# Radiative Asymptotic Symmetries of 3D Einstein–Maxwell Theory

Jorrit Bosma1, Marc Geiller2, Sucheta Majumdar2, Blagoje Oblak3,4

(
1Institute for Theoretical Physics, ETH Zürich, CH 8093 Zürich, Switzerland
2ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
3CPHT, CNRS, Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France
4Optique Non-linéaire Théorique, Université Libre de Bruxelles,
Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium)

###### Abstract

We study the null asymptotic structure of Einstein–Maxwell theory in three-
dimensional (3D) spacetimes. Although devoid of bulk gravitational degrees of
freedom, the system admits a massless photon and can therefore accommodate
electromagnetic radiation. We derive fall-off conditions for the Maxwell field
that contain both Coulombic and radiative modes with non-vanishing news. The
latter produces non-integrability and fluxes in the asymptotic surface
charges, and gives rise to a non-trivial 3D Bondi mass loss formula. The
resulting solution space is thus analogous to a dimensional reduction of 4D
pure gravity, with the role of gravitational radiation played by its
electromagnetic cousin. We use this simplified setup to investigate choices of
charge brackets in detail, and compute in particular the recently introduced
Koszul bracket. When the latter is applied to Wald–Zoupas charges, which are
conserved in the absence of news, it leads to the field-dependent central
extension found earlier in [1]. We also consider (Anti-)de Sitter asymptotics
to further exhibit the analogy between this model and 4D gravity with leaky
boundary conditions.

###### Contents

  1. 1 Introduction and summary
  2. 2 Pure Maxwell theory in 3D
    1. 2.1 Electromagnetic field from compact sources
    2. 2.2 Fall-offs in Bondi gauge
    3. 2.3 Electromagnetic memory
  3. 3 Setup for Einstein–Maxwell theory
    1. 3.1 Lagrangian and Noether currents
    2. 3.2 Fall-offs and radiative solution space
    3. 3.3 Zero-mode solutions and angular momentum
  4. 4 Asymptotic symmetries and charges
    1. 4.1 Symmetry generators and transformation laws
    2. 4.2 Asymptotic charges introduced
  5. 5 Brackets and charge algebras
    1. 5.1 Barnich–Troessaert bracket
    2. 5.2 Wald–Zoupas splits
    3. 5.3 Noether split and bracket
    4. 5.4 Koszul bracket
  6. 6 Turning on $\boldsymbol{\Lambda\neq 0}$
    1. 6.1 Solution space and analogy with the 4D vacuum case
    2. 6.2 Asymptotic symmetries and charges
  7. 7 Perspectives
  8. A Constructing the solution space
    1. A.1 Hypersurface equations
    2. A.2 Evolution equations
    3. A.3 Trivial equations

##  1 Introduction and summary

Despite its key role in physics, the behaviour of quantum gauge field theories
at large distances and low energies remains poorly understood. Indeed, the
bulk degrees of freedom of gauge theories can be quantized with standard
Hamiltonian methods [2], but their physical boundary states are exceedingly
sensitive to choices of fall-off conditions and thus require a separate
treatment. This subtlety is especially striking in cases that contain
propagating modes (as opposed to topological field theories), where the global
symmetries of boundary states—i.e. asymptotic symmetries—suffer from radiative
ambiguities. The present work is therefore devoted to a detailed classical
investigation of such issues in a simple gauge system: gravitation coupled to
electrodynamics in three spacetime dimensions (3D). In this introduction, we
first motivate the subject, and then present an overview of our main results
along with a plan of the paper.

#### Motivations.

The difficulties in quantizing the infrared sector of gauge theories are best
exemplified by gravitation, whose general covariance requires that conserved
quantities (say energy or momentum) be defined in terms of asymptotic fluxes
as opposed to local current densities [3, 4]. This is an old, general and
standard result whose counterpart for 4D gravitational radiation involves the
infinite-dimensional Bondi–Metzner–Sachs (BMS) group at null infinity, first
introduced half a century ago [5, 6, 7, 8, 9, 10, 11]. Such constructions are
indeed well known by now, but the full scope of their implications has only
recently become apparent thanks to new insights on holography in Minkowskian
spacetimes [12, 13, 14, 15] and the discovery of deep links between asymptotic
symmetries and the infrared sector of gauge theories [16, 17, 18, 19].
Following these ideas, asymptotic symmetries were studied in numerous
contexts, notably including electrodynamics and Yang–Mills theory [20, 21, 22,
23, 24, 25]. A common aspect of all these works is that one’s choice of fall-
off conditions for field components determines the phase space of available
field configurations, which in turn influences the soft sector of the theory
and the resulting symmetries.

The present paper is devoted to the asymptotic symmetries of 3D gravity
coupled to a Maxwell field. This setup was first studied in [1], but we shall
argue that a crucial sector of radiative field configurations was overlooked
in that reference. Indeed, our investigation is precisely motivated by the
(much more involved) issue of radiative ambiguities in the asymptotic
symmetries of 4D general relativity. In that case, the presence of
gravitational waves implies that the radiative phase space at null infinity is
an open system [26, 27, 28], which has important technical and conceptual
implications. The first is that asymptotic charges are not conserved in time
but obey instead flux-balance relations, such as the Bondi mass loss relating
the decrease of mass to the radiation crossing null infinity. The second is
that charges are generally non-integrable, as the associated symmetries fail
to be Hamiltonian due to the presence of symplectic flux. This hinders the
study of the corresponding asymptotic charge algebras, since no canonical
bracket can a priori be defined.

In fact, a tentative solution to this puzzle was put forward by Barnich and
Troessaert [13, 14, 12], yielding e.g. an asymptotic charge algebra extended
by a suggestive field-dependent cocycle [14]. But the prescription of [14] is
ultimately ambiguous, since it relies on an arbitrary splitting between
integrable terms and non-integrable fluxes when defining the very notion of
‘charge’. It is therefore essential to elucidate the origin and the extent of
this arbitrariness, the goal being to eventually isolate unambiguous boundary
symmetries. In the specific case of reference [14], the split ambiguity can
actually be fixed by the Wald–Zoupas (WZ) prescription [29, 30, 31] that
singles out the integrable charge by requiring that it be conserved in the
absence of radiation. However, it is unclear if an analogous way out exists in
general radiative cases. Radiative asymptotic symmetries are thus crucial for
quantum gravity and gauge theories as a whole, and for Minkowskian holography
in particular [32, 33, 34, 13, 35, 36, 37, 38, 39]. Note that similar
questions can also be raised in (Anti)-de Sitter [(A)dS] spacetimes, where
radiation and holography with porous boundary conditions were recently studied
in [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57].

Our focus on a 3D system is motivated by the desire to simplify matters, while
still retaining the essential features of realistic radiative systems. Indeed,
pure 3D gravity is a topological field theory (no propagating degrees of
freedom) that has often served as a fruitful testbed for asymptotic
symmetries, starting with the seminal work [58] on the boundary Virasoro
algebra of AdS3. The analogous symmetry of 3D Minkowskian spacetimes is the
BMS3 algebra [59, 60, 61]. In both AdS and flat cases, the absence of
radiation allows one to delve deep into holographic territory on a purely
group-theoretic basis. For instance, unitary representations of asymptotic
symmetries [62, 63, 64, 65, 66, 67, 68] and their relation to bulk metrics
through coadjoint orbits and geometric actions [69, 70, 71, 72, 73, 74, 75,
76, 77, 78] are well established in both situations, as is the Cardy-ology of
black holes and cosmological solutions [79, 80, 81]. No such control is
available in the 4D realm, where the study of coadjoint orbits [82] and
geometric actions [83, 84] is in its infancy, and mostly limited to the sector
without radiation.

Hence our interest in the 3D Einstein–Maxwell system: it is radiative (the
electromagnetic field has one local degree of freedom) but sits between 3D and
4D general relativity in terms of complexity, as it shares the geometrical
simplicity of the former while also describing radiative aspects relevant to
the latter. Its symmetries were studied in the aforementioned reference [1]
(see also [85]), where the authors identified a non-integrable contribution to
the charges, which they interpreted as being sourced by electromagnetic
‘news’. However, this interpretation should be done with care, as a slightly
more general setup reveals a key subtlety. This is actually the gist of our
work: we generalize the analysis of [1] by considering weaker fall-offs for
the Maxwell field. In particular, while the fall-offs of [1] include
logarithmic terms (as required for the 3D Coulomb solution) and integer powers
of the radius $r$, here we also allow for terms in half-integer powers of $r$.
This is because radiative fall-offs for the 3D Maxwell field in (retarded)
Bondi coordinates $(r,u,\phi)$ and in radial gauge take the form (see fig. 1)

$\displaystyle A_{\phi}=C\sqrt{r}+\dots,\qquad\qquad A_{u}=E(\ln r)+G+\dots.$
(1.1)

Here $C(u,\phi)$ is an unconstrained function on $\mathscr{I}^{+}$ and denotes
the radiative electromagnetic data, $E(\phi)$ is the electric charge aspect,
$G(u,\phi)$ is another unconstrained function, and the dots are subleading
terms (which we will analyse in detail). With such relaxed fall-offs, 3D
Einstein–Maxwell theory with electromagnetic radiation becomes analogous to a
dimensional reduction of 4D pure gravity with gravitational radiation [59,
86]. In particular, both the symplectic structure and the charges at null
infinity involve $C\delta N$, where

$\displaystyle
N(u,\phi)\coloneqq\partial_{u}C(u,\phi)\coloneqq\dot{C}(u,\phi)$ (1.2)

is the electromagnetic news. For comparison, recall that the news in 4D
gravity is a symmetric tracefree rank-2 tensor $N_{ab}$, while in 4D Maxwell
theory it is a vector $N_{a}$ in terms of coordinates $x^{a}$ on the celestial
2-sphere:

(1.6)

This highlights again that 3D Maxwell exhibits the simplest form of news.
Coupling it to Einstein gravity makes it possible to study the interplay
between this news and gravitational asymptotic symmetries. In particular, the
Bondi mass aspect $M$ then satisfies a mass loss formula $\dot{M}=-2N^{2}$,
analogous to its celebrated 4D version with gravitational news replaced by its
electromagnetic counterpart. No such structure occurs in the analysis of [1],
due to stronger fall-offs setting $C=0$.

$u$$r$$\phi$$\mathscr{I}^{+}$ Figure 1: A Penrose diagram of 3D Minkowski
spacetime, showing retarded Bondi coordinates $(r,u,\phi)$ along with light-
cones centred at the origin. Future null infinity is the region denoted
$\mathscr{I}^{+}$, where $r\to\infty$ at finite $(u,\phi)$. The presence of
electromagnetic radiation (wiggly lines) reaching $\mathscr{I}^{+}$ is
diagnosed by a non-zero news function (1.2). Adapted from [87, fig. 1].

As stated above, the new boundary conditions presented here provide an ideal
arena to study all the subtleties related to radiation, such as non-
integrability, flux-balance laws and ambiguities in one’s choice of charge
brackets. In addition, our goal is to contribute to the study of asymptotic
symmetries for general relativity coupled to matter fields. For 4D
Einstein–Maxwell theory, asymptotic symmetries were already partly studied in
[88, 89], where the first reference focusses on WZ charges. Given our
observations below on charge algebras and their field-dependent cocycles, it
would be interesting to also study the charge algebra in the 4D case. We hope
to come back to this in the future.

#### Outline and summary of results.

The paper is organized as follows. We start in section 2 by analysing pure
Maxwell theory in 3D Minkowski spacetime. From the electromagnetic field
produced by a compact source, we deduce the fall-offs to be used later in
Einstein–Maxwell theory in Bondi coordinates. We then compute the 3D
electromagnetic memory effect, which consists of a net change of angular
velocity caused by radiation on any charged test mass near null infinity. In
contrast to the 4D case [90], this is not directly related to a change of
U$(1)$ asymptotic charges; the mismatch ultimately stems from the difference
between Coulombic and radiative falls-off in any spacetime dimension other
than four.

Section 3 is devoted to the solution space of Einstein–Maxwell theory with
vanishing cosmological constant. We explain there how the field equations are
solved following the Bondi hierarchy, deferring the details to appendix A. The
upshot is that the solution space contains, among others, two functions
$M(u,\phi)$ and $L(u,\phi)$. These are respectively the mass and angular
momentum aspects in the metric, subject to the evolution equations (3.10), the
first of which is the Bondi mass loss sourced by the electromagnetic news
(1.2). In addition, the solution space contains two completely unconstrained
functions on $\mathscr{I}^{+}$, namely the electromagnetic shear $C(u,\phi)$
and a function $G(u,\phi)$ as in (1.1). Only $G$ was present in [1], where it
gave rise to non-integrable asymptotic charges and was thus identified as
‘news’. We will show instead that $G(u,\phi)$ should be considered as sourcing
a ‘spurious flux’, while the ‘true news’ is the quantity $\dot{C}$ sourcing
the Bondi mass loss through a flux of energy momentum at $\mathscr{I}^{+}$.

Note that the phase space thus obtained contains two free functions on
$\mathscr{I}^{+}$, namely $C(u,\phi)$ and $G(u,\phi)$. This may seem
perplexing given that the theory admits a single degree of freedom. Actually,
an analogous situations occurs in 4D pure gravity, where one can relax
boundary conditions so that the solution space contains other undetermined
functions of $u$ on top of the two degrees of freedom in the asymptotic shear
$C_{ab}$. This includes for instance the induced boundary metric on
$\mathscr{I}^{+}$ [53, 54] or the trace of $C_{ab}$ in Newman–Unti gauge [54,
91], and it allows one to describe physically relevant solutions with
$C_{ab}=0$, such as Robinson–Trautman spacetimes [92, 93, 94]. The metric on
$\mathscr{I}^{+}$ can similarly be relaxed in 3D pure gravity, resulting in a
phase space with three unspecified functions of $u$ even though the bulk
theory has no propagating degrees of freedom at all [95, 96]. In the case of
3D Einstein–Maxwell theory, the known radiative configurations are such that
$C\neq 0$ and $G=0$; in fact, $G$ may be seen as a gauge redundancy. It is
nevertheless possible that the theory admits interesting solutions with $C=0$
but $G\neq 0$, so we will initially keep both fields non-zero in our analysis
of symmetries and charges.

In section 4, we turn to the characterization of asymptotic symmetries. These
are labelled by an asymptotic Killing vector field $\xi$ and a U$(1)$ gauge
parameter $\varepsilon$ on $\mathscr{I}^{+}$, spanning a semi-direct sum
$\mathfrak{bms}_{3}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}}C^{\infty}(\mathscr{I}^{+})$. A striking aspect of this
construction is that the U$(1)$ parameter at leading order has an arbitrary
dependence on retarded time $u$, whose origin is the free $u$-dependent
function $G$ in the solution space (1.1). Indeed, one can see from the
transformation laws (4.4) that setting $G=0$ fixes the arbitrary time
dependence, reducing the aforementioned algebra to a semi-direct sum
$\mathfrak{bms}_{3}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}} \end{picture}}}}}C^{\infty}(S^{1})$.
Irrespective of this feature, we find that the (integrands of variations of)
asymptotic charges are given by (4.10) and exhibit two non-integrable terms
arising from supertranslations. The first such term is $N\delta C$, involving
the radiative data of (1.1) and the news (1.2); the second is $G\,\delta E$,
where $E$ is the electric charge aspect in (1.1). These non-integrable
contributions imply that a splitting prescription is required for the choice
of integrable parts and the bracket of surface charges. This is the (expected)
key subtlety anticipated in the motivations above.

Section 5 is therefore devoted to the study of the choice of bracket. We start
with the Barnich–Troessaert (BT) prescription [14] in the case $G\neq 0$,
whereupon the charge algebra reproduces that of asymptotic symmetries with a
field-dependent central extension (really a 2-cocycle) given in (5.6) below.
In the case $G=0$, which must be treated with care since it makes the U$(1)$
gauge parameter field-dependent, we define WZ charges that are conserved when
the news (1.2) vanishes. The resulting charge algebra features once again a
field-dependent 2-cocycle. Both such extensions were previously found in [1],
so they are unaffected by our weaker choice of fall-offs.

The presence of field-dependent cocycles is expected to affect the unitary
representations of charge algebras, hence quantum theories having these
algebras as symmetries. It is therefore important to ask whether field-
dependent cocycles are a generic feature, or if they can be removed with
suitable choices of integrable splits or brackets. To this end, we first
discuss the so-called Noether split and bracket introduced in [97, 98]; the
ensuing algebra closes with a vanishing cocycle (even the central extension of
[60] is absent!), but integrable charges fail to be conserved in the absence
of radiation. We therefore turn to the Koszul bracket111We thank Adrien
Fiorucci for encouraging us to study the Koszul bracket, and for sharing
preliminary results based on joint work with Glenn Barnich and Romain
Ruzziconi [99]. introduced in [99], whose key advantage is to be independent
of the choice of split between integrable and flux pieces. The Koszul bracket
gives a surprising result: when $G\neq 0$, the cocycle in the charge algebra
is non-zero but field-independent, as in pure 3D gravity [60]; but setting
$G=0$ yields the same field-dependent cocycle as with the BT bracket of WZ
charges. Moreover, this last cocycle turns out to be non-zero even in the
absence of electromagnetic news, suggesting that field-dependent extensions
are a genuine feature of 3D Einstein–Maxwell theory.

In section 6, we switch on a cosmological constant (with arbitrary sign).
Einstein–Maxwell theory in AdS3 was previously studied in [100, 101], albeit
in the Fefferman–Graham gauge that admits no well-defined flat limit. In Bondi
gauge, by contrast, introducing $\Lambda\neq 0$ is straightforward and the
flat limit is well-defined [61]. Interestingly, working in (A)dS3 deepens the
analogy between the present 3D model with electromagnetic radiation and pure
4D gravity with gravitational radiation. For instance, we find that
$\Lambda\neq 0$ allows no half-integer powers of $r$ in the fall-offs,
analogously to the condition of absence of logarithmic terms in (A)dS4 [53,
54]. Furthermore, when $\Lambda\neq 0$, initial conditions on a cut of
$\mathscr{I}^{+}$ need to be specified for only finitely many components of
the Maxwell field, while $\Lambda=0$ involves an infinite tower of data
subject to evolution equations. This is similar to what happens in 4D pure
gravity when studying the role of the subleading tensors in the angular
metric. We finally conclude in section 7 with some perspectives for future
work.

##  2 Pure Maxwell theory in 3D

This section sets the stage by spelling out basic facts on 3D pure
electrodynamics. We first solve Maxwell’s equations for a compact source to
motivate fall-off conditions in Lorenz gauge, before finding the analogous
fall-offs in Bondi gauge. We then briefly comment on electromagnetic memory
effects and point out the difference of their behavior with respect to the 4D
case.

###  2.1 Electromagnetic field from compact sources

Consider 3D Minkowski spacetime, choose some inertial coordinates
$x^{\mu}=(t,{\bf x})$, and let $j_{\mu}({\bf x},t)$ be some current density
that is compactly supported in space. The current acts as a source for an
electromagnetic field $A_{\mu}$ such that $\partial_{\mu}F^{\mu\nu}=-j^{\nu}$,
which yields $\square A_{\mu}=-j_{\mu}$ in the Lorenz gauge
($\partial_{\mu}A^{\mu}=0$). Our goal is to find the asymptotics of $A_{\mu}$
in Bondi coordinates.

The first step is to solve Maxwell’s equation $\square A_{\mu}=-j_{\mu}$.
Using the Green’s function of the 3D d’Alembert operator, the solution reads

$\displaystyle A_{\mu}({\bf x},t)=\frac{1}{2\pi}\int\text{d}^{2}{\bf
y}\,\int_{-\infty}^{t-|{\bf x}-{\bf y}|}\text{d}s\,\frac{j_{\mu}({\bf
y},s)}{\sqrt{(t-s)^{2}-|{\bf x}-{\bf y}|^{2}}}.$ (2.1)

In contrast to the more familiar 4D case, the Green’s function that gives
(2.1) is not localized on a light-cone emanating from the source. This may be
viewed as a violation of the Huygens principle in 3D [102], and it ultimately
results in (logarithmic) infrared divergences that are absent in 4D. Indeed,
consider (2.1) for a point charge. If the latter is static (say at the origin
$r=0$), its current density is $j_{0}({\bf y},s)=q\delta^{(2)}({\bf y})$ and
$j_{1}=j_{2}=0$. Plugging this in (2.1) yields $A_{1}=A_{2}=0$, but $A_{0}$
involves an integral that diverges owing to the infinite lower bound on $s$.
The key point, though, is that this divergence is a constant that depends on
none of the spacetime coordinates. To see this, fix the radius $r=|{\bf x}|$
and replace the lower integration bound in (2.1) by some cutoff time $-T$ such
that $T\gg r$ and $T\gg|t-r|$. This yields

$\displaystyle
A_{0}(r,t)=\frac{q}{2\pi}\int_{-T}^{t-r}\frac{\text{d}s}{\sqrt{(t-s)^{2}-r^{2}}}=\frac{q}{2\pi}\ln\Big{(}2\frac{T+t}{r}\Big{)}=-\frac{q}{2\pi}\ln(r)+\frac{q}{2\pi}\ln(2T)+\mathcal{O}(T^{-1})$
(2.2)

in the limit $T\to+\infty$, where the divergent term $\ln(2T)$ is indeed
independent of both $r$ and $t$. Thus the Coulombic electrostatic potential in
3D is

$\displaystyle A_{0}(r)=-\frac{q}{2\pi}\ln r$ (2.3)

up to an irrelevant additive constant, as was in fact expected from the
Green’s function of the 2D Laplacian. Note that the same behaviour holds near
future null infinity, where $r\to\infty$ and $t\to\infty$ with $u=t-r$ finite.

In 4D, Coulombic and radiative fall-offs coincide. This is not so in 3D, where
the Coulombic potential (2.3) has little to do with its radiative counterpart.
One can again verify this from (2.1), now letting $j_{\mu}$ be the current
density of a moving point charge so that $j_{\mu}({\bf
y},s)=qv_{\mu}(s)\delta^{(2)}({\bf y}-{\bf X}(s))$, where $v_{0}\coloneqq 1$
and $v_{i}\coloneqq\partial X_{i}/\partial s$ in terms of some timelike
worldline ${\bf X}(s)=(X_{1}(s),X_{2}(s))$. For simplicity, we shall assume
that the velocity ${\bf v}=(v_{1},v_{2})$ of the source is small compared with
the speed of light. Then the vector potential behaves as

$\displaystyle A_{i}({\bf
x},t)\sim\frac{q}{2\pi}\int_{-\infty}^{t-r}\frac{v_{i}(s)\,\text{d}s}{\sqrt{(t-s)^{2}-r^{2}}}$
(2.4)

at leading order in $|{\bf v}|$, and in the limit where $r$ and $t$ are large
but $t-r$ is finite. To make further progress, Fourier-transform the velocity
as

$\displaystyle
v_{j}(s)=\int_{-\infty}^{+\infty}\text{d}\omega\,\tilde{v}{}_{j}(\omega)\,e^{i\omega
s},$ (2.5)

where $\tilde{v}{}_{j}(\omega)=\mathcal{O}(\omega)$ as $\omega\to 0$ since
${\bf v}(s)$ is the time derivative of the position ${\bf X}(s)$. Plugging
(2.5) in the potential (2.4) and using the saddle-point approximation at large
$\omega r$ then yields

$\displaystyle A_{i}({\bf
x},t)\sim\frac{q}{2\pi}\sqrt{\frac{\pi}{2r}}\int_{-\infty}^{+\infty}\text{d}\omega\frac{\tilde{v}{}_{i}(\omega)}{\sqrt{\omega}}\,e^{i\omega
u+i\pi/4}.$ (2.6)

As for the scalar potential given by (2.1), it still satisfies (2.3) at
leading order. Note the stark contrast between this Coulombic $\mathcal{O}(\ln
r)$ fall-off and the radiative $\mathcal{O}(r^{-1/2})$ fall-off in (2.6). The
difference may be seen as a remnant of the saddle-point approximation, which
only holds at $\omega\neq 0$ and was therefore not valid in the Coulombic
case.

Equations (2.3) and (2.6) were written in inertial coordinates and in Lorenz
gauge, but moving to Bondi coordinates is a simple matter: using as usual
$x+iy=r\,e^{i\phi}$ and $t=u+r$, the Minkowski spacetime metric becomes

$\displaystyle\mathrm{d}s^{2}=-\mathrm{d}u^{2}-2\mathrm{d}u\,\mathrm{d}r+r^{2}\mathrm{d}\phi^{2}\qquad\text{(pure
Minkowski)}$ (2.7)

and Bondi components of the electromagnetic potential are related to its
inertial components by

$\displaystyle A_{u}=A_{0}=\mathcal{O}(\ln r),\qquad
A_{\phi}=xA_{y}-yA_{x}=\mathcal{O}(r^{1/2}),\qquad
A_{r}=A_{0}+\frac{1}{r}A_{i}x^{i}=\mathcal{O}(\ln r).$ (2.8)

This is already close to the fall-offs announced in (1.1); the only difference
is that Lorenz gauge and radial gauge differ. To move from the former to the
latter, perform a gauge transformation $A\to A+\text{d}\varepsilon$ that sets
$A_{r}+\partial_{r}\varepsilon=0$. This fixes $\varepsilon$ up to a
$(u,\phi)$-dependent integration function that we set to zero so as to leave
the components $A_{u}$ and $A_{\phi}$ unaffected. Note that $\varepsilon$ is
guaranteed to only depend on $r$ at leading order, since the leading part of
$A_{r}$ is the Coulombic field (2.1); as for subleading terms, one can
actually check that the radiative scalar potential in Lorenz gauge satisfies
$A_{0}\sim-\frac{q}{2\pi}\ln(r)-\frac{1}{r}A_{i}x^{i}+\mathcal{O}(r^{-3/2})$,
so that $A_{r}$ in (2.8) is independent of $(u,\phi)$ at least up to order
$r^{-3/2}$. This guarantees that the gauge transformation moving from Lorenz
gauge to radial gauge can indeed be chosen so as to leave $A_{u}$ and
$A_{\phi}$ untouched, at least at leading order.

###  2.2 Fall-offs in Bondi gauge

Having determined the electromagnetic fall-offs that will be used throughout
the paper, we now investigate them in detail and solve Maxwell’s equations in
Bondi coordinates, near null infinity, and in radial gauge $A_{r}=0$. We
stress that essentially identical results will hold even in the presence of
gravitation (see section 3.2), so this is a key prerequisite for all that
follows.

#### Fall-offs.

We start with the detailed form of the fall-off conditions (1.1) for the
Maxwell field:

$\displaystyle A_{u}$ $\displaystyle=\underbrace{\big{.}E\ln
r}_{\\!\\!\\!\\!\\!\\!\text{Coulombic}\\!\\!\\!\\!\\!\\!}~{}+\,G\,+\sum_{m\in\mathbb{N}/2}\,\sum_{n=0}^{\lceil
m\rceil}A^{m,n}_{u}\frac{(\ln r)^{n}}{r^{m}},$ (2.9a) $\displaystyle A_{\phi}$
$\displaystyle=A^{\ell}_{\phi}\ln
r+A^{0}_{\phi}+\sum_{m\in\mathbb{N}/2}\,\sum_{n=0}^{\lceil
m\rceil}A^{m,n}_{\phi}\frac{(\ln
r)^{n}}{r^{m}}+\underbrace{\big{.}C\sqrt{r}}_{\\!\\!\\!\\!\\!\\!\text{radiative}\\!\\!\\!\\!\\!\\!}.$
(2.9b)

Here the sum over $m$ includes both integer and half-integer powers of $r$,
$\lceil\cdot\rceil$ denotes the ceiling function,222Note that nothing requires
for now that the sums over $n$ in (2.9) stop at $n=\lceil m\rceil$. One could
a priori consider expansions more general than (2.9). However, Maxwell’s
equations turn out to impose $A_{u,\phi}^{m,n>\lceil m\rceil}=0$. We chose to
include this piece of information as a defining property in (2.9). and each
component of the form $A_{\times}^{\times}$ is a priori an arbitrary function
of $(u,\phi)$. Note the leading Coulombic data (electric charge aspect)
$E(u,\phi)$ in (2.9a): this mimics the Coulomb solution (2.3) and the fall-off
in (2.8), but now includes an angular dependence to account for the
possibility of Lorentz boosts (and ultimately superrotations, as we shall
see). Also note how $A_{\phi}$ in (2.9b) has an overleading term $C\sqrt{r}$
with respect to $A_{u}$, containing a field suggestively called $C(u,\phi)$ in
analogy with the Bondi shear of 4D gravity. Indeed, $C$ will turn out to
incorporate radiative degrees of freedom as in (2.8), and its time derivative
(1.2) will be the electromagnetic news.

A remark is in order regarding the field $G(u,\phi)$ in (2.9a). Namely, the
argument in $\ln r$ must be dimensionless, so an implicit radial constant
$r_{0}$ was set to unity; the first two terms in (2.9a) should thus be
understood as $E\ln(r/r_{0})+G$. One could be tempted to absorb $G$ in the
definition of $r_{0}$ by redefining the radial coordinate. However, we will
keep $G(u,\phi)$ as a separate, independent field because it will ultimately
appear in asymptotic charges.

#### Equations of motion.

Starting with the fall-offs (2.9), one can solve the vacuum Maxwell equations
$\nabla^{\mu}F_{\mu\nu}=0$. Note that these are always valid away from
sources. The radial Maxwell equation $\nabla^{\mu}F_{\mu r}=0$ then determines
the scalar potential $A_{u}^{m>0,n}$ in terms of the vector potential
$A_{\phi}$. More precisely, solving recursively in $1/r$, one gets the
leading-order behaviour

$\displaystyle\nabla^{\mu}F_{\mu r}=\mathcal{O}(r^{-3})\quad$
$\displaystyle\Rightarrow\quad\begin{cases}A_{u}^{1/2,1}=0,\\\
A_{u}^{1/2,0}=2C^{\prime},\end{cases}$ (2.10)

while subleading components are relegated to appendix A.1. As for the angular
Maxwell equation $\nabla^{\mu}F_{\mu\phi}=0$, it determines the time evolution
of the coefficients of $A_{\phi}$. Again solving recursively in $1/r$, one
finds

$\displaystyle\nabla^{\mu}F_{\mu\phi}=\mathcal{O}(r^{-2})\quad$
$\displaystyle\Rightarrow\quad\begin{cases}\dot{A}_{\phi}^{\ell}=E^{\prime},\\\
\dot{A}_{\phi}^{0}=E^{\prime}+G^{\prime},\\\ \dot{A}_{\phi}^{1/2,1}=0,\\\
\dot{A}_{\phi}^{1/2,0}=\displaystyle\frac{3}{2}\left(\frac{1}{4}C+C^{\prime\prime}\right).\end{cases}$
(2.11)

These equations, along with (2.10), show that the whole dynamics is fixed by
$C(u,\phi)$ along with the Coulombic data $E(u,\phi)$ and the function
$G(u,\phi)$. Finally turning to the remaining Maxwell equation, one obtains

$\displaystyle\lim_{r\to\infty}(r\nabla^{\mu}F_{\mu u})=0\quad$
$\displaystyle\Rightarrow\quad\dot{E}=0.$ (2.12)

Thus the Coulombic data is forced by dynamics to be time-independent:
$E=E(\phi)$. This is in fact the only contribution in the temporal Maxwell
equation. Indeed, since $\nabla^{\mu}F_{\mu r}=0=\nabla^{\mu}F_{\mu\phi}$ has
already been solved, one can deduce from the identity
$\nabla^{\mu}\nabla^{\nu}F_{\mu\nu}=0$ that $\partial_{r}(r\nabla^{\mu}F_{\mu
u})=0$, which means there is a single term in the radial expansion. Note that
the solutions (2.10), (2.11) and (2.12) are all linear in the Maxwell fields,
while the corresponding solutions in Einstein–Maxwell theory are non-linear
due to the backreaction of the gravitational field (see appendix A).

#### Gauge-invariant fields.

With the fall-offs (2.9), the components of the Faraday tensor are

$\displaystyle B\coloneqq F_{r\phi}$
$\displaystyle=\frac{C}{2\sqrt{r}}+\frac{A_{\phi}^{\ell}}{r}+\mathcal{O}(r^{-3/2}),$
(2.13a) $\displaystyle E_{r}\coloneqq F_{ur}$
$\displaystyle=-\frac{E}{r}+\frac{C^{\prime}}{r^{3/2}}+\mathcal{O}(r^{-2}),$
(2.13b) $\displaystyle E_{\phi}\coloneqq F_{u\phi}$
$\displaystyle=N\sqrt{r}+E^{\prime}+\mathcal{O}(r^{-1/2}),$ (2.13c)

where $N$ is the news (1.2). We stress again that the radiative and Coulombic
data in the radial electric field (2.13b) do not appear at the same order,
differently from what happens in 4D. The fact that the boundary conditions
(2.9) include both Coulombic and radiative data can then also be seen from the
Newman–Penrose formalism. Indeed, consider the triad of vectors

$\displaystyle\ell\coloneqq\partial_{r},\qquad\qquad
n\coloneqq-\partial_{u}+\frac{1}{2}\partial_{r},\qquad\qquad
m\coloneqq\frac{1}{r}\partial_{\phi}$ (2.14)

such that $g^{\mu\nu}=\ell^{\mu}n^{\nu}+\ell^{\nu}n^{\mu}+m^{\mu}m^{\nu}$ be
the inverse of the Minkowski metric (2.7) in Bondi coordinates. The triad is
thus null and normalized, and contracting its elements with the Faraday tensor
(2.13) yields the Maxwellian Newman–Penrose scalars

$\displaystyle\Phi_{0}$ $\displaystyle\coloneqq
F_{\mu\nu}\ell^{\mu}m^{\nu}=\frac{C}{2r^{3/2}}+\mathcal{O}(r^{-2}),$ (2.15a)
$\displaystyle\Phi_{1}$ $\displaystyle\coloneqq
F_{\mu\nu}n^{\mu}\ell^{\nu}\;=\frac{E}{r}+\mathcal{O}(r^{-3/2}),$ (2.15b)
$\displaystyle\Phi_{2}$ $\displaystyle\coloneqq
F_{\mu\nu}n^{\mu}m^{\nu}\\!=-\frac{N}{\sqrt{r}}+\mathcal{O}(r^{-1}),$ (2.15c)

where $\Phi_{1}$ and $\Phi_{2}$ respectively identify Coulombic and radiative
data.

For future use in the Einstein equations, let us compute the (symmetrized)
electromagnetic energy-momentum tensor

$\displaystyle
T_{\mu\nu}=2F_{\mu\alpha}{F_{\nu}}^{\alpha}-\frac{1}{2}g_{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}.$
(2.16)

Its components $T_{uu}$ and $T_{u\phi}$, written in Bondi coordinates,
respectively measure energy fluxes and angular-momentum fluxes. In the case at
hand, $g_{\mu\nu}$ is the Minkowski metric (2.7) and one finds

$\displaystyle T_{uu}=\frac{2N^{2}}{r}+\mathcal{O}(r^{-3/2}),\qquad\qquad
T_{u\phi}=-\frac{2NE}{\sqrt{r}}+\frac{2}{r}\big{(}NC^{\prime}-EE^{\prime}\big{)}+\mathcal{O}(r^{-3/2}).$
(2.17)

Note how the energy density $T_{uu}$ involves the square of the news; this
will eventually yield a Bondi mass loss formula for the dynamical metric. Also
note that the angular momentum density $T_{u\phi}$ depends not only on the
radiative field $C$, but also on the Coulombic data $E$. The same property
holds in 4D Maxwell theory, and is usually considered as a puzzle since one
naïvely expects no Coulombic data to contribute to the radiation of angular
momentum [103, 104, 88]. Another puzzle, albeit one that is specific to 3D, is
that $T_{u\phi}$ integrated on the asymptotic circle appears to be divergent
owing to the measure $r\mathrm{d}\phi$. A resolution of both puzzles will come
from the asymptotic charge (5.12a), which is finite and has a purely radiative
flux sourced by the news.

###  2.3 Electromagnetic memory

We close this section with a short discussion of the electromagnetic ‘kick’
memory effect [90], partly to illustrate how 3D and 4D differ. Consider a
particle with mass $m$ and charge $q$ whose path in spacetime traces some
worldline with proper 3-velocity $v$. The particle is subjected to some
external electromagnetic field $F_{\mu\nu}$, whereupon its proper acceleration
is given by the Lorentz force

$\displaystyle\dot{v}^{\mu}=\frac{q}{m}F^{\mu}{}_{\nu}\,v^{\nu}.$ (2.18)

To see how this leads to memory, write the proper velocity as
$v=\alpha\partial_{u}+\beta\partial_{r}+(\gamma/r)\partial_{\varphi}$. Using
the Faraday tensor (2.13), the equations of motion (2.18) become

$\displaystyle\dot{\alpha}\sim-\frac{E}{r}\alpha,\qquad\qquad\dot{\beta}\sim-\frac{\dot{C}}{\sqrt{r}}\gamma,\qquad\qquad\dot{\gamma}\sim-\frac{\dot{C}}{\sqrt{r}}\alpha$
(2.19)

at leading order in $1/r$. Thus, the angular acceleration $\dot{\gamma}$ is
determined by the news, as in 4D [90]; but in contrast to 4D, the effect is
suppressed by a factor $1/\sqrt{r}$. For a non-relativistic particle with
$\alpha\sim 1$, a finite burst of radiation thus yields a net change of
angular velocity given by

$\displaystyle\Delta\gamma\sim-\frac{\Delta C}{\sqrt{r}},$ (2.20)

where the right-hand side involves the integral of electromagnetic news:

$\displaystyle\Delta C(\phi)\coloneqq
C(u=+\infty,\phi)-C(u=-\infty,\phi)=\int\text{d}u\,N(u,\phi).$ (2.21)

The velocity change (2.20) is non-zero at finite $r$, but it decays as
$r\to\infty$. As a consequence, the memory effect (2.20) is unrelated to the
leading surface charges of Maxwell theory, which typically involve integrals
of the Coulombic (as opposed to radiative) data $E(\phi)$ over the celestial
circle: see (4.10) below. We will see nevertheless that subleading
electromagnetic charges involve contributions of the form
$C^{\prime}/\sqrt{r}$ (see (4.12) below). One may view this as yet another
manifestation of the mismatch between radiative and Coulombic fall-offs in 3D.
By contrast, 4D electromagnetic memory involves a net change in angular
velocity that is finite at infinity [90] and may be seen as a vacuum
transition under leading asymptotic symmetries.

##  3 Setup for Einstein–Maxwell theory

In this section, we gather the ingredients needed to study asymptotic
symmetries in Einstein–Maxwell theory. Starting from the Lagrangian, we
compute the Noether current and the equations of motion, which we then solve
with the fall-offs (2.9) in order to characterize the solution space. We
conclude with a discussion of simple zero-mode configurations.

###  3.1 Lagrangian and Noether currents

Consider the Lagrangian density for 3D Einstein–Maxwell theory:

$\displaystyle\mathcal{L}=\frac{1}{2}\sqrt{-g}\,\big{(}R-F^{\mu\nu}F_{\mu\nu}\big{)}.$
(3.1)

The equations of motion obtained by varying the corresponding action with
respect to $A^{\nu}$ and $g^{\mu\nu}$ are the standard Maxwell and Einstein
equations333The tensor $E_{\mu\nu}$ of the Einstein equation in (3.2) has
nothing to do with the electric charge aspect $E$ in (2.9a).

$\displaystyle\nabla^{\mu}F_{\mu\nu}=0,\qquad\qquad E_{\mu\nu}\coloneqq
G_{\mu\nu}-T_{\mu\nu}=0,$ (3.2)

with $G_{\mu\nu}$ the Einstein tensor and $T_{\mu\nu}$ the energy-momentum
tensor (2.16).

The pre-symplectic potential $\theta$ is identified as usual from the
variation of the Lagrangian. More precisely, we have
$\delta\mathcal{L}=(\text{EOM})\delta(A_{\mu},g_{\mu\nu})+\partial_{\mu}\theta^{\mu}$,
with

$\displaystyle\theta^{\mu}=\frac{1}{2}\sqrt{-g}\Big{(}g^{\alpha\beta}\delta\Gamma^{\mu}_{\alpha\beta}-g^{\mu\alpha}\delta\Gamma^{\beta}_{\alpha\beta}-4F^{\mu\nu}\delta
A_{\nu}\Big{)}.$ (3.3)

This is the starting point of the covariant phase space derivation of Noether
charges associated with gauge and diffeomorphism symmetries [105]. Indeed, any
infinitesimal diffeomorphism $\xi$ accompanied by a gauge transformation
$\varepsilon$ transforms the metric and the electromagnetic potential
according to $\delta_{\xi,\varepsilon}g_{\mu\nu}=\pounds_{\xi}g_{\mu\nu}$ and
$\delta_{\xi,\varepsilon}A_{\mu}=\pounds_{\xi}A_{\mu}+\partial_{\mu}\varepsilon$.
The corresponding transformation of the Lagrangian is
$\delta_{\xi,\varepsilon}\mathcal{L}=\pounds_{\xi}\mathcal{L}=\partial_{\mu}(\xi^{\mu}\mathcal{L})$,
and the ensuing off-shell Noether current is
$J^{\mu}=\theta^{\mu}[\delta_{\xi,\varepsilon}]-\xi^{\mu}\mathcal{L}$, where
$\theta$ is given by (3.3). In the case at hand, one has

$\displaystyle
J^{\mu}=\sqrt{-g}\,\xi_{\nu}\big{(}E^{\mu\nu}+2A^{\nu}\nabla_{\alpha}F^{\mu\alpha}\big{)}+\frac{1}{2}\sqrt{-g}\,\nabla_{\nu}\Big{(}\nabla^{\nu}\xi^{\mu}-\nabla^{\mu}\xi^{\nu}-4F^{\mu\nu}(\xi^{\alpha}A_{\alpha}+\varepsilon)\Big{)}.$
(3.4)

On-shell, the first term vanishes by virtue of the equations of motion (3.2).
The remaining surface term involves the integrand of the Noether charge,
namely the ‘Komar–Maxwell aspect’

$\displaystyle
K^{\mu\nu}_{\xi,\varepsilon}=\frac{1}{2}\sqrt{-g}\Big{(}\nabla^{\nu}\xi^{\mu}-\nabla^{\mu}\xi^{\nu}-4F^{\mu\nu}(\xi^{\alpha}A_{\alpha}+\varepsilon)\Big{)}.$
(3.5)

The latter will be used below to compute asymptotic charges.

###  3.2 Fall-offs and radiative solution space

This section is the Einstein–Maxwell analogue of section 2.2. We specify gauge
and fall-off conditions, then explain the construction of the corresponding
solution space.

#### Fall-offs and gauge conditions.

In order to solve the field equations, one first has to pick gauge and fall-
off conditions for the metric and the Maxwell field. For the latter, we choose
as before the radial gauge $A_{r}=0$ and the fall-off conditions (2.9). For
the metric, we work in Bondi coordinates and consider the standard line
element [106] that generalizes the pure Minkowski form (2.7):

$\displaystyle\mathrm{d}s^{2}=Ve^{2\beta}\mathrm{d}u^{2}-2e^{2\beta}\mathrm{d}u\,\mathrm{d}r+r^{2}(\mathrm{d}\phi-U\mathrm{d}u)^{2},$
(3.6)

where $(\beta,U,V)$ a priori depend on all three coordinates $(u,r,\phi)$. The
metric is thus written in Bondi gauge, with the three gauge conditions
$g_{rr}=0=g_{r\phi}$ and $g_{\phi\phi}=r^{2}$.

#### Bondi hierarchy of field equations.

Our goal is now to determine the perturbative large-$r$ behaviour of the
functions $(\beta,U,V)$ thanks to the equations of motion (3.2). This is
detailed in appendix A. The construction follows the so-called Bondi hierarchy
[6], where one first solves hypersurface constraint equations (which are
differential equations in $r$), before listing the true evolution equations in
retarded time $u$. This resolution of the field equations can be summarized as
follows.

First, the Einstein field equation $E_{rr}=0$ determines the radial expansion
of $\beta$ in terms of the components of $A_{\phi}$. The constraint
$E_{r\phi}=0$ then fixes the radial expansion of $U$ in terms of $A_{\phi}$
and $A_{u}$, up to a radial integration constant that is ultimately identified
with the angular momentum aspect $L(u,\phi)$. The radial Maxwell equation
$\nabla^{\mu}F_{\mu r}=0$ then yields the coefficients $A_{u}^{m>0,n}$ in
terms of $(E,A_{\phi})$. Finally, the remaining hypersurface constraint
$E_{ru}=0$ determines the radial expansion of $V$ up to a radial integration
constant, namely the mass aspect $M(u,\phi)$. The angular component
$\nabla^{\mu}F_{\mu\phi}=0$ of the Maxwell equation is then a true evolution
equation that determines the time evolution of $A_{\phi}$, aside from that of
the leading term $C$, while the equation $\nabla^{\mu}F_{\mu u}=0$ sets
$\dot{E}=0$ as in the earlier Minkowski-space result (2.12). Finally,
$E_{u\phi}=0$ and $E_{uu}=0$ determine the evolution of angular momentum and
mass, whereupon $E_{\phi\phi}=0$ is trivially satisfied.

A remark: in the solution for $\beta$ and $U$, we have set to zero the two
radial integration constants $\beta_{0}(u,\phi)$ and $U_{0}(u,\phi)$ appearing
at order $r^{0}$. These functions parametrize the induced boundary metric on
$\mathscr{I}^{+}$ and can be included in the solution space of 3D vacuum
gravity [96]. It would be interesting to investigate the effects of these
boundary fields in the case of Einstein–Maxwell theory, but this goes beyond
the scope of the present work.

#### Solution space.

At the end of the day, with the gauge and fall-off conditions (2.9)–(3.6), the
non-vanishing components of the metric in Bondi gauge are

$\displaystyle g_{uu}$ $\displaystyle=2E^{2}(\ln r)+2M+\mathcal{O}(r^{-1/2}),$
(3.7a) $\displaystyle g_{ur}$
$\displaystyle=-1+\frac{C^{2}}{2r}+\mathcal{O}(r^{-3/2}),$ (3.7b)
$\displaystyle g_{u\phi}$ $\displaystyle=\frac{8}{3}\sqrt{r}\,CE+2(\ln
r)EA_{\phi}^{\ell}+L+\mathcal{O}(r^{-1/2}),$ (3.7c) $\displaystyle
g_{\phi\phi}$ $\displaystyle=r^{2}.$ (3.7d)

The components of the Faraday tensor are still given by (2.13), as in
Minkowski space. The fact that electromagnetic aspects of the problem are so
weakly affected by gravity may be viewed as a consequence of the absence of
local gravitational degrees of freedom. Note, for comparison with reference
[1], that the dictionary between our notations and those of [1] is obtained
with the replacements

$\displaystyle E\to-\lambda,\qquad\quad A^{\ell}_{\phi}\to\alpha,\qquad\quad
C\to 0,\qquad\quad M\to\frac{\theta}{2},\qquad\quad
L\to\frac{1}{2}N_{\text{\small\cite[cite]{[\@@bibref{Number}{Barnich:2015jua}{}{}]}}}-\lambda\alpha.$
(3.8)

We stress again that the electromagnetic radiative data $C$ vanishes
identically in [1].

As in section 2.2, gauge-invariant fields can readily be deduced from the
results (2.13) and (3.7). Thus the $(uu)$ and $(u\phi)$ components of the
stress-energy tensor are given again by (2.17), and manifestly affect the
metric at leading order in $1/r$ since the components (3.7) only reduce to the
standard ones of pure gravity [13] when $E=C=0$. As for the Newman–Penrose
formalism, the triad (2.14) should now be replaced by the one appropriate for
the metric in Bondi gauge (3.6):

$\displaystyle\ell=\partial_{r},\qquad\qquad
n=-e^{-2\beta}\left(\partial_{u}+\frac{V}{2}\partial_{r}+U\partial_{\phi}\right),\qquad\qquad
m=\frac{1}{r}\partial_{\phi}.$ (3.9)

The corresponding Newman–Penrose Maxwell scalars have once again the expansion
(2.15), as in Minkowski space at leading order.

An important feature of the solution space concerns the nature of the free
data on $\mathscr{I}^{+}$. Indeed, the functions $C(u,\phi)$ and $G(u,\phi)$
in (2.9) do not satisfy any evolution equation in $u$: their time evolution on
$\mathscr{I}^{+}$ is unconstrained. As mentioned in the introduction, the
presence of two free functions where one expects a single one (since 3D
Maxwell theory has a single degree of freedom) may seem puzzling. From the
energy flux (2.17) however, it is clear that $C(u,\phi)$ is the
electromagnetic shear; by contrast, $G$ is a residual gauge degree of freedom
akin to those that can be unfrozen in 4D pure gravity [53, 54] or in vacuum 3D
gravity with a free boundary metric [95, 96]. Aside from this free data on
$\mathscr{I}^{+}$, the solution space contains an infinite amount of data
satisfying evolution equations in $u$: the mass $M(u,\phi)$, the angular
momentum $L(u,\phi)$, the charge aspect $E(\phi)$, and the components of
$A_{\phi}$ other than $C$. In particular, the mass and angular momentum
aspects, i.e. the terms of order $r^{0}$ in (3.7a) and (3.7c), satisfy the
evolution equations

$\displaystyle\dot{M}=-2N^{2},\qquad\qquad\dot{L}=M^{\prime}+EE^{\prime}+\frac{1}{2}\big{(}CN^{\prime}-3C^{\prime}N\big{)}.$
(3.10)

The time derivative (1.2), naturally identified as the news, is indeed the
quantity appearing in the flux of energy-momentum tensor
$rT_{uu}=2N^{2}+\mathcal{O}(r^{-1/2})$ given by (2.17). It is illuminating
that the evolution of the mass aspect in (3.10) takes a form similar to the 4D
Bondi mass loss of pure gravity:444The evolution of angular momentum does as
well, with the addition of the Coulombic term in $E$ here, but the comparison
with the 4D equations is perhaps less striking. it is sourced by the
electromagnetic news and exhibits that 3D Einstein–Maxwell theory with
radiative fall-offs is analogous to a dimensional reduction of 4D pure
gravity. At the difference with the 4D case however, one should note that the
fluxes on the right-hand side of the evolution equations (3.10) do not involve
the soft terms linear in the news.

The analogy with 4D pure gravity is further strengthened by the computation of
the on-shell symplectic potential (3.3). Its radial and temporal components
are indeed given by

$\displaystyle\theta^{r}$ $\displaystyle=\frac{1}{4}\delta\Big{[}4(\ln
r)E^{2}+2E^{2}+2M-CN\Big{]}+\underbrace{N\delta C+E\delta G}_{\text{non-
exact}}+\mathcal{O}(r^{-1/2}),$ (3.11a) $\displaystyle\theta^{u}$
$\displaystyle=\frac{C\delta C}{r}+\mathcal{O}(r^{-3/2}),$ (3.11b)

where $\theta^{r}$ contains a non-exact symplectic flux with two
contributions: the first, $N\delta C$, is the 3D analogue of the term
$N_{ab}\delta C^{ab}$ in 4D pure gravity; the second, $E\delta G$, is sourced
by the extra free field $G(u,\phi)$ in the solution space. This is the origin
of the spurious flux identified in [1], which will appear in the asymptotic
charges. For the time being, we keep both sources of flux non-vanishing; we
will set $G$ to zero only later, when studying surface charges.

###  3.3 Zero-mode solutions and angular momentum

Having characterized the solution space, it is illustrative to study ‘zero-
mode solutions’, i.e. configurations in which all fields on $\mathscr{I}^{+}$
are constant parameters. In pure AdS3 gravity with Brown–Henneaux boundary
conditions [58], such zero-modes span a 2D parameter space indexed by mass and
angular momentum, containing in particular black holes [107], conical deficits
and excesses, and certain spacetimes with closed time-like curves (see e.g.
the phase diagram in [65, fig. 8.5]). The flat limit of these vacuum solutions
respectively results in flat-space cosmologies, conical deficits and conical
excesses, while solutions with closed time-like curves are washed out [61].

The solution space built above contains purely gravitational data (i.e. data
that is also present in the vacuum case), namely the mass $M$ and the angular
momentum $L$. In addition, it includes an infinite amount of Maxwell data: the
shear $C$, the spurious field $G$, the Coulombic electric charge aspect $E$,
and all the components in the radial expansion of $A_{\phi}$. When searching
for zero-mode solutions, it is natural to switch this all off except for the
electric charge $E$. One thus seeks exact solutions labelled by three constant
parameters: the mass $M$, the angular momentum $L$, and the electric charge
$E$. We will indeed see in section 4.2 that these are the parameters conjugate
to time translations, rotations, and global U$(1)$ transformations
respectively.

One property of such zero-mode configurations deserves special
attention.555What follows is a digression on angular momentum that has few
implications for the rest of the paper. The hasty reader may thus go straight
so section 4. Namely, when all the Maxwell data is switched off except the
charge $E$, the (appended) equation of motion (A.16) imposes $EL=0$, implying
that the solution cannot have both non-zero angular momentum $L$ and non-zero
electric charge $E$. One might be tempted to conclude that a spinning charged
solution is necessarily radiative, but this is not so, since a consistent
solution space can be obtained even when $C=0$ [1]. Instead, all (A.16)
requires is that solutions having both $L\neq 0$ and $E\neq 0$ include other
modes of $A_{\phi}$; to the best of our knowledge, there is no closed-form
expression for such solutions.

The same conclusion can be reached using a flat limit of charged
Bañados–Teitelboim–Zanelli (BTZ) solutions in AdS3. Momentarily letting
$\Lambda=-1/\ell^{2}$, an exact solution to the field equations
$\nabla^{\mu}F_{\mu\nu}=0=G_{\mu\nu}+\Lambda g_{\mu\nu}-T_{\mu\nu}$ is indeed
given in coordinates $(t,r,\phi)$ by [108]

$\displaystyle A_{\mu}\mathrm{d}x^{\mu}$
$\displaystyle=\frac{Q}{\sqrt{2(1-L^{2}/\ell^{2})}}(\ln
r)\big{(}\mathrm{d}t-L\,\mathrm{d}\phi\big{)},$ (3.12a)
$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=\left(N_{r}\frac{r^{2}}{\rho^{2}}+\rho^{2}N_{\phi}^{2}\right)\mathrm{d}t^{2}-N_{r}^{-1}\mathrm{d}r^{2}+2\rho^{2}N_{\phi}\,\mathrm{d}t\,\mathrm{d}\phi+\rho^{2}\mathrm{d}\phi^{2},$
(3.12b)

where the radial coordinates $r$ and $\rho$ are related by

$\displaystyle\rho^{2}$
$\displaystyle=r^{2}+\frac{L^{2}}{1-L^{2}/\ell^{2}}\big{(}M+Q^{2}(\ln
r)\big{)}$ (3.13)

and the shift components are

$\displaystyle N_{r}=-\frac{r^{2}}{\ell^{2}}+M+Q^{2}(\ln r),\qquad\qquad
N_{\phi}=-\frac{L}{\rho^{2}(1-L^{2}/\ell^{2})}\big{(}M+Q^{2}(\ln r)\big{)}.$
(3.14)

The free parameters in (3.12) are $(M,L,Q)$, respectively fixing the mass, the
angular momentum and the charge of the black hole. In the neutral limit $Q=0$,
the metric (3.12b) reduces to that of a BTZ black hole [107, 109], whose
‘standard’ metric is obtained when using $\rho$ as the radial coordinate.
Taking instead the flat limit $\ell\to\infty$ yields a solution of (3.2) with

$\displaystyle A_{\mu}\mathrm{d}x^{\mu}$ $\displaystyle=\frac{Q}{\sqrt{2}}(\ln
r)\big{(}\mathrm{d}t-L\,\mathrm{d}\phi\big{)},$ (3.15a)
$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=N_{r}\,\mathrm{d}t^{2}-N_{r}^{-1}\mathrm{d}r^{2}-2LN_{r}\,\mathrm{d}t\,\mathrm{d}\phi+(r^{2}+L^{2}N_{r})\mathrm{d}\phi^{2},$
(3.15b)

and $N_{r}=M+Q^{2}(\ln r)$. In order to put this line element in Bondi gauge
(3.6), we introduce the new coordinate $u=t+f(r)-L\phi$ with
$\partial_{r}f(r)=N_{r}^{-1}$, whereupon

$\displaystyle A_{\mu}\mathrm{d}x^{\mu}=\frac{Q}{\sqrt{2}}(\ln
r)\big{(}\mathrm{d}u-N_{r}^{-1}\mathrm{d}r\big{)},\qquad\qquad\mathrm{d}s^{2}=N_{r}\,\mathrm{d}u^{2}-2\mathrm{d}u\,\mathrm{d}r+r^{2}\mathrm{d}\phi^{2}.$
(3.16)

Here the component $A_{r}$ can be trivially gauged away since it only depends
on $r$. As a result, in Bondi gauge, the flat limit of a charged BTZ black
hole has no angular momentum, and is parametrized only by the mass $M$ and the
electric charge $Q$. The angular momentum $L$ has indeed been reabsorbed from
(3.15) in the change of coordinates that puts the metric in Bondi gauge.

This result may seem surprising at first: had we set $Q=0$ from the outset,
the angular momentum would have been reabsorbed just as well, seemingly
contradicting the existence of flat-space cosmologies with non-zero angular
momentum. But in fact, the simplification agrees with the fact that all
solutions of 3D gravity are locally isometric to a maximally symmetric
spacetime, implying the existence of a diffeomorphism that indeed changes
angular momentum. To see this, start from the flat-space zero-mode metric with
mass $M$,

$\displaystyle\mathrm{d}s^{2}=M\mathrm{d}u^{2}-2\mathrm{d}u\,\mathrm{d}r+r^{2}\mathrm{d}\phi^{2},$
(3.17)

and change coordinates according to666We do not discuss the range of the
coordinates since we are only interested in producing a new solution.

$\displaystyle u\mapsto
u+\frac{1}{M}\left(L\phi-r+\sqrt{r^{2}-\frac{L^{2}}{M}}\right),\qquad
r\mapsto\sqrt{r^{2}-\frac{L^{2}}{M}},\qquad\phi\mapsto\phi-\frac{1}{\sqrt{M}}\text{arctanh}\left(\frac{r\sqrt{M}}{L}\right).$
(3.18)

Using this in (3.17) yields the new metric

$\displaystyle\mathrm{d}s^{2}=M\mathrm{d}u^{2}-2\mathrm{d}u\,\mathrm{d}r+2L\mathrm{d}u\,\mathrm{d}\phi+r^{2}\mathrm{d}\phi^{2},$
(3.19)

which is now a flat-space zero-mode with both mass $M$ and angular momentum
$L$. Crucially though, the transformation (3.18) is a ‘large’ diffeomorphism
that affects asymptotic charges. The steps leading from (3.12) to (3.16)
similarly involved a large diffeomorphism to bring the flat limit of the
charged BTZ black hole in Bondi gauge, thereby explaining why angular momentum
is ultimately absent from (3.16).

##  4 Asymptotic symmetries and charges

The solution space of a theory is a covariant version of its phase space
[110]. In the case at hand, we have just obtained the covariant phase space of
3D Einstein–Maxwell theory containing both radiative and Coulombic data. We
now investigate its (asymptotic) symmetries and derive the associated charges.
As we shall see, the presence of radiation entails non-integrable terms whose
influence on the charge algebra will be studied separately in section 5.

###  4.1 Symmetry generators and transformation laws

We start with the derivation of the asymptotic Killing vectors
$\xi^{\mu}=(\xi^{u},\xi^{r},\xi^{\phi})$, seen as generators of
diffeomorphisms that preserve the gauge and fall-off conditions used in
writing the metric (3.6). In particular, any such diffeomorphism must preserve
the Bondi gauge choices $g_{rr}=0=g_{r\phi}$ and $g_{\phi\phi}=r^{2}$.
Computing the relevant Lie derivatives thus imposes the following conditions:

$\displaystyle\pounds_{\xi}g_{rr}=0\quad$
$\displaystyle\Rightarrow\quad\xi^{u}=f,$ (4.1a)
$\displaystyle\pounds_{\xi}g_{r\phi}=0\quad$
$\displaystyle\Rightarrow\quad\xi^{\phi}=g-f^{\prime}\int_{r}^{\infty}\frac{e^{2\beta}}{\tilde{r}^{2}}\,\mathrm{d}\tilde{r}=g+\frac{f^{\prime}}{r}+\mathcal{O}(r^{-2}),$
(4.1b) $\displaystyle\pounds_{\xi}g_{\phi\phi}=0\quad$
$\displaystyle\Rightarrow\quad\xi^{r}=r\big{(}Uf^{\prime}-(\xi^{\phi})^{\prime}\big{)}=-rg^{\prime}+f^{\prime\prime}+\mathcal{O}(r^{-1/2}),$
(4.1c)

where $f=f(u,\phi)$ and $g=g(u,\phi)$ are free functions at this stage. An
additional constraint is that $\xi^{\mu}$ needs to preserve the fall-offs
(3.6). Namely, the condition $g_{ur}=-1+\mathcal{O}(r^{-1})$ requires
$\dot{f}=g^{\prime}$, and preserving $g_{u\phi}=\mathcal{O}(\sqrt{r})$ imposes
$\dot{g}=0$. In short,

$\displaystyle f=T+ug^{\prime},\qquad\qquad\dot{T}=0=\dot{g},$ (4.2)

where $T(\phi)$ and $g(\phi)$ are arbitrary functions on the celestial circle,
respectively generating supertranslations and superrotations [13]. The
remaining fall-off $g_{uu}=\mathcal{O}(\ln r)$ is automatically preserved.

Note that asymptotic Killing vector fields need not preserve the gauge and
fall-off conditions used in the Maxwell field (2.9), since any change due to a
diffeomorphism can be reabsorbed by a suitable gauge transformation. In
particular, one needs to ensure that the radial gauge $A_{r}=0$ is preserved
when acting on $A$ with the allowed symmetries
$\delta_{\xi,\varepsilon}A_{\mu}=\pounds_{\xi}A_{\mu}+\partial_{\mu}\varepsilon$.
Requiring $\pounds_{\xi}A_{r}+\partial_{r}\varepsilon=0$ with $\xi$ given by
(4.1)–(4.2) thus leads to the gauge parameter

$\displaystyle\varepsilon=\alpha(u,\phi)+f^{\prime}\int_{r}^{\infty}\frac{e^{2\beta}A_{\phi}}{\tilde{r}^{2}}\mathrm{d}\tilde{r}=\alpha(u,\phi)+\frac{2}{\sqrt{r}}f^{\prime}C+\mathcal{O}(r^{-1}).$
(4.3)

Here $\alpha(u,\phi)$ is a free function on $\mathscr{I}^{+}$, generating
‘large gauge transformations’ of the Maxwell field.777This function
$\alpha(u,\phi)$ has nothing to do with the $u$ component of the proper
velocity in section 2.3. We stress that its time dependence is unconstrained
(which will come back to haunt us when defining charges). The remaining fall-
off conditions in (2.9) are automatically satisfied. Asymptotic symmetry
generators are thus labelled by a $\mathfrak{bms}_{3}$ vector field $(T,g)$
and a function $\alpha$ on $\mathscr{I}^{+}$. We shall return to this
structure shortly.

#### Transformation laws.

The asymptotic symmetries obtained here were found off-shell, i.e. regardless
of the equations of motion of the theory. In practice, their action on
covariant phase space is obtained by computing Lie derivatives and gauge
transformations of the solution space of section 3.2. The resulting
transformation laws of the various fields of interest can then be organized as
follows. First, the transformations of the mass and angular momentum aspects
read

$\displaystyle\delta_{\xi}M$
$\displaystyle=f\dot{M}+\underbrace{\big{.}gM^{\prime}+2g^{\prime}M-g^{\prime\prime\prime}}_{\text{coadjoint
$\mathfrak{bms}_{3}$}}\underbrace{-g^{\prime}E^{2}}_{\text{Maxwell}},$ (4.4a)
$\displaystyle\delta_{\xi}L$
$\displaystyle=\underbrace{\Big{.}f\dot{L}+gL^{\prime}+2g^{\prime}L+2f^{\prime}M-f^{\prime\prime\prime}}_{\supset\text{~{}coadjoint
$\mathfrak{bms}_{3}$}}\underbrace{-2g^{\prime}EA_{\phi}^{\ell}-\frac{1}{4}g^{\prime\prime}C^{2}+\frac{1}{2}f^{\prime}CN}_{\text{Maxwell}},$
(4.4b)

where one recognizes the coadjoint representation of the $\mathfrak{bms}_{3}$
algebra with central charges $c_{1}=0$, $c_{2}\neq 0$ [60], supplemented by
corrections due to the time dependence of $M,L$ and the presence of
electromagnetic boundary data. Second, the transformations of the
electromagnetic shear and news read

$\displaystyle\delta_{\xi,\varepsilon}C=\Big{(}f\partial_{u}+g\partial_{\phi}+\frac{1}{2}g^{\prime}\Big{)}C,\qquad\qquad\delta_{\xi,\varepsilon}N=\Big{(}f\partial_{u}+g\partial_{\phi}+\frac{3}{2}g^{\prime}\Big{)}N,$
(4.5)

exhibiting the fact that $C$ and $N$ are $\mathfrak{bms}_{3}$ primary fields
with respective weights $1/2$ and $3/2$ under superrotations. Note the half-
integer labels due to half-integer powers in the fall-offs (2.9), reminiscent
of the half-integer superrotation weights that are ubiquitous in 4D gravity
[111, 82]. Finally, the transformation laws of the remaining leading
components of the Maxwell field are most instructive when imposing the
equations of motion. The logarithmic terms then transform as

$\displaystyle\delta_{\xi,\varepsilon}A_{\phi}^{\ell}$
$\displaystyle=\Big{(}f\partial_{u}+g\partial_{\phi}+g^{\prime}\Big{)}A_{\phi}^{\ell}+f^{\prime}E\approx\big{(}fE+gA_{\phi}^{\ell}\big{)}^{\prime},$
(4.6a) $\displaystyle\delta_{\xi,\varepsilon}E$
$\displaystyle=\Big{(}f\partial_{u}+g\partial_{\phi}+g^{\prime}\Big{)}E\approx\big{(}gE\big{)}^{\prime},$
(4.6b)

so they are both $\mathfrak{bms}_{3}$ primary fields with unit weight under
superrotations, and they involve no U$(1)$ gauge parameter. By contrast, the
$\mathcal{O}(1)$ terms transform as

$\displaystyle\delta_{\xi,\varepsilon}A_{\phi}^{0}$
$\displaystyle=\Big{(}f\partial_{u}+g\partial_{\phi}+g^{\prime}\Big{)}A_{\phi}^{0}-g^{\prime}A_{\phi}^{\ell}+f^{\prime}G+\alpha^{\prime}\approx\big{(}fG+gA_{\phi}^{0}\big{)}^{\prime}+fE^{\prime}-g^{\prime}A_{\phi}^{\ell}+\alpha^{\prime},$
(4.7a) $\displaystyle\delta_{\xi,\varepsilon}G$
$\displaystyle=\Big{(}f\partial_{u}+g\partial_{\phi}+g^{\prime}\Big{)}G-g^{\prime}E+\dot{\alpha},$
(4.7b)

so they also have unit weight under superrotations, but they now involve the
large gauge transformation $\alpha$, including its time derivative in $\delta
G$. We will see in section 5 that this time derivative plays an important role
in the reduced phase space where $G$ is set to zero.

#### Asymptotic symmetry algebra.

It is straightforward to compute the Lie bracket of asymptotic symmetries,
seen as vector fields on phase space. (The Poisson brackets of the
corresponding canonical charges are a whole other problem, treated separately
in section 4.2.) Indeed, in terms of the ‘modified bracket’ of [13], designed
to take into account field-dependent parameters, the asymptotic symmetry
generators satisfy the commutation relations
$\big{[}\delta_{\xi_{1},\varepsilon_{1}},\delta_{\xi_{2},\varepsilon_{2}}\big{]}=-\delta_{\xi_{12},\varepsilon_{12}}$
where

$\displaystyle\xi_{12}\coloneqq\big{[}\xi_{1},\xi_{2}\big{]}_{*}\coloneqq\big{[}\xi_{1},\xi_{2}\big{]}-\delta_{\xi_{1}}\xi_{2}+\delta_{\xi_{2}}\xi_{1},\qquad\varepsilon_{12}\coloneqq\pounds_{\xi_{1}}\varepsilon_{2}-\delta_{\xi_{1},\varepsilon_{1}}\varepsilon_{2}-(1\leftrightarrow
2).$ (4.8)

In the case at hand, all symmetry generators are field-independent, so the
$\delta$ pieces of (4.8) all vanish. The Lie bracket of asymptotic symmetry
generators thus reads

$\displaystyle f_{12}$
$\displaystyle=f_{1}g^{\prime}_{2}+g_{1}f^{\prime}_{2}-\delta_{\xi_{1}}f_{2}-(1\leftrightarrow
2),$ (4.9a) $\displaystyle g_{12}$
$\displaystyle=g_{1}g^{\prime}_{2}-\delta_{\xi_{1}}g_{2}-(1\leftrightarrow
2),$ (4.9b) $\displaystyle\alpha_{12}$
$\displaystyle=f_{1}\dot{\alpha}_{2}+g_{1}\alpha_{2}^{\prime}-\delta_{\xi_{1},\varepsilon_{1}}\alpha_{2}-(1\leftrightarrow
2),$ (4.9c)

showing that the asymptotic symmetry algebra is a semi-direct sum
$\mathfrak{bms}_{3}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}}C^{\infty}(\mathscr{I}^{+})$, where $\mathfrak{bms}_{3}$ acts
by Carrollian conformal transformations on functions on $\mathscr{I}^{+}$.
This is the same result as in [1], which is thus unaffected by our choice of
relaxed boundary conditions including $C$. However, the presence of $C$ does
affect asymptotic charges, as we now show.

###  4.2 Asymptotic charges introduced

Let $\xi$ be an asymptotic Killing vector field and let $\varepsilon$ be an
asymptotic U$(1)$ gauge parameter, both as described above. They generate
symmetries of the radiative phase space and should therefore define ‘canonical
generators’, meaning functions on phase space—or charges—that produce the
corresponding transformations in terms of the Poisson bracket. The functional
differential of these charges can be computed thanks to the Iyer–Wald formula
[112], starting from the symplectic potential (3.3) and the Komar–Maxwell
aspect (3.5). The upshot is the following variational one-form in field
space:888For compactness, we denote the charge simply by $\mathcal{Q}_{\xi}$
although it depends on both $\xi$ and the U$(1)$ parameter $\varepsilon$. The
notation $\not{\delta}\mathcal{Q}_{\xi}$ represents a one-form in field space
that may or may not be (and generally is not) exact.

$\displaystyle\not{\delta}\mathcal{Q}_{\xi}$ $\displaystyle=\delta
K^{ur}_{\xi}-K^{ur}_{\delta\xi}+\xi^{u}\theta^{r}-\xi^{r}\theta^{u}$ (4.10)
$\displaystyle\stackrel{{\scriptstyle\circ}}{{=}}f\big{(}\delta
M+\underbrace{2N\delta C\big{.}\\!}_{\\!\\!\\!\\!\\!\\!\\!\\!\text{non-
integrable}\\!\\!\\!\\!\\!\\!\\!\\!}\big{)}+g\delta\Bigl{(}L+\frac{1}{4}(C^{2})^{\prime}-E\big{(}A_{\phi}^{\ell}+2A_{\phi}^{0}\big{)}\Bigr{)}-2\big{(}\alpha+\underbrace{fG\big{)}\delta
E\big{.}}_{\\!\\!\\!\\!\\!\\!\\!\\!\text{non-
integrable}\\!\\!\\!\\!\\!\\!\\!\\!}\,+\,\mathcal{O}(r^{-1/2}).\qquad$ (4.11)

The same charge is found with the Barnich–Brandt formalism [113], so there is
no ambiguity for now. An aside on notation: for compactness, we will always
work with charge ‘aspects’ such as (4.10), as opposed to actual charges. The
latter are celestial integrals of the former, so this convention allows us to
omit the integral sign $\oint_{S^{1}}$ throughout. Since integrations by parts
in $\partial_{\phi}$ will nevertheless be needed, the symbol
$\stackrel{{\scriptstyle\circ}}{{=}}$ stands for equalities where all boundary
terms in $\partial_{\phi}$ have been discarded. The notation thus allows us to
integrate by parts without having to carry the integral sign.

Several comments are in order regarding the charge variation (4.10). First,
the asymptotic charge $\lim_{r\to\infty}\not{\delta}\mathcal{Q}_{\xi}$ is
finite. This is already a non-trivial result since (i) the fall-offs (2.9) are
weaker than those of [1] due to the radiative field $C$, and (ii) the
integrated angular momentum defined from the energy-momentum tensor (2.17) is
itself divergent (recall that a measure factor $r$ is required when computing
the integral). Second, assuming for now that $\delta f=\delta
g=\delta\alpha=0$ (i.e. all asymptotic symmetry generators are field-
independent), the charge (4.10) contains two non-integrable contributions, one
of which ($fG\delta E$) is unrelated to electromagnetic radiation. The charge
thus requires a more careful study, deferred to the next section, in order to
understand (i) how to extract its integrable part, (ii) whether the integrable
part is conserved or not, and (iii) how to compute the charge algebra.

Finally, a comment on the 3D electromagnetic memory effect. We saw in (2.20)
that the change in angular velocity is sourced by the radiative field $C$, but
it is clear from the charge (4.10) that this memory term is unrelated to
charges of large U$(1)$ gauge transformations, since the latter only involve
the Coulombic data $E$. There is thus no relation between memory and leading
charges. However, one can push the radial expansion of the charge variation
one order further to find

$\displaystyle\not{\delta}\mathcal{Q}_{\varepsilon}=-2\varepsilon\delta\big{(}\sqrt{-g}F^{ur}\big{)}=-2\alpha\delta
E+\frac{2}{\sqrt{r}}\big{(}\alpha\delta C^{\prime}-2f^{\prime}C\delta
E\big{)}+\mathcal{O}(r^{-1}),$ (4.12)

where the $r^{-1/2}$ term now involves a pairing $\alpha\delta C^{\prime}$
that is reminiscent of the change of velocity (2.20). One can then invert the
operator $\partial_{\phi}$ (say in the space of functions with vanishing mean)
and interpret the memory effect (2.20) as a transition between vacua whose
subleading U$(1)$ charges differ. A similar link between memory and vacuum
transitions exists in 4D, except that it crucially involves leading surface
charges [90]. Thus, 3D electromagnetic memory is related to surface charges,
but the link between them is less tight than in the 4D infrared triangle [17].
To the best of our knowledge, this is the first explicit (yet very simple)
example of such a mismatch. It begs the question of the possible matching
between asymptotic symmetries, memories and soft theorems in 3D (as opposed to
4D). We hope to come back to this issue in future work.

##  5 Brackets and charge algebras

We now turn to a detailed analysis of the charges (4.10) and their brackets,
i.e. the corresponding charge algebras. This involves several deep problems.
The first is that charges are not integrable in the presence of the free data
$C$ and $G$, so a prescription is required to choose their integrable part and
compute the bracket. The second is that (4.10) contains two sources of non-
integrability, and it is unclear if both should be called ‘fluxes’ since only
the contribution of shear and news ($N\delta C$) is identified as radiation.
The study of the charge therefore requires a proper treatment of the remaining
‘spurious flux’ ($G\delta E$).

To explore these issues, we will start by reviewing the commonly used
Barnich–Troessaert (BT) bracket [14] and pointing out its (known) split
ambiguities. We will then fix these, at least partly, by the Wald–Zoupas (WZ)
prescription, before briefly discussing the so-called Noether bracket [97] and
finally turning to the Koszul bracket [99].

Several possibilities and ambiguities will thus be covered, but the main
conclusion is as follows: the most reasonable approach seems to be to set
$G=0$ and use the Koszul bracket [99]. A virtue of this prescription is to
yield a charge algebra that does not depend on the split between integrable
part and flux, while the condition $G=0$ ensures one can define charges that
are conserved in the absence of news. With these properties, both physically
and mathematically well-motivated, the resulting algebra exhibits a field-
dependent cocycle (and is therefore strictly speaking a Lie algebroid). In
summary:

$\begin{array}[]{c}\text{Setting $G=0$, consider the integrable charges
$Q_{\xi}$ in \eqref{WZ int}.}\\\ \text{These are Wald--Zoupas charges
satisfying $\dot{Q}_{\xi}=0$ in the absence of news.}\\\ \text{Then using the
Koszul bracket \eqref{Koszul bracket}, which is crucially split-
independent,}\\\ \text{charges represent the symmetry algebra with the field-
dependent cocycle \eqref{WZ cocycle}.}\end{array}$

Let us now explore the various possibilities in detail.

###  5.1 Barnich–Troessaert bracket

The first proposal for charge brackets in the presence of fluxes was given by
Barnich and Troessaert in [14]. The computation of the eponymous bracket
requires a split of the charge between an integrable part $\delta Q_{\xi}$ and
a flux piece $\Xi_{\xi}[\delta]$, both of which are variational one-forms on
field space (with the integrable piece being the one that is exact). Such
splits will play a key role throughout this section. In the case at hand, the
most obvious split of the charge variation (4.10) is

$\displaystyle\not{\delta}\mathcal{Q}_{\xi}\stackrel{{\scriptstyle\circ}}{{=}}\delta
Q_{\xi}+\Xi_{\xi}[\delta]+\mathcal{O}(r^{-1/2}),$ (5.1)

with an integrable piece and a flux respectively given by999Technically, the
charge (5.2a) is the integrated version of an exact (i.e. integrable) one-form
in field space, but the ‘integrable’ terminology is more standard so we stick
to it.

$\displaystyle Q_{\xi}$
$\displaystyle=fM+g\left(L+\frac{1}{4}(C^{2})^{\prime}-E\big{(}A_{\phi}^{\ell}+2A_{\phi}^{0}\big{)}\right)-2\alpha
E$ $\displaystyle\text{(integrable charge)},$ (5.2a)
$\displaystyle\Xi_{\xi}[\delta]$ $\displaystyle=2f\big{(}N\delta C-G\delta
E\big{)}$ $\displaystyle\text{(flux)}.$ (5.2b)

Several comments are in order about this split. First, it reproduces the
choice made in [1] when $C=0$. Its integrable part (5.2a) exhibits the
familiar supertranslation charge aspect $fM$, the U$(1)$ charge aspect
$-2\alpha E$, and the superrotation charge aspect $gL$ supplemented by
contributions due to electromagnetic boundary data. Second, note that we are
being agnostic at this stage about the difference between the two non-
integrable contributions $N\delta C$ and $G\delta E$, treating them as fluxes
on equal footing. Finally, note for future reference that the very validity of
the split relies on the assumption that the asymptotic symmetry parameters
$(f,g,\alpha)$ are all field-independent; we will come back to this subtle
point when setting $G=0$ to define WZ charges.

Splits of charges between integrable pieces and fluxes form the starting point
of the charge algebra. Indeed, the BT bracket of charges is defined as [14]

$\displaystyle\big{\\{}Q_{\xi_{1}},Q_{\xi_{2}}\big{\\}}_{\text{BT}}\coloneqq\delta_{\xi_{2}}Q_{\xi_{1}}+\Xi_{\xi_{2}}[\delta_{\xi_{1}}]$
(5.3)

and satisfies the key property that the charge algebra reproduces the algebra
of asymptotic symmetry generators, possibly up to a 2-cocycle $K$:

$\displaystyle\big{\\{}Q_{\xi_{1}},Q_{\xi_{2}}\big{\\}}_{\text{BT}}=Q_{[\xi_{1},\xi_{2}]_{*}}+K_{\xi_{1},\xi_{2}}.$
(5.4)

In the case at hand, one can use (5.2) along with the transformation laws
(4.4)–(4.7) to separately evaluate the two contributions on the right-hand
side of (5.3). Using again equalities up to integration by parts on the
celestial angle, one finds

$\displaystyle\delta_{\xi_{2}}Q_{\xi_{1}}$
$\displaystyle\stackrel{{\scriptstyle\circ}}{{=}}Q_{[\xi_{1},\xi_{2}]_{*}}-(f_{1}g_{2}^{\prime\prime\prime}-f_{2}g_{1}^{\prime\prime\prime})-(f_{1}g_{2}^{\prime}-f_{2}g_{1}^{\prime})E^{2}+2(f_{1}\dot{\alpha}_{2}-f_{2}\dot{\alpha}_{1})E$
(5.5a) $\displaystyle\phantom{\
=}-2g_{1}E\big{(}f_{2}G\big{)}^{\prime}-2f_{1}f_{2}N^{2}-2g_{1}f_{2}C^{\prime}N-g_{1}^{\prime}f_{2}CN,$
(5.5b) $\displaystyle\Xi_{\xi_{2}}[\delta_{\xi_{1}}]$
$\displaystyle\stackrel{{\scriptstyle\circ}}{{=}}2f_{2}\big{(}N\delta_{\xi_{1}}C-G\delta_{\xi_{1}}E\big{)}$
(5.5c)
$\displaystyle\stackrel{{\scriptstyle\circ}}{{=}}2g_{1}E\big{(}f_{2}G\big{)}^{\prime}+2f_{1}f_{2}N^{2}+2g_{1}f_{2}C^{\prime}N+g_{1}^{\prime}f_{2}CN,$
(5.5d)

where the bracket of asymptotic symmetry generators in (5.5a) is given by
(4.9). This confirms that the charge algebra satisfies the expected relation
(5.4), with the centreless bracket (4.9) and a cocycle that turns out to be

$\displaystyle
K_{\xi_{1},\xi_{2}}\stackrel{{\scriptstyle\circ}}{{=}}-\underbrace{(T_{1}g_{2}^{\prime\prime\prime}-T_{2}g_{1}^{\prime\prime\prime})}_{\text{standard
$\mathfrak{bms}_{3}$}}-\underbrace{(f_{1}g_{2}^{\prime}-f_{2}g_{1}^{\prime})E^{2}+2(f_{1}\dot{\alpha}_{2}-f_{2}\dot{\alpha}_{1})E}_{\text{field-
dependent piece}}.$ (5.6)

Here one may recognize the standard central extension pairing superrotations
and supertranslations in $\mathfrak{bms}_{3}$ [60], and the $E$-dependent
cocycle is identical to that of [1]. This field-dependence is reminiscent of
that of the BMS4 charge algebra [14]. A key difference, however, is that the
cocycle of 4D gravity involves the gravitational shear, while here it is due
to the Coulombic data $E$.

The presence of field-dependent central extensions is typically expected to
affect physics—e.g. through representations (say coadjoint and their unitary
quantization) and the ensuing holographic dual. Accordingly, it is essential
to understand one’s leeway in writing down a cocycle such as (5.6). How is it
affected by the choice of split (5.1)? Are there choices that cancel the field
dependence? The remainder of this entire section is devoted to such questions.

#### Split ambiguities of charge algebras.

To probe how the BT cocycle is affected by the split (5.1), one can redefine
the latter by shifting the integrable piece as

$\displaystyle\delta
Q_{\xi}+\Xi_{\xi}[\delta]=\underbrace{\delta(Q_{\xi}+S_{\xi})}_{\displaystyle\delta\widetilde{Q}_{\xi}}+\underbrace{\Xi_{\xi}[\delta]-\delta
S_{\xi}}_{\displaystyle\widetilde{\Xi}_{\xi}[\delta]}.$ (5.7)

Here $S_{\xi}$ is any ‘shift’ functional on solution space that is linear in
the asymptotic symmetry parameters $(f,g,\alpha)$. Changing the split in this
way furnishes a new BT bracket (5.3):

$\displaystyle\big{\\{}\widetilde{Q}_{\xi_{1}},\widetilde{Q}_{\xi_{2}}\big{\\}}_{\text{BT}}=\delta_{\xi_{2}}\widetilde{Q}_{\xi_{1}}+\widetilde{\Xi}_{\xi_{2}}[\delta_{\xi_{1}}]=\widetilde{Q}_{[\xi_{1},\xi_{2}]_{*}}+\underbrace{K_{\xi_{1},\xi_{2}}-S_{\xi_{1},\xi_{2}}}_{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\displaystyle\text{shifted
cocycle~{}}\widetilde{K}_{\xi_{1},\xi_{2}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!}$
(5.8)

where the shift of the cocycle is given by
$S_{\xi_{1},\xi_{2}}\coloneqq\delta_{\xi_{1}}S_{\xi_{2}}-\delta_{\xi_{2}}S_{\xi_{1}}+S_{[\xi_{1},\xi_{2}]_{*}}$.
The latter can be deduced from the transformation laws (4.4) once the shift
$S_{\xi}$ is given. For example, consider the following list of shifts:

$\displaystyle S_{\xi}$ $\displaystyle=fEG$ $\displaystyle\Rightarrow$
$\displaystyle S_{\xi_{1},\xi_{2}}$
$\displaystyle\stackrel{{\scriptstyle\circ}}{{=}}-(f_{1}\dot{\alpha}_{2}-f_{2}\dot{\alpha}_{1})E+(f_{1}g_{2}^{\prime}-f_{2}g_{1}^{\prime})E^{2},$
(5.9a) $\displaystyle S_{\xi}$ $\displaystyle=fM$ $\displaystyle\Rightarrow$
$\displaystyle S_{\xi_{1},\xi_{2}}$
$\displaystyle\stackrel{{\scriptstyle\circ}}{{=}}(f_{1}g_{2}^{\prime\prime\prime}-f_{2}g_{1}^{\prime\prime\prime})+(f_{1}g_{2}^{\prime}-f_{2}g_{1}^{\prime})E^{2},$
(5.9b) $\displaystyle S_{\xi}$ $\displaystyle=TM$ $\displaystyle\Rightarrow$
$\displaystyle S_{\xi_{1},\xi_{2}}$
$\displaystyle\stackrel{{\scriptstyle\circ}}{{=}}(T_{1}g_{2}^{\prime\prime\prime}-T_{2}g_{1}^{\prime\prime\prime})+(T_{1}g_{2}^{\prime}-T_{2}g_{1}^{\prime})\big{(}E^{2}+2uN^{2}\big{)}.$
(5.9c)

Note that the last expression involves the pure supertranslation $T(\phi)$
defined by (4.2). Each of these changes the value of the cocycle (5.6).

The effect of cocycle shifts such as (5.9) can be dramatic, possibly even
cancelling any field-dependence. To see this, take the initial split (5.2) and
the cocycle (5.6) as a starting point. Then shift the integrable charge (5.2a)
by $S_{\xi}=f\big{(}M-2EG\big{)}$ as in (5.7). Using (5.9a)–(5.9b), the
resulting charge algebra takes the form (5.4) with a cocycle that differs from
(5.6) and reduces, instead, to
$\widetilde{K}_{\xi_{1},\xi_{2}}\stackrel{{\scriptstyle\circ}}{{=}}-2(T_{1}g_{2}^{\prime\prime\prime}-T_{2}g_{1}^{\prime\prime\prime})$.
This is no longer field-dependent, and even coincides with the standard
$\mathfrak{bms}_{3}$ central extension [60]. All the field-dependence of (5.6)
has been compensated by a mere redefinition of the charges! However, one
should keep in mind that the resulting integrable charge in (5.7) is no longer
conserved in the absence of news, even in the case $G=0$. In this sense,
asking for field-independent cocycles may be overly restrictive. Let us
therefore study more closely the conservation criteria that can be imposed on
integrable charges, regardless of whether the ensuing cocycles are field-
dependent or not.

###  5.2 Wald–Zoupas splits

Following the WZ prescription [29, 30, 31], it is natural to single out the
integrable part of the charge in (5.1) by the requirement that it be conserved
on suitable vacuum configurations. This was investigated in 4D
Einstein–Maxwell theory in [88], where the radiative vacuum can be defined
unambiguously in terms of gravitational and electromagnetic news. In the
present case, however, one has to face the issue of the presence of the two
fields $C$ and $G$. Only the former has a clear physical interpretation in
terms of electromagnetic radiation and gives rise to news $N=\dot{C}$. The
vacuum condition that should be used to construct WZ charges is thus unclear:
should one leave $G$ arbitrary, or instead set $G=0$? Here we explore both
options in turn.

#### Wald–Zoupas split with $\boldsymbol{G\neq 0}$.

If one insists on keeping $G$ arbitrary (and generally non-zero), an immediate
issue arises: the U$(1)$ symmetry generator (4.3) and the charge (4.10)
contain a function $\alpha(u,\phi)$ whose dependence on $u$ is arbitrary, so
that it cannot appear in WZ charges (which would prevent any possibility of
conservation). A way out is to exclude the contribution of $\alpha$ from the
definition of integrable charges, and define ‘vacua’ as being solutions with
vanishing news. This is indeed a consistent choice of vacuum, since the news’
transformation (4.5) is homogeneous. Thus, starting from (5.2a), one may
consider the shifted charge
$\widetilde{Q}_{\xi}=Q_{\xi}+2E(gA_{\phi}^{0}+\alpha)$ such that the
$\alpha$-dependent piece in (4.10) is, by definition, part of the flux
$\Xi_{\xi}$ in (5.1). Then the derivative $\partial_{u}\widetilde{Q}_{\xi}$ is
proportional to the news, so $\partial_{u}\widetilde{Q}_{\xi}=0$ when $N=0$.
This is a consistent choice and it can be used in the BT bracket (5.3),
leading in particular to a complicated field-dependent cocycle (which we do
not display but can be easily computed). The price to pay is that all
electromagnetic charges vanish, by construction. This is arguably unnatural in
a theory that purports to include electrically charged configurations. Let us
therefore explore an alternative route that constrains the time evolution of
$\alpha$ by setting $G=0$.

#### Wald–Zoupas split with $\boldsymbol{G=0}$.

Setting $G=0$ is physically acceptable since it does not prevent the solution
space from describing radiation or electric charge. As can be seen in (4.7b),
a consistency requirement with $G=0$ is to fix the time dependence of the
U$(1)$ gauge parameter to

$\displaystyle\alpha(u,\phi)=ug^{\prime}E+\alpha_{0}(\phi).$ (5.10)

Such a constrained time dependence only leaves out one arbitrary function
$\alpha_{0}(\phi)$, reducing the algebra
$\mathfrak{bms}_{3}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}} \end{picture}}}}}C^{\infty}(\mathscr{I}^{+})$
mentioned below (4.9) to its time-independent subalgebra,
$\mathfrak{bms}_{3}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}} \end{picture}}}}}C^{\infty}(S^{1})$. The
relevant brackets, however, are subtle: they become field-dependent even
without dealing with surface charges. Indeed, the bracket (4.8) now yields

$\displaystyle\alpha_{0,12}=g_{1}\alpha_{0,2}^{\prime}+f_{1}g_{2}^{\prime}E-(1\leftrightarrow
2),$ (5.11)

which agrees with [1, eq. (5.1)] and replaces (4.9c). The algebra
$\mathfrak{bms}_{3}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}} \end{picture}}}}}C^{\infty}(S^{1})$ is thus
endowed with an ‘exotic’ field-dependent bracket.

This complication extends to surface charges: since the relation (5.10) is
field-dependent, it cannot blindly be inserted in (5.2a). Instead, one must go
back to the variational expression (5.1) and properly integrate the term
$\alpha\delta E$. Doing so leads to a new split (5.1) with

$\displaystyle Q_{\xi}$
$\displaystyle=fM+g\Bigl{(}L+\frac{1}{4}(C^{2})^{\prime}-E\big{(}A_{\phi}^{\ell}+2A_{\phi}^{0}\big{)}\Bigr{)}-\big{(}ug^{\prime}E+2\alpha_{0}\big{)}E$
$\displaystyle\,\,\,\,\,\,\,\text{(integrable charge)},$ (5.12a)
$\displaystyle\Xi_{\xi}[\delta]$ $\displaystyle=2fN\delta C$
$\displaystyle\text{(flux)},$ (5.12b)

where the integrable part satisfies a WZ conservation property.101010One can
also view the integrable part as arising from the WZ formula $\delta
Q_{\xi}+\Xi_{\xi}[\delta]+\xi\lrcorner\,\vartheta$, where $\vartheta$ is a WZ
potential [29, 30, 31]. Choosing $\vartheta=(N\delta C)\partial_{r}$ yields
$\xi\lrcorner\,\vartheta\big{|}_{S^{1}_{\infty}}=2(\xi^{r}\vartheta^{u}-\xi^{u}\vartheta^{r})=-2\xi^{u}\vartheta^{r}=-2fN\delta
C$ on the celestial circle, so the WZ potential cancels the flux piece (5.12b)
and leaves out the WZ charge (5.12a). This is because its time derivative

$\displaystyle\dot{Q}_{\xi}\stackrel{{\scriptstyle\circ}}{{=}}-2N\delta_{\xi}C\stackrel{{\scriptstyle\circ}}{{=}}-2fN^{2}+g(CN^{\prime}-C^{\prime}N)\eqqcolon
F_{\xi}$ (5.13)

vanishes in the absence of news. Note that defining a bracket of
fluxes111111The same word ‘flux’ refers to two different objects: non-
integrable pieces in charge variations as in (5.1), and time derivatives of
integrable charges as in (5.13). In practice, there should be no confusion
since the meaning of the word is always clear from context. as
$\big{\\{}F_{\xi_{1}},F_{\xi_{2}}\big{\\}}\coloneqq\delta_{\xi_{2}}F_{\xi_{1}}$
yields

$\displaystyle\delta_{\xi_{2}}F_{\xi_{1}}\stackrel{{\scriptstyle\circ}}{{=}}F_{[\xi_{1},\xi_{2}]_{*}}-\partial_{u}\big{(}2f_{1}f_{2}N^{2}+2g_{1}f_{2}NC^{\prime}+g_{1}^{\prime}f_{2}NC\big{)},$
(5.14)

which implies that the integrated fluxes on $\mathscr{I}^{+}$ represent the
symmetry algebra, provided both field configurations at $\mathscr{I}^{+}_{+}$
and $\mathscr{I}^{+}_{-}$ (the endpoints of future null infinity) are vacua
with $N=0$. Such flux brackets are useful for celestial holography and soft
theorems [37, 114], but they will also provide a useful extra criterion to be
satisfied (or violated) by choices of splits (5.1). We will therefore return
to flux brackets shortly.

With the condition $G=0$ and the WZ split (5.12), the BT bracket represents
the asymptotic symmetry algebra in the sense of (5.4), now with the
aforementioned field-dependent bracket (5.11) and a new field-dependent
cocycle

$K_{\xi_{1},\xi_{2}}=-\underbrace{(T_{1}g_{2}^{\prime\prime\prime}-T_{2}g_{1}^{\prime\prime\prime})}_{\text{standard
$\mathfrak{bms}_{3}$}}+\underbrace{(T_{1}g_{2}^{\prime}-T_{2}g_{1}^{\prime})E^{2}}_{\text{field-
dependent piece}}.$ (5.15)

This coincides with the earlier result (5.6) upon imposing
$\dot{\alpha}=g^{\prime}E$. The same cocycle will occur with the Koszul
bracket in section 5.4, so we highlight it here to stress that it is a genuine
split-independent feature of 3D Einstein–Maxwell theory.

Again, the actual form of the cocycle is not robust under changes of the split
(5.1). For WZ charges, any shift (5.7) by a quantity which is conserved in the
absence of news is allowed, since it leaves conservation unaffected. Consider
for example the shift (5.9c) by $TM$, which is time-independent when $N=0$.
Then the WZ split (5.12) changes into

$\displaystyle Q_{\xi}$
$\displaystyle=(f+T)M+g\left(L+\tfrac{1}{4}(C^{2})^{\prime}-E\big{(}A_{\phi}^{\ell}+2A_{\phi}^{0}\big{)}\right)-\big{(}ug^{\prime}E+2\alpha_{0}\big{)}E$
$\displaystyle\,\,\,\,\,\,\,\text{(integrable)},$ (5.16a)
$\displaystyle\Xi_{\xi}[\delta]$ $\displaystyle=2fN\delta C-T\delta M$
$\displaystyle\text{(flux)}.$ (5.16b)

With this new split, the time derivative of integrable charges is

$\displaystyle\dot{Q}_{\xi}\stackrel{{\scriptstyle\circ}}{{=}}-2(f+T)N^{2}+g(CN^{\prime}-C^{\prime}N)\eqqcolon
F_{\xi},$ (5.17)

and vanishes as expected if $N=0$. The corresponding cocycle in (5.4) is now

$\displaystyle
K_{\xi_{1},\xi_{2}}=-2\underbrace{(T_{1}g_{2}^{\prime\prime\prime}-T_{2}g_{1}^{\prime\prime\prime})}_{\text{standard
$\mathfrak{bms}_{3}$}}-\underbrace{2uN^{2}(T_{1}g_{2}^{\prime}-T_{2}g_{1}^{\prime})}_{\text{field-
dependent piece}},$ (5.18)

which is remarkable: the cocycle is still field-dependent, but it becomes
field-independent in the absence of news. This is nearly all one could hope
for, but there is a catch: in contrast to the earlier fluxes (5.13), the new
fluxes (5.17) fail to represent the symmetry algebra. We conclude, at least in
the present case, that the following three requirements cannot be met
simultaneously:

  1. (i)

having integrable charges that are conserved in the absence of news;

  2. (ii)

having cocycles that are field-independent in the absence of news;

  3. (iii)

having fluxes that represent the symmetry algebra in the absence of news, as
in (5.14).

#### Changing slicings.

In the derivation of asymptotic symmetries in section 4, the parameters
$(f,g,\alpha)$ are functions of $(u,\phi)$ and appear as radial integration
constants. They can therefore be redefined in an arbitrary manner, possibly
even a field-dependent one. This freedom is known as a ‘change of slicing’,
and was studied at length in [115, 116, 117, 96, 95]. It can be used e.g. to
reabsorb ‘spurious fluxes’ or ‘fake news’ appearing because of overly loose
boundary conditions. For example, in [96] the authors studied vacuum 3D
gravity with a free boundary metric on $\mathscr{I}^{+}$, where Iyer–Wald
charges generally involve many non-integrable contributions despite the
absence of local degrees of freedom; field-dependent redefinitions of the
symmetry generators (i.e. a change of slicing) can then be used to cancel the
spurious non-integrability.

How does this freedom affect the present discussion? In the charge variation
(4.10), the two non-integrable contributions $fN\delta C$ and $fG\delta E$
stand on different footings: the former cannot be reabsorbed by a change of
slicing, but the latter can, namely by redefining the U$(1$) parameter as

$\displaystyle\alpha=\tilde{\alpha}-fG.$ (5.19)

One may view this as an indication that the news represents physical flux,
while $G$ is a spurious flux that can be removed. Assuming that the new
parameter $\tilde{\alpha}$ is field-independent ($\delta\tilde{\alpha}=0$),
the charge (4.10) splits into

$\displaystyle Q_{\xi}$
$\displaystyle=fM+g\left(L+\frac{1}{4}(C^{2})^{\prime}-E\big{(}A_{\phi}^{\ell}+2A_{\phi}^{0}\big{)}\right)-2\tilde{\alpha}E$
$\displaystyle\text{(integrable charge)},$ (5.20a)
$\displaystyle\Xi_{\xi}[\delta]$ $\displaystyle=2fN\delta C$
$\displaystyle\text{(flux)}.$ (5.20b)

Note that $\tilde{\alpha}$ here is generally time-dependent. Despite this, it
satisfies the field-dependent commutation relations (5.11) of the time-
independent case (with $\alpha_{0}$ replaced by $\tilde{\alpha}$). As for the
BT bracket (5.3), it satisfies again the algebra (5.4) with the same field-
dependent cocycle (5.15) as in the case $G=0$.

The prescription (5.20) thus leads to integrable charges in the absence of
news, with the same algebra as WZ charges in the time-independent phase space
with $G=0$. However, this does not solve the issue of the arbitrary time
dependence in $\tilde{\alpha}$. Indeed, the time evolution of the integrable
part (5.20a) reads

$\displaystyle\dot{Q}_{\xi}\stackrel{{\scriptstyle\circ}}{{=}}F_{\xi}\big{|}_{\eqref{WZ
evolution
flux}}+2E\Big{(}g\big{(}E+G\big{)}^{\prime}+g^{\prime}E-\dot{\tilde{\alpha}}\Big{)},$
(5.21)

whose right-hand side is non-zero even without news. One may imagine
cancelling the problematic second term by imposing the time evolution
constraint $\dot{\tilde{\alpha}}=g\big{(}E+G\big{)}^{\prime}+g^{\prime}E$, but
this reintroduces a field-dependence in $\tilde{\alpha}$, hence new sources of
non-integrability and spurious fluxes… In this sense, changes of slicing do
not cure the ambiguities present in the initial bare charge variation (4.10).
In particular, while they allow for the removal of the spurious flux, they do
not provide an unambiguous prescription for the time evolution of charges.

###  5.3 Noether split and bracket

For completeness, and in order to illustrate the proposal of [97] on a
concrete example, let us study the so-called Noether split and the associated
bracket. For this, we consider again the general case where $G\neq 0$ and the
time dependence of the gauge parameter $\alpha$ in (4.3) is completely
arbitrary. The Noether split consists in fixing the ambiguity in (5.1) by
choosing the integrable part to be the Noether charge. The latter actually
coincides with the Komar–Maxwell charge (3.5) for Einstein–Maxwell theory.
Thus, the split is chosen using the $r$-independent parts of the Noether
charge and flux,

$\displaystyle K^{ur}_{\xi}$
$\displaystyle\stackrel{{\scriptstyle\circ}}{{=}}-2fE^{2}(\ln
r)+Q_{\xi}+\mathcal{O}(r^{-1/2}),$ (5.22a)
$\displaystyle-K^{ur}_{\delta\xi}+\xi^{u}\theta^{r}-\xi^{r}\theta^{u}$
$\displaystyle\stackrel{{\scriptstyle\circ}}{{=}}+2fE^{2}(\ln
r)+\Xi_{\xi}[\delta]+\mathcal{O}(r^{-1/2}),$ (5.22b)

which explicitly yields

$\displaystyle Q_{\xi}$
$\displaystyle=f\left(\frac{1}{2}CN-E\big{(}E+2G\big{)}\right)+g\left(L+\frac{5}{4}(C^{2})^{\prime}-E\big{(}A_{\phi}^{\ell}+2A_{\phi}^{0}\big{)}\right)-2\alpha
E$ $\displaystyle\,\,\,\,\text{(integrable)},\\!\\!$ (5.23a)
$\displaystyle\\!\\!\\!\\!\\!\\!\Xi_{\xi}[\delta]$
$\displaystyle=f\Big{(}\delta M+\frac{3}{2}N\delta C-\frac{1}{2}C\delta
N+2E\delta\big{(}E+G\big{)}\Big{)}-g\delta(C^{2})^{\prime}$
$\displaystyle\text{(flux)}.$ (5.23b)

With this split, the BT bracket (5.3) satisfies the charge algebra (5.4), with
a homogeneous part given by (4.9) and a field-dependent cocycle

$\displaystyle
K_{\xi_{1},\xi_{2}}=2(f_{1}g_{2}^{\prime}-f_{2}g_{1}^{\prime})\big{(}E^{2}+CN\big{)}.$
(5.24)

This is remarkably simple, given the complicated shift that maps the initial
split (5.2) on the Noether split (5.23). Note in particular that all Virasoro-
type central extensions [involving third derivatives as in (5.6), (5.15) or
(5.18)] have now disappeared.

The simplification of cocycles is in fact a built-in property of brackets of
charges with the split (5.23), and it can be pushed further. Indeed, one can
check that the cocycle (5.24) involves the on-shell Lagrangian density, in the
sense that
$K_{\xi_{1},\xi_{2}}=\lim_{r\to\infty}(\xi^{r}_{1}\xi^{u}_{2}-\xi^{u}_{1}\xi^{r}_{2})\mathcal{L}$.
This is in fact a special case of a general result shown in [97]. Namely, when
using the Noether charge $K_{\xi}$ and the ‘Noetherian flux’ defined as
$\Xi_{\xi}[\delta]=-K_{\delta\xi}-\xi\lrcorner\,\theta$, the algebra (5.4)
given by the BT bracket (5.3) produces a cocycle121212A clarification: what we
call $K_{\xi}$ is the Noether charge (Komar) aspect (3.5), while
$K_{\xi_{1},\xi_{2}}$ is a cocycle in the charge algebra (5.4). The two
objects are completely different, despite their similar notations.

$\displaystyle
K_{\xi_{1},\xi_{2}}=\big{(}\xi_{1}\lrcorner\,\xi_{2}\lrcorner\,\mathcal{L}\big{)}\big{|}_{S^{1}_{\infty}}+a_{\xi_{1},\xi_{2}}=\lim_{r\to\infty}\big{(}\xi^{r}_{1}\xi^{u}_{2}-\xi^{u}_{1}\xi^{r}_{2}\big{)}\mathcal{L}+a_{\xi_{1},\xi_{2}},$
(5.25)

where $\mathcal{L}$ is the Lagrangian density and $a_{\xi_{1},\xi_{2}}$
accounts for possible anomalies [30], which happen to be absent in the present
setup. This leads to the so-called Noether bracket, defined in terms of the BT
bracket (5.3) as

$\displaystyle\big{\\{}Q_{\xi_{1}},Q_{\xi_{2}}\big{\\}}_{\text{N}}\coloneqq\big{\\{}Q_{\xi_{1}},Q_{\xi_{2}}\big{\\}}_{\text{BT}}-\big{(}\xi_{1}\lrcorner\,\xi_{2}\lrcorner\,\mathcal{L}\big{)}\big{|}_{S^{1}_{\infty}}-a_{\xi_{1},\xi_{2}}.$
(5.26)

Using the charge algebra (5.4) with the cocycle (5.25), it is then immediate
to show that the Noether bracket satisfies
$\big{\\{}Q_{\xi_{1}},Q_{\xi_{2}}\big{\\}}_{\text{N}}=Q_{[\xi_{1},\xi_{2}]_{*}}$.
This represents the symmetry algebra with a vanishing cocycle! Of course, the
simplification was bound to work, since (5.26) is just a BT bracket whose
cocycle has been included in the very definition of the bracket. What
crucially makes this possible is the explicit expression (5.25) of the
cocycle, valid only for the Noether split.

Although the Noether split has the advantage of being unambiguously defined
since it relies on the Noether charge, its drawback is to single out an
integrable charge (5.23a) that is not conserved in the absence of news. An
immediate way to see this is to note that no condition was imposed on the
gauge parameter $\alpha(u,\phi)$. Alternatively, fixing $G=0$ and the time-
dependence of $\alpha$ through (5.10) gives integrable Noether charges that
slightly differ from (5.23a), but still satisfy $\dot{Q}_{\xi}\neq 0$ in the
absence of news. The details are omitted here. Accordingly, we will not
consider the Noether split any further, and now turn instead to one last
choice of bracket.

###  5.4 Koszul bracket

Let us consider a proposal put forward and investigated by Barnich, Fiorucci,
and Ruzziconi [99], to which we shall refer as the Koszul bracket. Given a
split (5.1) between integrable part and flux, this bracket is defined as

$\displaystyle\big{\\{}Q_{\xi_{1}},Q_{\xi_{2}}\big{\\}}_{\text{K}}\coloneqq\big{\\{}Q_{\xi_{1}},Q_{\xi_{2}}\big{\\}}_{\text{BT}}-\int_{\gamma}\Big{(}\delta_{\xi_{1}}\Xi_{\xi_{2}}[\delta]-\delta_{\xi_{2}}\Xi_{\xi_{1}}[\delta]+\Xi_{[\xi_{1},\xi_{2}]_{*}}[\delta]\Big{)}.$
(5.27)

Here the first term is the BT bracket (5.3), while the second term involves
the integral, along a path $\gamma$, of a one-form in field space defined from
the flux $\Xi_{\xi}[\delta]$. The path is such that its endpoint is the point
in field space where the bracket (5.27) is meant to be evaluated. The starting
point is arbitrary, and the specific choice of path is irrelevant thanks to
the $\delta$-exactness of the integrand in (5.27). We will assume henceforth
that some starting point has been chosen once and for all, independently of
$\xi_{1},\xi_{2}$.

We will neither derive nor investigate the properties of the Koszul bracket;
suffice it to say that it is crucially independent of the choice of split
(5.1). This is because the quantity $S_{\xi_{1},\xi_{2}}$, produced by the BT
bracket under split changes (5.7), is exactly compensated by changes of the
integral along $\gamma$ in (5.27) when shifting
$\Xi_{\xi}[\delta]\to\Xi_{\xi}[\delta]-\delta S_{\xi}$. However, split-
independence of (5.27) does not mean that the Koszul bracket only captures
information about the genuine physical flux. Indeed, we will see that it is
still sensitive to reductions of the solution space (e.g. by setting $G=0$)
and to changes of slicing.

We have already computed the BT bracket for the charges (5.2) and (5.12),
respectively valid in the solution spaces with $G\neq 0$ and $G=0$. The former
leads to the cocycle (5.6), while the latter leads to (5.15), both of which
are field-dependent. One can therefore ask how the second term in (5.27)
affects the cocycle in these two cases. The answer turns out to be

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\Big{(}\delta_{\xi_{1}}\Xi_{\xi_{2}}[\delta]-\delta_{\xi_{2}}\Xi_{\xi_{1}}[\delta]+\Xi_{[\xi_{1},\xi_{2}]_{*}}[\delta]\Big{)}\Big{|}_{\eqref{BT
flux}}\hskip 4.23952pt$
$\displaystyle\\!=-\delta\Big{(}(f_{1}g_{2}^{\prime}-f_{2}g_{1}^{\prime})E^{2}-2(f_{1}\dot{\alpha}_{2}-f_{2}\dot{\alpha}_{1})E\Big{)},$
(5.28a)
$\displaystyle\\!\\!\\!\\!\\!\\!\Big{(}\delta_{\xi_{1}}\Xi_{\xi_{2}}[\delta]-\delta_{\xi_{2}}\Xi_{\xi_{1}}[\delta]+\Xi_{[\xi_{1},\xi_{2}]_{*}}[\delta]\Big{)}\Big{|}_{\eqref{WZ
flux}}$ $\displaystyle\\!=0,$ (5.28b)

showing that the change of cocycle due to (5.27) is entirely produced by the
contribution of the spurious flux $-2fG\delta E$: it is non-zero in (5.28a)
where $G\neq 0$, but vanishes in (5.28b) where $G=0$. In particular, the
correction (5.28a) exactly compensates the field-dependent part of the earlier
cocycle (5.6), which thus reduces to the standard $\mathfrak{bms}_{3}$ central
extension [60]. The Koszul bracket (5.27) entirely cancels the field-
dependence of the cocycle when $G\neq 0$.131313This holds up to a trivial
central extension due to the field space integral in (5.27) [99]. By contrast,
the cocycle change (5.28b) vanishes for WZ charges in the solution space where
$G=0$, so the earlier field-dependent cocycle (5.15) is unaffected.141414Note
that setting $G=0$ after having computed (5.28a) does not return the result
(5.28b), since one cannot change the phase space after having computed the
bracket. A third, mixed alternative is to consider the solution space $G\neq
0$ with the change of slicing (5.19) and flux (5.20b), whereupon (5.28b) holds
again and yields again the earlier cocycle (5.15). In short, the cocycle in
the Koszul bracket (5.27) only cares about whether $G$ is present or not in
the flux $\Xi_{\xi}[\delta]$, and not about whether it was removed by a change
of slicing or by the restriction $G=0$.

This result exhibits the sensitivity of the Koszul bracket to the choice of
slicing and to the ‘size’ of the solution space, despite its independence from
the split (5.1). In particular, when $G\neq 0$, the Koszul bracket with fluxes
(5.2b) yields a field-independent cocycle, while the change of slicing (5.19)
reintroduces the field-dependent cocycle (5.15). The latter also appears for
WZ charges with $G=0$, whose algebra is unchanged by the Koszul bracket. This
seems to indicate that the field-dependent cocycle (5.15) is a genuine feature
of 3D Einstein–Maxwell theory, as was indeed suggested in [1]. It also begs,
even more pressingly, the question of the interpretation of the field
$G(u,\phi)$, which is ultimately responsible for most ambiguities of
Einstein–Maxwell charge algebras.

##  6 Turning on $\boldsymbol{\Lambda\neq 0}$

In order to deepen the analogy between our analysis and 4D pure gravity [53,
54], let us extend the setup to include a non-zero cosmological constant
$\Lambda\neq 0$. The Bondi gauge is then well-adapted (at least in the vacuum
case) for the flat limit $\Lambda\to 0$ [61, 96]. We therefore begin by
solving the Einstein equations with cosmological constant $E_{\mu\nu}\coloneqq
G_{\mu\nu}+\Lambda g_{\mu\nu}-T_{\mu\nu}=0$, again with the fall-offs (2.9),
then discuss asymptotic symmetries and their charges. Differences and
similarities with the flat-space case $\Lambda=0$ are pointed out throughout.
We refer again to appendix A for some computational details.

###  6.1 Solution space and analogy with the 4D vacuum case

Here we solve the Einstein–Maxwell equations of motion with $\Lambda\neq 0$,
then highlight the similarity between the resulting solution space and that of
4D Einstein gravity.

#### Solution space.

Thanks to the Bondi gauge conditions $g_{rr}=0=g_{r\phi}$, the hypersurface
equations $E_{rr}=0$, $E_{r\phi}=0$ and $\nabla^{\mu}F_{\mu r}=0$ do not
depend on $\Lambda$. As a result, they lead to the same solutions (A.3), (A.6)
and (A.8) for $\beta$, $U$ and $A_{u}^{m>0,n}$ as in the flat case (recall
that we have set $\beta_{0}=0=U_{0}$). The first striking feature due to
$\Lambda\neq 0$ comes from the angular Maxwell equation, which now reads

$\displaystyle\nabla^{\mu}F_{\mu\phi}=-\frac{1}{4}\sqrt{r}\,\Lambda
C-\frac{\Lambda}{4\sqrt{r}}\left((\ln
r)A_{\phi}^{1/2,1}+\frac{3}{2}C^{3}+A_{\phi}^{1/2,0}-4A_{\phi}^{1/2,1}\right)+\mathcal{O}(r^{-1})=0.$
(6.1)

Note the leading term of this equation: it sets $C=0$ on shell. The first
subleading term then fixes $A_{\phi}^{1/2,1}=0$, hence $A_{\phi}^{1/2,0}=0$.
Continuing with the expansion at subleading orders, one also finds
$A_{\phi}^{3/2,0}=A_{\phi}^{3/2,1}=A_{\phi}^{3/2,2}=0$, etc., which in turn
implies [from (A.8)] that $A_{u}^{1/2,0}=A_{u}^{3/2,0}=A_{u}^{3/2,1}=0$, etc.
At the end of the day, the solution space with $\Lambda\neq 0$ contains no
half-integer powers of $r$, and the fall-offs (2.9) simply require
$m\in\mathbb{N}$ and $C=0$.

The second interesting feature of $\Lambda\neq 0$ appears in the dynamical
angular Maxwell equation. Indeed, expanding (6.1) further yields the
equations151515Note that we keep writing $C$ even though $C=0$ when
$\Lambda\neq 0$. This ensures we can still take the limit $\Lambda=0$ and
recover expressions that are valid in the flat case with $C\neq 0$. One should
merely keep in mind that $\Lambda C=0$.

$\displaystyle\dot{A}_{\phi}^{\ell}$ $\displaystyle=E^{\prime}+\Lambda
A_{\phi}^{1,1},$ (6.2a) $\displaystyle\dot{A}_{\phi}^{0}$
$\displaystyle=E^{\prime}+G^{\prime}+\Lambda A_{\phi}^{1,0},$ (6.2b)
$\displaystyle 3\dot{A}_{\phi}^{1,1}$
$\displaystyle=4\Lambda\Big{(}\big{(}A_{\phi}^{\ell}\big{)}^{3}+A_{\phi}^{2,1}\Big{)},$
(6.2c) $\displaystyle 3\dot{A}_{\phi}^{1,0}$
$\displaystyle=\frac{2\Lambda}{3}\Big{(}6A_{\phi}^{2,0}-2A_{\phi}^{2,1}+\big{(}A_{\phi}^{\ell}\big{)}^{3}\Big{)}-4MA_{\phi}^{\ell}+C^{2}E^{\prime}+2LE-7E(C^{2})^{\prime}+2\big{(}A_{\phi}^{\ell}\big{)}^{\prime\prime}.$
(6.2d)

When $\Lambda=0$, these are evolution equations for the components of
$A_{\phi}$ (apart from $C$, which is unconstrained), whose initial data is
free and must be specified at some time $u_{0}$. There is an infinite tower of
such equations, one for each $A_{\phi}^{m,n}$ in (2.9). But when $\Lambda\neq
0$, the meaning of these equations changes radically due to the terms
involving $\Lambda$ on the right-hand side of (6.2). These imply that,
starting from $A_{\phi}^{1,0}$ and $A_{\phi}^{1,1}$, all the coefficients
$A_{\phi}^{m,n}$ are algebraically determined by $A_{\phi}^{0}$ and
$A_{\phi}^{\ell}$, which now become completely free data to be specified for
all $u$. The temporal Maxwell equation then modifies the flat-space equation
(A.17) to $\dot{E}=-\Lambda\big{(}A_{\phi}^{\ell}\big{)}^{\prime}$, while $G$
remains completely free data as in the case $\Lambda=0$. Finally, the
evolution equations for mass and angular momentum take the form

$\displaystyle\dot{M}$
$\displaystyle=-2N^{2}-\Lambda^{2}A_{\phi}^{\ell}\big{(}2A_{\phi}^{1,0}+A_{\phi}^{1,1}\big{)}-\Lambda\Big{(}L^{\prime}+2A_{\phi}^{\ell}E^{\prime}-E\big{(}A_{\phi}^{\ell}\big{)}^{\prime}\Big{)},$
(6.3a) $\displaystyle\dot{L}$
$\displaystyle=M^{\prime}+EE^{\prime}+\frac{1}{2}\big{(}CN^{\prime}-3NC^{\prime}\big{)}+\Lambda
E\big{(}2A_{\phi}^{1,0}+A_{\phi}^{1,1}\big{)}-2\Lambda
A_{\phi}^{\ell}\big{(}A_{\phi}^{\ell}\big{)}^{\prime},$ (6.3b)

where again we have kept $C\neq 0$ for convenience in order to recover (3.10)
in the flat limit. Note here that one can replace $A_{\phi}^{1,0}$ and
$A_{\phi}^{1,1}$ by their value given by (6.2) in order to obtain the
evolution of mass and angular momentum in terms of free data
$(A_{\phi}^{\ell},A_{\phi}^{0},G)$.

Let us summarize. Starting with the fall-offs (2.9) where $C=0$ and
$m\in\mathbb{N}$, the solution space for $\Lambda\neq 0$ is completely
determined by (i) three arbitrary functions of $(u,\phi)$ in
$(A_{\phi}^{\ell},A_{\phi}^{0},G)$, along with (ii) the initial value at
$u_{0}$ for the data $(M,L,E)$, subject to evolution equations (6.3) in $u$.
There are thus two key differences with respect to $\Lambda=0$. First, the
free data on $\mathscr{I}^{+}$ is different: instead of $(C,G)|_{\Lambda=0}$,
it is now given by $(A_{\phi}^{\ell},A_{\phi}^{0},G)|_{\Lambda\neq 0}$.
Second, while $\Lambda=0$ involves infinitely many evolution equations for the
data in $A_{\phi}$, the case $\Lambda\neq 0$ only involves the three evolution
equations for $(M,L,E)$.

#### Analogy with 4D gravity.

The behaviour of 3D Einstein–Maxwell solution spaces at $\Lambda=0$ and
$\Lambda\neq 0$ is deeply reminiscent of the 4D vacuum case. A precise analogy
can indeed be established thanks to the following three observations:

  * •

First, the condition $\Lambda C=0$ enforced by (6.1) is analogous to the
condition obtained in 4D from the leading-order angular Einstein equations
(see [53] or [54, eq. (2.17a)]). There, the constraint on the shear takes the
form

$\displaystyle\Lambda
e^{2\beta_{0}}C_{ab}\propto(\partial_{u}-\partial_{u}\ln\sqrt{q})q_{ab}+D^{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}0}}_{\langle
a}U^{0}_{b\rangle},$ (6.4)

where the leading transverse metric $q_{ab}$ enters through its time
dependence, while $\beta_{0}$ and $U^{a}_{0}$ are 4D versions of the
integration constants that were discarded here.161616The notation
$D^{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}0}}_{\langle
a}U^{0}_{b\rangle}$ refers to the symmetric, trace-free part of the tensor
$D_{a}U^{0}_{b}$. The shear $C_{ab}$ thus depends on the induced metric on
$\mathscr{I}^{+}$ when $\Lambda\neq 0$, and it must actually vanish if
$U_{0}=0$ and if the time dependence of $q_{ab}$ is frozen, since then
$\Lambda C_{ab}=0$. The analogy with 3D Einstein–Maxwell is manifest and
reinforces the interpretation of $C$ as the 3D electromagnetic analogue of the
gravitational shear $C_{ab}$. Furthermore, it would be interesting to study
the generalization of the condition $\Lambda C=0$ with more relaxed 3D
boundary conditions [95, 96], where one precisely has $U_{0}\neq 0$ and a
time-dependent metric on the celestial circle.

  * •

Next, note the analogy between square-root terms in the 3D Einstein–Maxwell
system and logarithmic terms in 4D gravity. We saw here that $\Lambda\neq 0$
forces such square root terms to vanish. Similarly, subleading components of
the angular Einstein equations in 4D imply that all logarithmic terms vanish
[54], including both logarithmic terms in the angular metric, which can be
introduced by hand through fall-offs [118, 119, 120, 54], and logarithmic
terms that appear when solving for $U^{a}$ [121, 106].171717This absence of
logarithmic terms when $\Lambda\neq 0$ in 4D is a manifestation of the
Starobinsky–Fefferman–Graham theorem [122, 123].

  * •

The last part of the analogy involves the radial expansion of $A_{\phi}$ in 3D
and that of the angular metric in 4D. Indeed, the latter reads

$\displaystyle
g_{ab}=r^{2}q_{ab}+rC_{ab}+\frac{1}{4}q_{ab}C_{cd}C^{cd}+\sum_{n=1}^{\infty}\frac{g_{ab}^{(n)}}{r^{n}}$
(6.5)

where we chose a regular, logarithm-free expansion. Then the leading angular
Einstein equation constrains $C_{ab}$ as in (6.4), while the subleading terms
in the angular equations are exactly analogous to the angular Maxwell
equations (6.2) since they take the form [53]

$\displaystyle\dot{g}^{(n)}_{ab}=\Lambda g^{(n+1)}_{ab}+(\dots).$ (6.6)

When $\Lambda=0$, these are evolution equations for the coefficients
$g^{(n)}_{ab}$, whose value at some $u_{0}$ is required; there is an infinite
tower of such equations. By contrast, when $\Lambda\neq 0$, the constraints
(6.6) determine algebraically, and recursively, all the components
$g^{(n>1)}_{ab}$ in terms of $g^{(1)}_{ab}$ and its time derivative. Then
$g^{(1)}_{ab}$, rather than $C_{ab}$, is the free data that needs to be
specified for all $u$, exactly like $A_{\phi}^{\ell}$ and $A_{\phi}^{0}$
become free data instead of $C$ in the 3D Einstein–Maxwell case.

To go beyond this simple analogy, one may attempt to use it to gain insights
in 4D vacuum gravity. We saw in section 5 how the present 3D model allows one
to study charges in the presence of radiation, as a first step towards
holography with radiating sources. Another direction is to seek a
$w_{1+\infty}$ symmetry of 3D Einstein–Maxwell theory. In 4D gravity, recent
work has indeed shown the existence of such a structure [124, 125, 126]; it
can be related to the tower of metric fields (6.5) and their evolution
equations [127, 128]. Since 3D Einstein–Maxwell theory also admits such a
hierarchy, with the components of $A_{\phi}$ playing the role of the
subleading tensors in (6.5), it is natural to ask whether one can also exhibit
an underlying higher-spin structure in that setup. We leave this investigation
for future work.

###  6.2 Asymptotic symmetries and charges

We close this section with a brief discussion of asymptotic symmetries and
charges in the case $\Lambda\neq 0$. Previous work along these lines can be
found in [100, 101], albeit in Fefferman–Graham coordinates and using
Hamiltonian methods. We will work instead in Bondi coordinates and implement
once and for all the condition of vanishing square root terms in the fall-
offs, in particular setting $C=0$. After going through the construction of the
solution space, the metric is given by

$\displaystyle g_{uu}$ $\displaystyle=\Lambda r^{2}+2(\ln
r)\Big{(}E^{2}-\Lambda\big{(}A_{\phi}^{\ell}\big{)}^{2}\Big{)}+2M-\Lambda\big{(}A_{\phi}^{\ell}\big{)}^{2}+\mathcal{O}(r^{-1}),$
(6.7a) $\displaystyle g_{ur}$
$\displaystyle=-1+\frac{\big{(}A_{\phi}^{\ell}\big{)}^{2}}{r^{2}}+\mathcal{O}(r^{-3}),$
(6.7b) $\displaystyle g_{u\phi}$ $\displaystyle=2(\ln
r)EA_{\phi}^{\ell}+L+\mathcal{O}(r^{-1}),$ (6.7c) $\displaystyle g_{\phi\phi}$
$\displaystyle=r^{2},$ (6.7d)

where the mass and angular momentum satisfy the evolution equations (6.3) with
$C=0$. It is manifest that these components reduce to those in (3.7) when
$\Lambda=0$.

The components of asymptotic Killing vectors are once again given by (4.1).
Now, however, the functions $f$ and $g$ satisfy $\dot{f}=g^{\prime}$ and
$\dot{g}=-\Lambda f^{\prime}$ instead of the flat-space relations (4.2). Their
sum and difference $\sqrt{-\Lambda}f\pm g$ generate left- and right-moving
conformal transformations. The ensuing transformations of metric and
electromagnetic field components are as follows. First, mass and angular
momentum transform as

$\displaystyle\delta_{\xi}M$
$\displaystyle=\Big{(}f\partial_{u}+g\partial_{\phi}+2g^{\prime}\Big{)}M-g^{\prime\prime\prime}-2\Lambda
f^{\prime}L-g^{\prime}E^{2}+\Lambda
A_{\phi}^{\ell}\big{(}f^{\prime}E+g^{\prime}A_{\phi}^{\ell}\big{)},$ (6.8a)
$\displaystyle\delta_{\xi}L$
$\displaystyle=\Big{(}f\partial_{u}+g\partial_{\phi}+2g^{\prime}\Big{)}L-f^{\prime\prime\prime}+2f^{\prime}M-2g^{\prime}EA_{\phi}^{\ell}-\Lambda
f^{\prime}\big{(}A_{\phi}^{\ell}\big{)}^{2},$ (6.8b)

where one may recognize the coadjoint representation of the Virasoro algebra,
written in terms of $M\propto T+\bar{T}$ and $L\propto T-\bar{T}$ instead of
left and right stress tensor components $(T,\bar{T})$. This is accompanied by
new contributions due to the Maxwell field. The latter has leading logarithmic
components that transform as

$\displaystyle\delta_{\xi,\varepsilon}A_{\phi}^{\ell}$
$\displaystyle=\Big{(}f\partial_{u}+g\partial_{\phi}+g^{\prime}\Big{)}A_{\phi}^{\ell}+f^{\prime}E,$
(6.9a) $\displaystyle\delta_{\xi,\varepsilon}E$
$\displaystyle=\Big{(}f\partial_{u}+g\partial_{\phi}+g^{\prime}\Big{)}E-\Lambda
f^{\prime}A_{\phi}^{\ell},$ (6.9b)

while its $\mathcal{O}(1)$ components transform in a way that involves the
U$(1)$ gauge parameter:

$\displaystyle\delta_{\xi,\varepsilon}A_{\phi}^{0}$
$\displaystyle=\Big{(}f\partial_{u}+g\partial_{\phi}+g^{\prime}\Big{)}A_{\phi}^{0}-g^{\prime}A_{\phi}^{\ell}+f^{\prime}G+\alpha^{\prime},$
(6.10a) $\displaystyle\delta_{\xi,\varepsilon}G$
$\displaystyle=\Big{(}f\partial_{u}+g\partial_{\phi}+g^{\prime}\Big{)}G-g^{\prime}E-\Lambda
f^{\prime}A_{\phi}^{0}+\dot{\alpha}.$ (6.10b)

All these equations reduce to the flat-space transformations (4.4)–(4.7) when
$\Lambda=0$.

One can also study the algebra of asymptotic symmetry transformations. Under
the modified bracket (4.8), the algebra of the asymptotic Killing vectors is
$\xi_{12}=\big{[}\xi_{1},\xi_{2}\big{]}_{*}$, where one now has

$\displaystyle f_{12}$
$\displaystyle=f_{1}g^{\prime}_{2}+g_{1}f^{\prime}_{2}-\delta_{\xi_{1}}f_{2}-(1\leftrightarrow
2),$ (6.11a) $\displaystyle g_{12}$ $\displaystyle=g_{1}g^{\prime}_{2}-\Lambda
f_{1}f^{\prime}_{2}-\delta_{\xi_{1}}g_{2}-(1\leftrightarrow 2),$ (6.11b)
$\displaystyle\alpha_{12}$
$\displaystyle=f_{1}\dot{\alpha}_{2}+g_{1}\alpha_{2}^{\prime}-\delta_{\xi_{1},\varepsilon_{1}}\alpha_{2}-(1\leftrightarrow
2)$ (6.11c)

instead of the flat-space algebra (4.9). This is a semi-direct sum between two
commuting copies of the Witt algebra and an algebra of functions on the
cylinder or the plane. The flat limit ($\Lambda\to 0$) then behaves as an
İnönü–Wigner contraction that maps the algebra (6.11) to the aforementioned
structure
$\mathfrak{bms}_{3}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}}C^{\infty}(\mathscr{I}^{+})$.

The charge aspect corresponding to asymptotic symmetry generators satisfies
once again (5.1), albeit with a subleading correction $\mathcal{O}(r^{-1})$
instead of $\mathcal{O}(r^{-1/2})$. A simple choice of split is e.g.

$\displaystyle Q_{\xi}$
$\displaystyle=f\Bigl{(}M+\frac{\Lambda}{2}\big{(}A_{\phi}^{\ell}\big{)}^{2}\Bigr{)}+g\Bigl{(}L-E\big{(}A_{\phi}^{\ell}+2A_{\phi}^{0}\big{)}\Bigr{)}-2\alpha
E$ $\displaystyle\text{(integrable charge)},$ (6.12a)
$\displaystyle\Xi_{\xi}[\delta]$ $\displaystyle=2f\big{(}\Lambda
A_{\phi}^{\ell}\delta A_{\phi}^{0}-G\delta E\big{)}$
$\displaystyle\text{(flux)},$ (6.12b)

which reduces to the flat-space split (5.2) with $C=0$ when $\Lambda=0$. Using
this split, the BT bracket (5.3) satisfies (5.4) with the homogeneous algebra
(6.11) and the new field-dependent cocycle

$\displaystyle K_{\xi_{1},\xi_{2}}$
$\displaystyle=-(f_{1}g_{2}^{\prime\prime\prime}-f_{2}g_{1}^{\prime\prime\prime})-(f_{1}g_{2}^{\prime}-f_{2}g_{1}^{\prime})E^{2}+2(f_{1}\dot{\alpha}_{2}-f_{2}\dot{\alpha}_{1})E$
(6.13) $\displaystyle\phantom{\
=}+\Lambda\big{(}A_{\phi}^{\ell}\big{)}^{2}(f_{1}g_{2}^{\prime}-f_{2}g_{1}^{\prime})-2\Lambda\big{(}EA_{\phi}^{0}+A_{\phi}^{\ell}G\big{)}(f_{1}f_{2}^{\prime}-f_{2}f_{1}^{\prime}).$
(6.14)

The latter exhibits the standard Brown–Henneaux Virasoro extension [58], along
with numerous new electromagnetic contributions. It reduces again to its flat-
space counterpart (5.6) when $\Lambda=0$.

In short, the entire structure reproduces the results of [1] in the flat limit
$\Lambda=0$. More detailed studies of the charges (6.12) and the WZ
prescription are left for future work. This requires an understanding of
radiation in the case $\Lambda\neq 0$, which we expect to share features and
subtleties of the 4D vacuum case [52, 53, 54, 55, 56, 57].

Let us conclude by returning to the Koszul bracket (5.27), now applied to
surface charges (6.12) with $\Lambda\neq 0$. As in section 5.4, the Koszul
bracket changes the cocycle (6.13) by a term of the form

$\displaystyle\Big{(}\delta_{\xi_{1}}\Xi_{\xi_{2}}[\delta]-\delta_{\xi_{2}}\Xi_{\xi_{1}}[\delta]+\Xi_{[\xi_{1},\xi_{2}]_{*}}[\delta]\Big{)}\Big{|}_{\eqref{AdS
flux}}=-\delta K_{\xi_{1},\xi_{2}},$ (6.15)

where the variation $\delta$ only acts on the field-dependent terms in the
cocycle (6.13). This is highly non-trivial: the Koszul bracket apparently
cancels the entire field-dependence of the earlier extension (6.13), only
leaving out its field-independent Brown–Henneaux piece. The same behaviour was
observed above in the flat-space solution space with $G\neq 0$. In short, the
presence of extra unconstrained boundary degrees of freedom leaves no trace in
the Koszul bracket. The reason for this is unclear to us; we hope to return to
it in future work, along with a more detailed analysis of the (A)dS3
Einstein–Maxwell solution space.

##  7 Perspectives

This work was devoted to a detailed analysis of the asymptotic structure of 3D
Einstein–Maxwell theory with radiative boundary conditions. It contributes in
two specific ways to the already vast literature on asymptotic symmetries.
First, it provides a model where gravitational asymptotic symmetries interplay
in a non-trivial manner with radiative matter fields. Second, and most
important, it does so in the simplest possible setup where all the subtleties
related to radiative asymptotic symmetries are present, thereby providing the
most complete toy model one can hope for. We investigated this model along the
following lines:

In section 2, we studied pure Maxwell theory in order to obtain the fall-offs
(2.9) for Bondi-gauge radiative and Coulombic data, then solve the vacuum
Maxwell equations. This revealed a key difference with previous work on 3D
Einstein–Maxwell theory [1], namely that radiative data behaves as $\sqrt{r}$.
Equipped with these fall-offs and with the line element (3.6) in Bondi gauge,
we then solved the coupled equations of motion for Einstein–Maxwell theory in
section 3. This enabled us to characterize the solution space and identify, in
particular, the mass and angular momentum aspects with evolution equations
(3.10). The latter confirm that 3D Einstein–Maxwell has radiative features
akin to those of 4D pure gravity, albeit in a dimensionally-reduced setup.
This was in fact expected, based on the prescient analysis of [59] where the
first description of BMS3 symmetries was done precisely through a dimensional
reduction from the 4D theory. Here we supplemented this analysis with a
complete control over the matter sector (either seen as arising from
dimensional reduction, or put in by hand in 3D as we did).

Building on this setup, the core of our work in sections 4–5 consisted of an
analysis of asymptotic symmetries and their representations by surface
charges. For this, we first characterized the residual gauge transformations,
spanning an algebra
$\mathfrak{bms}_{3}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}} \end{picture}}}}}C^{\infty}(\mathscr{I}^{+})$
that consists of diffeomorphisms and U$(1)$ transformations. We then computed
the charge variation (4.10) using the Iyer–Wald prescription. This revealed,
as expected, that electromagnetic radiation described by the shear $C(u,\phi)$
gives rise to a non-integrable contribution, or flux. In addition, we found in
agreement with [1] that the field $G(u,\phi)$ also sources non-integrability.
The latter is somewhat spurious, so we treated $G$ as a gauge mode to be set
to zero. Doing so reduced the asymptotic symmetry algebra because the U$(1)$
gauge parameter acquires a fixed time dependence (5.10). It was then possible
to construct WZ charges (5.12) which are conserved in the absence of news
$N=\dot{C}$. As regards the algebra of such charges, we explored three
alternatives for the bracket: the BT bracket, the Noether bracket, and the
recently introduced Koszul bracket. The latter essentially extends the BT
bracket in such a way that it does not depend on the split (5.1) between an
integrable charge and a flux. Using this bracket, we showed that WZ charges
with $G=0$ represent the symmetry algebra with a field-dependent cocycle
(5.17). This confirms the result of [1] while exhibiting both some of its
robustness and some of its delicate dependence on the various apparently
arbitrary choices involved in radiative surface charges.

In summary, our study has shown that 3D Einstein–Maxwell theory captures most
of the subtleties related to radiative asymptotic symmetries. In addition to