aainstitutetext: Université Libre de Bruxelles and International Solvay
Institutes, ULB-Campus Plaine CP231, B-1050 Brussels, Belgiumbbinstitutetext:
Collège de France, 11 place Marcelin Berthelot, 75005 Paris,
Franceccinstitutetext: Haute-Ecole Robert Schuman, Rue Fontaine aux Mûres,
13b, B-6800, Belgium

# A note on the asymptotic symmetries of electromagnetism

Oscar Fuentealba a,b Marc Henneaux c and Cédric Troessaert
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We extend the asymptotic symmetries of electromagnetism in order to
consistently include angle-dependent $u(1)$ gauge transformations $\epsilon$
that involve terms growing at spatial infinity linearly and logarithmically in
$r$, $\epsilon\sim a(\theta,\varphi)r+b(\theta,\varphi)\ln
r+c(\theta,\varphi)$. The charges of the logarithmic $u(1)$ transformations
are found to be conjugate to those of the $\mathcal{O}(1)$ transformations
(abelian algebra with invertible central term) while those of the
$\mathcal{O}(r)$ transformations are conjugate to those of the subleading
$\mathcal{O}(r^{-1})$ transformations. Because of this structure, one can
decouple the angle-dependent $u(1)$ asymptotic symmetry from the Poincaré
algebra, just as in the case of gravity: the generators of these internal
transformations are Lorentz scalars in the redefined algebra. This implies in
particular that one can give a definition of the angular momentum which is
free from $u(1)$ gauge ambiguities. The change of generators that brings the
asymptotic symmetry algebra to a direct sum form involves non linear
redefinitions of the charges. Our analysis is Hamiltonian throughout and
carried at spatial infinity.

##  1 Introduction

In a recent paper Fuentealba:2022xsz , we extended the Hamiltonian formulation
of Einstein theory in the asymptotically flat context by allowing more
flexible boundary conditions that involved logarithmic terms. This
generalization led to an enlargement of the asymptotic symmetry, from the
original BMS4 algebra Bondi:1962px ; Sachs:1962wk ; Sachs:1962zza  to the log-
BMS4 algebra, which contains, besides the familiar angle-dependent
supertranslations, angle-dependent logarithmic supertranslations
Bergmann:1961zz ; AshtekarLog85 ; BeigSchmidt82 ; Beig:1983sw ;
Ashtekar:1990gc ; Compere:2011ve ; Troessaert:2017jcm . This extension was
carried out at spatial infinity following the approach developed in
Henneaux:2018hdj ; Henneaux:2019yax .

The presence in the asymptotic symmetry algebra of logarithmic
supertranslations had a dramatic impact, in that it enabled one to completely
disentangle the Poincaré subalgebra from the supertranslations (ordinary and
logarithmic), realizing at spatial infinity a mechanism similar to the one
described at null infinity in Mirbabayi:2016axw ; Bousso:2017dny ;
Javadinezhad:2018urv ; Javadinezhad:2022hhl  (see also Chen:2021szm ;
Chen:2021zmu ; Compere:2021inq  in that context).

The purpose of this note is to carry out the analogous construction in the
electromagnetic case, where angle-dependent $u(1)$ gauge transformations play
the role of the supertranslations He:2014cra ; Kapec:2015ena ;
Campiglia:2015qka ; Campiglia:2017mua ; Strominger:2017zoo . We consistently
enlarge the boundary conditions of Henneaux:2018gfi  in such a way that the
asymptotic symmetries contain angle-dependent gauge transformations that grow
at spatial infinity not only logarithmically in $r$, but also linearly in $r$,

$\epsilon\sim a(\theta,\varphi)r+b(\theta,\varphi)\ln
r+c(\theta,\varphi)+o(1).$

The term linear in $r$ is included because it is associated with the
subleading soft theorems Lysov:2014csa ; Campiglia:2016hvg ; Conde:2016csj .

In the standard approach where $\epsilon$ is restricted to take the form
$\epsilon\sim c(\theta,\varphi)+o(1)$, the angle-dependent $u(1)$ asymptotic
transformations of order one transform in a non-trivial representation of the
Poincaré algebra Barnich:2013sxa , Henneaux:2018gfi . As in the case of
gravity, we show that the enlargement of the symmetry enables one to
disentangle the internal asymptotic angle-dependent $u(1)$ symmetries from the
Poincaré algebra: the asymptotic symmetry algebra is the direct sum of the
two. In retrospect, this result is perhaps not too surprising as it is in the
line of the Coleman-Mandula theorem Coleman:1967ad  (even though the
hypotheses of this theorem are not all fulfilled). The fact that the internal
$u(1)$ improper gauge symmetries commute with the Lorentz transformations
leads to an angular momentum that is free from ambiguities under asymptotic
angle-dependent $u(1)$ transformations.

The change of generators that brings the asymptotic symmetry algebra to a
direct sum form involves redefinitions of the Poincaré generators by the
addition of field-dependent gauge transformations corresponding to a specific
non linear redefinitions of the charges.

Our paper is organized as follows. In Section 2, we give the form of the new,
more flexible, boundary conditions and verify the finiteness of the action. We
then describe the asymptotic gauge symmetries in Section 3. Poincaré
invariance is established in Section 4 and further discussed in Appendix A.
The algebra of the asymptotic symmetries and the redefinitions that bring it
to a direct sum form are successively analysed in Sections 5 and 6. In the
concluding Section 7, we outline some potential future developments.

##  2 Action and asymptotic conditions

###  2.1 Extended Hamiltonian action

In the formulation where the conjugate momentum to $A_{0}$ is kept, the phase
space of the Maxwell theory is spanned by the components $(A_{i},A_{0})$ of
the vector potential and their conjugate momenta $(\pi^{i},\pi^{0})$. These
are subject to the “primary constraint”

$\pi^{0}\approx 0\,,$ (1)

and Gauss’s law, which arises as a “secondary constraint”,

$\mathcal{G}=-\partial_{i}\pi^{i}\approx 0\,.$ (2)

It is customary to eliminate $\pi^{0}$ to obtain a reduced theory where the
canonical variables are $(A_{i},\pi^{i})$, the temporal component $A_{0}$
appearing then as a Lagrange multiplier for the (“secondary”) constraint
$\mathcal{G}\approx 0$. We shall refrain from doing so here because $A_{0}$
does carry degrees of freedom at infinity when one imposes boundary conditions
that are invariant under an angle-dependent $u(1)$ symmetry Henneaux:2018hdj ;
Henneaux:2018gfi . It is in that case useful to keep its conjugate momentum in
the Hamiltonian description.

The extended Hamiltonian action of Maxwell theory is given by

$\displaystyle I_{H}[A_{i},\pi^{i},A_{0},\pi^{0};\psi,\lambda]=$
$\displaystyle\int dt\int
d^{\text{3}}x\big{[}\pi^{i}\dot{A}_{i}+\pi^{0}\dot{A}_{0}-(\mathcal{H}+\tilde{\epsilon}\mathcal{G}+\tilde{\mu}\pi^{0})-\psi\mathcal{G}-\lambda\pi^{0}\big{]}+\mathfrak{B}\,,$
(3) $\displaystyle\mathcal{H}=$
$\displaystyle\frac{1}{2\sqrt{g}}\pi^{i}\pi_{i}+\frac{\sqrt{g}}{4}F^{ij}F_{ij}+A_{0}\mathcal{G}-\partial^{i}\pi^{0}A_{i}\,,$
(4)

where $\psi$ and $\lambda$ are the respective Lagrange multipliers for the
constraints (2) and (1) and where the Hamiltonian contains constraint terms
besides the usual energy density $\frac{1}{2}(\mathbf{E}^{2}+\mathbf{B}^{2})$,
which we have included following Henneaux:2018hdj ; Henneaux:2018gfi  for
later convenience. We have set

$\displaystyle\tilde{\epsilon}$ $\displaystyle=\frac{\ln
r}{r}\tilde{\epsilon}_{\log}^{(1)}+\frac{1}{r}\tilde{\epsilon}^{(1)}+o\left(r^{-1}\right)\,,$
(5) $\displaystyle\tilde{\mu}$ $\displaystyle=\frac{\ln
r}{r^{2}}\tilde{\mu}_{\log}^{(1)}+\frac{1}{r^{2}}\tilde{\mu}^{(1)}+o\left(r^{-2}\right)\,,$
(6)

with

$\displaystyle\tilde{\mu}_{\log}^{(1)}-\tilde{\mu}^{(1)}$
$\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{A}+3\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\,,$ (7)
$\displaystyle\tilde{\epsilon}_{\log}^{(1)}-\tilde{\epsilon}^{(1)}$
$\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}\,.$ (8)

Here, the functions of the angles $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{A}$, $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}$ and  $\Psi$
are the $\mathcal{O}(1)$ coefficients appearing in the asymptotic expansion of
the vector potential $A_{\mu}$, see (15)-(18) below. The functions
$\tilde{\mu}_{\log}^{(1)}$, $\tilde{\mu}^{(1)}$,
$\tilde{\epsilon}_{\log}^{(1)}$ and $\tilde{\epsilon}^{(1)}$ are not
completely determined by the equations (7) and (8) but only the combinations
$\tilde{\mu}_{\log}^{(1)}-\tilde{\mu}^{(1)}$ and
$\tilde{\epsilon}_{\log}^{(1)}-\tilde{\epsilon}^{(1)}$ are physically relevant
(see below). With the inclusion of the constraint terms, the Hamiltonian
density $\mathcal{H}+\tilde{\epsilon}\mathcal{G}+\tilde{\mu}\pi^{0}$ coincides
with the density of the Poincaré generator of time translations discussed
below. Finally, the term $\mathfrak{B}$ is the integral over time of a surface
term which we will write explicitly once we have given the boundary
conditions.

The constraint function $\mathcal{G}$ generates the gauge transformation

$\delta A_{i}=\partial_{i}\epsilon\,,$ (9)

while the other constraint function $\pi^{0}$ generates the gauge
transformations

$\delta A_{0}=\mu\,.$ (10)

These are proper in the sense of Benguria:1976in  if $\epsilon$ and $\mu$
decrease sufficiently fast at infinity, i.e.

$\epsilon=o\left(r^{-1}\right)\,,\qquad\mu=o\left(r^{-2}\right)\,,$ (11)

as we shall explicitly show in Section 3 below. We will assume that these
fall-off conditions are fulfilled until we discuss improper gauge
transformations, which can only be meaningfully done after the boundary
conditions have been made precise, a task which we have not achieved yet.

Since the Lagrange multipliers parametrize the gauge transformation performed
in the course of the evolution (on top of the evolution generated by the
Hamiltonian), we take for them the same asymptotic decay

$\psi=o\left(r^{-1}\right)\,,\qquad\lambda=o\left(r^{-2}\right)\,.$ (12)

Again, more general asymptotic behaviours can be considered once one has full
control of the improper gauge symmetries.

The action is invariant under the gauge transformations (9) and (10) provided
one transforms at the same time the Lagrange multipliers as

$\delta\psi=\dot{\epsilon}-\mu-\delta\tilde{\epsilon}\,,\qquad\delta\lambda=\dot{\mu}-\triangle\epsilon-\delta\tilde{\mu}$
(13)

($\delta\tilde{\epsilon}$ and $\delta\tilde{\mu}$ preserve the asymptotic
decays of $\psi$ and $\lambda$ because
$\delta\tilde{\mu}_{\log}^{(1)}=\delta\tilde{\mu}^{(1)}=\delta\tilde{\epsilon}_{\log}^{(1)}=\delta\tilde{\epsilon}^{(1)}=0$).
The action (3) is called the “extended action” because the gauge parameters
$\epsilon$, $\mu$ of the constraints are taken to be independent Dirac1967 ;
Henneaux:1992ig . This is the form that exhibits most explicitly the symmetry.
The formulation that emerges from the Maxwell Lagrangian is characterized by
$\psi=-\tilde{\epsilon}$ (a condition that is permissible once one allows for
more general asymptotic behaviours of the Lagrange multipliers), which relates
$\epsilon$ and $\mu$ through $\mu=\dot{\epsilon}$ so that $\delta
A_{\mu}=\partial_{\mu}\epsilon$. The two formulations are physically
equivalent because $\dot{\epsilon}$ and $\epsilon$ are independent at any
given time Dirac1967 ; Henneaux:1992ig .

###  2.2 Asymptotic conditions

The gauge transformations (9), (10) and (11) are abelian so that their finite
form coincides with their infinitesimal ones. The boundary conditions on the
canonical variables are taken to differ from those of Henneaux:2018hdj ;
Henneaux:2018gfi  by gauge transformation terms $\Delta
A_{i}=\partial_{i}\Theta$, $\Delta A_{0}=\Xi$ with finite gauge parameter
($\Theta$, $\Xi$) that contain contributions that are of order
$\mathcal{O}(r)$ and $\mathcal{O}(\ln r)$ with respect to the leading orders
present in Henneaux:2018hdj ; Henneaux:2018gfi , i.e.

$\Theta=r\Phi_{\text{lin}}+\ln r\,\Phi_{\log}+\Phi+\frac{\ln
r}{r}\Phi_{\log}^{(1)}+o\left(\frac{\ln r}{r}\right),$
$\Xi=\Psi_{\text{lin}}+\frac{\ln
r}{r}\Psi_{\log}+\frac{1}{r}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}+\frac{\ln
r}{r^{2}}\Psi_{\log}^{(1)}+o\left(\frac{\ln r}{r^{2}}\right).$

The terms $\Phi_{\text{lin}}$, $\Phi_{\log}$, $\Phi^{(1)}_{\log}$,
$\Psi_{\text{lin}}$, $\Psi^{(1)}_{\log}$ and $\Psi_{\log}$ are absent in
Henneaux:2018hdj ; Henneaux:2018gfi .

These gauge transformations only affect the components of the vector potential
since the momenta are gauge invariant. In polar coordinates,

$ds^{2}=-dt^{2}+dr^{2}+g_{AB}dx^{A}dx^{B}\,,\qquad{g_{AB}}=r^{2}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}\,,$ (14)

where $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}_{AB}$ ($A,B=1,2$) is the round metric of the unit sphere, the
asymptotic conditions read explicitly

$\displaystyle A_{r}$
$\displaystyle=\Phi_{\text{lin}}+\frac{1}{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}+\frac{\ln
r}{r^{2}}A_{r}^{\log(2)}+\frac{1}{r^{2}}A_{r}^{(2)}+o\left(r^{-2}\right)\,,$
(15) $\displaystyle A_{A}$ $\displaystyle=r\partial_{A}\Phi_{\text{lin}}+\ln
r\,\partial_{A}\Phi_{\log}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{A}+\frac{\ln
r}{r}A_{A}^{\log(2)}+\frac{1}{r}A_{A}^{(2)}+o\left(r^{-2}\right)\,,$ (16)
$\displaystyle\quad A_{r}^{\log(2)}=-\Phi_{\log}^{(1)},\quad
A_{A}^{\log(2)}=\partial_{A}\left(\Phi_{\log}^{(1)}\right)\quad\Leftrightarrow\quad\partial_{A}A_{r}^{\log(2)}+A_{A}^{\log(2)}=0\,,$
(17) $\displaystyle A_{0}$ $\displaystyle=\Psi_{\text{lin}}+\frac{\ln
r}{r}\Psi_{\log}+\frac{1}{r}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}+\frac{\ln
r}{r^{2}}\Psi_{\log}^{(1)}+\frac{1}{r^{2}}\Psi^{(1)}+o\left(r^{-2}\right)\,,$
(18)

for the vector potential and

$\displaystyle\pi^{r}$ $\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}+\frac{1}{r}\pi_{(2)}^{r}+\frac{1}{r^{2}}\pi_{(3)}^{r}+\mathcal{O}\left(r^{-3}\right)\,,$
(19) $\displaystyle\pi^{A}$ $\displaystyle=\frac{1}{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}+\frac{1}{r^{2}}\pi_{(2)}^{A}+\frac{1}{r^{3}}\pi_{(3)}^{A}+\mathcal{O}\left(r^{-4}\right)\,,$
(20) $\displaystyle\pi^{0}$
$\displaystyle=\frac{1}{r^{2}}\pi_{\Psi}^{(2)}+o\left(r^{-2}\right)\,,$ (21)

for the momenta, which carry density weight one. Here, the coefficients in the
expansion of $r$ and $\ln r$ depend only on the angles and are subject to the
following parity conditions,

$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}$
$\displaystyle=\text{odd}\,,\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}=\text{even}\,,\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{A}=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}})_{\text{even}}+\partial_{A}\Phi\,\;\;(\Phi=\text{even})\,,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}=\text{odd}\,,$ (22) $\displaystyle\Phi_{\text{lin}}$
$\displaystyle=\text{odd}\,,\quad\Phi_{\log}=\text{odd}\,,\quad\Psi_{\text{lin}}=\text{even}\,,\quad\Psi_{\text{$\log$}}=\text{even}\,.$
(23)

The parity conditions (22) reduce to those of Henneaux:2018hdj ;
Henneaux:2018gfi . We will mention as we proceed where the parity conditions
(23) on the new terms are needed. The parity conditions (23) are simply
enforced by allowing only $\Theta$ and $\Xi$ in $\Delta
A_{i}=\partial_{i}\Theta$, $\Delta A_{0}=\Xi$ that fulfill these parity
conditions. Note that if $\Phi$ had an odd part, it could be absorbed in
$(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}})_{\text{even}}$ so that we can assume that $\Phi$ is even.

We have kept the terms of order $\mathcal{O}(r^{-2})$ in $A_{r}$ and
$\mathcal{O}(r^{-1})$ in $A_{A}$ because they now contribute to the charges.
Similarly, $\pi_{(2)}^{r}$ becomes physically relevant. Note that the
intermediate terms $A_{r}^{\log(2)}$ and $A_{A}^{\log(2)}$, absent in
Henneaux:2018hdj ; Henneaux:2018gfi , have their entire origin in the gauge
transformation with gauge parameter $\epsilon=\frac{\ln
r}{r}\Phi_{\log}^{(1)}$, hence (17).

Once the boundary conditions have been specified, one can write down
explicitly the boundary terms in the action. The constructive procedure that
we have followed for doing so is explained in Appendix A. It leads to

$\mathfrak{B}=\int dt\oint_{S_{2}^{\infty}}d^{2}x\mathcal{S}-\int
dt\oint_{S_{2}^{\infty}}d^{2}x\mathcal{B}\,,\\\ $ (24)

where $\oint d^{2}x\mathcal{S}$ is a surface integral which is linear in the
time derivatives of the canonical variables,

$\mathcal{S}=\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\Big{(}-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\dot{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}}+\Psi_{\log}\dot{\Phi}-A_{r}^{(2)}\dot{\Psi}_{\text{lin}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}^{(1)}\dot{\Phi}_{\text{lin}}\Big{)}\,,\qquad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}^{(1)}\equiv\Psi_{\log}^{(1)}-\Psi^{(1)}\,,$ (25)

while $\mathcal{B}$ does not depend on the time derivatives of the canonical
variables and is given by

$\mathcal{B}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Pi$\kern 0.0pt}}}^{r}\Psi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\Phi_{\text{lin}}-2\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\Phi_{\text{lin}}\,,$ (26)

with

$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Pi$\kern
0.0pt}}}^{r}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\Psi_{\log}\,.$ (27)

It should be noted that the zero mode of $\Phi$, which drops from the
potential $A_{i}$, is present in the action through the boundary kinetic term.
Its conjugate is non trivial even in the absence of charged fields, and given
by the zero mode of $\Psi_{\log}$.

The boundary term $\oint d^{2}x\mathcal{S}$ completes the boundary term
introduced in Henneaux:2018gfi  in a way that integrability of the boost
charges, which was the very reason for introducing it there, is maintained
with our more general boundary conditions. Because $\mathcal{S}$ involves the
time derivatives of the fields, it contributes to the symplectic form, which
reads explicitly

$\displaystyle\Omega$ $\displaystyle=\int
d^{3}x\big{(}d_{V}\pi^{i}d_{V}A_{i}+d_{V}\pi^{0}d_{V}A_{0}\big{)}$
$\displaystyle\quad-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\left[d_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}d_{V}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern
0.0pt}}}-d_{V}\Psi_{\log}d_{V}\Phi+d_{V}A_{r}^{(2)}d_{V}\Psi_{\text{lin}}-d_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}^{(1)}d_{V}\Phi_{\text{lin}}\right]\,.$ (28)

This symplectic form is non-degenerate in the sense that if $X$ is a phase
space vector field such that $\iota_{X}\Omega=0$, then $X=0$.

The symplectic form pairs $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}$ with  $\Psi$ , as in
Henneaux:2018gfi , and also introduces a surface conjugate $\Psi_{\log}$ to
$\Phi$, which had none with the earlier boundary conditions since there was no
logarithmic term $\ln r/r$ in $A_{0}$. The new variables $\Psi_{\text{lin}}$
and $\Phi_{\text{lin}}$, which would be absent if we had limited the extension
of the gauge transformations of Henneaux:2018gfi  to gauge transformations
blowing only logarithmically at infinity (with no linear term), are naturally
paired with the subleading terms in the expansion of the components of the
potential. This canonical structure will be reflected in the Poisson brackets
of the improper gauge charges. In fact, the brackets of these charges
conversely restrict the symplectic structure to the above form, which provides
an independent argument (besides integrability of the boost charges) for
extending the $\mathcal{S}$ of Henneaux:2018gfi  as in (25).

As a side final comment to this subsection, we note that the pair
$(A_{0},\pi^{0})$ was denoted $(\Psi,\pi_{\Psi})$ in Henneaux:2018hdj ;
Henneaux:2018gfi , and that the Lagrange multiplier $\psi$ was denoted
$A_{0}$. This was due to the peculiar constructive way they were arrived at by
studying the dynamics at the boundary. We revert here to more familiar
notations.

###  2.3 Asymptotic form of the constraints

We shall also require that the constraints decay one order faster than the one
prescribed in Henneaux:2018gfi . This yields (21) for $\pi^{0}$ as well as
$\mathcal{G}=\mathcal{O}(r^{-3})$, which implies, since

$\mathcal{G}=\frac{1}{r}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}+\frac{1}{r^{2}}\left(\partial_{A}\pi_{(2)}^{A}-\pi_{(2)}^{r}\right)+\frac{1}{r^{3}}\left(\partial_{A}\pi_{(3)}^{A}-2\pi_{(3)}^{r}\right)+\mathcal{O}\left(r^{-4}\right)\,,$
(29)

the conditions

$\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern
0.0pt}}}^{A}=0\quad\text{and}\quad\partial_{A}\pi_{(2)}^{A}-\pi_{(2)}^{r}=0\,.$
(30)

These conditions are part of the asymptotic conditions on the canonical
variables.

###  2.4 Finiteness of the kinetic term

It is clear that the only potentially divergent term in the action is the bulk
kinetic term $\int dtd^{3}x\,\pi^{i}\dot{A}_{i}$. We check in this subsection
that it is finite. One finds by direct substitution

$\displaystyle\int dt\,dr\,d^{2}x\,\pi^{i}\dot{A}_{i}$ $\displaystyle=\int
dt\,dr\,d^{2}x\Big{[}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}\dot{\Phi}_{\text{lin}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}\partial_{A}\dot{\Phi}_{\text{lin}}+\frac{\ln
r}{r}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}\partial_{A}\dot{\Phi}_{\log}$ (31)
$\displaystyle\quad+\frac{1}{r}\left(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}\dot{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}}_{A}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}\dot{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}}_{r}+\pi_{(2)}^{r}\dot{\Phi}_{\text{lin}}+\pi_{(2)}^{A}\partial_{A}\dot{\Phi}_{\text{lin}}\right)+\mathcal{O}\left(r^{-2}\right)\Big{]}\,.$
(32)

i) For the linear divergence, we note immediately that the first term is zero
because $\Phi_{\text{lin}}$ is _odd_ , while the second vanishes by virtue of
the condition $\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}=0$.

ii) The term proportional to $r^{-1}\ln r$ vanishes by virtue of
$\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{A}=0$.

iii) The terms proportional to the logarithmic divergence reduce to (after
considering parity conditions)

$\oint d^{2}x\left[-\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}\dot{\Phi}-\left(\partial_{A}\pi_{(2)}^{A}-\pi_{(2)}^{r}\right)\dot{\Phi}_{\text{lin}}\right]\,,$
(33)

which vanish by virtue of the conditions in (30).

The bulk kinetic term and hence the action are finite. The parity conditions
on $\Phi_{\text{lin}}$ and on the asymptotic fields appearing already in
Henneaux:2018hdj ; Henneaux:2018gfi , as well as the fast asymptotic decay of
the constraint functions, are key for this result.

###  2.5 Equations of motion

The equations of motion that follow from the action can be analyzed as
follows.

  * •

Variation with respect to the Lagrange multipliers yields the constraints
$\mathcal{G}\approx 0$ and $\pi^{0}\approx 0$.

  * •

Variation with respect to the field $A_{0}$ yields, besides the bulk equation
of motion $\dot{\pi}^{0}+\mathcal{G}=0$ which is a consequence of the
constraints, the following conditions on the asymptotic fields,

$\displaystyle\dot{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}}_{r}$ $\displaystyle=\dot{\Phi}_{\text{lin}}=0\,,$ (34)
$\displaystyle\dot{A}_{r}^{(2)}$
$\displaystyle=\frac{1}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\left(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,\Psi_{\log}\right)\,,$ (35) $\displaystyle\dot{\Phi}$
$\displaystyle=\Psi_{\text{lin}}\,.$ (36)

  * •

Variation with respect to $\pi^{0}$ yields the bulk equation of motion

$\dot{A}_{0}-\lambda-\partial^{i}A_{i}-\tilde{\mu}=0$ (37)

(without surface term contribution because $\pi^{0}$ decreases sufficiently
fast at infinity). This equation implies

$\displaystyle\dot{\Psi}_{\text{lin}}$ $\displaystyle=0\,,$ (38)
$\displaystyle\dot{\Psi}_{\log}$ $\displaystyle=0\,,$ (39)
$\displaystyle\dot{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Phi_{\text{lin}}+2\Phi_{\text{lin}}\,,$ (40)
$\displaystyle\dot{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}}^{(1)}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Phi_{\log}+2\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\,,$ (41)

to the leading orders to which $\lambda=o\left(r^{-2}\right)$ does not
contribute. The next orders can then be used to express the subleading terms
in $\dot{A}_{0}$ to the Lagrange multiplier $\lambda$.

  * •

Variation of the action with respect to the conjugate momenta $\pi^{i}$ yields
the equation

$\dot{A}_{i}-\partial_{i}\left(A_{0}+\psi+\tilde{\epsilon}\right)-\frac{\pi_{i}}{\sqrt{g}}=0\,,$
(42)

or equivalently

$\pi_{i}=\sqrt{g}\left(\partial_{t}A_{i}-\partial_{i}\left(A_{0}+\psi+\tilde{\epsilon}\right)\right)\,.$
(43)

The asymptotic form of the field equations yields the following conditions

$\displaystyle\dot{\Phi}_{\text{lin}}$ $\displaystyle=0\,,$ (44)
$\displaystyle\dot{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}}_{r}$ $\displaystyle=0\,,$ (45)
$\displaystyle\dot{A}_{r}^{\log(2)}+\Psi_{\log}$ $\displaystyle=0\,,$ (46)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}$ $\displaystyle=\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\left(\dot{A}_{r}^{(2)}-\Psi_{\log}\right)\,,$ (47)
$\displaystyle\pi_{(2)}^{r}$ $\displaystyle=\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\left(\dot{A}_{r}^{(3)}+2\Psi^{(1)}-\Psi^{(1)}_{\log}+2\psi^{(2)}-\psi^{(2)}_{\log}+2\tilde{\epsilon}^{(2)}-\tilde{\epsilon}^{(2)}_{\log}\right)\,,$
(48)

which are consistent with the equations (34) and (35) obtained previously.

  * •

Variation of the action with respect to the gauge field $A_{i}$ yields for the
bulk term (up to a total time derivative):

$\delta I_{\text{bulk}}=\int dt\Big{[}-\int
d^{3}x\Big{(}\dot{\pi}^{i}+\sqrt{g}\nabla_{j}F^{ij}-\partial^{i}\pi^{0}\Big{)}\delta
A_{i}-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern
0.0pt}}}_{r}-\Phi_{\log}\Big{)}\delta\Phi_{\text{lin}}\Big{]}\,,$ (49)

and for the boundary term

$\displaystyle\delta I_{\text{boundary}}=-\int dt\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\Big{\\{}$ $\displaystyle\Big{[}\dot{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}}-\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\triangle$\kern
0.0pt}}}\,\Phi_{\text{lin}}+2\Phi_{\text{lin}}\Big{)}\Big{]}\delta\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}+\dot{\Psi}_{\log}\delta\Phi$ (50)
$\displaystyle+\dot{\Psi}_{\text{lin}}\delta
A_{r}^{(2)}+\Big{[}\dot{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}}^{(1)}-\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}+2\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\Big{)}\Big{]}\delta\Phi_{\text{lin}}\Big{\\}}\,.$
(51)

This implies the equations

$\displaystyle\dot{\pi}^{i}$
$\displaystyle=-\sqrt{g}\nabla_{j}F^{ij}+\partial^{i}\pi^{0}\,,$ (52)
$\displaystyle\dot{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}}^{(1)}$ $\displaystyle=-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}-\Phi_{\log}\Big{)}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}+2\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Phi_{\log}+2\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\,,$ (53)
$\displaystyle\dot{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Phi_{\text{lin}}+2\Phi_{\text{lin}}\,,$ (54)
$\displaystyle\dot{\Psi}_{\log}$ $\displaystyle=\dot{\Psi}_{\text{lin}}=0\,,$
(55)

which are consistent with the dynamical equations for the asymptotic
components of $A_{0}$ derived above.

The time derivatives of the first three leading orders of the conjugate
momentum are then given by

$\displaystyle\dot{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}}^{r}$ $\displaystyle=0\,,$ (56)
$\displaystyle\dot{\pi}_{(2)}^{r}$ $\displaystyle=\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\triangle$\kern 0.0pt}}}\,\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}-\Phi_{\log}\Big{)}\,,$ (57)
$\displaystyle\dot{\pi}_{(3)}^{r}$ $\displaystyle=\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\Big{[}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\triangle$\kern 0.0pt}}}\,A_{r}^{(2)}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\Big{(}A^{(2)A}-A^{\log(2)A}\Big{)}\Big{]}\,.$ (58)

Note that while the leading “monopole” term $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}$ is
conserved, the subsequent terms are not. This is of course well known, but
will not prevent us from defining a conserved quantity involving the
subleading term $\pi_{(2)}^{r}$ and generating gauge transformations that blow
up at infinity, as achieved in Seraj:2016jxi  along different lines.

This concludes the discussion of the equations of motion.

##  3 Improper gauge symmetries

If one allows the parameters $\epsilon$, $\mu$ in (9) and (10) to decrease
slowlier at infinity, in a way compatible with the asymptotic form of the
gauge potential,

$\displaystyle\epsilon$ $\displaystyle=r\epsilon_{\text{lin}}+\ln
r\epsilon_{\log}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}+\frac{\ln
r}{r}\epsilon_{\log}^{(1)}+\frac{1}{r}\epsilon^{(1)}+o\left(r^{-1}\right)\,,$
(59) $\displaystyle\mu$ $\displaystyle=\mu_{\text{lin}}+\frac{\ln
r}{r}\mu_{\log}+\frac{1}{r}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}+\frac{\ln
r}{r^{2}}\mu_{\log}^{(1)}+\frac{1}{r^{2}}\mu^{(1)}+o\left(r^{-2}\right)\,,$
(60)

with

$\epsilon_{\text{lin}}=\text{odd}\,,\quad\epsilon_{\log}=\text{odd}\,,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}=\text{even}\,,\quad\mu_{\text{lin}}=\text{even}\,,\quad\mu_{\text{$\log$}}=\text{even}\,,$
(61)

one finds that the transformations are still symmetries of the action but now
of the “improper gauge type” Benguria:1976in . The bulk part of the
corresponding generator is again given by the combination $\int
d^{3}x\left(\mu\pi^{0}+\epsilon\mathcal{G}\right)$ of the constraints, but
there are in addition non-vanishing surface terms at infinity.

The complete generator $Q_{X}$ of the transformation generated by the phase
space vector field $X$ is obtained through the general formula

$\iota_{X}\Omega=-d_{V}Q_{X}\,,$ (62)

which receives in our case non trivial contributions both from the bulk part
of the symplectic form and from its surface part $\mathcal{S}$
Henneaux:2018gfi . (If $\Omega$ reduced to its canonical bulk part, the rule
(62) would reproduce the integrability condition of Regge:1974zd .)

Applying this general rule to the transformations (9) and (10), which imply
the following transformations of the asymptotic fields

$\displaystyle\delta_{\epsilon}\Phi_{\text{lin}}=\epsilon_{\text{lin}}\,,\quad\delta_{\epsilon}\Phi_{\log}=\epsilon_{\log}\,,\quad\delta_{\epsilon}\Phi=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}\,,\quad\delta_{\epsilon}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}=\epsilon_{\log}\,,\quad\delta_{\epsilon}A_{r}^{(2)}=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}^{(1)}\,,\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}^{(1)}\equiv\epsilon_{\log}^{(1)}-\epsilon^{(1)}\,,$ (63)
$\displaystyle\delta_{\mu}\Psi_{\text{lin}}=\mu_{\text{lin}}\,,\quad\delta_{\mu}\Psi_{\log}=\mu_{\log}\,,\quad\delta_{\mu}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\mu$\kern 0.0pt}}}\,,\quad\delta_{\mu}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}^{(1)}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\mu$\kern 0.0pt}}}^{(1)}\,,\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}^{(1)}\equiv\mu_{\log}^{(1)}-\mu^{(1)}\,,$ (64)

one finds that the canonical generator of the improper gauge symmetries reads

$G_{\mu,\epsilon}=\int
d^{3}x\left(\mu\pi^{0}+\epsilon\mathcal{G}\right)+Q_{\epsilon_{\text{lin}}}+Q_{\epsilon_{\log}}+Q_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}}+Q_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\epsilon$\kern
0.0pt}}}^{(1)}}+Q_{\mu_{\text{lin}}}+Q_{\mu_{\log}}+Q_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}}+Q_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\mu$\kern 0.0pt}}}^{(1)}}\,,$ (65)

with

$\displaystyle Q_{\epsilon_{\text{lin}}}$ $\displaystyle=\oint
d^{2}x\epsilon_{\text{lin}}\left[\pi_{(2)}^{r}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\left(\Psi_{\log}^{(1)}-\Psi^{(1)}\right)\right]\,,$ (66)
$\displaystyle Q_{\epsilon_{\log}}$ $\displaystyle=\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,\epsilon_{\log}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}\,,$ (67) $\displaystyle
Q_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}}$ $\displaystyle=\oint d^{2}x\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}\left(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\Psi_{\log}\right)\,,$ (68)
$\displaystyle Q_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}}$ $\displaystyle=\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}\Psi_{\text{lin}}\,,$
(69)

and

$\displaystyle Q_{\mu_{\text{lin}}}$ $\displaystyle=-\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,\mu_{\text{lin}}A_{r}^{(2)}\,,$ (70) $\displaystyle
Q_{\mu_{\log}}$ $\displaystyle=-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\mu_{\log}\,\Phi\,,$ (71) $\displaystyle Q_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}}$
$\displaystyle=-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\,,$ (72) $\displaystyle Q_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}^{(1)}}$
$\displaystyle=-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}^{(1)}\Phi_{\text{lin}}\,.$ (73)

In the charge $Q_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}}$, only the even part of
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}$ contributes since the field
$\Psi_{\text{lin}}$ is even. The odd part of $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}$
drops out and defines a proper gauge symmetry. To emphasize this point, one
sometimes writes $Q_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}_{\text{even}}}$. Therefore, among the
charges generating the improper gauge symmetries parametrized by $\epsilon$,
there are two charges characterized by parameters that are even functions on
the sphere ($Q_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}}$ and $Q_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}_{\text{even}}}$), and
two charges characterized by odd parameters ($Q_{\epsilon_{\text{lin}}}$ and
$Q_{\epsilon_{\log}}$).

Similarly, among the charges generating the improper gauge symmetries
parametrized by $\mu$, there are two charges characterized by parameters that
are odd functions on the sphere ($Q_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}_{\text{odd}}}$ and
$Q_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}^{(1)}_{\text{odd}}}$), and two charges characterized by even
parameters ($Q_{\mu_{\text{lin}}}$ and $Q_{\mu_{\log}}$). This is because
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}$ and $\Phi_{\text{lin}}$ are odd funtions on the sphere, so that
$\mu_{\text{even}}$ and $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}^{(1)}_{\text{even}}$ drop out
from the charges and define proper gauge symmetries.

Furthermore, if $\epsilon_{\log}^{(1)}$ and $\epsilon^{(1)}$ are such that
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}=0$, the corresponding gauge transformation
is proper since it has zero charge. A similar property holds for
$\mu_{\log}^{(1)}$ and $\mu^{(1)}$.

There is thus a total of eight non trivial improper gauge charges, each
characterized by a definite parity under the sphere antipodal map. The
brackets among these charges are easily computed. They are found to form a
centrally extended Abelian algebra with the following non-zero central charges

$\displaystyle\\{G_{\epsilon_{\text{lin}}},G_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}^{(1)}}\\}$
$\displaystyle=-\\{G_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}^{(1)}},G_{\epsilon_{\text{lin}}}\\}=\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,\epsilon_{\text{lin}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}^{(1)}\,,$ (74)
$\displaystyle\\{G_{\epsilon_{\log}},G_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}}\\}$
$\displaystyle=-\\{G_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}},G_{\epsilon_{\log}}\\}=\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,\epsilon_{\log}\,\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}\,,$ (75)
$\displaystyle\\{G_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}},G_{\mu_{\log}}\\}$
$\displaystyle=-\\{G_{\mu_{\log}},G_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}}\\}=\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}\,\mu_{\log}\,,$ (76)
$\displaystyle\\{G_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}},G_{\mu_{\text{lin}}}\\}$
$\displaystyle=-\\{G_{\mu_{\text{lin}}},G_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}^{(1)}}\\}=\oint d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}^{(1)}\,\mu_{\text{lin}}\,.$ (77)

In the paper Henneaux:2018gfi  where the boundary conditions are more
restrictive, only the improper gauge charges $G_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}}$ and
$G_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}}$ are present. These commute, $\\{G_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}},G_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\mu$\kern 0.0pt}}}}\\}=0$ so that the central terms are absent. The
improper gauge charges associated with logarithmic gauge transformations
($G_{\epsilon_{\log}}$ and $G_{\mu_{\log}}$) are conjugate to these charges.
The improper gauge transformations linear in $r$ bring in two additional new
charges ($G_{\epsilon_{\text{lin}}}$ and $G_{\mu_{\text{lin}}}$) as expected,
but at the same time they turn on the subleading gauge transformations, which
become non-trivial and also bring in two additional non trivial charges. These
form canonically conjugate pairs. It is because of the central terms present
in the extended improper gauge algebra that one can make the improper gauge
generators commute with the Poincaré generators in the asymptotic symmetry
algebra, as in the gravity case Fuentealba:2022xsz , and as we shall show
explicitly below.

Among the improper gauge charges, $G_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}}$ and
$G_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}}$ combine to form the angle-dependent $u(1)$ charge seen at null
infinity Henneaux:2018gfi . The charge $G_{\epsilon_{\text{lin}}}$ involves
the $1/r^{3}$ component of the electric field with an odd gauge parameter and
is related to the electric dipole moment for the $\ell=1$ spherical harmonic.
Together with its companion $G_{\mu_{\text{lin}}}$, we expect that it should
be connected to the charges that underlie the subleading soft photon theorems
Lysov:2014csa ; Campiglia:2016hvg ; Conde:2016csj .

Note that the integrands of the charges $Q_{\epsilon_{\text{lin}}}$ and
$Q_{\epsilon_{\log}}$ are not conserved in time since it follows from the
equations of motion that

$\partial_{t}\left(\pi_{(2)}^{r}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}^{(1)}\right)=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}+2\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\,,\qquad\partial_{t}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\triangle$\kern 0.0pt}}}\,\Phi_{\text{lin}}+2\Phi_{\text{lin}}\,.$ (78)

This expresses the fact that $Q_{\epsilon_{\text{lin}}}$ and
$Q_{\epsilon_{\log}}$ do not commute with the generator $H$ of time
translations and hence are conserved only at the price of an explicit time
dependence. Indeed the Poisson brackets with the Poincaré generators (prior to
decoupling) imply, as we shall see,

$\\{G_{\epsilon_{\text{lin}}},H\\}=G_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}}\,,\qquad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}=-(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\triangle$\kern 0.0pt}}}\,\epsilon_{\text{lin}}+2\epsilon_{\text{lin}})$
(79)

and

$\\{G_{\epsilon_{\log}},H\\}=G_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}^{(1)}}\,,\qquad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}^{(1)}=-(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\epsilon_{\log}+2\epsilon_{\log})\,.$ (80)

Similarly,

$\\{G_{\mu_{\text{lin}}},H\\}=G_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}}\,,\qquad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}=-\mu_{\text{lin}}$ (81)

and

$\\{G_{\mu_{\log}},H\\}=G_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}^{(1)}}\,,\qquad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}=-\mu_{\log}\,,$ (82)

in agreement with (35) and (36).

The equations of motion have been derived in Subsection 2.5 assuming that the
Lagrange multipliers $\psi$ and $\lambda$ decrease sufficiently fast at
infinity so that they define proper gauge symmetries. The surface term in the
action was adjusted under this assumption. In fact, if we had not known that
$\psi=o(r^{-1})$ and $\lambda=o(r^{-2})$, we would have derived from the
extremization of the action (3) (with that boundary term) that consistency of
the equations of motion implies that $\psi$ and $\lambda$ should define proper
gauge transformations ($\psi_{\text{lin}}=0=\psi_{\log}=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\psi$\kern
0.0pt}}}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\psi$\kern 0.0pt}}}^{(1)}$ and
$\lambda_{\text{lin}}=0=\lambda_{\log}=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}^{(1)}$).

A more general behaviour of the Lagrange multipliers $\psi$ and $\lambda$ can
be accomodated. One can allow them to define improper gauge symmetries
provided one includes in the action the corresponding surface integral. This
would be in particular needed if one wanted to impose the Lorenz gauge
$\dot{A}_{0}=\partial^{i}A_{i}$ which requires from (37) that $\lambda$ should
be equal to $-\tilde{\mu}$.

##  4 Poincaré invariance

###  4.1 Poincaré transformations

The analysis of Poincaré invariance proceeds along the lines of
Henneaux:2018gfi , adapted to the less stringent boundary conditions
considered in this paper. We shall therefore give only the final results and
check their consistency. A more constructive approach, which was actually the
approach we followed to arrive at the results presented below, is given in
Appendix A.

The $10$-dimensional Poincaré symmetry is generated by the vector fields
conveniently parametrized in spherical coordinates as

$\displaystyle\xi$ $\displaystyle=br+T\,,$ (83) $\displaystyle\xi^{r}$
$\displaystyle=W\,,$ (84) $\displaystyle\xi^{A}$
$\displaystyle=Y^{A}+\frac{1}{r}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}W\,,$ (85)

where

$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}b+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}b=0\,,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}W+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}W=0\,,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}Y_{B}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}Y_{A}=0\,,\quad\partial_{A}T=0\,.$ (86)

The boost function $b(x^{A})$ and the spatial translation function $W(x^{A})$
each belong to the three-dimensional space of the spin-$1$ representation of
the rotation group. The vector field $Y^{A}$ generates infinitesimal
rotations. It is a Killing vector on the sphere and depends also on three
parameters. The last parameter of the Poincaré group is given by $T$, which is
constant.

The Poincaré transformations of the fields are given by

$\displaystyle\delta_{\xi,\xi^{i}}A_{i}$
$\displaystyle=\frac{\xi\pi_{i}}{\sqrt{g}}+\partial_{i}\left(\xi
A_{0}\right)+\mathcal{L}_{\xi}A_{i}+\delta_{\epsilon_{(T,W)}}A_{i}\,,$ (87)
$\displaystyle\delta_{\xi,\xi^{i}}\pi^{i}$
$\displaystyle=\sqrt{g}\nabla_{m}\big{(}F^{mi}\xi\big{)}+\xi\partial^{i}\pi^{0}+\mathcal{L}_{\xi}\pi^{i}\,,$
(88) $\displaystyle\delta_{\xi,\xi^{i}}A_{0}$
$\displaystyle=\nabla_{i}\left(\xi
A^{i}\right)+\mathcal{L}_{\xi}A_{0}+\delta_{\mu_{(b,T,W)}}A_{0}\,,$ (89)
$\displaystyle\delta_{\xi,\xi^{i}}\pi^{0}$
$\displaystyle=\xi\partial_{i}\pi^{i}+\mathcal{L}_{\xi}\pi^{0}\,,$ (90)

These transformations take the same form as in Henneaux:2018gfi , except for
the additional correction terms $\delta_{\epsilon_{(T,W)}}A_{i}$ and
$\delta_{\mu_{(b,T,W)}}A_{0}$ that must be added in order to preserve
integrability of the Poincaré charges (see below). These transformations are
improper gauge symmetries with parameters

$\epsilon_{(T,W)}=\frac{\ln
r}{r}\epsilon_{\log(T,W)}^{(1)}+\frac{1}{r}\epsilon_{(T,W)}^{(1)}+o\left(r^{-1}\right)\,,$
(91)

and

$\mu_{(b,T,W)}=\frac{\ln
r}{r^{2}}\mu_{\log(b,T,W)}^{(1)}+\frac{1}{r^{2}}\mu_{(b,T,W)}^{(1)}+o\left(r^{-2}\right)\,,$
(92)

with

$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}_{(T,W)}^{(1)}=T\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{0}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{A}-W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\,,$ (93)

and

$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}_{(b,T,W)}^{(1)}=4bA_{r}^{(2)}+T\left(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}^{A}+3\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\right)-\left(W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}-\partial_{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}\right)-\left[-\partial_{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}+W\left(\Psi_{\log}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}\right)\right]\,.$ (94)

Asymptotically expanding (87)-(90) leads to the following transformation laws:

  * •

For the leading orders:

$\displaystyle\delta_{\xi,\xi^{i}}\Phi_{\text{lin}}$
$\displaystyle=\mathcal{L}_{Y}\Phi_{\text{lin}}+b\Psi_{\text{lin}}\,,$ (95)
$\displaystyle\delta_{\xi,\xi^{i}}\Phi_{\log}$
$\displaystyle=\mathcal{L}_{Y}\Phi_{\log}+b\Psi_{\log}\,,$ (96)
$\displaystyle\delta_{\xi,\xi^{i}}\Psi_{\text{lin}}$
$\displaystyle=\mathcal{L}_{Y}\Psi_{\text{lin}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\Phi_{\text{lin}}\right)+3b\Phi_{\text{lin}}\,,$
(97) $\displaystyle\delta_{\xi,\xi^{i}}\Psi_{\log}$
$\displaystyle=\mathcal{L}_{Y}\Psi_{\text{$\log$}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\Phi_{\log}\right)\,,$ (98)

  * •

For the subleading orders:

$\displaystyle\delta_{\xi,\xi^{i}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}$
$\displaystyle=\mathcal{L}_{Y}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}+\frac{b}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}_{r}+b\Psi_{\log}\,,$ (99)
$\displaystyle\delta_{\xi,\xi^{i}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{A}$
$\displaystyle=\mathcal{L}_{Y}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{A}+\frac{b}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}_{A}+\partial_{A}\left(W\Phi_{\text{lin}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{B}W\partial_{B}\Phi_{\text{lin}}+b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}+T\Psi_{\text{lin}}\right)\,,$ (100)
$\displaystyle\delta_{\xi,\xi^{i}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}$
$\displaystyle=\mathcal{L}_{Y}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\left[b\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}\left(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}-\Phi_{\log}\right)\right]\,,$ (101)
$\displaystyle\delta_{\xi,\xi^{i}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}$
$\displaystyle=\mathcal{L}_{Y}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}-\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}^{AB}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{C}\big{(}b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$F$\kern 0.0pt}}}_{BC}\big{)}\,,$
(102) $\delta_{\xi,\xi^{i}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}=\mathcal{L}_{Y}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W\partial_{A}\Psi_{\text{lin}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\big{(}b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}^{A}\big{)}+2b\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}+T\left(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Phi_{\text{lin}}+2\Phi_{\text{lin}}\right)\,,$ (103)

which implies that

$\delta_{\xi,\xi^{i}}\Phi=\mathcal{L}_{Y}\Phi+W\Phi_{\text{lin}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{B}W\partial_{B}\Phi_{\text{lin}}+b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}^{\text{odd}}+T\Psi_{\text{lin}}\,.$ (104)

  * •

For the sub-subleading orders:

$\displaystyle\delta_{\xi,\xi^{i}}A_{r}^{(2)}$
$\displaystyle=\mathcal{L}_{Y}A_{r}^{(2)}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W\left(\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{A}\right)-W\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}$
$\displaystyle\quad+\frac{b}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}\pi_{(2)}^{r}+b\left(\Psi^{\log(1)}-\Psi^{(1)}\right)+\frac{T}{\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}+T\left(\Psi_{\log}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}\right)+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}_{(T,W)}^{(1)}\,,$ (105)
$\displaystyle\delta_{\xi,\xi^{i}}A_{A}^{(2)}$
$\displaystyle=\mathcal{L}_{Y}A_{A}^{(2)}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{B}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}W\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}^{B}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{B}W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{B}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{A}+\partial_{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}$
$\displaystyle\quad+W\partial_{A}\Phi_{\log}+\frac{b}{\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}\pi_{(2)A}+\frac{T}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}_{A}\,,$ (106)
$\displaystyle\delta_{\xi,\xi^{i}}\pi_{(2)}^{r}$
$\displaystyle=\mathcal{L}_{Y}\pi_{(2)}^{r}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}\right)-\partial_{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}$
$\displaystyle\quad+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\big{[}b\big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}A_{r}^{(2)}+A^{(2)A}-A^{\log(2)A}\big{)}\big{]}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\left[T\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}\left(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}-\Phi_{\log}\right)\big{)}\right]\,,$ (107)
$\displaystyle\delta_{\xi,\xi^{i}}\pi_{(2)}^{A}$
$\displaystyle=\mathcal{L}_{Y}\pi_{(2)}^{A}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{B}\left(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{B}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}\right)+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{B}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{B}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}-W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}$
$\displaystyle\quad-\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}^{AB}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{C}\big{(}bF_{BC}^{(2)}\big{)}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}^{AB}b\left(\partial_{B}A_{r}^{(2)}+A_{B}^{(2)}\right)$
$\displaystyle\quad-\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}^{AB}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{C}\big{(}T\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$F$\kern
0.0pt}}}_{BC}\big{)}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,T\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\,,$ (108)
$\displaystyle\delta_{\xi,\xi^{i}}\Psi_{\log}^{(1)}$
$\displaystyle=\mathcal{L}_{Y}\Psi_{\log}^{(1)}+\partial_{A}W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\Psi_{\log}-W\Psi_{\log}$ $\displaystyle\quad+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(bA^{\log(2)A}\right)+bA_{r}^{\log(2)}+T\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Phi_{\text{log}}+\mu_{\log(b,T,W)}^{(1)}\,,$ (109)
$\displaystyle\delta_{\xi,\xi^{i}}\Psi^{(1)}$
$\displaystyle=\mathcal{L}_{Y}\Psi^{(1)}+\partial_{A}W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}+W\left(\Psi_{\log}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}\right)$
$\displaystyle\quad+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern
0.0pt}}}_{A}\left(bA^{(2)A}\right)+b\left(A_{r}^{(2)}+A_{r}^{\log(2)}\right)+T\left(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}^{A}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\right)+\mu_{(b,T,W)}^{(1)}\,,$
(110)
$\displaystyle\delta_{\xi,\xi^{i}}\Phi_{\log}^{(1)}=\mathcal{L}_{Y}\Phi_{\log}^{(1)}-\partial_{A}W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\Phi_{\log}+b\Psi^{(1)}+T\Psi_{\log}-\epsilon^{(1)}_{\log(T,W)}\,.$
(111)

###  4.2 Poincaré generators

The justification of the above definitions of the Poincaré transformations of
the fields and of the symplectic structure is that the latter is invariant
under the former,

$\mathcal{L}_{X_{\xi,\xi^{i}}}\Omega=0$ (112)

with $X_{\xi,\xi^{i}}$ the phase space vector field defined by (87)-(90). The
verification of (112) is somewhat cumbersome and involves the following key
ingredients:

  * •

The divergent terms in $\mathcal{L}_{X_{\xi,\xi^{i}}}\Omega$ are zero thanks
to the parity conditions and the equation (17). In particular, the parity
conditions (23) on the new terms in the asymptotic expansion of the vector
potential are needed.

  * •

The surface term in the symplectic form has been adjusted so that the
remaining terms in $\mathcal{L}_{X_{\xi,\xi^{i}}}\Omega$, which are finite
surface terms (with no bulk contribution), exactly cancel, taking into account
the contributions coming from the correcting gauge transformation terms
$\delta_{\epsilon_{(T,W)}}A_{i}$ and $\delta_{\mu_{(b,T,W)}}A_{0}$. As in
Henneaux:2018gfi , it is actually the requirement that
$\mathcal{L}_{X_{\xi,\xi^{i}}}\Omega$ should vanish for boosts ($\xi=br$,
$\xi^{i}=0$) that fixes the form of the surface terms to be added to the
standard canonical bulk part of $\Omega$.

Since
$\mathcal{L}_{X_{\xi,\xi^{i}}}\Omega=d_{V}\iota_{X_{\xi,\xi^{i}}}\Omega$, one
can now compute the Poincaré generators through the formula

$\iota_{X_{\xi,\xi^{i}}}\Omega=-d_{V}P_{\xi,\xi^{i}}\,.$ (113)

One gets

$P_{\xi,\xi^{i}}=\int
d^{3}x\left(\xi\mathcal{H}+\xi^{i}\mathcal{H}_{i}+\epsilon_{(T,W)}\mathcal{G}+\mu_{(b,T,W)}\pi^{0}\right)+\mathcal{B}_{\xi,\xi^{i}}\,,$
(114)

where

$\mathcal{H}=\frac{\sqrt{g}}{4}F^{ij}F_{ij}+\frac{1}{2\sqrt{g}}\pi^{i}\pi_{i}-\partial^{i}\pi^{0}A_{i}-\partial_{i}\pi^{i}A_{0}\,,\\\
$ (115)

(as above) and

$\displaystyle\mathcal{H}_{i}$
$\displaystyle=F_{ij}\pi^{j}-\partial_{j}\pi^{j}A_{i}+\pi^{0}\partial_{i}A_{0}\,,$
(116) $\displaystyle\mathcal{B}_{\xi,\xi^{i}}$ $\displaystyle=\oint
d^{2}x\Big{\\{}b\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Pi$\kern 0.0pt}}}^{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}+\Pi_{(2)}^{r}\Psi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{A}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\partial_{A}A_{r}^{(2)}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\Phi_{\text{lin}}-\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,A_{r}^{(2)}\Phi_{\text{lin}}\Big{)}$
$\displaystyle\quad+Y^{A}\left(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Pi$\kern 0.0pt}}}^{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{A}+\Pi_{(2)}^{r}\partial_{A}\Phi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\Psi_{\text{lin}}\partial_{A}A_{r}^{(2)}\right)$
$\displaystyle\quad+T\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Pi$\kern
0.0pt}}}^{r}\Psi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\partial_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\Phi_{\text{lin}}-2\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\Phi_{\text{lin}}\Big{)}$
$\displaystyle\quad+W\Big{[}\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Pi$\kern
0.0pt}}}^{r}+3\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Pi$\kern 0.0pt}}}^{r}\Big{)}\Phi_{\text{lin}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Pi$\kern
0.0pt}}}^{r}\partial_{A}\Phi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Psi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\partial_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\Psi_{\text{lin}}\Big{]}\Big{\\}}\,,$ (117)

with

$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Pi$\kern 0.0pt}}}^{r}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,\Psi_{\log}\,,$ (118) $\displaystyle\Pi_{(2)}^{r}$
$\displaystyle=\pi_{(2)}^{r}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\left(\Psi_{\log}^{(1)}-\Psi^{(1)}\right)\,.$ (119)

##  5 Asymptotic symmetry algebra

A direct computation shows that the algebra of the asymptotic symmetries as
defined above is the semi-direct sum of the Poincaré algebra with the above
Abelian set of improper gauge symmetries, endowed with non-trivial central
charges. Indeed, the Poisson brackets of the generators are given by

$\displaystyle\big{\\{}P_{\xi_{1},\xi_{1}^{i}},P_{\xi_{2},\xi_{2}^{i}}\big{\\}}$
$\displaystyle=P_{\hat{\xi},\hat{\xi}^{i}}\,,$ (120)
$\displaystyle\big{\\{}G_{\mu,\epsilon},P_{\xi,\xi^{i}}\big{\\}}$
$\displaystyle=G_{\hat{\mu},\hat{\epsilon}}\,,$ (121)
$\displaystyle\big{\\{}G_{\mu_{1},\epsilon_{1}},G_{\mu_{2},\epsilon_{2}}\big{\\}}$
$\displaystyle=C_{\\{\mu_{1},\epsilon_{1};\mu_{2},\epsilon_{2}\\}}\,,$ (122)

where

$\displaystyle\hat{\xi}$
$\displaystyle=\xi_{1}^{i}\partial_{i}\xi_{2}-\xi_{2}^{i}\partial_{i}\xi_{1}\,,$
(123) $\displaystyle\hat{\xi}^{i}$
$\displaystyle=\xi_{1}^{j}\partial_{j}\xi_{2}^{i}-\xi_{2}^{j}\partial_{j}\xi_{1}^{i}+g^{ij}\left(\xi_{1}\partial_{j}\xi_{2}-\xi_{2}\partial_{j}\xi_{1}\right)\,,$
(124)

and

$\displaystyle\hat{\mu}_{\text{lin}}$
$\displaystyle=-Y^{A}\partial_{A}\mu_{\text{lin}}-3b\epsilon_{\text{lin}}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\epsilon_{\text{lin}}\right)\,,$ (125)
$\displaystyle\hat{\epsilon}_{\text{lin}}$
$\displaystyle=-Y^{A}\partial_{A}\epsilon_{\text{lin}}-b\mu_{\text{lin}}\,,$
(126) $\displaystyle\hat{\mu}_{\log}$
$\displaystyle=-Y^{A}\partial_{A}\mu_{\log}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\epsilon_{\log}\right)\,,$ (127)
$\displaystyle\hat{\epsilon}_{\log}$
$\displaystyle=-Y^{A}\partial_{A}\epsilon_{\log}-b\mu_{\log}\,,$ (128)
$\displaystyle\hat{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\mu$\kern 0.0pt}}}}$ $\displaystyle=-Y^{A}\partial_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}\right)-T\left(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\epsilon_{\text{lin}}+2\epsilon_{\text{lin}}\right)-\partial_{A}W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\mu_{\text{lin}}\,,$ (129) $\displaystyle\hat{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}}$
$\displaystyle=-Y^{A}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}-b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}-T\mu_{\text{lin}}-W\epsilon_{\text{lin}}-\partial_{A}W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\epsilon_{\text{lin}}\,,$ (130)
$\displaystyle\hat{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\mu$\kern 0.0pt}}}}^{(1)}$
$\displaystyle=-Y^{A}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}^{(1)}+3b\epsilon^{(1)}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\epsilon^{(1)}\right)-T\left(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\epsilon_{\log}+2\epsilon_{\log}\right)+3W\mu_{\log}-\partial_{A}W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\mu_{\log}\,,$ (131) $\displaystyle\hat{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}}^{(1)}$
$\displaystyle=-Y^{A}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}-b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}^{(1)}-T\mu_{\log}-\partial_{A}\left(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W\epsilon_{\log}\right)\,.$ (132)

in addition to (74)-(77).

##  6 Algebraic decoupling of the $u(1)$ charges

The presence of invertible central terms in the algebra of the improper gauge
symmetries can be used to redefine the Poincaré generators in such a way that
the new form of the algebra has a direct sum structure: the improper gauge
charges commute with the new Poincaré generators. The redefinition is non-
linear and is obtained by just applying the general formulas derived in
Fuentealba:2022xsz .

It is in our case explicitly achieved by adding to the Poincaré
transformations the following field-dependent improper gauge transformations

$\displaystyle\mu_{\text{lin}}^{(b,Y)}$
$\displaystyle=-\mathcal{L}_{Y}\Psi_{\text{lin}}-3b\Phi_{\text{lin}}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\Phi_{\text{lin}}\right)\,,$ (133)
$\displaystyle\epsilon_{\text{lin}}^{(b,Y)}$
$\displaystyle=-\mathcal{L}_{Y}\Phi_{\text{lin}}-b\Psi_{\text{lin}}\,,$ (134)
$\displaystyle\mu_{\log}^{(b,Y)}$
$\displaystyle=-\frac{1}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\mathcal{L}_{Y}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Pi$\kern
0.0pt}}}^{r}-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\right)\,,$
(135) $\displaystyle\epsilon_{\log}^{(b,Y)}$
$\displaystyle=-\mathcal{L}_{Y}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}-\frac{b}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Pi$\kern 0.0pt}}}^{r}\,,$ (136)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\mu$\kern 0.0pt}}}^{(b,Y,T,W)}$
$\displaystyle=-\mathcal{L}_{Y}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}(b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}^{A})-T\left(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Phi_{\text{lin}}+2\Phi_{\text{lin}}\right)-\partial_{A}W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\Psi_{\text{lin}}\,,$ (137) $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}^{(b,Y,T,W)}$ $\displaystyle=-\mathcal{L}_{Y}\Phi-b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}-T\Psi_{\text{lin}}-W\Phi_{\text{lin}}-\partial_{A}W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\Phi_{\text{lin}}\,,$ (138) $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}_{(b,Y,T,W)}^{(1)}$ $\displaystyle=-\frac{1}{\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}\mathcal{L}_{Y}\Pi_{(2)}^{r}-3bA_{r}^{(2)}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}A_{r}^{(2)}\right)$
$\displaystyle\quad-T\left(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}+2\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\right)+\frac{1}{\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}\left(-3W\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Pi$\kern 0.0pt}}}^{r}+\partial_{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Pi$\kern 0.0pt}}}^{r}\right)\,,$
(139) $\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}_{(b,Y,T,W)}^{(1)}$
$\displaystyle=-\mathcal{L}_{Y}A_{r}^{(2)}-\frac{b}{\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}\Pi_{(2)}^{r}-\frac{T}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Pi$\kern
0.0pt}}}^{r}+2W\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}-\partial_{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\,.$ (140)

The parameters of these transformations depend on the charges themselves and
preserve integrability. The charges associated with these gauge
transformations are given by

$\displaystyle Q_{\mu,\epsilon}^{\text{extra}}$ $\displaystyle=-\oint
d^{2}x\Big{\\{}b\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Pi$\kern 0.0pt}}}^{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}+\Pi_{(2)}^{r}\Psi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{A}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\partial_{A}A_{r}^{(2)}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\Phi_{\text{lin}}-\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,A_{r}^{(2)}\Phi_{\text{lin}}\Big{)}$
$\displaystyle\quad+Y^{A}\left(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Pi$\kern 0.0pt}}}^{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{A}+\Pi_{(2)}^{r}\partial_{A}\Phi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\Psi_{\text{lin}}\partial_{A}A_{r}^{(2)}\right)$
$\displaystyle\quad+T\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Pi$\kern
0.0pt}}}^{r}\Psi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\partial_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\Phi_{\text{lin}}-2\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\Phi_{\text{lin}}\Big{)}$
$\displaystyle\quad+W\Big{[}\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Pi$\kern
0.0pt}}}^{r}+3\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Pi$\kern 0.0pt}}}^{r}\Big{)}\Phi_{\text{lin}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Pi$\kern
0.0pt}}}^{r}\partial_{A}\Phi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Psi_{\text{lin}}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\partial_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\Psi_{\text{lin}}\Big{]}\Big{\\}}\,.$ (141)

These boundary terms are supplemented by weakly vanishing bulk terms with
gauge parameters $\epsilon^{\text{extra}}_{(b,Y,T,W)}$ and
$\mu^{\text{extra}}_{(b,Y,T,W)}$, the fall-off of which is determined by the
above parameters.

In fact, the above transformations were derived from these charges, which were
constructed according to the general formulas of Fuentealba:2022xsz , so that
integrabilty was not really an issue.

The new Poincaré generators read

$\tilde{P}_{\xi,\xi^{i}}=P_{\xi,\xi^{i}}+G_{\mu,\epsilon}^{\text{extra}}\,,$
(142)

Because

$\mathcal{B}_{\xi,\xi^{i}}+Q_{\mu,\epsilon}^{\text{extra}}=0\,,$ (143)

the new Poincaré generators are pure bulk

$\tilde{P}_{\xi,\xi^{i}}=\int
d^{3}x\left[\xi\mathcal{H}+\xi^{i}\mathcal{H}_{i}+(\epsilon_{(T,W)}+\epsilon^{\text{extra}}_{(b,Y,T,W)})\mathcal{G}+(\mu_{(b,T,W)}+\mu^{\text{extra}}_{(b,Y,T,W)})\pi^{0}\right]\,,$
(144)

without surface term. One can then check that their brackets with the $u(1)$
gauge charges are weakly zero:

$\big{\\{}G_{\mu,\epsilon},\tilde{P}_{\xi,\xi^{i}}\big{\\}}=0\,.$ (145)

This implies in particular that they are conserved with no explicit time
dependence. One can also check that the new Poincaré generators satisfies the
Poincaré algebra:

$\big{\\{}\tilde{P}_{\xi_{1},\xi_{1}^{i}},\tilde{P}_{\xi_{2},\xi_{2}^{i}}\big{\\}}=\tilde{P}_{\hat{\xi},\hat{\xi}^{i}}\,.$
(146)

The computation is direct. This result is actually guaranteed to hold by the
general argument of Fuentealba:2022xsz .

##  7 Conclusions

In this paper, we have consistently extended the asymptotic symmetries of
electromagnetism in four spacetime dimensions by allowing improper gauge
transformations parametrized by coefficients that blow up at infinity like
$\ln r$ and $r$. We have also shown that the structure of the algebra can be
used to redefine the Poincaré generators in such a way that there is no $u(1)$
ambiguity in the definition of the Lorentz generators (angular momentum and
boost generators (center of mass)). Even though parametrized by a function of
the angles, the $u(1)$ charges transform in the trivial representation of the
redefined Lorentz algebra. The necessary redefinitions are non-linear and can
be performed in a direct manner because we have a well-defined Hamiltonian
formulation (see Fuentealba:2022yqt  for a more complete discussion of this
point in the context of Poisson manifolds). In particular, integrability of
the redefined transformations is never an issue.

Similar results, which go beyond the analysis of Henneaux:2019yqq  by adopting
more flexible boundary conditions, can be established in higher dimensions as
we shall show in future work.

One might wonder whether one could continue the construction and include gauge
transformations that asymptotically blow up like $r^{2}$, or $r^{3}$ etc.
While we have not made a systematic analysis of this question, preliminary
investigations lead to a negative conclusion, because there are too many
divergences in the formalism that cannot be cancelled by suitable choices of
the parity conditions, which can only be of two types (even or odd). This
agrees with the conclusion of Campiglia:2016hvg  (note that the multipole
charges of Seraj:2016jxi  generically diverge in a time-dependent context,
since one might then have higher spherical harmonic terms in, say the $1/r$
order of the fields).

The terms linear in $r$ in the gauge parameters are somewhat analogous to the
superrotations or the Diff$(S^{2})$ transformations considered in the works
Barnich:2009se ; Barnich:2010eb ; Campiglia:2014yka ; Campiglia:2015yka . Our
study gives some hope that these can be consistently included at spatial
infinity, but the non-linear complexity of Einstein theory calls for caution
in drawing conclusions that might be premature. Further work is clearly needed
to settle satisfactorily this issue.

Finally, it would be of great interest to investigate how our work is
explicitly translated at null infinity. To achieve that task one should first
go back to the standard ‘non-extended’ formalism where the Lagrange multiplier
$\psi$ is fixed in terms of the other fields since it is the formulation that
is directly connected with the standard Lagrangian formulation. Work along
these lines is in progress.

Note Added. After this paper was completed, we became aware of the recent
preprint Peraza:2023ivy  where the canonical formulation of gauge
transformations that blow up at infinity is considered along different lines.

## Acknowledgments

We thank Hernán González, Alfredo Pérez, David Tempo and Ricardo Troncoso for
useful discussions. M.H. is grateful to the Universidad Adolfo Ibáñez for kind
hospitality while this work was completed. The research of O.F. was partially
supported by a Marina Solvay Fellowship. This work was partially supported by
FNRS-Belgium (conventions FRFC PDRT.1025.14 and IISN 4.4503.15), as well as by
funds from the Solvay Family.

##  Appendix A Poincaré invariance

In this appendix, we provide details on how the kinetic term (25) and the
asymptotic form of the transformations of the fields under Poincaré
transformation were arrived at.

###  A.1 Non-integrability of the boosts and time translations with standard
symplectic form

We start with the familiar description of the canonical formalism of gravity
in terms of the spatial components $A_{i}$ of the vector potential and its
conjugate momoentum $\pi^{i}$ (electric field). The corresponding symplectic
form is pure bulk and reads

$\Omega=\int d^{3}xd_{V}\pi^{i}d_{V}A_{i}\,.$ (147)

The change of the symplectic form $\Omega$ under the Poincaré diffeomorphism
$\xi=br+T$ normal to the equal time hyperplanes is given by

$\mathcal{L}_{\xi}\Omega=\oint d^{2}x\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\left(br+T\right)\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}^{AB}d_{V}F_{rA}d_{V}A_{B}\,.$
(148)

Taking into account the fall-off of the mixed radial-angular component of the
curvature

$F_{rA}=-\frac{1}{r}\left(\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}-\partial_{A}\Phi_{\log}\right)-\frac{\ln
r}{r^{2}}\left(\partial_{A}A_{r}^{\log(2)}+A_{A}^{\log(2)}\right)-\frac{1}{r^{2}}\left(\partial_{A}A_{r}^{(2)}+A_{A}^{(2)}-A_{A}^{\log(2)}\right)+o\left(r^{-2}\right)\,,$
(149)

and of the angular component of the gauge field, we obtain the following
expression,

$\displaystyle\mathcal{L}_{\xi}\Omega$ $\displaystyle=-\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern
0.0pt}}}}\,\left(br+T\right)\left(\partial_{A}d_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}-\partial_{A}d_{V}\Phi_{\log}\right)\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}d_{V}\Phi_{\text{lin}}$ $\displaystyle\quad-\ln r\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern
0.0pt}}}}\,b\left(\partial_{A}d_{V}A_{r}^{\log(2)}+d_{V}A_{A}^{\log(2)}\right)\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}d_{V}\Phi_{\text{lin}}$ $\displaystyle-\ln r\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,b\left(\partial_{A}d_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}-\partial_{A}d_{V}\Phi_{\log}\right)D^{A}d_{V}\Phi_{\log}$
$\displaystyle\quad-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,b\left(\partial_{A}d_{V}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}-\partial_{A}d_{V}\Phi_{\log}\right)d_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{A}$
$\displaystyle-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,b\left(\partial_{A}d_{V}A_{r}^{(2)}+d_{V}A_{A}^{(2)}-d_{V}A_{A}^{\log(2)}\right)\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}d_{V}\Phi_{\text{lin}}\,,$ (150)

We now make use of the fact that the components $A_{i}^{\log(2)}$ are pure
gauge, then $\partial_{A}A_{r}^{\log(2)}+A_{A}^{\log(2)}=0$, which takes care
of the first logarithmic divergence. The remaining two divergent terms (in $r$
and $\ln r$) vanish by assuming that $\Phi_{\log}$ is _odd_. After integration
by parts and using that

$\displaystyle\Phi_{\text{lin}}$ $\displaystyle=\text{odd}\,,$ (151)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{A}$ $\displaystyle=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{A})_{\text{even}}+\partial_{A}\Phi,$ (152)

the finite terms can be re-written as

$\displaystyle\mathcal{L}_{\xi}\Omega$ $\displaystyle=\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,d_{V}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\left[\partial_{A}\left(bd_{V}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{A}\right)+T\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,d_{V}\Phi_{\text{lin}}\right]-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\partial_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}d_{V}\Phi_{\log}\right)d_{V}\Phi$ $\displaystyle\quad+\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,d_{V}A_{r}^{(2)}\partial_{A}\left(b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}d_{V}\Phi_{\text{lin}}\right)+\oint d^{2}x\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\partial_{A}\left(bd_{V}A^{(2)A}\right)d_{V}\Phi_{\text{lin}}$
$\displaystyle\quad-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\left[\partial_{A}\left(bd_{V}A^{\log(2)A}\right)+T\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,d_{V}\Phi_{\text{log}}\right]d_{V}\Phi_{\text{lin}}\,.$ (153)

These are not zero and hence, with the relaxed boundary conditions for
$A_{i}$, Poincaré transformations are not canonical transformations for (147).

###  A.2 New symplectic form and invariance under boosts and time translations

In order to obtain an invariant symplectic form, we include $A_{0}$ and its
conjugate $\pi^{0}$ with the asymptotic behaviour described in the text and
adopt as new symplectic form

$\displaystyle\Omega^{\text{new}}$ $\displaystyle=\int
d^{3}x\big{(}d_{V}\pi^{i}d_{V}A_{i}+d_{V}\pi^{0}d_{V}A_{0}\big{)}$
$\displaystyle\quad-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\left[d_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}d_{V}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern
0.0pt}}}-d_{V}\Psi_{\log}d_{V}\Phi+d_{V}A_{r}^{(2)}d_{V}\Psi_{\text{lin}}-\left(d_{V}\Psi_{\log}^{(1)}-d_{V}\Psi^{(1)}\right)d_{V}\Phi_{\text{lin}}\right]\,,$
(154)

(with $\pi^{0}\approx 0$; $\Omega^{\text{new}}$ is just denoted $\Omega$ in
the main text since it is the only one appearing there). The bulk term
$d_{V}\pi^{0}d_{V}A_{0}$ is the standard canonical one, the surface term
$d_{V}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}d_{V}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}$ has been introduced in Henneaux:2018gfi  while the
idea behind the introduction of the other surface terms is that if one takes
$\delta_{\xi}\Psi_{\text{lin}}=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\Phi_{\text{lin}}\right)$,
$\delta_{\xi}\Psi_{\log}=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\Phi_{\log}\right)$, $\delta_{\xi}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\big{(}b\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{A}\big{)}+T\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,d_{V}\Phi_{\text{lin}}$,
$\delta_{\xi}\Psi_{\log}^{(1)}=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(bA^{\log(2)A}\right)+T\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}}\,\Phi_{\text{log}}$ and
$\delta_{\xi}\Psi^{(1)}=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\left(bA^{(2)A}\right)$, one
eliminates the non-integrable terms (153). However, this introduces other non-
integrable terms (since there are additional variations coming from the new
terms), so that the approach is not yet complete.

To get integrable boosts, we proceed as in Henneaux:2018gfi  and adopt the
transformation laws of the bulk fields shown there to work, namely,

$\displaystyle\delta_{\xi}A_{0}$ $\displaystyle=\nabla_{i}\left(\xi
A^{i}\right)\,,$ (155) $\displaystyle\delta_{\xi}\pi^{0}$
$\displaystyle=\xi\partial_{i}\pi^{i}\,,$ (156)
$\displaystyle\delta_{\xi}A_{i}$
$\displaystyle=\frac{\xi\pi_{i}}{\sqrt{g}}+\partial_{i}\left(\xi\Psi\right)\,,$
(157) $\displaystyle\delta_{\xi}\pi^{i}$
$\displaystyle=\sqrt{g}\nabla_{m}\big{(}F^{mi}\xi\big{)}+\xi\partial^{i}\pi^{0}\,.$
(158)

This implies the following transformation laws for the asymptotic fields

$\displaystyle\delta_{\xi}\Psi_{\text{lin}}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\Phi_{\text{lin}}\right)+3b\Phi_{\text{lin}}\,,$
(159) $\displaystyle\delta_{\xi}\Psi_{\log}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\Phi_{\log}\right)\,,$ (160)
$\displaystyle\delta_{\xi}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}$
$\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\big{(}b\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{A}\big{)}+2b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}+T\left(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\triangle$\kern
0.0pt}}}\,\Phi_{\text{lin}}+2\Phi_{\text{lin}}\right)\,,$ (161)
$\displaystyle\delta_{\xi}\Psi_{\log}^{(1)}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\left(bA^{\log(2)A}\right)+bA_{r}^{\log(2)}+T\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern
0.0pt}}}\,\Phi_{\text{log}}\,,$ (162) $\displaystyle\delta_{\xi}\Psi^{(1)}$
$\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern
0.0pt}}}_{A}\left(bA^{(2)A}\right)+b\left(A_{r}^{(2)}+A_{r}^{\log(2)}\right)+T\left(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}^{A}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\right)\,.$ (163)
$\displaystyle\delta_{\xi}\Phi_{\text{lin}}$
$\displaystyle=b\Psi_{\text{lin}}\,,$ (164) $\displaystyle\delta_{\xi}\Phi$
$\displaystyle=b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}^{\text{odd}}+T\Psi_{\text{lin}}\,,$ (165)
$\displaystyle\delta_{\xi}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}$
$\displaystyle=\frac{b}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}_{r}+b\Psi_{\log}\,,$ (166) $\displaystyle\delta_{\xi}A_{r}^{(2)}$
$\displaystyle=\frac{b}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}\pi_{(2)}^{r}+b\left(\Psi^{\log(1)}-\Psi^{(1)}\right)+\frac{T}{\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}+T\left(\Psi_{\log}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}\right)\,.$
(167)

These contain additional terms besides the new ones written below (154).

The change in the bulk part of the symplectic form is found to be

$\displaystyle\mathcal{L}_{\xi}\left(\Omega^{\text{new}}\right)_{\text{bulk}}$
$\displaystyle=\int d^{2}S_{i}\xi d_{V}F^{ij}d_{V}A_{j}+r\oint
d^{2}xbd_{V}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}d_{V}\Psi_{\text{lin}}+\ln r\oint
d^{2}xbd_{V}\pi_{(2)}^{r}d_{V}\Psi_{\log}$ $\displaystyle\quad+\oint
d^{2}x\left[b\left(d_{V}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}d_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}+d_{V}\pi_{(2)}^{r}d_{V}\Psi_{\text{\text{lin}}}\right)+Td_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}d_{V}\Psi_{\text{lin}}\right]\,.$ (168)

In order to eliminate the divergent terms we impose the following parity
conditions

$\Psi_{\text{lin}}=\text{even}\qquad\text{and}\qquad\Psi_{\log}=\text{even}\,.$
(169)

The variation of the boundary term of the symplectic form is given by

$\displaystyle\mathcal{L}_{\xi}\left(\Omega^{\text{new}}\right)_{\text{boundary}}$
$\displaystyle=\mathcal{L}_{\xi}\Omega^{\text{previous}}-\oint
d^{2}x\left[b\left(d_{V}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}d_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}+d_{V}\pi_{(2)}^{r}d_{V}\Psi_{\text{\text{lin}}}\right)+Td_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}d_{V}\Psi_{\text{lin}}\right]$ $\displaystyle\text{$\quad-\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern
0.0pt}}}}\,\left\\{\left[4bd_{V}A_{r}^{(2)}+Td_{V}\left(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}^{A}+3\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\right)\right]d_{V}\Phi_{\text{lin}}-Td_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}d_{V}\Psi_{\text{lin}}\right\\}$},$ (170)

where $\mathcal{L}_{\xi}\Omega^{\text{previous}}$ is given in (153), and is
identically cancelled by the first surface integral in (168).

Putting everything together, one finds that the variation of the new
symplectic form reduces to the expression

$\mathcal{L}_{\xi}\Omega^{\text{new}}=-\oint d^{2}x\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\left\\{\left[4bd_{V}A_{r}^{(2)}+Td_{V}\left(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}^{A}+3\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\right)\right]d_{V}\Phi_{\text{lin}}-Td_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}d_{V}\Psi_{\text{lin}}\right\\}\,.$ (171)

These terms can be eliminated by performing the following field dependent
subleading gauge transformations

$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}_{(T)}^{(1)}$ $\displaystyle=T\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}\,,$ (172)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\mu$\kern 0.0pt}}}_{(b,T)}^{(1)}$
$\displaystyle=4bA_{r}^{(2)}+T\left(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}^{A}+3\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\right)\,,$ (173)

where $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}^{(1)}=\epsilon_{\log}^{(1)}-\epsilon^{(1)}$ and
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}^{(1)}=\mu_{\log}^{(1)}-\mu^{(1)}$. Note that these correcting terms
do not vanish even if the boosts are zero since they contain a contribution
involving $T$.

###  A.3 Invariance of the new symplectic form under spatial translations

The change of the bulk term of the symplectic form under a spatial
diffeomorphism is given by

$\mathcal{L}_{\xi^{i}}\left(\Omega^{\text{new}}\right)_{\text{bulk}}=\int
d^{2}S_{i}\xi^{i}\left(d_{V}\pi^{j}d_{V}A_{j}+d_{V}\pi^{0}d_{V}A_{0}\right)\,,$
(174)

which reduces to the non-vanishing boundary term :

$\mathcal{L}_{\xi^{i}}\left(\Omega^{\text{new}}\right)_{\text{bulk}}=\oint
d^{2}x\left(Wd_{V}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}-\partial_{A}Wd_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}\right)d_{V}\Phi_{\text{lin}}\,.$ (175)

for our relaxed asymptotic conditions. On the other hand, the variation of the
boundary term reads

$\displaystyle\mathcal{L}_{\xi^{i}}\left(\Omega^{\text{new}}\right)_{\text{boundary}}$
$\displaystyle=$ $\displaystyle\oint d^{2}x\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\Big{\\{}\left[-\partial_{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}d_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}+Wd_{V}\left(\Psi_{\log}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}\right)\right]d_{V}\Phi_{\text{lin}}$ (176)
$\displaystyle\qquad+\left(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}Wd_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{A}-Wd_{V}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}\right)d_{V}\Psi_{\text{lin}}\Big{\\}}\,.$

The sum of these terms can be cancelled by corrective gauge transformations
generated by the subleading parameters

$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}_{(W)}^{(1)}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{A}-W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{r}\,,$ (177)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\mu$\kern 0.0pt}}}_{(W)}^{(1)}$
$\displaystyle=-\left(W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}-\partial_{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}\right)-\left[-\partial_{A}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}+W\left(\Psi_{\log}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}\right)\right]\,,$ (178)

which involve only spatial translations.

This completes the discussion of Poincaré invariance of the theory.

## References

  * (1) O. Fuentealba, M. Henneaux and C. Troessaert, “Logarithmic supertranslations and supertranslation-invariant Lorentz charges,” [arXiv:2211.10941 [hep-th]].
  * (2) H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, “Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc. Lond. A 269 (1962), 21-52 doi:10.1098/rspa.1962.0161
  * (3) R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,” Proc. Roy. Soc. Lond. A 270 (1962), 103-126 doi:10.1098/rspa.1962.0206
  * (4) R. Sachs, “Asymptotic symmetries in gravitational theory,” Phys. Rev. 128 (1962) 2851.
  * (5) P. G. Bergmann, “’Gauge-Invariant’ Variables in General Relativity,” Phys. Rev. 124 (1961), 274-278 doi:10.1103/PhysRev.124.274
  * (6) A. Ashtekar, “Logarithmic Ambiguities in the Description of Spatial Infinity,” Found. Phys. 15 (1985), 419
  * (7) R. Beig and B. Schmidt, “Einstein’s equations near spatial infinity,” Commun. Math. Phys. 87 (1982) 65.
  * (8) R. Beig, “Integration Of Einstein’s Equations Near Spatial Infinity,” Proc. Royal Soc. A 1801 (1984) 295–304.
  * (9) A. Ashtekar, L. Bombelli and O. Reula, “The covariant phase space of asymptotically flat gravitational fields," PRINT-90-0318 (Syracuse), published in “Mechanics, Analysis and Geometry: 200 Years After Lagrange”, ed. Mauro Francaviglia, North-Holland Delta Series, Elsevier (Amsterdam: 1991)
  * (10) G. Compère and F. Dehouck, “Relaxing the Parity Conditions of Asymptotically Flat Gravity,” Class. Quant. Grav. 28 (2011), 245016 [erratum: Class. Quant. Grav. 30 (2013), 039501] doi:10.1088/0264-9381/28/24/245016 [arXiv:1106.4045 [hep-th]].
  * (11) C. Troessaert, “The BMS4 algebra at spatial infinity,” Class. Quant. Grav. 35 (2018) no.7, 074003 doi:10.1088/1361-6382/aaae22 [arXiv:1704.06223 [hep-th]].
  * (12) M. Henneaux and C. Troessaert, “Hamiltonian structure and asymptotic symmetries of the Einstein-Maxwell system at spatial infinity,” JHEP 1807 (2018) 171 [arXiv:1805.11288 [gr-qc]].
  * (13) M. Henneaux and C. Troessaert, “The asymptotic structure of gravity at spatial infinity in four spacetime dimensions,” Proc. Steklov Inst. Math. 309 (2020) 127-149 [arXiv:1904.04495 [hep-th]].
  * (14) M. Mirbabayi and M. Porrati, “Dressed Hard States and Black Hole Soft Hair,” Phys. Rev. Lett. 117, no.21, 211301 (2016) doi:10.1103/PhysRevLett.117.211301 [arXiv:1607.03120 [hep-th]].
  * (15) R. Bousso and M. Porrati, “Soft Hair as a Soft Wig,” Class. Quant. Grav. 34 (2017) no.20, 204001 doi:10.1088/1361-6382/aa8be2 [arXiv:1706.00436 [hep-th]].
  * (16) R. Javadinezhad, U. Kol and M. Porrati, “Comments on Lorentz Transformations, Dressed Asymptotic States and Hawking Radiation,” JHEP 01 (2019), 089 doi:10.1007/JHEP01(2019)089 [arXiv:1808.02987 [hep-th]].
  * (17) R. Javadinezhad, U. Kol and M. Porrati, “Supertranslation-invariant dressed Lorentz charges,” JHEP 04 (2022), 069 doi:10.1007/JHEP04(2022)069 [arXiv:2202.03442 [hep-th]].
  * (18) P. N. Chen, M. T. Wang, Y. K. Wang and S. T. Yau, “Supertranslation invariance of angular momentum,” Adv. Theor. Math. Phys. 25 (2021) no.3, 777-789 doi:10.4310/ATMP.2021.v25.n3.a4 [arXiv:2102.03235 [gr-qc]].
  * (19) P. N. Chen, J. Keller, M. T. Wang, Y. K. Wang and S. T. Yau, “Evolution of Angular Momentum and Center of Mass at Null Infinity,” Commun. Math. Phys. 386 (2021) no.1, 551-588 doi:10.1007/s00220-021-04053-7 [arXiv:2102.03221 [gr-qc]].
  * (20) G. Compère and D. A. Nichols, “Classical and Quantized General-Relativistic Angular Momentum,” [arXiv:2103.17103 [gr-qc]].
  * (21) T. He, P. Mitra, A. P. Porfyriadis and A. Strominger, “New Symmetries of Massless QED,” JHEP 1410 (2014) 112 [arXiv:1407.3789 [hep-th]].
  * (22) D. Kapec, M. Pate and A. Strominger, “New Symmetries of QED,” Adv. Theor. Math. Phys. 21 (2017), 1769-1785 doi:10.4310/ATMP.2017.v21.n7.a7 [arXiv:1506.02906 [hep-th]].
  * (23) M. Campiglia and A. Laddha, “Asymptotic symmetries of QED and Weinberg’s soft photon theorem,” JHEP 07 (2015), 115 doi:10.1007/JHEP07(2015)115 [arXiv:1505.05346 [hep-th]].
  * (24) M. Campiglia and R. Eyheralde, “Asymptotic $U(1)$ charges at spatial infinity,” JHEP 1711 (2017) 168 [arXiv:1703.07884 [hep-th]].
  * (25) A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” arXiv:1703.05448 [hep-th].
  * (26) M. Henneaux and C. Troessaert, “Asymptotic symmetries of electromagnetism at spatial infinity,” JHEP 05, 137 (2018) doi:10.1007/JHEP05(2018)137 [arXiv:1803.10194 [hep-th]].
  * (27) V. Lysov, S. Pasterski and A. Strominger, “Low’s Subleading Soft Theorem as a Symmetry of QED,” Phys. Rev. Lett. 113 (2014) no.11, 111601 [arXiv:1407.3814 [hep-th]].
  * (28) M. Campiglia and A. Laddha, “Subleading soft photons and large gauge transformations,” JHEP 1611 (2016) 012 [arXiv:1605.09677 [hep-th]].
  * (29) E. Conde and P. Mao, “Remarks on asymptotic symmetries and the subleading soft photon theorem,” Phys. Rev. D 95 (2017) no.2, 021701 [arXiv:1605.09731 [hep-th]].
  * (30) G. Barnich and P. H. Lambert, “Einstein-Yang-Mills theory: Asymptotic symmetries,” Phys. Rev. D 88 (2013), 103006 doi:10.1103/PhysRevD.88.103006 [arXiv:1310.2698 [hep-th]].
  * (31) S. R. Coleman and J. Mandula, “All Possible Symmetries of the S Matrix,” Phys. Rev. 159 (1967), 1251-1256 doi:10.1103/PhysRev.159.1251
  * (32) R. Benguria, P. Cordero and C. Teitelboim, “Aspects of the Hamiltonian Dynamics of Interacting Gravitational Gauge and Higgs Fields with Applications to Spherical Symmetry,” Nucl. Phys. B 122 (1977), 61-99 doi:10.1016/0550-3213(77)90426-6
  * (33) P. A. M. Dirac, “Lectures on Quantum Mechanics,” Yeshiva University, Academic Press (New York: 1967)
  * (34) M. Henneaux and C. Teitelboim, “Quantization of gauge systems,” Princeton University Press (Princeton: 1992)
  * (35) A. Seraj, “Multipole charge conservation and implications on electromagnetic radiation,” JHEP 06 (2017), 080 doi:10.1007/JHEP06(2017)080 [arXiv:1610.02870 [hep-th]].
  * (36) T. Regge and C. Teitelboim, “Role of Surface Integrals in the Hamiltonian Formulation of General Relativity,” Annals Phys. 88 (1974), 286 doi:10.1016/0003-4916(74)90404-7
  * (37) O. Fuentealba, M. Henneaux, J. Matulich and C. Troessaert, “Asymptotic structure of the gravitational field in five spacetime dimensions: Hamiltonian analysis,” JHEP 07 (2022), 149 doi:10.1007/JHEP07(2022)149 [arXiv:2206.04972 [hep-th]].
  * (38) M. Henneaux and C. Troessaert, “Asymptotic structure of electromagnetism in higher spacetime dimensions,” Phys. Rev. D 99 (2019) no.12, 125006 doi:10.1103/PhysRevD.99.125006 [arXiv:1903.04437 [hep-th]].
  * (39) 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]].
  * (40) G. Barnich and C. Troessaert, “Aspects of the BMS/CFT correspondence,” JHEP 05 (2010), 062 doi:10.1007/JHEP05(2010)062 [arXiv:1001.1541 [hep-th]].
  * (41) M. Campiglia and A. Laddha, “Asymptotic symmetries and subleading soft graviton theorem,” Phys. Rev. D 90 (2014) no.12, 124028 doi:10.1103/PhysRevD.90.124028 [arXiv:1408.2228 [hep-th]].
  * (42) M. Campiglia and A. Laddha, “New symmetries for the Gravitational S-matrix,” JHEP 04 (2015), 076 doi:10.1007/JHEP04(2015)076 [arXiv:1502.02318 [hep-th]].
  * (43) J. Peraza, “Renormalized electric and magnetic charges for $O(r^{n})$ large gauge symmetries,” [arXiv:2301.05671 [hep-th]].