}}=\bar{\mathcal{J}}\mathcal{Y}_{1}\mathcal{Y}_{2}+(\mathcal{J}\mathcal{Y}_{1}\bar{\mathcal{Y}}_{2})+\mathcal{P}\mathcal{Y}_{1}\mathcal{T}_{2}+\frac{1}{2}\mathcal{P}\mathcal{T}_{1}\mathcal{Y}_{2}.$
(4.47)

(v) In the case when there is no non-degenerate pairing, it is still true that
the vector space of elements $([\mathcal{J}],[\bar{\mathcal{J}}],\mathcal{P})$
forms a representation under $\mathfrak{bms}_{4}$. This representation
$\rho_{(\mathcal{Y},\bar{\mathcal{Y}},\mathcal{T})}$, which is no longer the
coadjoint representation, is defined by replacing ${\rm
ad}^{*}_{(\mathcal{Y},\bar{\mathcal{Y}},\mathcal{T})}$ by
$\rho_{(\mathcal{Y},\bar{\mathcal{Y}},\mathcal{T})}$ in the left hand side of
(4.40).

####  4.2.2 Conformal fields

The above definitions are expressed in terms of weighted scalars. The
analogous definitions in terms of the associated conformal fields are obtained
by a direct rewriting that consists in adding tilde’s on all the scalars and
replacing $\mathcal{D}$ by $D$ and $\bar{\mathcal{D}}$ by $\bar{D}$.

In particular, under the mapping (4.16), the representation (4.36) becomes

$\tilde{\mathcal{Y}}\cdot\phi_{h,\bar{h}}=\tilde{\mathcal{Y}}{D}\phi_{h,\bar{h}}+h{D}\tilde{\mathcal{Y}}\phi_{h,\bar{h}},\quad\tilde{\bar{\mathcal{Y}}}\cdot\phi_{h,\bar{h}}=\tilde{\bar{\mathcal{Y}}}\bar{{D}}\phi_{h,\bar{h}}+\bar{h}\bar{D}\tilde{\bar{\mathcal{Y}}}\phi_{h,\bar{h}}.$
(4.48)

The Weyl covariant derivatives $D,\bar{D}$ may again be replaced by the
ordinary derivatives $\partial,\bar{\partial}$ in these representations
because the additional terms cancel.

The pairing is given by

$\langle([\tilde{\mathcal{J}}],[\tilde{\bar{\mathcal{J}}}],\tilde{\mathcal{P}}),(\tilde{\mathcal{Y}},\tilde{\bar{\mathcal{Y}}},\tilde{\mathcal{T}})\tilde{\rangle}=\int_{\mathcal{S}}\tilde{d\mu}\,[\tilde{\bar{\mathcal{J}}}\tilde{\mathcal{Y}}+\tilde{\mathcal{J}}\tilde{\bar{\mathcal{Y}}}+\tilde{\mathcal{P}}\tilde{\mathcal{T}}],$
(4.49)

where the measure has conformal dimensions $(-1,-1)$, Weyl weights $(0,0)$ and
is associated to the measure (4.39) through
$d\mu=(P\bar{P})^{-1}\tilde{d\mu}$, so that

$\tilde{d\mu}=iCd\xi\wedge d\bar{\xi}.$ (4.50)

###  4.3 Coadjoint representation of the group

####  4.3.1 Conformal fields

Consider conformal coordinate transformations,
$(\xi^{\prime}(\xi),\bar{\xi}^{\prime}(\bar{\xi}))=(g(\xi),\bar{g}(\bar{\xi}))$,
such that $\frac{\partial g}{\partial\xi}>0$,
$\frac{\partial\bar{g}}{\partial\bar{\xi}}>0$. They form a group $G$ under
composition. For a conformal field ${\phi_{h,\bar{h}}}(x)$ of dimensions
$(h,\bar{h})$ (and vanishing Weyl weights), a representation of $G$ is defined
through

$\big{(}(g,\bar{g})\cdot{\phi}_{h,\bar{h}}\big{)}(x^{\prime})=\big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\big{)}^{h}\big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\big{)}^{\bar{h}}{\phi}_{h,\bar{h}}(x).$
(4.51)

The BMS4 group is determined by elements $(g,\bar{g},\tilde{\mathcal{T}})$
with multiplication

$(g_{1},\bar{g}_{1},\tilde{\mathcal{T}}_{1})\cdot(g_{2},\bar{g}_{2},\tilde{\mathcal{T}}_{2})=\big{(}g_{1}\circ
g_{2},\bar{g}_{1}\circ\bar{g}_{2},\tilde{\mathcal{T}}_{1}+(g_{1},\bar{g}_{1})\cdot\tilde{\mathcal{T}}_{2}\big{)}.$
(4.52)

Elements of the form $(g,\bar{g},0)$ form a subgroup isomorphic to $G$. In the
notation of section 3, we thus have $f=(g,\bar{g})$,
$X=(\tilde{\mathcal{Y}},\tilde{\bar{\mathcal{Y}}})$ and
$\alpha=\tilde{\mathcal{T}}$ with

$\big{(}\sigma_{f}(\alpha)\big{)}(x^{\prime})=\big{(}(g,\bar{g})\cdot\tilde{\mathcal{T}}\big{)}(x^{\prime})=\big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\big{)}^{-\frac{1}{2}}\big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\big{)}^{-\frac{1}{2}}\tilde{\mathcal{T}}(x).$
(4.53)

For the adjoint action, defined in equation (3.3), we get

$\big{(}{\rm
Ad}_{f}X\big{)}(x^{\prime})=\big{(}(g,\bar{g})\cdot(\mathcal{Y},\bar{\mathcal{Y}})\big{)}(x^{\prime})=\Big{(}\big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\big{)}^{-1}\tilde{\mathcal{Y}},\big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\big{)}^{-1}\tilde{\bar{\mathcal{Y}}}\Big{)}(x),$
(4.54) $\big{(}\Sigma_{{\rm
Ad}_{f}X}\alpha\big{)}(x^{\prime})=\big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\big{)}^{-\frac{1}{2}}\big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\big{)}^{-\frac{1}{2}}(\tilde{\mathcal{Y}}D\tilde{\mathcal{T}}-\frac{1}{2}D\tilde{\mathcal{Y}}\tilde{\mathcal{T}}+{\rm
c.c.})(x),$ (4.55)

whereas definition (3.10) for the coadjoint representation gives

$\big{(}{\rm
Ad}^{*}_{f}\tilde{\mathcal{J}}\big{)}(x^{\prime})=\big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\big{)}\big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\big{)}^{2}\tilde{\mathcal{J}}(x),$
(4.56)
$\big{(}\sigma^{*}_{f}\tilde{\mathcal{P}}\big{)}(x^{\prime})=\big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\big{)}^{\frac{3}{2}}\big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\big{)}^{\frac{3}{2}}\tilde{\mathcal{P}}(x),$
(4.57)
$\big{(}\tilde{\mathcal{T}}\times\sigma^{*}_{f}\tilde{\mathcal{P}}\big{)}(x^{\prime})=\Big{(}\big{(}\frac{\partial\xi}{\partial\xi^{\prime}}\big{)}\big{(}\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}}\big{)}^{2}(\frac{1}{2}\tilde{\mathcal{T}}\bar{D}\tilde{\mathcal{P}}+\frac{3}{2}\bar{D}\tilde{\mathcal{T}}\tilde{\mathcal{P}}),{\rm
c.c.}\Big{)}(x),$ (4.58)

where ${\rm c.c.}$ denotes the complex conjugate of the expression to the left
of the comma.

Remarks:

(i) As usual for diffeomorphisms, the adjoint and coadjoint representations of
the algebra discussed previously are the differentials of those of the group
discussed in this section up to an overall minus sign.

(ii) As usual in conformal field theory, on the level of the algebra, we will
consider not only the Lie algebra of the globally well-defined conformal
transformations but also the algebra of infinitesimal local conformal
transformations.

(iii) The formulas for the group can also be used to understand how the
coadjoint representation behaves under conformal mappings. We briefly discuss
the standard map from the punctured plane to the cylinder below.

####  4.3.2 Weighted scalars

The description in terms of weighted scalars is very similar. As compared to
the previous section, one simply removes the tilde’s and replaces $D$,
$\bar{D}$ by $\mathcal{D}$, $\bar{\mathcal{D}}$, while at the same time
replacing
$(\frac{\partial\xi}{\partial\xi^{\prime}})^{h}(\frac{\partial\bar{\xi}}{\partial\bar{\xi}^{\prime}})^{\bar{h}}$
by $e^{wE_{R}(x^{\prime})}e^{-isE_{I}(x^{\prime})}$ using table 1.

For future reference, let us nevertheless provide explicit formulas. In this
case, the BMS4 group is determined by elements $(g,\bar{g},{\mathcal{T}})$
with multiplication

$(g_{1},\bar{g}_{1},{\mathcal{T}}_{1})\cdot(g_{2},\bar{g}_{2},{\mathcal{T}}_{2})=\big{(}g_{1}\circ
g_{2},\bar{g}_{1}\circ\bar{g}_{2},{\mathcal{T}}_{1}+(g_{1},\bar{g}_{1})\cdot{\mathcal{T}}_{2}\big{)},$
(4.59)

where the representation of $G$ on a weighted scalar $\eta^{s,w}$ is defined
through

$\big{(}(g,\bar{g})\cdot\eta^{s,w}\big{)}(x^{\prime})=e^{wE_{R}(x^{\prime})}e^{-isE_{I}(x^{\prime})}\eta^{s,w}(\xi,\bar{\xi}).$
(4.60)

In the notation of section 3, we now have $f=(g,\bar{g})$,
$X=(\mathcal{Y},\bar{\mathcal{Y}})$, and $\alpha=\mathcal{T}$ with

$\big{(}\sigma_{f}(\alpha)\big{)}(x^{\prime})=\big{(}(g,\bar{g})\cdot{\mathcal{T}}\big{)}(x^{\prime})=e^{E_{R}(x^{\prime})}{\mathcal{T}}(x).$
(4.61)

For the adjoint representation, we get

$\big{(}{\rm
Ad}_{f}X\big{)}(x^{\prime})=\big{(}(g,\bar{g})\cdot(\mathcal{Y},\bar{\mathcal{Y}})\big{)}(x^{\prime})=\left(e^{E_{R}(x^{\prime})}e^{iE_{I}(x^{\prime})}{\mathcal{Y}}(x),e^{E_{R}(x^{\prime})}e^{-iE_{I}(x^{\prime})}{\bar{\mathcal{Y}}}(x)\right),$
(4.62) $\big{(}\Sigma_{{\rm
Ad}_{f}X}\alpha\big{)}(x^{\prime})=e^{E_{R}(x^{\prime})}({\mathcal{Y}}\mathcal{D}{\mathcal{T}}-\frac{1}{2}\mathcal{D}{\mathcal{Y}}{\mathcal{T}}+{\rm
c.c.})(x),$ (4.63)

whereas for the coadjoint representation, we get

$\big{(}Ad^{*}_{f}{\mathcal{J}}\big{)}(x^{\prime})=e^{-3E_{R}(x^{\prime})}e^{iE_{I}(x^{\prime})}{\mathcal{J}}(x),$
(4.64)
$\big{(}\sigma^{*}_{f}{\mathcal{P}}\big{)}(x^{\prime})=e^{-3E_{R}(x^{\prime})}{\mathcal{P}}(x),$
(4.65)
$\big{(}{\mathcal{T}}\times\sigma^{*}_{f}{\mathcal{P}}\big{)}(x^{\prime})=\left(e^{-3E_{R}(x^{\prime})}e^{iE_{I}(x^{\prime})}(\frac{1}{2}{\mathcal{T}}\bar{\mathcal{D}}{\mathcal{P}}+\frac{3}{2}\bar{\mathcal{D}}{\mathcal{T}}{\mathcal{P}})(x),{\rm
c.c.}\right).$ (4.66)

###  4.4 Weyl invariance

The structure described above is covariant with respect to conformal
coordinate transformations combined with complex Weyl rescalings since it is
defined in terms of suitable covariant derivatives.

In particular, the above descriptions are valid for all conformal factors $P$
and $\bar{P}$ and all two-dimensional surfaces $\mathcal{S}$ such that total
$\mathcal{D},\bar{\mathcal{D}}$ derivatives are annihilated and the pairing is
non-degenerate. In the remainder of the paper, we mainly focus on two
particular cases: $(i)$ the sphere $S^{2}$ of radius $R$ with
$P_{S}=\frac{1+\xi\bar{\xi}}{R\sqrt{2}}=\bar{P}_{S}$, and $(ii)$ the punctured
complex plane $\mathbb{C}-\\{0\\}=\mathbb{C}_{0}$ with $P=1=\bar{P}$.

##  5 Realization on the sphere

###  5.1 Generalities

If $\xi=\cot\frac{\theta}{2}e^{-i\phi}$, the standard metric on the sphere of
radius $R$ is

$ds^{2}=-2(P_{S}\bar{P}_{S})^{-1}d\xi d\bar{\xi},\quad
P_{S}=\frac{1+\xi\bar{\xi}}{R\sqrt{2}}.$ (5.1)

The globally well-defined conformal coordinate transformations for the sphere
are the fractional linear unimodular transformations

$\xi^{\prime}=\frac{a\xi+b}{c\xi+d},\quad ad-bc=1,\quad a,b,c,d\in\mathbb{C}.$
(5.2)

In particular,

$\frac{\partial\xi}{\partial\xi^{\prime}}=(c\xi+d)^{2}.$ (5.3)

Under combined conformal coordinate transformations and Weyl rescalings, the
metric takes the standard form in the new coordinates if one freezes the Weyl
transformations as in equation (4.12). For $P_{F}=P_{S}$, we have [5, 15]

$e^{E(x^{\prime})}=\frac{P_{S}(x)}{P_{S}(x^{\prime})}\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial\xi^{\prime}}{\partial\xi}$}\kern
1.0pt\iff
e^{E_{R}(x^{\prime})}=\frac{1+\xi\bar{\xi}}{|a\xi+b|^{2}+|c\xi+d|^{2}},\quad
e^{iE_{I}(x^{\prime})}=\frac{\bar{c}\bar{\xi}+\bar{d}}{c\xi+d}.$ (5.4)

In this context, $w$ is referred to as the boost weight.

The derivative operators (4.28) now take the explicit form (cf. section 4.15
of [17])

$\eth\eta^{s,w}=P_{S}^{1-s}\partial(P_{S}^{s}\eta^{s,w}),\quad\bar{\eth}\eta^{s,w}=P_{S}^{1+s}\bar{\partial}(P_{S}^{-s}\eta^{s,w}).$
(5.5)

The pairing on the sphere is defined between scalars $\eta^{s,w}$ of weights
$[s,w]$ and $\kappa^{s,-w-2}$ of weights $[s,-w-2]$ as follows,

$\langle\kappa^{s,-w-2},\eta^{s,w}\rangle=\frac{1}{4\pi
R^{2}}\int_{S^{2}}\frac{id\xi\wedge d\bar{\xi}}{P_{S}\bar{P}_{S}}\
\widebar{\kappa^{s,-w-2}}\eta^{s,w},$ (5.6)

where the normalization $C=(4\pi R^{2})^{-1}$ is chosen so that

$\frac{1}{4\pi R^{2}}\int_{S^{2}}\frac{id\xi\wedge
d\bar{\xi}}{P_{S}\bar{P}_{S}}=\frac{1}{2\pi}\int_{S^{2}}\frac{id\xi\wedge
d\bar{\xi}}{(1+\xi\bar{\xi})^{2}}=\frac{1}{4\pi}\int_{0}^{\pi}d\theta\sin\theta\int^{2\pi}_{0}d\phi=1.$
(5.7)

When compared to (4.38), we thus have

$\langle([\mathcal{J}],[\bar{\mathcal{J}}],\mathcal{P}),(\mathcal{Y},\bar{\mathcal{Y}},\mathcal{T})\rangle=\langle\mathcal{J},\mathcal{Y}\rangle+\langle\bar{\mathcal{J}},\bar{\mathcal{Y}}\rangle+\langle\mathcal{P},\mathcal{T}\rangle,\quad
d\mu(\xi,\bar{\xi})=\frac{id\xi\wedge d\bar{\xi}}{4\pi
R^{2}P_{S}\bar{P}_{S}}.$ (5.8)

This pairing has all the required properties.

###  5.2 Adjoint and coadjoint representations of the group

In terms of spin-weighted scalars, the (co)adjoint representation of the BMS4
group on the sphere is described by the general formulas established in
subsection 4.3.2, where now $f=(g,\bar{g})$ are given by general linear
fractional transformations of (5.2) (and the associated transformation of the
complex conjugate variable) and the factors $e^{E_{R}(x^{\prime})}$ and
$e^{iE_{I}(x^{\prime})}$ are given in (5.4). Let us write them out explicitly,
with $x=\xi,\bar{\xi}$.

For the adjoint representation, under a combined transformation $f$ and a
supertranslation $\alpha$ with which one acts and a supertranslation $\beta$
on which one acts, (where $\alpha,\beta$ are two different supertranslation
fields with the same weights than $\mathcal{T}$ ),

$\begin{split}&\mathcal{Y}^{\prime}(x^{\prime})=e^{E_{R}(x^{\prime})}e^{iE_{I}(x^{\prime})}\mathcal{Y}(x),\\\
&\bar{\mathcal{Y}}^{\prime}(x^{\prime})=e^{E_{R}(x^{\prime})}e^{-iE_{I}(x^{\prime})}\bar{\mathcal{Y}}(x),\\\
&\beta^{\prime}(x^{\prime})=e^{E_{R}(x^{\prime})}\Big{(}\beta-\big{(}{\mathcal{Y}}\eth\alpha-\frac{1}{2}\alpha\eth\mathcal{Y}+{\rm
c.c.}\big{)}\Big{)}(x).\end{split}$ (5.9)

For the coadjoint representation, if we denote by $\mathcal{T}$ instead of
$\alpha$ the supertranslation with which one acts,

$\begin{split}&\mathcal{J}^{\prime}(x^{\prime})=e^{-3E_{R}(x^{\prime})}e^{iE_{I}(x^{\prime})}\Big{(}\mathcal{J}+(\frac{1}{2}{\mathcal{T}}\bar{\eth}{\mathcal{P}}+\frac{3}{2}\bar{\eth}{\mathcal{T}}{\mathcal{P}})\Big{)}(x)\\\
&\bar{\mathcal{J}}^{\prime}(x^{\prime})=e^{-3E_{R}(x^{\prime})}e^{-iE_{I}(x^{\prime})}\Big{(}\bar{\mathcal{J}}+(\frac{1}{2}{\mathcal{T}}\eth{\mathcal{P}}+\frac{3}{2}\eth{\mathcal{T}}{\mathcal{P}})\Big{)}(x)\\\
&\mathcal{P}^{\prime}(x^{\prime})=e^{-3E_{R}(x^{\prime})}\mathcal{P}(x).\end{split}$
(5.10)

Not surprisingly, when using the associated conformal fields, these
transformations simplify. The formulas of section 4.3.1 apply. The Jacobians
$\partial\xi/\partial\xi^{\prime},\partial\bar{\xi}/\partial\bar{\xi}^{\prime}$
are explicitly given by (5.3) and its complex conjugate. In this case, the
integration measure is

$\tilde{d\mu}=\frac{id\xi\wedge d\bar{\xi}}{4\pi R^{2}},$ (5.11)

and

$\begin{split}&\tilde{\mathcal{Y}}^{\prime}(\xi^{\prime})=(c\xi+d)^{-2}\tilde{\mathcal{Y}}(\xi),\\\
&\tilde{\bar{\mathcal{Y}}}^{\prime}(\bar{\xi}^{\prime})=(\bar{c}\bar{\xi}+\bar{d})^{-2}\tilde{\bar{\mathcal{Y}}}(\xi),\\\
&\tilde{\beta}^{\prime}(x^{\prime})=(c\xi+d)^{-1}(\bar{c}\bar{\xi}+\bar{d})^{-1}\Big{(}\tilde{\beta}-\big{(}\tilde{\mathcal{Y}}\partial\tilde{\alpha}-\frac{1}{2}\tilde{\alpha}\partial\tilde{\mathcal{Y}}+{\rm
c.c.}\big{)}\Big{)}(x).\end{split}$ (5.12)
$\begin{split}&\tilde{\mathcal{J}}^{\prime}(x^{\prime})=(c\xi+d)^{2}(\bar{c}\bar{\xi}+\bar{d})^{4}\Big{(}\tilde{\mathcal{J}}(x)+(\frac{1}{2}\tilde{\mathcal{T}}\bar{\partial}\tilde{\mathcal{P}}+\frac{3}{2}\bar{\partial}\tilde{\mathcal{T}}\tilde{\mathcal{P}})\Big{)}(x)\\\
&\bar{\mathcal{J}}^{\prime}(x^{\prime})=(c\xi+d)^{4}(\bar{c}\bar{\xi}+\bar{d})^{2}\Big{(}\tilde{\bar{\mathcal{J}}}+(\frac{1}{2}\tilde{\mathcal{T}}\partial\tilde{\mathcal{P}}+\frac{3}{2}\partial\tilde{\mathcal{T}}\tilde{\mathcal{P}})\Big{)}(x)\\\
&\tilde{\mathcal{P}}^{\prime}(x^{\prime})=(c\xi+d)^{3}(\bar{c}\bar{\xi}+\bar{d})^{3}\tilde{\mathcal{P}}(x).\end{split}$
(5.13)

###  5.3 Expansions

####  5.3.1 Spin-weighted spherical harmonics

We now decompose the relevant spin-weighted scalars in terms of spin-weighted
spherical harmonics (see appendix A for conventions).

For the $\mathfrak{bms}_{4}$ Lie algebra, we have
$\bar{\mathcal{D}}\mathcal{Y}=\bar{\eth}\mathcal{Y}=0$ with $\mathcal{Y}$ of
weights $[-1,1]$. It follows from array (4.15.60) of [17] that $\mathcal{Y}$
belongs to the three-dimensional vector space of spherical harmonics with spin
weight $s=-1$ and $j=1$. Hence, defining

$\mathcal{Y}_{m}=-R\sqrt{2}~{}{}_{-1}Z_{1,m},\quad m=-1,0,1,$ (5.14)

gives rise to the decomposition

$\bar{\eth}\mathcal{Y}=0\iff\mathcal{Y}=\sum_{m=-1}^{1}y_{m}\mathcal{Y}_{m}.$
(5.15)

In the same way,

$\bar{\mathcal{Y}}_{m}=(-1)^{m}R\sqrt{2}~{}{}_{1}Z_{1,-m}\Longrightarrow\bar{\mathcal{Y}}=\sum_{m=-1}^{1}\bar{y}_{m}\bar{\mathcal{Y}}_{m},$
(5.16)

while

$\mathcal{T}_{j,m}={}_{0}Z_{j,m}\Longrightarrow\mathcal{T}=\sum_{j,|m|\leq
j}t_{j,m}\mathcal{T}_{j,m},$ (5.17)

where we impose $\bar{t}_{j,m}=(-1)^{m}t_{j,-m}$ since $\mathcal{T}$ is real.

When taking into account the pairing (5.6) and the normalization of the
${}_{s}Z_{jm}$ in (A.4), this choice of basis for the algebra implies the
following choice for the associated dual basis of the coadjoint
representation,

$\begin{split}\mathcal{Y}_{*}^{m}&=\frac{-6}{R\sqrt{2}(1+m)!(1-m)!}\
{}_{-1}Z_{1,m},\\\
\bar{\mathcal{Y}}_{*}^{m}&=\frac{(-1)^{m}6}{R\sqrt{2}(1+m)!(1-m)!}\
{}_{1}Z_{1,-m},\\\
\mathcal{T}^{j,m}_{*}&=\frac{(2j+1)!(2j)!}{j!j!(j+m)!(j-m)!}\
{}_{0}Z_{j,m},\end{split}$ (5.18)

and thus also the following expansions,

$\mathcal{J}=\sum_{m=-1}^{1}j_{m}\mathcal{Y}_{*}^{m},\quad\bar{\mathcal{J}}=\sum_{m=-1}^{1}\bar{j}_{m}\bar{\mathcal{Y}}_{*}^{m},\quad\mathcal{P}=\sum_{j,|m|\leq
j}p_{j,m}\mathcal{T}_{*}^{j,m},$ (5.19)

where $\bar{p}_{j,m}=(-1)^{m}p_{j,-m}$ since $\mathcal{P}$ is real.

Note that the explicit expressions for ${}_{-1}Z_{1,m}$ in (A.2) gives

$\tilde{\mathcal{Y}}_{m}=\mathcal{Y}_{m}P_{S}=\xi^{1-m},\quad\tilde{\bar{\mathcal{Y}}}_{m}={\bar{\mathcal{Y}}}_{m}\bar{P}_{S}={\bar{\xi}}^{1-m}.$
(5.20)

Remarks:

(i) From the discussion of the behavior of spin-weighted spherical harmonics
under Lorentz transformations in section 4.15 of [17], it follows that, if
$w\geq|s|$ then $\eth^{w-s+1}\eta^{s,w}$, $\bar{\eth}^{w+s+1}\eta^{s,w}$ have
definite spin and boosts weights given by $[w+1,s-1]$ and $[-w-1,-s-1]$
respectively. This is the case for

$\eth\bar{\mathcal{Y}},\ \bar{\eth}\mathcal{Y},\ \eth^{3}\mathcal{Y},\
\bar{\eth}^{3}\bar{\mathcal{Y}}.$ (5.21)

As shown there, the equations $\bar{\eth}\mathcal{Y}=0$ and
$\eth^{3}\mathcal{Y}=0$ on the one hand, and $\eth\bar{\mathcal{Y}}=0$ and
$\bar{\eth}^{3}\bar{\mathcal{Y}}=0$ on the other, define the same Lorentz
invariant three-dimensional subspaces described above.

For the dual situation where $w\leq-|s|-2$, $\eth^{s-w-1}\kappa^{w+1,s-1}$ and
$\bar{\eth}^{-s-w-1}\kappa^{-w-1,-s-1}$ have definite spin and boost weights
$[s,w]$, it is shown that equivalence classes
$[\eta^{s,w}],\eta^{s,w}\sim\eta^{s,w}+\eth^{s-w-1}\kappa^{w+1,s-1}$ or
$\eta^{s,w}\sim\eta^{s,w}+\bar{\eth}^{-s-w-1}\kappa^{-w-1,-s-1}$ define
Lorentz invariant subspaces. This is the case for

$\bar{\mathcal{J}}\sim\bar{\mathcal{J}}+\bar{\eth}\bar{\mathcal{L}},\quad\bar{\mathcal{J}}\sim\bar{\mathcal{J}}+\eth^{3}\mathcal{M},$
(5.22)

where $\bar{\mathcal{L}}:[2,-2]$ and $\mathcal{M}:[-2,0]$ and both equivalence
classes define the same three-dimensional Lorentz invariant subspaces.
Similarly, by complex conjugation

$\mathcal{J}\sim\mathcal{J}+\eth\mathcal{L},\quad\mathcal{J}\sim\mathcal{J}+\bar{\eth}^{3}\bar{\mathcal{M}}$
(5.23)

where $\bar{\mathcal{L}}:[-2,-2]$ and $\bar{\mathcal{M}}:[2,0]$.

(ii) The (well-known) coadjoint representation of the Poincaré group may be
discussed from the perspective developed here by imposing in addition the
conditions $\eth^{2}\mathcal{T}=0=\bar{\eth}^{2}\mathcal{T}$ reducing super to
ordinary translations. Again, these equations define a four-dimensional
Lorentz invariant subspace because $\mathcal{T}$ has the required weights. At
the same time, one should consider equivalence classes
$\mathcal{P}\sim\mathcal{P}+\eth^{2}\mathcal{N}+\bar{\eth}^{2}\bar{\mathcal{N}}$,
where $\mathcal{N}:[-2,-1]$, $\bar{\mathcal{N}}:[2,-1]$ have the required
weights and which also defines a Lorentz invariant four-dimensional subspace.

####  5.3.2 Overcomplete set of functions

The representation of the generators $\mathcal{Y}_{m},\bar{\mathcal{Y}}_{m}$
of the Lorentz algebra on weighted scalars $\eta^{s,w}$ is explicitly given by

$\begin{split}\mathcal{Y}_{m}\cdot\eta^{s,w}&=\xi^{-m}\big{(}\xi\partial\eta^{s,w}+(\frac{s-w}{2}(1-m)+w\frac{\xi\bar{\xi}}{1+\xi\bar{\xi}})\eta^{s,w}\big{)},\\\
\bar{\mathcal{Y}}_{m}\cdot\eta^{s,w}&=\bar{\xi}^{-m}\big{(}\bar{\xi}\bar{\partial}\eta^{s,w}+(-\frac{s+w}{2}(1-m)+w\frac{\xi\bar{\xi}}{1+\xi\bar{\xi}})\eta^{s,w}\big{)}.\end{split}$
(5.24)

This follows from using (4.36) written in terms of $\eth$ and $\bar{\eth}$
together with (5.20).

For the associated conformal field $\phi_{h,\bar{h}}=P_{S}^{w}\eta^{s,w}$,
this simplifies to

$\begin{split}\tilde{\mathcal{Y}}_{m}\cdot\phi_{h,\bar{h}}&=\xi^{-m}\big{(}\xi\partial\phi_{h,\bar{h}}+h(1-m)\phi_{h,\bar{h}}\big{)},\\\
\tilde{\bar{\mathcal{Y}}}_{m}\cdot\phi_{h,\bar{h}}&=\bar{\xi}^{-m}\big{(}\bar{\xi}\bar{\partial}\phi_{h,\bar{h}}+\bar{h}(1-m)\phi_{h,\bar{h}}\big{)},\end{split}$
(5.25)

where $s=h-\bar{h},w=-h-\bar{h}$.

Rather than expanding the spin-weighted scalar $\eta^{s,w}$ in terms of
(unnormalized) spin-weighted spherical harmonics, one may also work with
suitable sets of over-complete functions. We follow [14], section 4.C (up to
conventions). Let $|s|\leq L$. For a fixed $L\in\mathbb{N}$, there is an
invertible matrix that relates the spin-weighted spherical harmonics
${}_{s}Y_{j,m}$ with $j\leq L$ to the functions

${}_{s}Z^{L}_{m_{1},m_{2}}=(1+\xi\bar{\xi})^{-L}\xi^{L-s-
m_{1}}\bar{\xi}^{L+s-m_{2}},\quad 0\leq m_{1}\leq L-s,\quad 0\leq m_{2}\leq
L+s.$ (5.26)

Depending on the conformal weight $w$, one may label these same functions as

${}_{h,\bar{h}}Z^{\tilde{L}}_{k,l}=P_{S}^{-w}{}_{h,\bar{h}}\tilde{Z}^{\tilde{L}}_{k,l},\quad{}_{h,\bar{h}}\tilde{Z}^{\tilde{L}}_{k,l}=(R\sqrt{2})^{h+\bar{h}}(1+\xi\bar{\xi})^{-\tilde{L}}\xi^{\tilde{L}-h-k}\bar{\xi}^{\tilde{L}-\bar{h}-l},$
(5.27)

where

$\tilde{L}=L+h+\bar{h},\quad k=m_{1}+h,\quad l=m_{2}+\bar{h},\quad h\leq
k\leq\tilde{L}-h,\quad\bar{h}\leq l\leq\tilde{L}-\bar{h}.$ (5.28)

In particular, if $h,\bar{h}$ are half-integer, so are $k,l$.

When taking for $\eta^{s,w}$ one of the functions
${}_{s,w}Z^{\tilde{L}}_{k,l}$, it follows that

$\begin{split}\mathcal{Y}_{m}\cdot{}_{h,\bar{h}}Z^{\tilde{L}}_{k,l}&=-(hm+k){}_{h,\bar{h}}Z^{\tilde{L}+1}_{k+m,l}+(\tilde{L}-(hm+k)){}_{h,\bar{h}}Z^{\tilde{L}+1}_{k+m+1,l+1},\\\
\bar{\mathcal{Y}}_{m}\cdot{}_{h,\bar{h}}Z^{\tilde{L}}_{k,l}&=-(\bar{h}m+l){}_{h,\bar{h}}Z^{\tilde{L}+1}_{k,l+m}+(\tilde{L}-(\bar{h}m+l)){}_{h,\bar{h}}Z^{\tilde{L}+1}_{k+1,l+m+1},\end{split}$
(5.29)

where the following (elementary) relations have been used,

${}_{h,\bar{h}}Z^{\tilde{L}}_{k,l}={}_{h,\bar{h}}Z^{\tilde{L}+1}_{k,l}+{}_{h,\bar{h}}Z^{\tilde{L}+1}_{k+1,l+1}.$
(5.30)

By construction, when taking for the conformal fields $\phi_{h,\bar{h}}$ the
functions ${}_{h,\bar{h}}\tilde{Z}^{\tilde{L}}_{k,l}$, the relations (5.29)
hold with the substitutions $\mathcal{Y}_{m}\to\tilde{\mathcal{Y}}_{m}$,
$\bar{\mathcal{Y}}_{m}\to\tilde{\bar{\mathcal{Y}}}_{m}$,
${}_{h,\bar{h}}Z^{\tilde{L}}_{k,l}\to{}_{h,\bar{h}}\tilde{Z}^{\tilde{L}}_{k,l}$.

When taking into account that

$\xi=\cot\frac{\theta}{2}e^{-i\phi},\quad\mu=\cos\theta,\quad\xi\bar{\xi}=\frac{1+\mu}{1-\mu}=\cot^{2}\frac{\theta}{2},\quad
1+\xi\bar{\xi}=\frac{2}{1-\mu},$ (5.31)

it follows that

$\begin{split}\langle{}_{s}Z^{L}_{m^{\prime}_{1},m^{\prime}_{2}},{}_{s}Z^{L}_{m_{1},m_{2}}\rangle&=\delta^{m_{1}+m_{2}^{\prime}}_{m}\delta^{m_{1}^{\prime}+m_{2}}_{m}\frac{1}{2}\int^{1}_{-1}d\mu(\frac{1-\mu}{2})^{2L}(\frac{1+\mu}{1-\mu})^{2L-m}\\\
&=\delta^{m_{1}+m_{2}^{\prime}}_{m}\delta^{m_{1}^{\prime}+m_{2}}_{m}\frac{m!(2L-m)!}{(2L+1)!}\\\
&=\delta^{m_{1}+m_{2}^{\prime}}_{m}\delta^{m_{1}^{\prime}+m_{2}}_{m}\beta(m+1,2L-m+1),\end{split}$
(5.32)

where $0\leq m\leq 2L$. Instead of reverting to angular variables for the
integrals, they may also be worked out directly in complex coordinates:

$\langle{}_{s}Z^{L}_{m^{\prime}_{1},m^{\prime}_{2}},{}_{s}Z^{L}_{m_{1},m_{2}}\rangle=\frac{i}{2\pi}\int
d\xi\wedge d\bar{\xi}\
(1+\xi\bar{\xi})^{-2L-2}\xi^{2L-m_{1}-m^{\prime}_{2}}\bar{\xi}^{2L-m^{\prime}_{1}-m_{2}}\\\
=\frac{1}{2L+1}\frac{1}{2\pi i}\int d\xi\wedge d\bar{\xi}\
\partial\big{(}(1+\xi\bar{\xi})^{-2L-1}\bar{\xi}^{2L-m^{\prime}_{1}-m_{2}-1}\big{)}\xi^{2L-m_{1}-m^{\prime}_{2}},$
(5.33)

where $0\leq m^{\prime}_{1}+m_{2}\leq 2L$, $0\leq m_{1}+m^{\prime}_{2}\leq
2L$. If $m_{1}+m^{\prime}_{2}=2L$, one then proceeds by using Stokes’ theorem
together with a kind of Cauchy residue theorem for the remaining line integral
(see for instance [35] in the current context). If $m_{1}+m^{\prime}_{2}<2L$,
one makes an integration by parts to lower the degree of $\xi$, and applies
the same reasoning for all integrals that involve a total $\partial$ with
expressions that have poles in $\bar{\xi}$.

Because of the weights $[s,w]$ and $[s,-w-2]$ of the spin-weighted spherical
harmonics involved in (5.6), the relevant integrals pair
${}_{h,\bar{h}}Z^{\tilde{L}}_{k,l}$ on the right with
${}_{h^{\prime},\bar{h}^{\prime}}Z^{\tilde{L}^{\prime}}_{k^{\prime},l^{\prime}}$
on the left, where $h^{\prime}=-\bar{h}+1$, $\bar{h}^{\prime}=-h+1$ and
$\tilde{L}^{\prime}=\tilde{L}-2h-2\bar{h}+2$,

$\langle{}_{-\bar{h}+1,-h+1}Z^{\tilde{L}-2h-2\bar{h}+2}_{k^{\prime},l^{\prime}},{}_{h,\bar{h}}Z^{\tilde{L}}_{k,l}\rangle=\delta^{k+l^{\prime}-1}_{m}\delta^{k^{\prime}+l-1}_{m}\beta(m+1,2(\tilde{L}-h-\bar{h})-m+1).$
(5.34)

By construction, the associated conformal fields
${}_{h,\bar{h}}\tilde{Z}^{\tilde{L}}_{k,l}$ have the same integrals when using
the measure $\tilde{d\mu}$ given in (5.11),

$\langle{}_{-\bar{h}+1,-h+1}\tilde{Z}^{\tilde{L}-2h-2\bar{h}+2}_{k^{\prime},l^{\prime}},{}_{h,\bar{h}}\tilde{Z}^{\tilde{L}}_{k,l}\tilde{\rangle}=\delta^{k+l^{\prime}-1}_{m}\delta^{k^{\prime}+l-1}_{m}\beta(m+1,2(\tilde{L}-h-\bar{h})-m+1).$
(5.35)

###  5.4 Structure constants

When taking as generators for supertranslations the unnormalized spherical
harmonics (5.17), one can now work out the structure constants of the
$\mathfrak{bms}_{4}$ algebra (4.34)-(4.35) by using properties (A.7) and
(A.10). For the first part of the algebra, they can be read off from the
commutation relations

$[\mathcal{Y}_{m},\mathcal{Y}_{n}]=(m-n)\mathcal{Y}_{m+n},\quad[\bar{\mathcal{Y}}_{m},\bar{\mathcal{Y}}_{n}]=(m-n)\bar{\mathcal{Y}}_{m+n},\quad[\mathcal{Y}_{m},\bar{\mathcal{Y}}_{n}]=0,$
(5.36)

while those involving supertranslation generators can be obtained from

$[\mathcal{Y}_{-1},\mathcal{T}_{j,m}]=+\frac{(j+2)(j+m)(j+m-1)}{4(2j+1)(2j-1)}\mathcal{T}_{j-1,m-1}-\frac{(j+m)}{2}\mathcal{T}_{j,m-1}\\\
+(j-1)\mathcal{T}_{j+1,m-1},$ (5.37)
$\displaystyle[\mathcal{Y}_{0},\mathcal{T}_{j,m}]=-\frac{(j+2)(j+m)(j-m)}{4(2j+1)(2j-1)}\mathcal{T}_{j-1,m}-\frac{m}{2}\mathcal{T}_{j,m}+(j-1)\mathcal{T}_{j+1,m},$
(5.38)
$[\mathcal{Y}_{1},\mathcal{T}_{j,m}]=+\frac{(j+2)(j-m)(j-m-1)}{4(2j+1)(2j-1)}\mathcal{T}_{j-1,m+1}+\frac{(j-m)}{2}\mathcal{T}_{j,m+1}\\\
+(j-1)\mathcal{T}_{j+1,m+1},$ (5.39)

The commutation relations involving $\bar{\mathcal{Y}}_{m}$ and
$\mathcal{T}_{j,m}$ may then be obtained by complex conjugation. They are
explicitly given by

$[\bar{\mathcal{Y}}_{-1},\mathcal{T}_{j,m}]=-\frac{(j+2)(j-m)(j-m-1)}{4(2j+1)(2j-1)}\mathcal{T}_{j-1,m+1}+\frac{(j-m)}{2}\mathcal{T}_{j,m+1}\\\
-(j-1)\mathcal{T}_{j+1,m+1},$ (5.40)
$[\bar{\mathcal{Y}}_{0},\mathcal{T}_{j,m}]=-\frac{(j+2)(j+m)(j-m)}{4(2j+1)(2j-1)}\mathcal{T}_{j-1,m}+\frac{m}{2}\mathcal{T}_{j,m}+(j-1)\mathcal{T}_{j+1,m},$
(5.41)
$[\bar{\mathcal{Y}}_{1},\mathcal{T}_{j,m}]=-\frac{(j+2)(j+m)(j+m-1)}{4(2j+1)(2j-1)}\mathcal{T}_{j-1,m-1}-\frac{(j+m)}{2}\mathcal{T}_{j,m-1}\\\
-(j-1)\mathcal{T}_{j+1,m-1}.$ (5.42)

Finally, the supertranslations generators commute with each other,

$[\mathcal{T}_{j,m},\mathcal{T}_{j^{\prime},m^{\prime}}]=0.$ (5.43)

In order to establish the relation to the commutation relations of section 2,
one defines

$l_{m}=\tilde{\mathcal{Y}}_{m}\partial,\quad\bar{l}_{m}=\tilde{{\bar{\mathcal{Y}}}}_{m}\bar{\partial},$
(5.44)

and takes into account equation (5.20) together with
$\xi=\cot\frac{\theta}{2}e^{-i\phi}$. If one makes the identification of the
generators as in (2.8) at $r\to\infty$ and $u=0$, it follows that

$\displaystyle
L_{z}=-i(l_{0}-\bar{l}_{0})=-i(\xi\partial-\bar{\xi}\bar{\partial}),$
$\displaystyle
K_{z}=-(l_{0}+\bar{l}_{0})=-(\xi\partial+\bar{\xi}\bar{\partial}),$
$\displaystyle
L^{+}=+(l_{1}+\bar{l}_{-1})=\partial+\bar{\xi}^{2}\bar{\partial},$
$\displaystyle L^{-}=+(\bar{l}_{1}+l_{-1})=\bar{\partial}+\xi^{2}\partial,$
(5.45) $\displaystyle
K^{+}=-(\bar{l}_{-1}-l_{1})=\partial-\bar{\xi}^{2}\bar{\partial},$
$\displaystyle K^{-}=-(l_{-1}-\bar{l}_{1})=\bar{\partial}-\xi^{2}\partial.$

This allows one to explicitly relate the commutation relations for the Lorentz
algebra in (5.36) to those in (2.2), respectively to the first part of (2.4).
The Poincaré generators are represented by

$H=1,\quad P_{z}=\frac{1-\xi\bar{\xi}}{2(1+\xi\bar{\xi})},\quad
P^{+}=-\frac{\bar{\xi}}{{1+\xi\bar{\xi}}},\quad
P^{-}=\frac{\xi}{{1+\xi\bar{\xi}}}.$ (5.46)

For functions $f$ on the sphere, we now get instead of (2.9)

$\begin{split}[L_{z},f]&=L_{z}(f),\quad[L^{\pm},f]=L^{\pm}(f),\quad[K_{z},f]=K_{z}(f)+\frac{1-\xi\bar{\xi}}{1+\xi\bar{\xi}}f,\\\
[K^{+},f]&=K^{+}(f)+\frac{2\bar{\xi}}{1+\xi\bar{\xi}}f,\quad[K^{-},f]=K^{-}(f)+\frac{2\xi}{1+\xi\bar{\xi}}f.\end{split}$
(5.47)

When applied to the four Poincaré generators in (5.46), this reproduces the
second part of (2.4).

In order to relate the action of the Lorentz generators on the
supertranslation generators (5.17) given in (5.37)-(5.42) to the more standard
form (2.19)-(2.22), we may start from the (first equality in the) relations
(5.45) and use (5.37)-(5.42) to show that

$\begin{split}[L_{z},\mathcal{T}_{j,m}]=&im\mathcal{T}_{j,m},\quad[L^{\pm},\mathcal{T}_{j,m}]=\pm(j\mp
m)\mathcal{T}_{j,m\pm 1},\\\
[K_{z},\mathcal{T}_{j,m}]=&-2(j-1)\mathcal{T}_{j+1,m}+\frac{(j+2)(j+m)(j-m)}{2(2j+1)(2j-1)}\mathcal{T}_{j-1,m},\\\
[K^{\pm},\mathcal{T}_{j,m}]=&\pm 2(j-1)\mathcal{T}_{j+1,m\pm
1}\pm\frac{(j+2)(j\mp m)(j\mp m-1)}{2(2j+1)(2j-1)}\mathcal{T}_{j-1,m\pm
1}.\end{split}$ (5.48)

When taking the normalization (A.3) into account, we then recover the
commutation relations (2.19)-(2.22). Note that the commutation relations of
$L_{z}$ with the supertranslations generators are particularly simple since
the latter are expressed in terms of (unnormalized) spherical harmonics.

The choice of basis for the Lorentz algebra in (5.36) is adapted to the
$\mathfrak{sl}(2,\mathbb{R})\times\mathfrak{sl}(2,\mathbb{R})$ decomposition.
It is thus useful to organize the supertranslation generators, or more
generally, the functions on the sphere, accordingly. This will also allow us
to compare directly with the realization on the punctured complex plane to be
discussed below. Hence, instead of providing the commutation relation between
the Lorentz and supertranslation generators, one may replace the latter by the
overcomplete set of functions adapted to $\mathcal{T}$ of weights $[0,1]$,

$\mathcal{T}^{\tilde{L}}_{k,l}={}_{-\frac{1}{2},-\frac{1}{2}}Z^{\tilde{L}}_{k,l}.$
(5.49)

One then finds

$\begin{split}[\mathcal{Y}_{m},\mathcal{T}^{\tilde{L}}_{k,l}]=(\frac{m}{2}-k)\mathcal{T}^{\tilde{L}+1}_{k+m,l}+(\tilde{L}+\frac{m}{2}-k)\mathcal{T}^{\tilde{L}+1}_{k+m+1,l+1},\\\
[\bar{\mathcal{Y}}_{m},\mathcal{T}^{\tilde{L}}_{k,l}]=(\frac{m}{2}-l)\mathcal{T}^{\tilde{L}+1}_{k,l+m}+(\tilde{L}+\frac{m}{2}-l)\mathcal{T}^{\tilde{L}+1}_{k+1,l+m+1},\end{split}$
(5.50)
$[\mathcal{T}^{\tilde{L}}_{k,l},\mathcal{T}^{\tilde{L}}_{k^{\prime},l^{\prime}}]=0.$
(5.51)

###  5.5 Coadjoint representation of the algebra

The coadjoint representation may now be written explicitly using (3.13).
Alternatively, it can be derived using the results of subsection 4.2 together
with (A.7) and (A.10). One finds,

${\rm
ad}^{*}_{\mathcal{Y}_{m}}\mathcal{Y}^{n}_{*}=(-2m+n)\mathcal{Y}^{n-m}_{*},\quad{\rm
ad}^{*}_{\mathcal{Y}_{m}}\bar{\mathcal{Y}}^{n}_{*}=0,$ (5.52) ${\rm
ad}^{*}_{\bar{\mathcal{Y}}_{m}}\bar{\mathcal{Y}}^{n}_{*}=(-2m+n)\bar{\mathcal{Y}}^{n-m}_{*},\quad{\rm
ad}^{*}_{\bar{\mathcal{Y}}_{m}}\mathcal{Y}^{n}_{*}=0,$ (5.53)
$\displaystyle{\rm ad}^{*}_{\mathcal{Y}_{-1}}\mathcal{T}^{j,m}_{*}=$
$\displaystyle-\frac{(j+3)(j+m+2)(j+m+1)}{4(2j+3)(2j+1)}\mathcal{T}^{j+1,m+1}_{*}$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{(j+m+1)}{2}\mathcal{T}^{j,m+1}_{*}-(j-2)\mathcal{T}^{j-1,m+1}_{*},$
(5.54) $\displaystyle{\rm ad}^{*}_{\mathcal{Y}_{0}}\mathcal{T}^{j,m}_{*}=$
$\displaystyle\frac{(j+3)(j+m+1)(j-m+1)}{4(2j+3)(2j+1)}\mathcal{T}^{j+1,m}_{*}+\frac{m}{2}\mathcal{T}^{j,m}_{*}-(j-2)\mathcal{T}^{j-1,m}_{*},$
(5.55) $\displaystyle{\rm ad}^{*}_{\mathcal{Y}_{1}}\mathcal{T}^{j,m}_{*}=$
$\displaystyle-\frac{(j+3)(j-m+2)(j-m+1)}{4(2j+3)(2j+1)}\mathcal{T}^{j+1,m-1}_{*}$
$\displaystyle\qquad\qquad\qquad\qquad-\frac{(j-m+1)}{2}\mathcal{T}^{j,m-1}_{*}-(j-2)\mathcal{T}^{j-1,m-1}_{*},$
(5.56) $\displaystyle{\rm
ad}^{*}_{\bar{\mathcal{Y}}_{-1}}\mathcal{T}^{j,m}_{*}=$
$\displaystyle+\frac{(j+3)(j-m+2)(j-m+1)}{4(2j+3)(2j+1)}\mathcal{T}^{j+1,m-1}_{*}$
$\displaystyle\qquad\qquad\qquad\qquad-\frac{(j-m+1)}{2}\mathcal{T}^{j,m-1}_{*}+(j-2)\mathcal{T}^{j-1,m-1}_{*},$
(5.57) $\displaystyle{\rm
ad}^{*}_{\bar{\mathcal{Y}}_{0}}\mathcal{T}^{j,m}_{*}=$
$\displaystyle+\frac{(j+3)(j+m+1)(j-m+1)}{4(2j+3)(2j+1)}\mathcal{T}^{j+1,m}_{*}-\frac{m}{2}\mathcal{T}^{j,m}_{*}-(j-2)\mathcal{T}^{j-1,m}_{*},$
(5.58) $\displaystyle{\rm
ad}^{*}_{\bar{\mathcal{Y}}_{1}}\mathcal{T}^{j,m}_{*}=$
$\displaystyle+\frac{(j+3)(j+m+2)(j+m+1)}{4(2j+3)(2j+1)}\mathcal{T}^{j+1,m+1}_{*}$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{(j+m+1)}{2}\mathcal{T}^{j,m+1}_{*}+(j-2)\mathcal{T}^{j-1,m+1}_{*},$
(5.59)

${\rm ad}^{*}_{\mathcal{T}_{j,m}}\mathcal{Y}^{p}_{*}=0={\rm
ad}^{*}_{\mathcal{T}_{j,m}}\bar{\mathcal{Y}}^{p}_{*},$ (5.60)

$\begin{split}&{\rm
ad}^{*}_{\mathcal{T}_{j,m}}\mathcal{T}_{*}^{j^{\prime},m^{\prime}}=\\\
&\big{(}-\frac{(j+2)(j+m)(j+m-1)}{4(2j+1)(2j-1)}\delta^{j^{\prime}}_{j-1}+\frac{(j+m)}{2}\delta^{j^{\prime}}_{j}-(j-1)\delta^{j^{\prime}}_{j+1}\big{)}\delta^{m^{\prime}}_{m-1}\mathcal{Y}_{*}^{-1}\\\
&+\big{(}-\frac{(j+2)(j+m)(j-m)}{4(2j+1)(2j-1)}\delta^{j^{\prime}}_{j-1}-\frac{m}{2}\delta^{j^{\prime}}_{j}+(j-1)\delta^{j^{\prime}}_{j+1}\big{)}\delta^{m^{\prime}}_{m}\mathcal{Y}_{*}^{0}\\\
&+\big{(}-\frac{(j+2)(j-m)(j-m-1)}{4(2j+1)(2j-1)}\delta^{j^{\prime}}_{j-1}-\frac{(j-m)}{2}\delta^{j^{\prime}}_{j}-(j-1)\delta^{j^{\prime}}_{j+1}\big{)}\delta^{m^{\prime}}_{m+1}\mathcal{Y}_{*}^{1}\\\
&+\big{(}+\frac{(j+2)(j-m)(j-m-1)}{4(2j+1)(2j-1)}\delta^{j^{\prime}}_{j-1}-\frac{(j-m)}{2}\delta^{j^{\prime}}_{j}+(j-1)\delta^{j^{\prime}}_{j+1}\big{)}\delta^{m^{\prime}}_{m+1}\bar{\mathcal{Y}}^{-1}_{*}\\\
&+\big{(}-\frac{(j+2)(j+m)(j-m)}{4(2j+1)(2j-1)}\delta^{j^{\prime}}_{j-1}+\frac{m}{2}\delta^{j^{\prime}}_{j}+(j-1)\delta^{j^{\prime}}_{j+1}\big{)}\delta^{m^{\prime}}_{m}\bar{\mathcal{Y}}^{0}_{*}\\\
&+\big{(}+\frac{(j+2)(j+m)(j+m-1)}{4(2j+1)(2j-1)}\delta^{j^{\prime}}_{j-1}+\frac{(j+m)}{2}\delta^{j^{\prime}}_{j}+(j-1)\delta^{j^{\prime}}_{j+1}\big{)}\delta^{m^{\prime}}_{m-1}\bar{\mathcal{Y}}^{1}_{*}.\end{split}$
(5.61)

In terms of the overcomplete sets of functions, if one uses
${}_{\frac{3}{2},\frac{3}{2}}Z^{\tilde{L}+4}_{k,l}$ rather than
$\mathcal{T}_{*}^{j,m}$ for the expansion of $\mathcal{P}$, one may use (5.29)
to replace equations (5.54)-(5.59) through

$\begin{split}\mathcal{Y}_{m}\cdot{}_{\frac{3}{2},\frac{3}{2}}Z^{\tilde{L}+4}_{k,l}&=-(\frac{3}{2}m+k)\mathcal{}_{\frac{3}{2},\frac{3}{2}}Z^{\tilde{L}+5}_{k+m,l}+(\tilde{L}+4-(\frac{3}{2}m+k)){}_{\frac{3}{2},\frac{3}{2}}Z^{\tilde{L}+5}_{k+m+1,l+1},\\\
\bar{\mathcal{Y}}_{m}\cdot{}_{\frac{3}{2},\frac{3}{2}}Z^{\tilde{L}+4}_{k,l}&=-(\frac{3}{2}m+l){}_{\frac{3}{2},\frac{3}{2}}Z^{\tilde{L}+5}_{k,l+m}+(\tilde{L}+4-(\frac{3}{2}m+l)){}_{\frac{3}{2},\frac{3}{2}}Z^{\tilde{L}+5}_{k+1,l+m+1},\end{split}$
(5.62)

while equations (5.60) become

${\rm ad}^{*}_{\mathcal{T}^{\tilde{L}}_{k,l}}\mathcal{Y}^{p}_{*}=0={\rm
ad}^{*}_{\mathcal{T}^{\tilde{L}}_{k,l}}\bar{\mathcal{Y}}^{p}_{*}.$ (5.63)

Finally, it also follows from

$\langle{}_{\frac{3}{2},\frac{3}{2}}Z^{\tilde{L}+4}_{k^{\prime},l^{\prime}},\mathcal{T}^{\tilde{L}}_{k,l}\rangle=\delta^{k^{\prime}+l-1}_{m}\delta^{k+l^{\prime}-1}_{m}\beta(m+1,2\tilde{L}+3-m),$
(5.64)

the commutation relations (5.50), (5.51) and the definition of the coadjoint
representation that

${\rm ad}^{*}_{\mathcal{T}^{\tilde{L}}_{k,l}}\
{}_{\frac{3}{2},\frac{3}{2}}Z^{\tilde{L}+5}_{k^{\prime},l^{\prime}}=\big{(}\frac{k^{\prime}-3k+l-l^{\prime}}{2}\beta(k^{\prime}+l,2\tilde{L}+6-k^{\prime}-l)\\\
+(\tilde{L}+\frac{k^{\prime}-3k+l-l^{\prime}}{2})\beta(k^{\prime}+l+1,2\tilde{L}+5-k^{\prime}-l)\big{)}\mathcal{Y}_{*}^{k^{\prime}-k+l-l^{\prime}}\\\
+\big{(}\frac{l^{\prime}-3l+k-k^{\prime}}{2}\beta(l^{\prime}+k,2\tilde{L}+6-l^{\prime}-k)\\\
+(\tilde{L}+\frac{l^{\prime}-3l+k-k^{\prime}}{2})\beta(l^{\prime}+k+1,2\tilde{L}+5-l^{\prime}-k)\big{)}\bar{\mathcal{Y}}_{*}^{l^{\prime}-l+k-k^{\prime}}.$
(5.65)

##  6 Realization on the punctured complex plane

###  6.1 Generalities

Since the whole structure is Weyl invariant, one may start from the sphere
with radius $R$ and perform a Weyl rescaling as in (4.10) with

$e^{-E(\xi,\bar{\xi})}=\frac{\sqrt{2}}{1+\xi\bar{\xi}},$ (6.1)

followed by the (conformal) coordinate transformations that consists of a
simple rescaling $\xi=R^{-1}z$, $\bar{\xi}=R^{-1}\bar{z}$, so that the metric
becomes

$ds^{2}=-2dzd\bar{z}.$ (6.2)

The next step is to remove the points at infinity and at the origin to go to
the 1-punctured complex plane $\mathbb{C}_{0}$. This changes the allowed space
of functions. Conformal coordinate transformations are of the form

$z^{\prime}=z^{\prime}(z),\quad\bar{z}^{\prime}=\bar{z}^{\prime}(\bar{z}),$
(6.3)

where the globally well-defined ones that are connected to the identity are
$z^{\prime}=az,a\in\mathbb{C},a\neq 0$. The derivative operators $\eth$ and
$\bar{\eth}$ defined in (4.28) simply become $\partial$ and $\bar{\partial}$,
respectively. There is no difference between conformal fields and weighted
scalars. Indeed, freezing the conformal factor as in (4.12) with
$P_{F}=1=\bar{P}_{F}$ yields

$e^{E(x^{\prime})}=\frac{\partial z^{\prime}}{\partial
z}~{}\Longleftrightarrow~{}e^{E_{R}(x^{\prime})}=\left(\frac{\partial
z^{\prime}}{\partial
z}\frac{\partial\bar{z}^{\prime}}{\partial\bar{z}}\right)^{\frac{1}{2}},~{}e^{iE_{I}(x^{\prime})}=\left(\frac{\partial
z^{\prime}/\partial
z}{\partial\bar{z}^{\prime}/\partial\bar{z}}\right)^{\frac{1}{2}},$ (6.4)

which implies that conformal fields (of vanishing Weyl weights) and their
associated weighted scalars through the map (4.16) are equal and transform in
the same way (compare (4.15) and (4.18) by taking (6.4) into account). In the
following, we use the notation for conformal fields $\phi_{h,\bar{h}}$.

We assume here that conformal fields on the punctured complex plane may be
expanded in series as

$\phi_{h,\bar{h}}(z,\bar{z})=\sum_{k,l}a_{k,l}\
{}_{h,\bar{h}}\tilde{Z}_{k,l},\quad{}_{h,\bar{h}}\tilde{Z}_{k,l}=z^{-h-k}\bar{z}^{-\bar{h}-l},$
(6.5)

where the coefficients $a_{k,l}\in\mathbb{C}$ and satisfy suitable conditions
that we will not discuss in detail here (see e.g. [36] for more details). We
also assume that $h,\bar{h}$ are either integer or half-integer. In the former
case $k,l\in\mathbb{Z}$, whereas in the latter case
$k,l\in\frac{1}{2}+\mathbb{Z}$. Other choices are also possible. The reason we
are choosing Neveu-Schwarz conditions here is that, up to factors of
$(1+z\bar{z})$, the functions that appear here then include those that have
appeared naturally in the case of the sphere.

Residues with respect to $z$ and $\bar{z}$ are defined as

$\text{Res}_{z}[\phi_{h,\bar{h}}](\bar{z})=\sum_{l}a_{1-h,l}\bar{z}^{-\bar{h}-l},\quad\text{Res}_{\bar{z}}[\phi_{h,\bar{h}}](z)=\sum_{k}a_{k,1-\bar{h}}z^{-h-k}.$
(6.6)

This allows one to define pairing

$\langle\psi_{-\bar{h}+1,-h+1},\phi_{h,\bar{h}}\tilde{\rangle}=\text{Res}_{z}\text{Res}_{\bar{z}}[\overline{\psi_{-\bar{h}+1,-h+1}}\phi_{h,\bar{h}}].$
(6.7)

This pairing is non-degenerate, and since
$\text{Res}_{z}[\partial\phi]=0=\text{Res}_{\bar{z}}[\bar{\partial}\phi]$, it
annihilates total derivatives $\partial$ and $\bar{\partial}$, as it should.
The pairing can then be defined as

$\langle([\tilde{\mathcal{J}}],[\tilde{\bar{\mathcal{J}}}],\tilde{\mathcal{P}}),(\tilde{\mathcal{Y}},\tilde{\bar{\mathcal{Y}}},\tilde{\mathcal{T}})\tilde{\rangle}=\langle\tilde{\mathcal{J}},\tilde{\mathcal{Y}}\tilde{\rangle}+\langle\tilde{\bar{\mathcal{J}}},\tilde{\bar{\mathcal{Y}}}\tilde{\rangle}+\langle\tilde{\mathcal{P}},\tilde{\mathcal{T}}\tilde{\rangle}.$
(6.8)

###  6.2 Adjoint and coadjoint representations of the group

The formulas for the adjoint and coadjoint representations of the group are
the same than those for the conformal fields on the sphere, except for the
general Jacobians $\partial z/\partial
z^{\prime},\partial\bar{z}/\partial\bar{z}^{\prime}$,

$\begin{split}&\tilde{\mathcal{Y}}^{\prime}(z^{\prime})=\big{(}\frac{\partial
z}{\partial z^{\prime}}\big{)}^{-1}\tilde{\mathcal{Y}}(z),\\\
&\tilde{\bar{\mathcal{Y}}}^{\prime}(\bar{z}^{\prime})=\big{(}\frac{\partial\bar{z}}{\partial\bar{z}^{\prime}}\big{)}^{-1}\tilde{\bar{\mathcal{Y}}}(z),\\\
&\tilde{\beta}^{\prime}(x^{\prime})=\big{(}\frac{\partial z}{\partial
z^{\prime}}\big{)}^{-\frac{1}{2}}\big{(}\frac{\partial\bar{z}}{\partial\bar{z}^{\prime}}\big{)}^{-\frac{1}{2}}\Big{(}\tilde{\beta}-\big{(}\tilde{\mathcal{Y}}\partial\tilde{\alpha}-\frac{1}{2}\tilde{\alpha}\partial\tilde{\mathcal{Y}}+{\rm
c.c.}\big{)}\Big{)}(x),\end{split}$ (6.9)
$\begin{split}&\tilde{\mathcal{J}}^{\prime}(x^{\prime})=\big{(}\frac{\partial
z}{\partial
z^{\prime}}\big{)}^{1}\big{(}\frac{\partial\bar{z}}{\partial\bar{z}^{\prime}}\big{)}^{2}\Big{(}\tilde{\mathcal{J}}+(\frac{1}{2}\tilde{\mathcal{T}}\bar{\partial}\tilde{\mathcal{P}}+\frac{3}{2}\bar{\partial}\tilde{\mathcal{T}}\tilde{\mathcal{P}})\Big{)}(x)\\\
&\tilde{\bar{\mathcal{J}}}^{\prime}(x^{\prime})=\big{(}\frac{\partial
z}{\partial
z^{\prime}}\big{)}^{2}\big{(}\frac{\partial\bar{z}}{\partial\bar{z}^{\prime}}\big{)}^{1}\Big{(}\tilde{\bar{\mathcal{J}}}+(\frac{1}{2}\tilde{\mathcal{T}}\partial\tilde{\mathcal{P}}+\frac{3}{2}\partial\tilde{\mathcal{T}}\tilde{\mathcal{P}})\Big{)}(x)\\\
&\tilde{\mathcal{P}}^{\prime}(x^{\prime})=\big{(}\frac{\partial z}{\partial
z^{\prime}}\big{)}^{\frac{3}{2}}\big{(}\frac{\partial\bar{z}}{\partial\bar{z}^{\prime}}\big{)}^{\frac{3}{2}}\tilde{\mathcal{P}}(x).\end{split}$
(6.10)

###  6.3 Expansions

In terms of the basis functions defined in (6.5),

$\langle{}_{-\bar{h}+1,-h+1}\tilde{Z}_{k^{\prime},l^{\prime}},{}_{h,\bar{h}}\tilde{Z}_{k,l}\tilde{\rangle}=\delta_{l^{\prime}+k}^{0}\delta_{k^{\prime}+l}^{0}.$
(6.11)

In particular, the dual becomes

$({}_{h,\bar{h}}\tilde{Z}_{k,l})^{*}={}_{-\bar{h}+1,-h+1}\tilde{Z}_{-l,-k}\iff(z^{-h-k}\bar{z}^{-\bar{h}-l})^{*}=z^{\bar{h}-1+l}\bar{z}^{h-1+k}.$
(6.12)

The basis for the conformal fields relevant for the algebra is

$\tilde{\mathcal{Y}}_{m}={}_{-1,0}\tilde{Z}_{m,0}=z^{1-m},\quad\tilde{\bar{\mathcal{Y}}}_{m}={}_{0,-1}\tilde{Z}_{0,m}=\bar{z}^{1-m},\quad\mathscr{\tilde{T}}_{k,l}={}_{-\frac{1}{2},-\frac{1}{2}}\tilde{Z}_{k,l}=z^{\frac{1}{2}-k}\bar{z}^{\frac{1}{2}-l},$
(6.13)

where $m,k+\frac{1}{2},l+\frac{1}{2}\in\mathbb{Z}$. We have

$\tilde{\mathcal{Y}}=\sum_{m\in\mathbb{Z}}\tilde{y}_{m}\tilde{\mathcal{Y}}_{m},\quad\tilde{\bar{\mathcal{Y}}}=\sum_{m\in\mathbb{Z}}\tilde{\bar{y}}_{m}\tilde{\bar{\mathcal{Y}}}_{m},\quad\tilde{\mathcal{T}}=\sum_{k,l\in\frac{1}{2}+\mathbb{Z}}\tilde{t}_{k,l}\mathscr{\tilde{T}}_{k,l},$
(6.14)

where $\bar{\tilde{t}}_{k,l}=\tilde{t}_{l,k}$ since $\tilde{\mathcal{T}}$ is
real. For the coadjoint representation, one finds from (6.12) (or from the
definition with equivalence classes when taking into account that
$z^{-1},\bar{z}^{-1}$ are not equivalent to zero because they are not the
derivative of a monomial but of the logarithm),

$\tilde{\mathcal{Y}}^{m}_{*}=z^{-1}{\bar{z}}^{-2+m},\quad\tilde{\bar{\mathcal{Y}}}_{*}^{m}={z}^{-2+m}\bar{z}^{-1},\quad\mathscr{\tilde{T}}^{k,l}_{*}={z}^{-\frac{3}{2}+l}{\bar{z}}^{-\frac{3}{2}+k},$
(6.15)

we have

$\tilde{\mathcal{J}}=\sum_{m\in\mathbb{Z}}\tilde{j}_{m}\tilde{\mathcal{Y}}^{m}_{*},\quad\tilde{\bar{\mathcal{J}}}=\sum_{m\in\mathbb{Z}}\tilde{\bar{j}}_{m}\tilde{\bar{\mathcal{Y}}}_{*}^{m},\quad\tilde{\mathcal{P}}=\sum_{k,l\in\frac{1}{2}+\mathbb{Z}}\tilde{p}_{k,l}\mathscr{\tilde{T}}^{k,l}_{*},$
(6.16)

where $\bar{\tilde{p}}_{k,l}=p_{l,k}$ since $\tilde{\mathcal{P}}$ is real.

In terms of basis elements, the representation (4.48) becomes

$\tilde{\mathcal{Y}}_{m}\cdot{}_{h,\bar{h}}\tilde{Z}_{k,l}=-(hm+k)\
{}_{h,\bar{h}}\tilde{Z}_{k+m,l},\quad\tilde{\bar{\mathcal{Y}}}_{m}\cdot{}_{h,\bar{h}}\tilde{Z}_{k,l}=-(\bar{h}m+l)\
{}_{h,\bar{h}}\tilde{Z}_{k,l+m},$ (6.17)

while

$\begin{split}\tilde{\mathcal{Y}}_{m}\cdot({}_{h,\bar{h}}\tilde{Z}_{k,l})^{*}&=[(\bar{h}-1)m+l]\
({}_{h,\bar{h}}\tilde{Z}_{k,l-m})^{*},\\\
\tilde{\bar{\mathcal{Y}}}_{m}\cdot({}_{h,\bar{h}}\tilde{Z}_{k,l})^{*}&=[(h-1)m+k]\
({}_{h,\bar{h}}\tilde{Z}_{k-m,l})^{*}.\end{split}$ (6.18)

###  6.4 Structure constants

As discussed in subsection 4.2, all the results stated there can be readily
expressed in terms of conformal fields. In particular, the
$\mathfrak{bms}_{4}$ algebra (4.34)-(4.35) simplifies to

$[(\tilde{\mathcal{Y}}_{1},\tilde{\bar{\mathcal{Y}}}_{1},\tilde{\mathcal{T}}_{1}),(\tilde{\mathcal{Y}}_{2},\tilde{\bar{\mathcal{Y}}}_{2},\tilde{\mathcal{T}}_{2})]=(\hat{\tilde{\mathcal{Y}}},\hat{\tilde{\bar{\mathcal{Y}}}},\hat{\tilde{\mathcal{T}}}),$
(6.19)

where

$\left\\{\begin{aligned}
\hat{\tilde{\mathcal{Y}}}&=\tilde{\mathcal{Y}}_{1}\partial\tilde{\mathcal{Y}}_{2}-\tilde{\mathcal{Y}}_{2}\partial\tilde{\mathcal{Y}}_{1}\,,\\\
\hat{\tilde{\mathcal{T}}}&=\tilde{\mathcal{Y}}_{1}\partial\tilde{\mathcal{T}}_{2}-\frac{1}{2}\partial\tilde{\mathcal{Y}}_{1}\tilde{\mathcal{T}}_{2}-(1\leftrightarrow
2)+{\rm c.c.}\,.\\\ \end{aligned}\right.$ (6.20)

In the basis (6.13), the commutation relations become

$\begin{split}&[\tilde{\mathcal{Y}}_{m},\tilde{\mathcal{Y}}_{n}]=(m-n)\tilde{\mathcal{Y}}_{m+n},\quad[\tilde{\bar{\mathcal{Y}}}_{m},\tilde{\bar{\mathcal{Y}}}_{n}]=(m-n)\tilde{\bar{\mathcal{Y}}}_{m+n},\\\
&[\tilde{\mathcal{Y}}_{m},\mathscr{\tilde{T}}_{k,l}]=(\frac{1}{2}m-k)\mathscr{\tilde{T}}_{m+k,l},\quad[\tilde{\bar{\mathcal{Y}}}_{m},\mathscr{\tilde{T}}_{k,l}]=(\frac{1}{2}m-l)\mathscr{\tilde{T}}_{k,m+l}\\\
&[\tilde{\mathcal{Y}}_{m},\tilde{\bar{\mathcal{Y}}}_{n}]=0=[\mathscr{\tilde{T}}_{k,l},\mathscr{\tilde{T}}_{r,s}].\end{split}$
(6.21)

###  6.5 Coadjoint representation of the algebra

The coadjoint representation in the basis (6.15) may be obtained from the
structure constants of the algebra contained in (6.21) using (3.13).
Alternatively, it can be derived using the results of subsection 4.2 together
with the explicit expressions of the generators and their duals (6.15), and
also from (6.18). Explicitly,

$\begin{split}&{\rm
ad}^{*}_{\tilde{\mathcal{Y}}_{m}}\tilde{\mathcal{Y}}^{n}_{*}=(-2m+n)\tilde{\mathcal{Y}}^{n-m}_{*},\quad{\rm
ad}^{*}_{\tilde{\bar{\mathcal{Y}}}_{m}}\tilde{\bar{\mathcal{Y}}}^{n}_{*}=(-2m+n)\tilde{\bar{\mathcal{Y}}}_{*}^{n-m},\\\
&{\rm
ad}^{*}_{\tilde{\mathcal{Y}}_{m}}\mathscr{\tilde{T}}^{k,l}_{*}=(-\frac{3}{2}m+k)\mathscr{\tilde{T}}^{k-m,l}_{*},\quad{\rm
ad}^{*}_{\tilde{\bar{\mathcal{Y}}}_{m}}\mathscr{\tilde{T}}^{k,l}_{*}=(-\frac{3}{2}m+l)\mathscr{\tilde{T}}^{k,l-m}_{*},\\\
&{\rm
ad}^{*}_{\mathscr{\tilde{T}}_{k,l}}\mathscr{\tilde{T}}^{r,s}_{*}=(\frac{r-3k}{2})\delta^{s}_{l}\tilde{\mathcal{Y}}_{*}^{r-k}+(\frac{s-3l}{2})\delta^{r}_{k}\tilde{\bar{\mathcal{Y}}}^{s-l}_{*},\\\
&{\rm
ad}^{*}_{\tilde{\mathcal{Y}}_{m}}\tilde{\bar{\mathcal{Y}}}^{n}_{*}=0={\rm
ad}^{*}_{\tilde{\bar{\mathcal{Y}}}_{m}}\tilde{\mathcal{Y}}^{n}_{*},\quad{\rm
ad}^{*}_{\mathscr{\tilde{T}}_{k,l}}\tilde{\mathcal{Y}}^{m}_{*}=0={\rm
ad}^{*}_{\mathscr{\tilde{T}}_{k,l}}\tilde{\bar{\mathcal{Y}}}^{m}_{*}.\end{split}$
(6.22)

##  7 Comments on the cylinder

The mapping from the punctured plane to the vertical cylinder is standard in
the context of conformal field theory. It is defined through

$z=e^{-i\frac{2\pi}{L_{1}}w},\quad w=w_{1}+iw_{2},\quad w_{1}\sim
w_{1}+L_{1}.$ (7.1)

According to (4.51), conformal fields on the cylinder are related to those on
the punctured plane through

$\phi^{C_{V}}_{h,\bar{h}}(w,\bar{w})=\big{(}-i\frac{2\pi}{L_{1}}z\big{)}^{h}\big{(}i\frac{2\pi}{L_{1}}\bar{z}\big{)}^{\bar{h}}\
\phi_{h,\bar{h}}(z,\bar{z}).$ (7.2)

When naively substituting the expansion adapted to the punctured plane (6.5),
the associated expansion on the cylinder is

$\phi^{C_{V}}_{h,\bar{h}}(w,\bar{w})=\sum_{k,l}\ a_{k,l}\
{}_{h,\bar{h}}Z^{C_{V}}_{k,l},\quad{}_{h,\bar{h}}Z^{C_{V}}_{k,l}=i^{\bar{h}-h}\big{(}\frac{2\pi}{L_{1}}\big{)}^{h+\bar{h}}e^{i\frac{2\pi}{L_{1}}kw}\
e^{-i\frac{2\pi}{L_{1}}l\bar{w}},$ (7.3)

with $k,l$ semi-integer when $h,\bar{h}$ are semi-integer. As usual for Neveu-
Schwarz boundary conditions, it follows that for half-integer conformal
weights, holomorphic or anti-holomorphic fields on the cylinder are anti-
periodic.

The generators (6.13) of the $\mathfrak{bms}_{4}$ algebra become

$\mathcal{Y}^{C_{V}}_{m}=i\big{(}\frac{2\pi}{L_{1}}\big{)}^{-1}e^{i\frac{2\pi}{L_{1}}mw},\
{\bar{\mathcal{Y}}}^{C_{V}}_{m}=-i\big{(}\frac{2\pi}{L_{1}}\big{)}^{-1}e^{-i\frac{2\pi}{L_{1}}m\bar{w}},\
{\mathcal{T}}^{C_{V}}_{k,l}=\big{(}\frac{2\pi}{L_{1}}\big{)}^{-1}e^{i\frac{2\pi}{L_{1}}kw}e^{-i\frac{2\pi}{L_{1}}l\bar{w}},$
(7.4)

while those of the coadjoint representation (6.15) become

$\mathcal{Y}^{m}_{C_{V}*}=i\big{(}\frac{2\pi}{L_{1}}\big{)}^{3}e^{i\frac{2\pi}{L_{1}}m\bar{w}},\
{\bar{\mathcal{Y}}}^{m}_{C_{V}*}=-i\big{(}\frac{2\pi}{L_{1}}\big{)}^{3}e^{-i\frac{2\pi}{L_{1}}mw},\
{\mathcal{T}}^{k,l}_{C_{V}*}=\big{(}\frac{2\pi}{L_{1}}\big{)}^{3}e^{-i\frac{2\pi}{L_{1}}lw}e^{i\frac{2\pi}{L_{1}}k\bar{w}}.$
(7.5)

By construction, the commutation relations of the elements in (7.4) are
unchanged: they are obtained from (6.21) by adding a superscript $C_{V}$ to
the generators.

For the coadjoint representation, matters are more subtle. It remains true
that the vector space generated by the elements of (7.5) is a representation
of $\mathfrak{bms}_{4}$ in the sense of Remark (v) of section 4.2, which is
explicitly given by adding a superscript, respectively subscript, $C_{V}$ to
the generators of (6.22). This is not however the coadjoint representation
since there are issues with the pairing on the infinite cylinder. Indeed,

$\langle\psi^{C_{V}}_{-\bar{h}+1,-h+1},\phi^{C_{V}}_{h,\bar{h}}\rangle_{C_{V}}\equiv\frac{1}{8\pi^{2}}\int
idw\wedge d\bar{w}\ \big{[}\overline{\psi^{C_{V}}_{-\bar{h}+1,-h+1}}\
\phi^{C_{V}}_{h,\bar{h}}\big{]}\\\
=\frac{1}{4\pi^{2}}\int_{0}^{L_{1}}dw_{1}\int^{\infty}_{-\infty}dw_{2}\
\big{[}\overline{\psi^{C_{V}}_{-\bar{h}+1,-h+1}}\
\phi^{C_{V}}_{h,\bar{h}}\big{]},$ (7.6)

and in particular,

$\langle{}_{-\bar{h}+1,-h+1}Z^{C_{V}}_{k^{\prime},l^{\prime}},{}_{h,\bar{h}}Z^{C_{V}}_{k,l}\rangle_{C_{V}}=\frac{1}{L_{1}}\delta^{k+l^{\prime}}_{m}\delta^{k^{\prime}+l}_{m}\quad\int^{+\infty}_{-\infty}dw_{2}\
e^{-\frac{4\pi}{L_{1}}w_{2}m}\\\
=\frac{1}{2}\delta^{k+l^{\prime}}_{m}\delta^{k^{\prime}+l}_{m}\quad\int^{+\infty}_{-\infty}\frac{d\kappa}{2\pi}\,e^{\kappa
m}.$ (7.7)

The remaining integral does not impose $m=0$, as in (6.11) in the context of
the coadjoint representation on the punctured plane. For the infinite vertical
cylinder, the functions

${}_{h,\bar{h}}Z^{C_{V}}_{k,l}=i^{\bar{h}-h}\big{(}\frac{2\pi}{L_{1}}\big{)}^{h+\bar{h}}e^{i\frac{2\pi}{L_{1}}(k-l)w_{1}}\
e^{-\frac{2\pi}{L_{1}}(k+l)w_{2}},$ (7.8)

are not appropriate for expansions. One should rather use

$\phi^{C_{V}}_{h,\bar{h}}(w,\bar{w})=\sum_{m}\int^{+\infty}_{-\infty}d\kappa\
a_{m}(\kappa)Z^{C_{V}}_{m}(\kappa),\quad
Z^{C_{V}}_{m}(\kappa)=i^{\bar{h}-h}\big{(}\frac{2\pi}{L_{1}}\big{)}^{h+\bar{h}}e^{i\frac{2\pi}{L_{1}}mw_{1}}e^{i\frac{2\pi}{L_{1}}\kappa
w_{2}},$ (7.9)

which satisfy

$\langle{}_{-\bar{h}+1,-h+1}Z^{C_{V}}_{m^{\prime}}(\kappa^{\prime}),{}_{h,\bar{h}}Z^{C_{V}}_{m}(\kappa)\rangle_{C_{V}}=\delta^{m^{\prime}}_{m}\delta(\kappa^{\prime}-\kappa).$
(7.10)

##  8 Identification in non-radiative asymptotically flat spacetimes

We limit ourselves in this section to the case of the sphere. Asymptotically
flat space-times in the Newman-Penrose-Unti sense are for instance defined in
[18], end of section 9.8. Here we consider the case with the Maxwell field
turned off, $\varphi_{1}=0=\varphi_{2}$. Non-radiative spacetimes correspond
to the subset of solutions with $u$-independent asymptotic part of the shear,

$\partial_{u}\sigma^{0}=0,$ (8.1)

(as well as its complex conjugate and all higher order $u$ derivatives), so
that the news and also $\Psi^{0}_{3},\Psi^{0}_{4}$ vanish. It follows that

$\Psi^{0}_{2}-\bar{\Psi}^{0}_{2}=\bar{\eth}^{2}\sigma^{0}-\eth^{2}\bar{\sigma}^{0},$
(8.2)

while the evolution equations imply that

$\partial_{u}\Psi^{0}_{2}=0,\quad\Psi^{0}_{1}=\Psi^{0}_{1}(\xi,\bar{\xi})+u\eth\Psi^{0}_{2}.$
(8.3)

Such non-radiative space-times are completely characterized by specifying, at
the cut $u=0$ of ${\mycal I}^{+}$, the free data

$\Psi^{0}_{2}+\bar{\Psi}^{0}_{2},\Psi^{0}_{1},\sigma^{0},$ (8.4)

together with the different orders $\Psi^{n}_{0}$ in a $1/r$ expansion of
$\Psi_{0}$,

$\Psi_{0}=\sum_{n\geq 0}\Psi^{n}_{0}(\xi,\bar{\xi})\,r^{-5-n}$ (8.5)

Besides the linear dependence $u$ dependence of $\Psi^{0}_{1}$ in (8.3), there
are also evolution equations that govern the $u$-dependence of $\Psi^{n}_{0}$
which do not concern us here.

The transformation of this data under BMS symmetries has been worked out in
different ways and under various assumptions in [37, 38, 22, 39]111The arxiv
version of the last reference is preferable to the published one on account of
typesetting issues in the latter. In the case of the sphere, $P=P_{S}$ and the
scalar curvature is $R_{S}=2$. Furthermore, the solutions
$\mathcal{Y},\bar{\mathcal{Y}}$ to the conformal Killing equation on the
sphere are given by (5.15) and (5.16). This implies in particular that

$\eth^{3}\mathcal{Y}=0\quad\eth R_{S}=0,$ (8.6)

together with the complex conjugate relations222Note that in the
considerations below the value of $R_{S}=2$ on the sphere is never needed,
only the second of (8.6) is used. In the non-radiating case and at $u=0$, the
infinitesimal transformations reduce to333Up to a conventional overall sign
that we have changed here.

$\begin{split}\delta_{s}\Psi^{0}_{2}&=[\mathcal{Y}\eth+\bar{\mathcal{Y}}\bar{\eth}+\frac{3}{2}\eth\mathcal{Y}+\frac{3}{2}\bar{\eth}\bar{\mathcal{Y}}]\Psi^{0}_{2},\\\
\delta_{s}\Psi^{0}_{1}&=[\mathcal{Y}\eth+\bar{\mathcal{Y}}\bar{\eth}+2\eth\mathcal{Y}+\bar{\eth}\bar{\mathcal{Y}}]\Psi^{0}_{1}+\mathcal{T}\eth\Psi_{2}^{0}+3\eth\mathcal{T}\Psi^{0}_{2},\\\
\delta_{s}\sigma^{0}&=[\mathcal{Y}\eth+\bar{\mathcal{Y}}\bar{\eth}+\frac{3}{2}\eth\mathcal{Y}-\frac{1}{2}\bar{\eth}\bar{\mathcal{Y}}]\sigma^{0}-\eth^{2}\mathcal{T},\\\
\delta_{s}\Psi^{0}_{0}&=[\mathcal{Y}\eth+\bar{\mathcal{Y}}\bar{\eth}+\frac{5}{2}\eth\mathcal{Y}+\frac{1}{2}\bar{\eth}\bar{\mathcal{Y}}]\Psi^{0}_{0}+\mathcal{T}\eth\Psi_{1}^{0}+3\mathcal{T}\sigma^{0}\Psi_{2}^{0}+4\eth\mathcal{T}\Psi^{0}_{1},\\\
\delta_{s}\Psi^{1}_{0}&=\big{[}\mathcal{Y}\eth+\bar{\mathcal{Y}}\bar{\eth}+3\eth\mathcal{Y}+\bar{\eth}\bar{\mathcal{Y}}\big{]}\Psi^{1}_{0}-\overline{\eth}\big{[}5\eth\mathcal{T}\Psi^{0}_{0}+\mathcal{T}\eth\Psi^{0}_{0}+4\mathcal{T}\Psi^{0}_{1}\sigma^{0}\big{]}.\end{split}$
(8.7)

There are increasingly complicated transformations laws for the higher
$\Psi^{n}_{0}$, $n\geq 2$, that are not relevant for our purpose here.

When expressing the first two of the equations in (8.7) in terms of the free
data by taking the constraint (8.2) into account, one finds (trivially) that

$\delta_{s}(\Psi^{0}_{2}+\bar{\Psi}^{0}_{2})=[\mathcal{Y}\eth+\bar{\mathcal{Y}}\bar{\eth}+\frac{3}{2}\eth\mathcal{Y}+\frac{3}{2}\bar{\eth}\bar{\mathcal{Y}}](\Psi^{0}_{2}+\bar{\Psi}^{0}_{2}),$
(8.8)

and

$\delta_{s}\Psi^{0}_{1}=[\mathcal{Y}\eth+\bar{\mathcal{Y}}\bar{\eth}+2\eth\mathcal{Y}+\bar{\eth}\bar{\mathcal{Y}}]\Psi^{0}_{1}+\frac{1}{2}\mathcal{T}\eth(\Psi_{2}^{0}+\bar{\Psi}^{0}_{2}+\bar{\eth}^{2}\sigma^{0}-\eth^{2}\bar{\sigma}^{0})\\\
+\frac{3}{2}\eth\mathcal{T}(\Psi_{2}^{0}+\bar{\Psi}^{0}_{2}+\bar{\eth}^{2}\sigma^{0}-\eth^{2}\bar{\sigma}^{0}).$
(8.9)

Following the analysis in three dimensions, one fixes the normalization by
computing the surface charge algebra, directly related to linear super
momentum and angular momentum of the system. In the non-radiating case, this
has been discussed for instance in section 4.2 of [38] (see also [39]). Let us
summarize the relevant part of those results in the notation and conventions
adopted here. Let

$f=\mathcal{T}+\frac{1}{2}u(\eth\mathcal{Y}+\bar{\eth}\bar{\mathcal{Y}}),$
(8.10)

and consider the $2$-form

$J_{s}=\frac{i}{R^{2}}\big{[}(P_{S}\bar{P}_{S})^{-1}\mathcal{J}^{u}_{s}d\xi\wedge
d\bar{\xi}+P_{S}^{-1}\mathcal{J}_{s}^{\bar{\xi}}du\wedge
d\xi-\bar{P}_{S}^{-1}\mathcal{J}_{s}^{\xi}du\wedge d\bar{\xi}],$ (8.11)

with $\mathcal{J}^{\xi}_{s}=\mathcal{J}_{s}$,
$\mathcal{J}^{\bar{\xi}}=\bar{\mathcal{J}}$ and

$\begin{split}\mathcal{J}^{u}_{s}&=-\frac{1}{8\pi
G}\big{[}(\Psi^{0}_{2}+\bar{\Psi}^{0}_{2})f+\Psi^{0}_{1\bar{J}}\mathcal{Y}+\bar{\Psi}^{0}_{1\bar{J}}\bar{\mathcal{Y}}\big{]},\\\
\mathcal{J}_{s}&=\frac{1}{8\pi
G}\big{[}\Psi^{0}_{2}\mathcal{Y}+\frac{1}{2}\eth\bar{\sigma}^{0}(\eth\mathcal{Y}-\bar{\eth}\bar{\mathcal{Y}})-\frac{1}{2}\bar{\sigma}^{0}\eth(\eth\mathcal{Y}-\bar{\eth}\bar{\mathcal{Y}})\big{]},\\\
\Psi^{0}_{1\bar{J}}&=\Psi^{0}_{1}+\sigma^{0}\eth\bar{\sigma}^{0}+\frac{1}{2}\eth(\sigma^{0}\bar{\sigma}^{0}).\end{split}$
(8.12)

The transformation law of $\Psi^{0}_{1\bar{J}}$ turns out to be

$\delta_{s}\Psi^{0}_{1\bar{J}}=[\mathcal{Y}\eth+2\eth\mathcal{Y}]\Psi^{0}_{1\bar{J}}+\bar{\eth}(\bar{\mathcal{Y}}\Psi^{0}_{1\bar{J}})+\frac{1}{2}\mathcal{T}\eth(\Psi^{0}_{2}+\bar{\Psi}^{0}_{2})+\frac{3}{2}\eth\mathcal{T}(\Psi^{0}_{2}+\bar{\Psi}^{0}_{2})\\\
+\frac{1}{2}\bar{\eth}(\mathcal{T}\bar{\eth}\eth\sigma^{0}-\bar{\eth}\mathcal{T}\eth\sigma^{0}+3\eth\mathcal{T}\bar{\eth}\sigma^{0}-3\bar{\eth}\eth\mathcal{T}\sigma^{0}-\frac{3}{2}R_{S}\mathcal{T}\sigma^{0})-\frac{1}{2}\eth^{3}(\mathcal{T}\bar{\sigma}^{0}),$
(8.13)

where the terms on the second line are irrelevant when multiplied by
$\mathcal{Y}$ and integrated over the sphere (cf. Remark (i) in section
5.3.1).

When taking the retarded time-dependence of $\Psi^{0}_{1}$ in (8.3) and the
constraint (8.2) into account, this 2-form is closed,

$dJ_{s}=0\iff\partial_{u}\mathcal{J}^{u}_{s}+\eth\mathcal{J}_{s}+\bar{\eth}\bar{\mathcal{J}_{s}}=0.$
(8.14)

Furthermore,

$\delta_{s_{1}}\mathcal{J}^{u}_{s_{2}}=-\mathcal{J}^{u}_{[s_{1},s_{2}]}+\eth\mathcal{L}_{s_{2},s_{1}}+\bar{\eth}\overline{\mathcal{L}_{s_{2},s_{1}}},$
(8.15)

where the concrete expression for $\mathcal{L}_{s_{2},s_{1}}$ is not needed
here. This transformation law is in line with (4.46). The charges defined by

$Q_{s}=\int_{S^{2},u=u_{0}}J^{u}_{s},$ (8.16)

with $u_{0}$ constant, are conserved in the sense that they do not depend on
$u$ and

$\delta_{s_{1}}Q_{s_{2}}=-Q_{[s_{1},s_{2}]}.$ (8.17)

More precisely, the polynomial algebra $\mathcal{F}$ generated by the free
data
$\Psi^{0}_{2}+\bar{\Psi}^{0}_{2},\Psi^{0}_{1},\sigma^{0},\bar{\Psi}^{0}_{1},\bar{\sigma}^{0}$
carries a representation $\delta_{s}$ of the BMS4 algebra. It then follows
from the identification at $u=0$,

$Q_{s}=\langle([J],[\bar{J}],\mathcal{P}),(\mathcal{Y},\bar{\mathcal{Y}},\mathcal{T})\rangle,$
(8.18)

that the pre-moment map $\mu:\mathcal{F}\to\mathfrak{bms}^{*}_{4}$ defined by

$\mu(-\frac{1}{2G}[\Psi^{0}_{2}+\bar{\Psi}^{0}_{2}])=\mathcal{P},\quad\mu(-\frac{1}{2G}\Psi^{0}_{1\bar{J}})=[\bar{\mathcal{J}}],\quad\mu(-\frac{1}{2G}\bar{\Psi^{0}_{1\bar{J}}})=[\mathcal{J}],$
(8.19)

is compatible with the representation,

$\mu\circ\delta_{s}={\rm ad}^{*}_{s}\circ\mu.$ (8.20)

The transformation law of the asymptotic part of the shear implies in
particular that

$\delta_{s}(\bar{\eth}^{2}\sigma^{0}-\eth^{2}\bar{\sigma}^{0})=[\mathcal{Y}\eth+\bar{\mathcal{Y}}\bar{\eth}+\frac{3}{2}\eth\mathcal{Y}+\frac{3}{2}\bar{\eth}\bar{\mathcal{Y}}](\bar{\eth}^{2}\sigma^{0}-\eth^{2}\bar{\sigma}^{0}).$
(8.21)

This means that constraining the asymptotic part of the shear to be electric
[5],

$\bar{\eth}^{2}\sigma^{0}_{e}=\eth^{2}\bar{\sigma}^{0}_{e},$ (8.22)

is a BMS invariant condition. In this case, the transformation law (8.9)
simplifies and suggests

$\mu(-\frac{1}{2G}\Psi^{0}_{1})=\bar{\mathcal{J}}\sim\Psi^{0}_{1},\quad\mu(-\frac{1}{2G}\bar{\Psi}^{0}_{1})=\mathcal{J}.$
(8.23)

That this is compatible with the previous identification can be seen as
follows. The electric condition is solved by a real field
$\chi_{e}=\bar{\chi}_{e}$ with the same weights $s=0,w=1$ than $\mathcal{T}$,

$\sigma^{0}_{e}=\eth^{2}\chi_{e},\quad\bar{\sigma}^{0}=\bar{\eth}^{2}\chi_{e},\quad\delta_{s}\chi_{e}=[\mathcal{Y}\eth+\bar{\mathcal{Y}}\bar{\eth}-\frac{1}{2}\eth\mathcal{Y}-\frac{1}{2}\bar{\eth}\bar{\mathcal{Y}}]\chi_{e}-\mathcal{T}+\sum_{j\leq
1,m}\lambda^{jm}{}_{0}Z_{j,m},$ (8.24)

where $\lambda^{jm}\in\mathbb{R}$. Inserting this solution into
$\Psi^{0}_{1\bar{J}}$ one finds that it indeed agrees with $\Psi^{0}_{1}$ up
to terms that are projected to zero by the map,

$\Psi^{0}_{1\bar{J}}=\Psi^{0}_{1}+\frac{1}{2}\bar{\eth}(\eth^{3}\chi_{e}\bar{\eth}\chi_{e}+3\eth^{2}\chi_{e}\eth\bar{\eth}\chi_{e}-\frac{3}{4}R_{S}\eth\chi_{e}\eth\chi_{e})-\frac{1}{4}\eth^{3}(\bar{\eth}\chi_{e}\bar{\eth}\chi_{e}).$
(8.25)

Relevant formulas for the group can be found in [22] and will not be repeated
here. On the punctured plane, in order to have room for the Witt algebra, one
cannot limit oneself to non-radiative spacetimes since turning off the news
requires
$\partial^{3}\tilde{\mathcal{Y}}=0=\bar{\partial}^{3}\bar{\tilde{\mathcal{Y}}}$.
In the presence of news, currents are no longer conserved. Current algebra is
broken both by flux terms and by a field dependent central extension discussed
in more details in [27]. In this case, the last term in (8.13) is no longer
trivial and becomes the associated (field-dependent) Souriau cocyle.

##  9 Discussion and perspectives

For the generalized BMS group on the sphere introduced in [40] (see also [41,
42, 43, 44] for further considerations), the coadjoint representation is
obtained from the approach developed here simply by removing the conformal
Killing equation on infinitesimal superrotations
$\bar{\eth}\mathcal{Y}=0=\eth\bar{\mathcal{Y}}$ and the associated equivalence
relations on super-angular momentum $\mathcal{J},\bar{\mathcal{J}}$. All
fields should then simply be expanded in terms of spin-weighted spherical
harmonics according to their weights.

A detailed recent study of the coadjoint representation of closely related
semi-direct product groups involving diffeomeorphisms on the sphere, along the
lines of our analysis in three dimensions [25, 26] (see also [45] for a
review), has recently appeared in [46].

For the $\mathfrak{bms}_{4}$ algebra on the punctured plane, a discussion of
central extensions can be found in [21], whereas deformations have been
studied in detail in [47].

After having set up the basics in this paper, the next steps are to classify
the coadjoint orbits, to re-discuss unitary irreducible representations [48,
49, 50, 51, 52, 10] from the viewpoint of the orbit method [53] and to
construct the associated geometric actions [54, 55], as in the three
dimensional case [56] (see [57] in this context). One could also explore
whether some aspects of positive energy theorems for the Bondi mass [58, 59,
60, 61, 62, 63, 64] might be understood from such a perspective, again as in
three dimensions [65].

The most interesting question is to understand in detail how such effective
actions for the sector captured by the coadjoint representation interacts with
the radiative degrees of freedom, as described in [66, 67] and more recently
in [68], see also [69, 70] in this context.

Another more technical question is to extend the considerations in section 8
to a full-fledged momentum map at null infinity, as recently constructed at
spatial infinity [71, 72], by starting from [73] and also [74].

In the case of celestial scattering amplitudes and soft theorems, the relevant
surface is neither the (Riemann) sphere nor the plane, but rather two Riemann
spheres with punctures related by an antipodal map. On each of these surfaces,
the superrotation part of the extended algebra is given by the Witt algebra
only if there are two particles/punctures. In this context, complementary
aspects of the BMS group have been discussed in [75, 76, 77, 78, 79, 80, 81,
82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93].

In this exposition here, we have followed the general relativity route from
the sphere to the punctured plane. In conformal field theory, one travels in
the opposite direction. For more punctures, the appropriate algebra should
presumably be Krichever-Novikov algebras [94, 95].

## Acknowledgments

The authors thank P. Mao for collaboration on related questions and are
grateful to L. Ciambelli, Y. Herfray, M. Petropolous, B. Oblak, C. Marteau and
L. Szabados for insightful discussions. The work of G.B. is supported by the
F.R.S.-FNRS, Belgium, convention FRFC PDR T.1025.14 and convention IISN
4.4514.08. R.R. is supported by the Austrian Science Fund (FWF), project P
32581-N.

##  Appendix A Spin-weighted spherical harmonics

We follow the conventions of [17], section 4.15. Instructive alternative
presentations and perspectives can be found in [96], [97], [98], [99], section
1.10, [100], [101].

Let

$\begin{pmatrix}\alpha&\beta\\\
\gamma&\delta\end{pmatrix}=\frac{i}{\sqrt{1+\xi\bar{\xi}}}\begin{pmatrix}-1&\xi\\\
\bar{\xi}&1\end{pmatrix}.$ (A.1)

Let also $s$ be integer or half-integer, $j\pm s\in\mathbb{N}$, $j\pm
m\in\mathbb{N}$, $|m|\leq j$, $|s|\leq j$, and consider

${}_{s}Z_{j,m}=\sum_{r}\frac{(j+m)!(j-m)!(j+s)!(j-s)!\alpha^{r}\beta^{j-m-r}\gamma^{j+s-r}\delta^{r+m-s}}{(2j)!r!(j-m-r)!(j+s-r)!(r+m-s)!},$
(A.2)

where the summation extends over integer values of $r$ in the range
$\max(0,s-m)\leq r\leq\min(j-m,j+s)$.

The spin-weighted spherical harmonics ${}_{s}Y_{j,m}$ are then defined by

${}_{s}Y_{j,m}=(-1)^{j+m}~{}_{s}Z_{j,m}\sqrt{\frac{(2j+1)!(2j)!}{4\pi(j+s)!(j-s)!(j+m)!(j-m)!}}.$
(A.3)

For $s=0$, one recovers the usual spherical harmonics functions, i.e.,
${}_{0}Y_{j,m}$. The following properties hold:

  * •

For each $s$, the ${}_{s}Z_{j,m}$ form an orthogonal basis for the spin-
weighted scalars $\eta^{s}$ on the sphere with

$\langle{}_{s}Z_{j^{\prime},m^{\prime}},{}_{s}Z_{j,m}\rangle=\frac{(j+s)!(j-s)!(j+m)!(j-m)!}{(2j+1)!(2j)!}\delta_{jj^{\prime}}\delta_{mm^{\prime}},$
(A.4)

for the pairing (5.6). The ${}_{s}Y_{j,m}$ form an orthonormal basis with

$4\pi\langle{}_{s}Y_{j^{\prime},m^{\prime}},{}_{s}Y_{j,m}\rangle=\delta_{jj^{\prime}}\delta_{mm^{\prime}}.$
(A.5)

  * •

The behavior under complex conjugation is

$\overline{{}_{s}Z_{j,m}}=(-1)^{m+s}~{}_{-s}Z_{j,-m},\quad\overline{{}_{s}Y_{j,m}}=(-1)^{3m+s}~{}_{-s}Y_{j,-m}.$
(A.6)

  * •

The action of the operators $\eth$ and $\bar{\eth}$ defined in (5.5) is
explicitly given by

$\eth\
{}_{s}Z_{j,m}=-\left(\frac{j-s}{R\sqrt{2}}\right){}_{s+1}Z_{j,m},\quad\bar{\eth}\
{}_{s}Z_{j,m}=\left(\frac{j+s}{R\sqrt{2}}\right){}_{s-1}Z_{j,m}.$ (A.7)
$\begin{split}\eth\
_{s}Y_{j,m}&=-\sqrt{\frac{(j+s+1)(j-s)}{2R^{2}}}{}_{s+1}Y_{j,m},\\\
\bar{\eth}\
{}_{s}Y_{j,m}&=\sqrt{\frac{(j-s+1)(j+s)}{2R^{2}}}{}_{s-1}Y_{j,m}.\end{split}$
(A.8)

  * •

The ${}_{s}Z_{j,m}$ and ${}_{s}Y_{j,m}$ are eigenfunctions of the operator
$\bar{\eth}\eth$:

$\bar{\eth}\eth_{s}Z_{j,m}=-(j+s+1)(j-s)\frac{1}{2}~{}_{s}Z_{j,m},\quad\bar{\eth}\eth_{s}Y_{j,m}=-(j+s+1)(j-s)\frac{1}{2}~{}_{s}Y_{j,m}.$
(A.9)

  * •

Products of spin-weighted spherical harmonics can be decomposed as

$\begin{split}{}_{s_{1}}Z_{j_{1},m_{1}}~{}_{s_{2}}Z_{j_{2},m_{2}}&=\sqrt{\frac{(j_{1}+s_{1})!(j_{1}-s_{1})!(j_{1}+m_{1})!(j_{1}-m_{1})!}{(2j_{1})!(2j_{1})!}}\\\
&\times\sqrt{\frac{(j_{2}+s_{2})!(j_{2}-s_{2})!(j_{2}+m_{2})!(j_{2}-m_{2})!}{(2j_{2})!(2j_{2})!}}\\\
&\times\sum_{j}(-1)^{j_{1}+j_{2}+j}~{}_{(s_{1}+s_{2})}Z_{j,(m_{1}+m_{2})}\\\
&\times\sqrt{\frac{(2j)!(2j)!}{(j+s_{1}+s_{2})!(j-s_{1}-s_{2})!(j+m_{1}+m_{2})!(j-m_{1}-m_{2})!}}\\\
&\times\langle j_{1},s_{1};j_{2},s_{2}|j,(s_{1}+s_{2})\rangle~{}\langle
j_{1},m_{1};j_{2},m_{2}|j,(m_{1}+m_{2})\rangle,\end{split}$ (A.10)

or

$\begin{split}{}_{s_{1}}Y_{j_{1},m_{1}}~{}_{s_{2}}Y_{j_{2},m_{2}}=&\sum_{j}\sqrt{\frac{(2j_{1}+1)(2j_{2}+1)}{4\pi(2j+1)}}~{}_{(s_{1}+s_{2})}Y_{j,(m_{1}+m_{2})}\\\
&\times\langle j_{1},s_{1};j_{2},s_{2}|j,(s_{1}+s_{2})\rangle~{}\langle
j_{1},m_{1};j_{2},m_{2}|j,(m_{1}+m_{2})\rangle,\end{split}$ (A.11)

where the summation extends over integer values of $j$ in the range
$\max(|j_{1}-j_{2}|,|s_{1}+s_{2}|,|m_{1}+m_{2}|)\leq j\leq j_{1}+j_{2}$, and
where $\langle j_{1},m_{1};j_{2},m_{2}|j,(m_{1}+m_{2})\rangle$ is a Clebsch-
Gordan coefficient of the rotation group (see e.g. [15, 100] in this context).

## References

  * [1] H. Bondi, M. G. van der Burg, and A. W. Metzner, “Gravitational waves in general relativity. 7. Waves from axi-symmetric isolated systems,” Proc. Roy. Soc. Lond. A 269 (1962) 21.
  * [2] R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-time,” Proc. Roy. Soc. Lond. A 270 (1962) 103.
  * [3] R. K. Sachs, “Asymptotic symmetries in gravitational theory,” Phys. Rev. 128 (1962) 2851–2864.
  * [4] R. Penrose, “Asymptotic properties of fields and space-times,” Phys. Rev. Lett. 10 no. 2, (1963) 66–68.
  * [5] E. T. Newman and R. Penrose, “Note on the Bondi-Metzner-Sachs Group,” J. Math. Phys. 7 (1966) 863–870. http://link.aip.org/link/?JMP/7/863/1.
  * [6] J. Foster, “Conformal structure of i+ and asymptotic symmetry. I. Definitions and local theory,” Journal of Physics A: Mathematical and General 11 no. 1, (1978) 93. http://stacks.iop.org/0305-4470/11/i=1/a=012.
  * [7] J. Foster, “Asymptotic symmetry and the global structure of future null infinity,” International Journal of Theoretical Physics 26 (1987) 1107–1124. http://dx.doi.org/10.1007/BF00669365. 10.1007/BF00669365.
  * [8] E. T. Newman, “A possible connexion between the gravitational field and elementary particle physics,” Nature 206 no. 4986, (05, 1965) 811–812. http://dx.doi.org/10.1038/206811a0.
  * [9] A. Komar, “Quantized gravitational theory and internal symmetries,” Phys. Rev. Lett. 15 no. 2, (Jul, 1965) 76–78.
  * [10] P. J. McCarthy, “The Bondi-Metzner-Sachs Group in the Nuclear Topology,” Royal Society of London Proceedings Series A 343 (May, 1975) 489–523.
  * [11] J. H. Rawnsley, “Representations of a semi-direct product by quantization,” Mathematical Proceedings of the Cambridge Philosophical Society 78 (9, 1975) 345–350. http://journals.cambridge.org/article_S0305004100051793.
  * [12] T. Fulton, F. Rohrlich, and L. Witten, “Conformal invariance in physics,” Rev. Mod. Phys. 34 no. 3, (Jul, 1962) 442–457.
  * [13] E. D’Hoker and D. Phong, “The Geometry of String Perturbation Theory,” Rev.Mod.Phys. 60 (1988) 917.
  * [14] J. N. Goldberg, A. J. Macfarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan, “Spin-s Spherical Harmonics and eth,” Journal of Mathematical Physics 8 no. 11, (1967) 2155–2161. http://link.aip.org/link/?JMP/8/2155/1.
  * [15] A. Held, E. T. Newman, and R. Posadas, “The Lorentz Group and the Sphere,” Journal of Mathematical Physics 11 no. 11, (1970) 3145–3154. http://link.aip.org/link/?JMP/11/3145/1.
  * [16] R. P. Geroch, A. Held, and R. Penrose, “A space-time calculus based on pairs of null directions,” J. Math. Phys. 14 (1973) 874–881.
  * [17] R. Penrose and W. Rindler, Spinors and Space-Time, Volume 1: Two-spinor Calculus and Relativistic Fields.  Cambridge University Press, 1984.
  * [18] R. Penrose and W. Rindler, Spinors and Space-Time, Volume 2: Spinor and Twistor Methods in Space-Time Geometry.  Cambridge University Press, 1986.
  * [19] G. Barnich and C. Troessaert, “Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited,” Phys.Rev.Lett. 105 (2010) 111103, arXiv:0909.2617 [gr-qc].
  * [20] G. Barnich and C. Troessaert, “Aspects of the BMS/CFT correspondence,” JHEP 05 (2010) 062, arXiv:1001.1541 [hep-th].
  * [21] G. Barnich and C. Troessaert, “Supertranslations call for superrotations,” PoS CNCFG2010 (2010) 010, arXiv:1102.4632 [gr-qc].
  * [22] G. Barnich and C. Troessaert, “Finite BMS transformations,” JHEP 03 (2016) 167, arXiv:1601.04090 [gr-qc].
  * [23] A. Ashtekar, J. Bicak, and B. G. Schmidt, “Asymptotic structure of symmetry-reduced general relativity,” Phys. Rev. D 55 no. 2, (Jan, 1997) 669–686.
  * [24] G. Barnich and G. Compère, “Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions,” Class. Quant. Grav. 24 (2007) F15, gr-qc/0610130. Corrigendum: ibid 24 (2007) 3139.
  * [25] G. Barnich and B. Oblak, “Notes on the BMS group in three dimensions: I. Induced representations,” JHEP 1406 (2014) 129, arXiv:1403.5803 [hep-th].
  * [26] G. Barnich and B. Oblak, “Notes on the BMS group in three dimensions: II. Coadjoint representation,” JHEP 1503 (2015) 033, arXiv:1502.00010 [hep-th].
  * [27] G. Barnich, “Centrally extended BMS4 Lie algebroid,” JHEP 06 (2017) 007, arXiv:1703.08704 [hep-th].
  * [28] I. E. Antoniou and B. Misra, “Characterization of semidirect sum Lie algebras,” Journal of Mathematical Physics 32 no. 4, (1991) 864–868. https://doi.org/10.1063/1.529344.
  * [29] M. Carmeli, Group theory and general relativity: Representations of the Lorentz group and their applications to the gravitational field.  Imperial College Press, 2000.
  * [30] F. Alessio and G. Esposito, “On the structure and applications of the Bondi–Metzner–Sachs group,” Int. J. Geom. Meth. Mod. Phys. 15 no. 02, (2018) 1830002, arXiv:1709.05134 [gr-qc].
  * [31] R. Geroch, “Asymptotic structure of space-time,” in Symposium on the asymptotic structure of space-time, P. Esposito and L. Witten, eds., pp. 1–105.  Plenum, New York, 1977.
  * [32] C. Duval, G. Gibbons, and P. Horvathy, “Conformal Carroll groups and BMS symmetry,” Class.Quant.Grav. 31 (2014) 092001, arXiv:1402.5894 [gr-qc].
  * [33] C. Duval, G. Gibbons, and P. Horvathy, “Conformal Carroll groups,” J.Phys. A47 (2014) 335204, arXiv:1403.4213 [hep-th].
  * [34] I. Gradshteyn and I. Ryzhik, “8-9 \- special functions,” in Table of Integrals, Series, and Products, I. GRADSHTEYN and I. RYZHIK, eds., pp. 904 – 1080.  Academic Press, 1980.  http://www.sciencedirect.com/science/article/pii/B9780122947605500209.
  * [35] J. Porter, “Green’s Functions Associated with the Edth Operators,” Gen.Rel.Grav. 13 (1981) 531–539.
  * [36] V. Kac, Vertex algebras for beginners.  University Lecture Series, Volume 10. American Mathematical Society, 2nd ed., 1997\.
  * [37] G. Barnich and P.-H. Lambert, “A note on the Newman-Unti group and the BMS charge algebra in terms of Newman-Penrose coefficients,” J.Phys.Conf.Ser. 410 (2013) 012142.
  * [38] G. Barnich and C. Troessaert, “Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity,” JHEP 1311 (2013) 003, arXiv:1309.0794 [hep-th].
  * [39] G. Barnich, P. Mao, and R. Ruzziconi, “BMS current algebra in the context of the Newman-Penrose formalism,” Class. Quant. Grav. 37 no. 9, (2020) 095010, arXiv:1910.14588 [gr-qc].
  * [40] M. Campiglia and A. Laddha, “Asymptotic symmetries and subleading soft graviton theorem,” Phys. Rev. D90 no. 12, (2014) 124028, arXiv:1408.2228 [hep-th].
  * [41] G. Compère, A. Fiorucci, and R. Ruzziconi, “Superboost transitions, refraction memory and super-Lorentz charge algebra,” JHEP 11 (2018) 200, arXiv:1810.00377 [hep-th]. [Erratum: JHEP 04, 172 (2020)].
  * [42] E. E. Flanagan, K. Prabhu, and I. Shehzad, “Extensions of the asymptotic symmetry algebra of general relativity,” JHEP 01 002, 1910.04557.
  * [43] M. Campiglia and J. Peraza, “Generalized BMS charge algebra,” Phys. Rev. D 101 no. 10, 104039, 2002.06691.
  * [44] G. Compère, A. Fiorucci, and R. Ruzziconi, “The $\Lambda$-BMS4 charge algebra,” JHEP 10 205, 2004.10769.
  * [45] B. Oblak, BMS Particles in Three Dimensions.  PhD thesis, Brussels U., 2016.  1610.08526.
  * [46] W. Donnelly, L. Freidel, S. F. Moosavian, and A. J. Speranza, “Gravitational Edge Modes, Coadjoint Orbits, and Hydrodynamics,” 2012.10367.
  * [47] H. R. Safari and M. M. Sheikh-Jabbari, “BMS4 algebra, its stability and deformations,” JHEP 04 068, 1902.03260.
  * [48] V. Cantoni, “A Class of Representations of the Generalized Bondi-Metzner Group,” Journal of Mathematical Physics 7 no. 8, (1966) 1361–1364. http://link.aip.org/link/?JMP/7/1361/1.
  * [49] P. J. McCarthy, “Representations of the Bondi-Metzner-Sachs Group. I. Determination of the Representations,” Royal Society of London Proceedings Series A 330 (Nov., 1972) 517–535.
  * [50] P. J. McCarthy, “Representations of the Bondi-Metzner-Sachs Group. II. Properties and Classification of the Representations,” Royal Society of London Proceedings Series A 333 (May, 1973) 317–336.
  * [51] P. J. McCarthy and M. Crampin, “Representations of the Bondi-Metzner-Sachs Group. III. Poincare Spin Multiplicities and Irreducibility,” Royal Society of London Proceedings Series A 335 (Nov., 1973) 301–311.
  * [52] L. Girardello and G. Parravicini, “Continuous spins in the Bondi-Metzner-Sachs group of asymptotic symmetry in general relativity,” Phys.Rev.Lett. 32 (1974) 565–568.
  * [53] A. A. Kirillov, Lectures on the orbit method.  American Mathematical Society, 2004.
  * [54] A. Alekseev, L. Faddeev, and S. L. Shatashvili, “Quantization of symplectic orbits of compact Lie groups by means of the functional integral,” J.Geom.Phys. 5 (1988) 391–406.
  * [55] A. Alekseev and S. Shatashvili, “Path integral quantization of the coadjoint orbits of the virasoro group and 2-d gravity,” Nuclear Physics B 323 no. 3, (1989) 719 – 733. http://www.sciencedirect.com/science/article/pii/0550321389901302.
  * [56] G. Barnich, H. A. Gonzalez, and P. Salgado-Rebolledo, “Geometric actions for three-dimensional gravity,” Class. Quant. Grav. 35 no. 1, (2018) 014003, arXiv:1707.08887 [hep-th].
  * [57] K. Nguyen and J. Salzer, “The effective action of superrotation modes,” JHEP 02 108, 2008.03321.
  * [58] W. Israel and J. M. Nester, “Positivity of the Bondi gravitational mass,” Physics Letters A 85 no. 5, (1981) 259 – 260. http://www.sciencedirect.com/science/article/pii/0375960181909518.
  * [59] M. Ludvigsen and J. Vickers, “The positivity of the Bondi mass,” J.Phys. A14 (1981) L389–L391.
  * [60] R. Schoen and S.-T. Yau, “Proof That the Bondi Mass is Positive,” Phys.Rev.Lett. 48 (1982) 369–371.
  * [61] G. T. Horowitz and M. J. Perry, “Gravitational Energy Cannot Become Negative,” Phys.Rev.Lett. 48 (1982) 371.
  * [62] G. T. Horowitz and P. Tod, “A relation between local and total energy in general relativity,” Communications in Mathematical Physics 85 (Sept., 1982) 429–447.
  * [63] A. Ashtekar and G. T. Horowitz, “Energy-momentum of isolated systems cannot be null,” Physics Letters A 89 no. 4, (1982) 181 – 184. http://www.sciencedirect.com/science/article/pii/0375960182902031.
  * [64] O. Reula and K. P. Tod, “Positivity of the Bondi energy,” Journal of Mathematical Physics 25 (Apr., 1984) 1004–1008.
  * [65] G. Barnich and B. Oblak, “Holographic positive energy theorems in three-dimensional gravity,” Class.Quant.Grav. 31 (2014) 152001, arXiv:1403.3835 [hep-th].
  * [66] A. Ashtekar and M. Streubel, “Symplectic geometry of radiative modes and conserved quantities at null infinity,” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 376 no. 1767, (1981) 585–607. http://rspa.royalsocietypublishing.org/content/376/1767/585.abstract.
  * [67] A. Ashtekar, “Asymptotic Quantization: Based on 1984 Naples Lectures,”. Naples, Italy: Bibliopolis (1987) 107 p. (Monographs and textbooks in physical science, 2).
  * [68] T. Adamo, E. Casali, and D. Skinner, “Perturbative gravity at null infinity,” Class. Quant. Grav. 31 no. 22, (2014) 225008, arXiv:1405.5122 [hep-th].
  * [69] C. Troessaert, “Hamiltonian surface charges using external sources,” J. Math. Phys. 57 no. 5, 053507, 1509.09094.
  * [70] W. Wieland, “Null infinity as an open Hamiltonian system,” 2012.01889.
  * [71] C. Troessaert, “The BMS4 algebra at spatial infinity,” Class. Quant. Grav. 35 no. 7, 074003, 1704.06223.
  * [72] M. Henneaux and C. Troessaert, “BMS Group at Spatial Infinity: the Hamiltonian (ADM) approach,” JHEP 03 147, 1801.03718.
  * [73] C. G. Torre, “Null Surface Geometrodynamics,” Class. Quant. Grav. 3 (1986) 773.
  * [74] R. Oliveri and S. Speziale, “Boundary effects in General Relativity with tetrad variables,” Gen. Rel. Grav. 52 no. 8, 83, 1912.01016.
  * [75] A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 1407 (Dec., 2014) 152, 1312.2229.
  * [76] T. He, V. Lysov, P. Mitra, and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP 05 (2015) 151, arXiv:1401.7026 [hep-th].
  * [77] E. Conde and P. Mao, “BMS Supertranslations and Not So Soft Gravitons,” JHEP 05 (2017) 060, arXiv:1612.08294 [hep-th].
  * [78] C. Cheung, A. de la Fuente, and R. Sundrum, “4D scattering amplitudes and asymptotic symmetries from 2D CFT,” JHEP 01 112, 1609.00732.
  * [79] S. Banerjee, “Symmetries of free massless particles and soft theorems,” Gen. Rel. Grav. 51 no. 9, 128, 1804.06646.
  * [80] J. Distler, R. Flauger, and B. Horn, “Double-soft graviton amplitudes and the extended BMS charge algebra,” JHEP 08 021, 1808.09965.
  * [81] L. Donnay, A. Puhm, and A. Strominger, “Conformally Soft Photons and Gravitons,” JHEP 01 (2019) 184, arXiv:1810.05219 [hep-th].
  * [82] S. Stieberger and T. R. Taylor, “Symmetries of celestial amplitudes,” Phys. Lett. B 793 (2019) 141–143, arXiv:1812.01080 [hep-th].
  * [83] T. Adamo, L. Mason, and A. Sharma, “Celestial amplitudes and conformal soft theorems,” Class. Quant. Grav. 36 no. 20, 205018, 1905.09224.
  * [84] A. Fotopoulos and T. R. Taylor, “Primary Fields in Celestial CFT,” JHEP 10 167, 1906.10149.
  * [85] A. Fotopoulos, S. Stieberger, T. R. Taylor, and B. Zhu, “Extended bms algebra of celestial cft,” JHEP 03 (2020) 130, arXiv:1912.10973 [hep-th].
  * [86] W. Fan, A. Fotopoulos, S. Stieberger, and T. R. Taylor, “On Sugawara construction on Celestial Sphere,” JHEP 09 139, 2005.10666.
  * [87] L. Donnay, S. Pasterski, and A. Puhm, “Asymptotic Symmetries and Celestial CFT,” JHEP 09 176, 2005.08990.
  * [88] L. Donnay, G. Giribet, and F. Rosso, “Quantum BMS transformations in conformally flat space-times and holography,” JHEP 12 102, 2008.05483.
  * [89] H. A. González, A. Puhm, and F. Rojas, “Loop corrections to celestial amplitudes,” Phys. Rev. D 102 no. 12, 126027, 2009.07290.
  * [90] S. Banerjee, S. Ghosh, and R. Gonzo, “BMS symmetry of celestial OPE,” JHEP 04 130, 2002.00975.
  * [91] S. Banerjee, S. Ghosh, and P. Paul, “MHV graviton scattering amplitudes and current algebra on the celestial sphere,” JHEP 02 176, 2008.04330.
  * [92] E. Himwich, S. A. Narayanan, M. Pate, N. Paul, and A. Strominger, “The Soft $\mathcal{S}$-Matrix in Gravity,” JHEP 09 129, 2005.13433.
  * [93] A. Atanasov, A. Ball, W. Melton, A.-M. Raclariu, and A. Strominger, “$(2,2)$ Scattering and the Celestial Torus,” 2101.09591.
  * [94] I. M. Krichever and S. P. Novikov, “Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons,” Functional Analysis and Its Applications 21 no. 2, (04, 1987) 126–142. http://dx.doi.org/10.1007/BF01078026.
  * [95] I. M. Krichever and S. P. Novikov, “Virasoro-type algebras, riemann surfaces and strings in minkowsky space,” Functional Analysis and Its Applications 21 no. 4, (1987) 294–307. http://dx.doi.org/10.1007/BF01077803.
  * [96] T. T. Wu and C. N. Yang, “Dirac Monopole Without Strings: Monopole Harmonics,” Nucl. Phys. B107 (1976) 365.
  * [97] K. S. Thorne, “Multipole expansions of gravitational radiation,” Rev. Mod. Phys. 52 no. 2, (Apr, 1980) 299–339.
  * [98] M. Eastwood and P. Tod, “Edth-a differential operator on the sphere,” Mathematical Proceedings of the Cambridge Philosophical Society 92 (1982) 317–330.
  * [99] A. S. Galperin, Harmonic Superspace.  New York : Cambridge University Press, 2011.
  * [100] F. Beyer, B. Daszuta, J. Frauendiener, and B. Whale, “Numerical evolutions of fields on the 2-sphere using a spectral method based on spin-weighted spherical harmonics,” Class. Quant. Grav. 31 (2014) 075019, arXiv:1308.4729 [physics.comp-ph].
  * [101] M. Boyle, “How should spin-weighted spherical functions be defined?,” J. Math. Phys. 57 no. 9, (2016) 092504, arXiv:1604.08140 [gr-qc].