aainstitutetext: Facultad de Ciencias, Universidad de la República,
Iguá 4225, esq. Mataojo, 11400 Montevideo, Uruguay.

# Renormalized electric and magnetic charges for $O(r^{n})$ large gauge
symmetries

Javier Peraza<EMAIL_ADDRESS>
###### Abstract

In this work we present the construction of a renormalized symplectic form on
an extended phases space where the higher order large gauge transformations
(LGT) act canonically. The expressions of the subn-leading electric charges
associated with each $O(r^{n})$ LGT are then obtained, in agreement with the
expressions previously proposed in Campiglia:2018dyi  by means of the tree-
level subn-leading formulas. We also present the duality extension of the
extended phase space, computing the full electromagnetic charge algebra,
showing a tower of central extensions.

##  1 Introduction

Over the past years, the understanding of asymptotic symmetries in gravity and
gauge theories has been deepened due to several results that relate them to
soft theorems in field theory. The seminal works of Strominger and
collaborators (e.g., Strominger:2013lka ; He:2017fsb ; He:2014laa ;
Cachazo:2014fwa ; Lysov:2014csa ; He:2014cra ; He:2015zea ; Kapec:2015ena )
showed that the well known Weinberg’s soft theorem Weinberg:1965nx  can be
understood as a Ward identity associated to an infinite dimensional symmetry
group. The group is constructed via large gauge transformations (LGT) at null
infinity. It implies infinite conservation laws in scattering processes from
the past to the future asymptotic regions.

In the case of Quantum Electrodynamics (QED), it was shown in Hamada:2018vrw
and Li:2018gnc  that for tree-level amplitudes, there exist an infinite number
of soft theorems, each of them implying a conservation law for the tree level
scattering process. Weinberg’s soft photon theorem corresponds to the first
level in the hierarchy, while Low’s subleading soft photon theorem Low:1954kd
; Low:1958sn  corresponds to the second level.

The conserved quantities found in (Strominger:2013lka, ) for the S-matrix
constitute thus the first level in an infinite hierarchy of soft theorems.
Seraj made a first approach towards higher orders in Seraj:2016jxi , where an
infinite number of conserved quantities are shown at spatial infinity,
proportional to the multipole moments, and generated by specific large gauge
transformations of order $O(r^{n})$. At null infinity, Campiglia and Laddha
showed in Campiglia:2018dyi  that (for tree-level scattering and restricting
the radiative data space to a suitable subset) exists an infinite tower of
conservation laws, such that at each level there is an infinite dimensional
family of conserved charges, $Q^{n}_{\epsilon}$, labeled by functions on the
sphere. The authors also presented evidence that the Ward identities
associated with the level $n$ of the charges are equivalent to sub-$n$ soft
photon theorems, along with the conservation laws within the classical theory.
The non-abelian case is substantially harder since the charges up to level $n$
of the hierarchy do not form a closed algebra, as in the abelian case. In
(Campiglia:2021oqz, ), a first step towards a classical derivation of the
charge hierarchy in the non-abelian case is suggested. Some recent
developments in celestial holography using Operator Product Expansion (OPE)
tools Pasterski:2021rjz ; Guevara:2021tvr ; Guevara:2021abz  seem to be
promising avenues in the study of asymptotic symmetries and the role of CCFT
in flat holography for Yang-Mills and gravity.

Working in terms of retarded coordinates $(u,r,x^{1},x^{2})$, the massless
fields at the asymptotic region are determined by the limit
$t:=r+u\rightarrow+\infty$ at constant $u$, where $t$ is the usual Minkowski
time. This limit moves the Cauchy slices to a well-defined manifold, called
the future null infinity and denoted by $\mathcal{I}^{+}$. 111This convergence
is point-wise equivalent to the limit $r\rightarrow+\infty$ at $u=cnt$, but
taking $t\rightarrow+\infty$ is more natural since we are defining the charges
in terms of Cauchy slices. The topology of $\mathcal{I}^{+}$ is that of
$\mathbb{R}\times S^{2}$ and its boundaries at $u=\pm\infty$ are denoted by
$\mathcal{I}^{+}_{\pm}$ (they are diffeomorphic to $S^{2}$).

The $r$-expansion of the LGTs at the bulk establishes a hierarchy of charges
at the asymptotic region. $O(1)$ LGTs correspond to leading charges (for
instance, by imposing a constant LGT we obtain the total electric charge of
the system, (Strominger:2017zoo, )), while $O(r)$ LGT corresponds to sub-
leading charges, see (Lysov:2014csa, ; Campiglia:2016hvg, ).

The canonical derivation of conserved quantities at null infinity in the
context of the classical theory at the leading and subleading imposes the
following question: can the infinite tower of charges, associated with subn-
leading soft theorems, be canonically derived within the classical theory? One
of the main problems that arise when studying $O(r)$ LGT is the divergent
formulas for the charges when calculated from the usual phase space structure,
both at null (e.g., Campiglia:2016hvg ) and spatial (e.g., Seraj:2016jxi )
infinities. In particular, the expressions for the symplectic form evaluated
on an LGT at level $n$ (and therefore the charges) diverge in the
$t\rightarrow+\infty$ and $u\rightarrow-\infty$ limits.

In this paper we provide a renormalization procedure that removes both
divergences. Following ideas from Freidel:2019ohg , we show that suitable
boundary and corner terms exist for the symplectic form that renormalize the
divergences while not changing the dynamics of the fields. This
renormalization is minimal, in the sense that it cancels all the divergent
terms while keeping unchanged the finite ones. We define a subset of the
radiative space and an extended phase space that contains all LGTs up to
arbitrary order. This extended space has a symplectic structure, allowing us
to calculate the electric-type charges. Finally, allowing the duality symmetry
to act and extending the phase space with extra boundary gauge fields (e.g.,
Hosseinzadeh:2018dkh ; Freidel:2018fsk ; Geiller:2021gdk ), the magnetic
analog of the electric hierarchy is also presented, as well as the full
electromagnetic charge algebra.

The paper is organized as follows. In section 2 we review the asymptotic
structure of Maxwell theory at null infinity. For simplicity, the charged
matter consist of a massless complex scalar field coupled to the $U(1)$ gauge
field. We also review the structure of the LGT for arbitrary order. In section
3, we revisit the derivation of asymptotic charges associated with leading and
sub-leading soft photon theorems, defining an extended phase space and
calculating the leading and sub-leading charges. Our derivation is along the
lines of Campiglia:2018dyi , but we place special emphasis on the symplectic
structure, which will be used later. Section 4 contains the main result: we
can renormalize the symplectic potential to have a finite value for every
$O(r^{n})$ LGT. Section 5 contains the derivation of magnetic charges, and the
algebra of electromagnetic charges is presented. Finally, in section 6, we
discuss the results and possible future directions.

##  2 Preliminaries

This section reviews previous results on the asymptotic expansion of Maxwell
fields at null infinity.

###  2.1 Radiative phase space

Consider retarded coordinates $(u,r,x^{a})$, in terms of which the Minkowski
metric is

$ds^{2}=-du^{2}-2dudr+r^{2}q_{ab}dx^{a}dx^{b}.$ (1)

Indices $a,b,c,...$ will indicate sphere coordinates, while Greek indices
$\mu,\nu,\sigma,...$ will indicate spacetime coordinates. The metric $q_{ab}$
is the standard round metric with constant curvature in the sphere $S^{2}$,
with connection $D$. The limit $r+u=:t\rightarrow+\infty$ at fixed $u$ defines
$\mathcal{I}^{+}$, “scri plus”, a null hypersurface with the topology of
$\mathbb{R}\times S^{2}$. Its boundaries are defined by the limits
$u\to\pm\infty$, denoted by $\mathcal{I}^{+}_{\pm}$ respectively, and have the
topology of a sphere.

We consider a massless charged scalar field $\phi$ coupled to the Maxwell
field $\mathcal{A}_{\mu}$ in Minkowski spacetime, with lagrangian

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\mathcal{D}_{\mu}\phi\overline{\mathcal{D}^{\mu}\phi},$
(2)

and satisfying the field equations,

$\displaystyle\nabla^{\nu}F_{\mu\nu}$ $\displaystyle=$ $\displaystyle
j_{\mu},$ (3) $\displaystyle\mathcal{D}_{\mu}\mathcal{D}^{\nu}\phi$
$\displaystyle=$ $\displaystyle 0,$ (4)

where $j_{\mu}=ie\phi\overline{\mathcal{D}_{\mu}\phi}+c.c.$, with
$\mathcal{D}_{\mu}\phi:=\partial_{\mu}\phi-ie\mathcal{A}_{\mu}\phi$, the gauge
covariant derivative and $\nabla$ the metric covariant derivative. In retarded
coordinates, Maxwell equations are

$\displaystyle r^{2}j_{r}$ $\displaystyle=$
$\displaystyle-\partial_{r}(r^{2}F_{ru})+D^{a}F_{ra},$ (5) $\displaystyle
r^{2}j_{u}$ $\displaystyle=$
$\displaystyle-\partial_{r}(r^{2}F_{ru})+\partial_{u}(r^{2}F_{ru})+D^{a}F_{ua},$
(6) $\displaystyle j_{a}$ $\displaystyle=$
$\displaystyle\partial_{r}(F_{ua}-F_{ra})+\partial_{u}F_{ra}+\frac{1}{r^{2}}D^{b}F_{ab}.$
(7)

Bianchi identities, $0=\partial_{[a}F_{bc]}$, are the integrability conditions
for the electromagnetic strength tensor: there exists a one-form
$\mathcal{A}_{\mu}$ such that $F_{\mu\nu}=\partial_{[\mu}\mathcal{A}_{\nu]}$.
We will work in the harmonic gauge, $\nabla^{\mu}\mathcal{A}_{\mu}=0$ 222We
leave the study of the renormalization procedure in other gauges for future
work. In particular, the light-cone gauge in the self-dual sector of Yang-
Mills theory seems a promising avenue to extend the present results to non-
abelian theories, Nagy:2022xxs , which in this particular coordinates implies

$r^{2}\partial_{u}\mathcal{A}_{u}+\partial_{r}(r^{2}\mathcal{A}_{r})+r^{2}D^{a}\mathcal{A}_{a}=0.$
(8)

We are interested in the symplectic structure and charges at $\mathcal{I}$, so
we will need to take the $t\to+\infty$ limit, fixed $u$. Therefore, we need
the $1/r$-expansion of the fields. The usual fall-offs for the electromagnetic
tensor are (see Strominger:2017zoo  and Campiglia:2021oqz ):

$F_{ru}=\frac{1}{r^{2}}F^{(-2)}_{ru}+o(r^{-2}),\quad F_{ar}=o(r^{-1}),\quad
F_{au}=O(1),\quad F_{ab}=F^{(0)}_{ab}+o(1)$ (9)

where it is understood that all the coefficients in the expansions are
functions of $u$ and $x^{a}$. The fall-off for the scalar field is,

$\phi=\frac{\phi^{(-1)}}{r}+o(r^{-1}).$ (10)

These expressions imply the following fall off’s on the charge current:

$j_{u}=\frac{j_{u}^{(-2)}}{r^{2}}+o(r^{-2}),\quad
j_{a}=\frac{j_{A}^{(-2)}}{r^{2}}+o(r^{-2}),\quad
j_{r}=\frac{j_{r}^{(-2)}}{r^{4}}+o(r^{-4}).$ (11)

Fall off’s for $\mathcal{A}_{\mu}$ compatible with the expansion above and the
harmonic gauge condition are:

$\mathcal{A}_{a}=\mathcal{A}_{a}^{(0)}+o(1),\quad\mathcal{A}_{u}=\mathcal{A}_{u}^{(-1,\ln)}\frac{\ln
r}{r}+o(r^{-1}),\quad\mathcal{A}_{r}=o(r^{-1}).$ (12)

The previous asymptotic behaviors are consistent with the field equations and
the harmonic gauge condition.

Using Maxwell equations, the scalar field equation, Bianchi identities and the
harmonic gauge condition, we can solve all the components of the
electromagnetic tensor and the scalar field in terms of
$\mathcal{A}_{a}^{(0)}$ and $\phi^{(-1)}$ (see appendix A of Campiglia:2021oqz
for Yang-Mills case). These functions are the free data for the gauge field
and the scalar field, respectively. To simplify notation, we will refer
$\mathcal{A}_{a}^{(0)}$ as $A_{a}$ and $\phi^{(-1)}$ as $\varphi$,
respectively.

The hypothesis of “tree-level” decays for $A_{a}$ in the limits
$u\rightarrow\pm\infty$,

$\partial_{u}A_{a}(u,x^{1},x^{2})=O(1/|u|^{\infty}),$ (13)

that is, its decay is faster than that of any power $1/|u|^{n}$, implies the
following fall-offs for the radiative data of a generic solution of Maxwell’s
equations Ashtekar:1981bq , Campiglia:2018dyi ,

$F^{(-2)}_{ru}(u,x^{1},x^{2})=F^{-2,0}_{ru}(x^{1},x^{2})+O(1/|u|^{\infty}).$
(14)

For the massless charged scalar field, we assume a consistent fall off with
(14) and the equations of motion, ((5), (6) and (7))

$\varphi(u,x^{1},x^{2})=O(1/|u|^{\infty}).$ (15)

This condition is of a technical nature, and it is imposed only for the sake
of consistency of the equations. Our radiative phase space is thus defined in
terms of the functions $A_{a}$ and $\varphi$,

$\mathcal{F}_{0}=\\{\left(A_{A}(u,x^{a}),\varphi(u,x^{a})\right):\partial_{u}A_{a}(u,x^{1},x^{2}),\varphi(u,x^{1},x^{2})=O(1/|u|^{\infty})\\}.$
(16)

###  2.2 $u$-expansions for fields

From Maxwell equations and Bianchi identities, we can obtain recursive
formulas for the coefficients in both $F_{ru}$ and $\epsilon^{ab}F_{ab}$
expansions in $r$ and $u$, where $\epsilon^{ab}$ is the area form of the
sphere. By Bianchi identity $\partial_{[a}F_{ru]}=0$, contracting with $D^{a}$
and the first two Maxwell equations, we have

$\Delta
F_{ru}+\partial_{r}(\partial_{r}(r^{2}F_{ru})-2r^{2}\partial_{u}F_{ru})=r^{2}\partial_{u}j_{r}-\partial_{r}(r^{2}j_{u})$
(17)

where $\Delta$ denotes the Laplacian operator on the sphere, with metric
$q_{ab}$. We assume that $F_{ru}$ can be expanded in an $r$-series,
$F_{ru}=\frac{1}{r^{2}}\sum_{k=0}^{\infty}\frac{F_{ru}^{(-2-k)}}{r^{k}}$. By
direct substitution in (17),

$2(k+1)\partial_{u}F_{ru}^{(-2-k-1)}+\left(\Delta+k(k+1)\right)F_{ru}^{(-2-k)}=\partial_{u}j_{r}^{(-2-k)}+kj_{r}^{(-2-k)}.$
(18)

From the assumed fall off (14), and equation (18), it is clear that the
behaviour of $F_{ru}^{(-2-n)}$ in the limit $u\to-\infty$ is

$F^{(-2-n)}_{ru}=\sum_{j=0}^{n}u^{j}F_{ru}^{(-2-n,j)}(x^{a})+r_{n}(u,x^{a}),$
(19)

where each of the $F_{ru}^{(-2-n,0)}(x^{a})$ is a function on the sphere, and
$r_{n}$ some function with an $O(1/u^{\infty})$ decay (analogous expansion can
be done in the limit $u\rightarrow+\infty$). We can solve order by order
recursively in terms of the current and these free functions. As a reference,
the full expression for $F_{ru}$ is

$r^{2}F_{ru}=\sum_{k\geq
0}\frac{1}{r^{k}}\sum_{j=0}^{k}u^{j}F_{ru}^{(-2-k,j)}(x^{a})+...$ (20)

The same analysis can be carried out for the function $\epsilon^{ab}F_{ab}$,
obtaining the following equation,

$2\epsilon^{ab}D_{a}j_{b}=2\partial_{u}\partial_{r}\epsilon^{ab}F_{ab}-\partial_{r}\partial_{r}\epsilon^{ab}F_{ab}-\frac{1}{r^{2}}(\Delta\epsilon^{ab}F_{ab}+4\epsilon^{ab}F_{ab}),$
(21)

and by performing the $r-$ and $u-$expansions we obtain a recursive formula
for the coefficients in $\epsilon^{ab}F_{ab}$ (see Appendix A for details). In
section (5), we will use these results.

###  2.3 Variation space

We now turn to the large gauge transformations (LGT) on the variation space.
The usual formulas for the gauge symmetries,

$\mathcal{A}_{\mu}\mapsto\mathcal{A}_{\mu}+\partial_{\mu}\epsilon,\quad\phi\mapsto
e^{-ie\epsilon}\phi$ (22)

establish the following action for variations of the fields,

$\delta_{\epsilon}\mathcal{A}_{\mu}=\partial_{\mu}\epsilon,\quad\delta_{\epsilon}\phi=-ie\epsilon\phi.$
(23)

The variations allowed in our radiative phase space are tangent to
$\mathcal{F}_{0}$, i.e., that maintain the fall-offs of the fields. By the
definition of finite symmetry, given a gauge symmetry generator $\epsilon$ we
see that $\partial_{\mu}\epsilon$ must have the same fall offs as
$\mathcal{A}_{\mu}$:

$\partial_{a}\epsilon=O(1),\quad\partial_{u}\epsilon=o(1),\quad\partial_{r}\epsilon=o(r^{-1})$
(24)

We study the global symmetries as arising from the residual LGT, and by the
choice of harmonic gauge, are solutions to the wave equation,

$\square\epsilon=0.$ (25)

This equation can be solved up to order $O(r^{-1})$ (see Appendix A in
Campiglia:2016hvg ),

$\epsilon(u,r,x^{1},x^{2})=\epsilon_{0}(x^{1},x^{2})+O(\ln(r)/r).$ (26)

###  2.4 Higher order LGT

We are interested in relating the LGTs containing higher orders in $r$ with
the charges that arise from sub${}^{n}-$leading soft photons theorems. The
usual mode expansion reasoning in the soft theorem derivation suggests that
for a sub${}^{n}-$leading soft photon, we need to look for an LGT $\Lambda$
whose $O(1)$ in the $r-$expansion behaves as $u^{n}$. This asymptotic behavior
of the gauge generator must be compatible with the harmonic gauge and,
therefore, implies an $O(r^{n})$ leading behavior, as we show below by solving
$\square\Lambda=0$.

Consider the following $r-$expansion for a $O(r^{n})$ large gauge parameter,

$\Lambda(u,x^{a})=r^{n}\epsilon^{(n)}+\sum_{k=0}^{n-1}r^{k}\epsilon^{(k)}+\frac{\ln
r}{r}\epsilon^{ln}+O(r^{-1}),$ (27)

where $\epsilon^{(i)}=\epsilon^{(i)}(u,x^{1},x^{2})$. We have
$\square\Lambda=0$, which in retarded coordinates reads,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-6r^{n-1}\partial_{u}\epsilon^{(n)}+\sum_{k=-1}^{n-2}r^{k}\left(\Delta\epsilon^{(k+2)}-2(k+2)\partial_{u}\epsilon^{(k+1)}+(k+2)(k+3)\epsilon^{(k+2)}\right)$
(28) $\displaystyle+\frac{\ln
r}{r^{3}}\Delta\epsilon^{(ln)}+\frac{2}{r^{2}}(\Delta\epsilon^{(0)}-\partial_{u}\epsilon^{(ln)})+\frac{1}{r^{3}}\epsilon^{(ln)}+....$

The first term in (28) implies that $\epsilon^{(n)}$ is a free function on the
sphere. Next, we have a recursive equation on the successive coefficients:

$2(k+1)\partial_{u}\epsilon^{(k)}=\Delta\epsilon^{(k+1)}+(k+1)(k+2)\epsilon^{(k+1)}$
(29)

Integrating (29) and fixing each integration constant to zero in each step
gives an LGT of order $O(r^{n})$ generated by $\epsilon\equiv\epsilon^{(n)}$,
which we will call $\Lambda^{n}_{\epsilon}$. If the integration constants are
non-zero, each one of them will be a free $S^{2}$ function that contributes
linearly with an LGT of corresponding order:

$\Lambda_{\alpha}=\Lambda^{n}_{\epsilon_{n}}+\Lambda^{n-1}_{\epsilon_{(n-1)}}+...,$
(30)

where $\alpha=\\{\epsilon_{j}\\}_{j}$ is the sequence of integration constants
$\epsilon_{j}$ in the equation (29), that are free $S^{2}-$functions, each one
generating an $O(r^{j})$ $LGT$. We will call an LGT “pure” if only one free
function generates it. When using the notation $\Lambda^{m}_{f}$, subscripts
indicate the generating function or sequence of functions, and superscripts
indicate the leading term in the $r-$expansion, if the generating function is
not a sequence.

Some remarks are in order. First, one implication of equation (29) for a pure
$O(r^{n})$ LGT is the following property:

$\epsilon^{(n-1)}=O(u),\quad...,\quad\epsilon^{(k)}=O(u^{n-k})$ (31)

This shows that the order $O(r^{n})$ is necessary for a $u^{n}$ asymptotic
behavior at order $O(r^{0})$ for the LGT, as was stated at the beginning of
the section. Second, the term $\ln(r)/r$ is needed for the $O(r^{0})$ to be
consistent; otherwise, we would get $\Delta\epsilon^{(0)}=0$, and since we are
in a sphere, that would give a trivial function. Third, the non-trivial fact
that equation (29) resembles the form of equation (18), but it presents
crucial differences in the constants multiplying the functions. This
similarity between the recursive expressions is useful when showing the Ward
identity equivalence with the sub${}^{n}-$ soft theorems.

##  3 Leading and Subleading charges

In this section, we review the phase space construction and the symplectic
charges in the case where the LGTs are $O(r)$. We leave the renormalization
procedure for the next section, focusing exclusively on the first step of the
phase space extension and the recovery of the charges.

###  3.1 Extended phase space

The usual phase space, (16), contains the physical information regarding the
leading order charges, restricted to $O(r^{0})$ LGT. Their usual expressions
are (Strominger:2013lka ,Strominger:2017zoo ):

$Q_{\epsilon_{0}}=\int_{S^{2}}\sqrt{q}\epsilon_{0}\int_{\mathbb{R}}\partial_{u}F^{(-2)}_{ru}du,$
(32)

where $\epsilon_{0}$ is a function on the sphere. The fall-offs (12) are not
preserved by an $O(r^{1})$ LGT as soon as we allow higher order LGT (through
its action (22)) and therefore, the variations are no longer tangent to the
radiative phase space $\mathcal{F}_{0}$. We expand the phase space in these
extra directions by first extending the vector potential sector in (16).
Consider the following space:

$\mathcal{F}_{1}=\mathcal{F}_{0}\times\\{\psi(x^{1},x^{2}):\psi\in
C^{\infty}(S^{2})\\}$ (33)

We define the new vector potential as
$\hat{\mathcal{A}}_{\mu}=\mathcal{A}_{\mu}+\partial_{\mu}\Lambda^{1}_{\psi}$,
where $\mathcal{A}_{\mu}$ is the vector potential that has $A_{a}$ as initial
data (from section (2)) and $\Lambda^{1}_{\psi}$ is the pure $O(r)$ LGT
generated by $\psi$. Observe that this definition is indeed consistent, since
$\partial_{[\mu}\partial_{\nu]}\Lambda^{1}_{\psi}=0$ and thus $a$ makes no
contribution to the electromagnetic tensor, i.e. $\hat{F}=F$. 333This feature
in the abelian case is in sharp contrast to the non-abelian case, where the
linear extension was studied in Campiglia:2021oqz  Observe also that the
harmonic gauge condition is trivially satisfied for the extended
electromagnetic potential 444It is left for future works to study the phase
space extension in more general gauges, and whether it changes the structure..

Given a general $O(r)$ LGT, $\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}$, the
variations generated by it on $\mathcal{F}_{1}$ are split in terms of the
$S^{2}$ free functions $\epsilon_{1}$ and $\epsilon_{0}$ corresponding to
order $O(r)$ and order $O(1)$ in the $r-$expansion respectively (see (30)):

$\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}=r\epsilon_{1}+(\epsilon_{0}+u\frac{1}{2}(\Delta+2)\epsilon_{1})+O(\frac{\ln
r}{r})$ (34)

The action on the phase space $\mathcal{F}_{1}$ comes from the identity
$\delta_{\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}}\hat{\mathcal{A}}_{\mu}=\partial_{\mu}\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}$,
which after the splitting reads:

$\displaystyle\delta_{\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}}A_{a}$
$\displaystyle=$ $\displaystyle\partial_{a}\epsilon_{0},$ (35)
$\displaystyle\delta_{\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}}\psi$
$\displaystyle=$ $\displaystyle\epsilon_{1}.$ (36)

Allowing a $O(r)$ LGT in the massless field sector also implies a change in
the massless field $\phi$. The equations of motion are invariant under the
simultaneous change

$\mathcal{A}_{\mu}\mapsto\hat{\mathcal{A}}=\mathcal{A}_{\mu}+\partial_{\mu}\Lambda^{1}_{\psi},\quad\phi\mapsto\hat{\phi}=e^{-ie\Lambda^{1}_{\psi}}\phi.$
(37)

Since the finite gauge symmetry involves a product
$e^{-ie\Lambda^{1}_{\psi}}\phi$, we can define an extended field
$\hat{\phi}=e^{-ie\Lambda^{1}_{\psi}}\phi$, where $\psi$ is the free $S^{2}$
function now generating a phase for the scalar field, while $\phi$ is the
massless field with the usual fall off, with $\varphi\in\mathcal{F}_{0}$ as
free data. The covariant gauge derivative is given by

$\hat{\mathcal{D}}_{\mu}:=\partial_{\mu}-ie\hat{\mathcal{A}}_{\mu},$ (38)

from where we have that the new current $\hat{j}_{\mu}$ maintain its original
form,

$\hat{j}_{\mu}=ie\hat{\phi}(\hat{\mathcal{D}}_{\mu}\hat{\phi})^{*}+c.c.=ie\phi(\mathcal{D}_{\mu}\phi)^{*}+c.c.,$
(39)

The consistency of the action of the $O(r)$ LGT action on $\hat{\phi}$ with
the splitting of the extended phase space implies

$\delta_{\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}}\varphi=-ie\epsilon_{0}\varphi.$
(40)

This type of extension of the phase space and the dressing of the fields is
part of a more general procedure, using “Goldstone modes” on the boundary,
that has been introduced both in the context of gauge theories and gravity
(see Donnelly:2016auv ; Freidel:2018fsk ; Ciambelli:2021nmv ;
Campiglia:2021oqz  and references therein).

###  3.2 Calculation of leading and subleading charges

This subsection reviews the covariant phase space procedure for calculating
charges associated with a gauge transformation generated by $\epsilon$.
Consider the Lagrangian (2), in our extended phase space we have the usual
symplectic potential current,

$\theta^{\mu}(\delta)=\sqrt{g}\left(\hat{F}^{\mu\nu}\delta\hat{\mathcal{A}}_{\nu}+\hat{\mathcal{D}}^{\mu}\hat{\phi}\delta\bar{\hat{\phi}}+c.c.\right),$
(41)

and the symplectic current by taking the exterior derivative in the phase
space,

$\omega^{\mu}(\delta,\delta^{\prime})=\delta\theta^{\mu}(\delta^{\prime})-\delta^{\prime}\theta^{\mu}(\delta)-\theta([\delta,\delta^{\prime}]).$
(42)

The symplectic form is obtained by integrating the symplectic current over
$\Sigma_{t}$,
$\Omega(\delta,\delta^{\prime})=\int_{\Sigma_{t}}\omega^{\mu}(\delta,\delta^{\prime})dS_{\mu}$.
We evaluate it on a variation generated by a general LGT
($\Lambda_{\epsilon_{1},\epsilon_{0}}$) and an admissible variation (denoted
by $\delta$), obtaining an expression for the charge,

$\delta
Q_{\Lambda_{\epsilon_{1},\epsilon_{0}}}=\Omega(\delta,\delta_{\Lambda_{\epsilon_{1},\epsilon_{0}}})=\int_{\Sigma_{t}}\omega^{\mu}(\delta,\delta_{\Lambda_{\epsilon_{1},\epsilon_{0}}})dS_{\mu}$
(43)

where the integrals are taken over a $t=cnt$ surface. As it was shown in
Campiglia:2016hvg , one could find the leading and subleading charges
(consistent with the Ward identities) by taking the limit
$t=r+u\rightarrow+\infty$ at constant $u$,

$Q_{\Lambda_{\epsilon_{1},\epsilon_{0}}}=\lim_{t\rightarrow\infty}\int_{\Sigma_{t}}(\partial_{r}-\partial_{u})(r^{2}\Lambda_{\epsilon_{1},\epsilon_{0}}\hat{F}_{ru})dx^{2}du,$
(44)

and considering the finite part in the limit. By counting orders in $t$, it is
straightforward to see that the expression (44) contains divergent terms;
therefore, the definition of the charge at the limit is ill-defined. In what
follows, we drop the hat $\hat{\textbullet}$ in $F_{ru}$ since it is the same
field as in the radiative space.

As we previously mentioned, the main result of this paper is that we can
define a procedure to renormalize the symplectic potential and get rid of the
divergent terms in (44) for any arbitrary higher order $O(r^{n})$. This will
be the content of the next section, while in the remainder of this section, we
motivate the renormalization in the particular case of the extension for
$n=1$.

Since we can trace back the divergences to the symplectic potential, due to
varying with $\delta_{\Lambda_{\epsilon_{1},\epsilon_{0}}}$, our starting
point is to compute the symplectic potential on the hypersurfaces
$\Sigma_{t}$,

$\theta^{t}(\delta)=\underbrace{\sqrt{q}\left(r^{2}F_{ru}(\delta A_{r}-\delta
A_{u})+q^{bc}F_{ub}\delta
A_{c}\right)}_{\theta_{0}^{t}}+\underbrace{\sqrt{q}(\partial_{r}-\partial_{u})(r^{2}F_{ru}\delta\Lambda^{1}_{\psi})}_{\theta_{1}^{t}},$
(45)

where we did not write the total derivative
$r^{2}D_{c}(\sqrt{q}q^{bc}F_{ub}\delta\Lambda_{\psi}^{1})$, since it vanishes
after integration on $\Sigma_{t}$. The first term can be regarded as the
radiative phase space symplectic potential, $\theta_{0}^{t}$, while the second
term is the new extended term, which we will call $\theta_{1}^{t}$.

The term $\theta_{0}^{t}(\delta)$ will contribute to the symplectic form (by
integrating by parts and using the equations of motion) as usual,

$\omega_{0}^{t}(\delta,\delta^{\prime})=\sqrt{q}q^{bc}\delta
F_{ub}\wedge\delta^{\prime}\mathcal{A}_{c}+\sqrt{q}r^{2}\delta
F_{ru}\wedge\delta^{\prime}(\mathcal{A}_{r}-\mathcal{A}_{u}),$ (46)

The term $\theta_{1}^{t}(\delta)$ presents the divergence: the action of
$\partial_{u}$ on $\delta\Lambda_{\psi}^{1}$ leaves an $O(r)$ term, which in
turns imply a $t$ factor when changing variables from $(u,r,x^{1},x^{2})$ to
$(t,r,x^{1},x^{2})$. In the next section we show a systematic approach for
renormalizing such terms while keeping the finite ones unchanged (i.e., the
minimal subtraction to make the expressions finite). For now, we assume that
we can drop the divergent term and that the expression we get also has a
finite limit $u\to-\infty$. Assuming the above, we find the following
expression for the renormalization of $\theta_{1}^{t}(\delta)$,

$\theta_{1}^{ren,t}(\delta)=\sqrt{q}\left(D^{a}j_{a}^{(0)}-\frac{u}{2}\Delta\partial_{u}F_{ru}^{(-2)}\right)\delta\psi,$
(47)

where $ren$ stands for “renormalized”. The symplectic current is split then,

$\omega^{ren,t}(\delta,\delta^{\prime})=\omega_{0}^{t}(\delta,\delta^{\prime})+\omega_{1}^{ren,t}(\delta,\delta^{\prime}),$
(48)

where the last term comes from the exterior derivative of
$\theta_{1}^{ren,t}(\delta)$, and the total symplectic form on
$\mathcal{I}^{+}$ is well defined (by taking $t\rightarrow+\infty$) ,

$\Omega^{ren}(\delta,\delta^{\prime})=\int_{\mathcal{I}}\omega^{ren}(\delta,\delta^{\prime})=\int_{\mathcal{I}}\omega_{0}(\delta,\delta^{\prime})+\int_{S^{2}}\sqrt{q}(\delta
F_{ru}^{(-3,0)}\wedge\delta^{\prime}\psi).$ (49)

The last term comes from the value of $F_{ru}^{(-3,0)}$ in (19), which can be
seen as the value of the following limit (see Campiglia:2018dyi  for details):

$F_{ru}^{(-3,0)}=\lim_{u\to-\infty}F_{ru}^{(-3)}-uF_{ru}^{(-3,1)}=\int_{\mathbb{R}}\left(D^{a}j_{a}^{(0)}-\frac{u}{2}\Delta\partial_{u}F_{ru}^{(-2)}\right)du,$
(50)

where the contribution from $u=+\infty$ in integral zero due to the absence of
massive charges ($F^{(2)}_{ru}(u=+\infty,x^{1},x^{2})=0$). Since
$\partial_{u}F_{ru}^{(-2)}$ decays faster than any polynomial in $u$, the
above integral is convergent. Observe that $F_{ru}^{(-3,0)}$ is the canonical
conjugate to $\psi$.

Next, we compute the leading and subleading charges. Taking $\delta^{\prime}$
to be a large gauge transformation, and $\delta$ any arbitrary admissible
variation (compatible with $\mathcal{F}_{1}$), we calculate the charge
associated to any LGT $\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}$ by equation
(43). Since $F_{\mu\nu}$ is invariant under
$\delta_{\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}}$ and
$\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}$ is not affected by $\delta$
555This again is in contrast with the non-abelian case, where the harmonic
gauge condition implies a field dependent LGT’s., the calculation is
straightforward,

$Q_{\Lambda_{\\{\epsilon_{1},\epsilon_{0}\\}}}=\int_{S^{2}}\sqrt{q}\left(\epsilon_{0}F_{ru}^{(-2,0)}+\epsilon_{1}F_{ru}^{(-3,0)}\right)dx^{2}=:Q^{0}_{\epsilon_{0}}+Q^{1}_{\epsilon_{1}},$
(51)

where we also used (19) in the radiative space sector, and
$Q^{i}_{\epsilon_{i}}$, with $i=0,1$, denotes the leading and subleading
charges, respectively.

The charge $Q^{0}_{\epsilon_{0}}$ is the usual for a $O(1)$ large gauge
symmetry, while the second term is the one obtained in Campiglia:2016hvg  and
Campiglia:2018dyi . In both cases, we obtained “corner” charges. They depend
on the values of the fields at the boundary of $\mathcal{I}$, which is by
itself the boundary of the domain we started with (as in Donnelly:2016auv ;
Freidel:2018fsk ).

##  4 Tower of asymptotic charges

In this section, we derive an infinite hierarchy of charges from a symplectic
form in an extended phase space that contains enough degrees of freedom to
allow for $O(r^{n})$ LGTs for arbitrary $n$. Certain difficulties in defining
the symplectic potential arise, in particular, the appearance of several
divergent integrals, as shown in the previous section. The renormalization
procedure we apply is based on Freidel:2019ohg .

First, we define the extended phase space and show the type of divergences we
have, both in the $t\rightarrow+\infty$ and $u\rightarrow+\infty$ limits
inside the expression (44). Then, we proceed to prescribe a renormalization on
the symplectic potential that will lead to the correct expression for the
charges while the symplectic form remains finite.

###  4.1 Extended phase space and charges

Let $\mathcal{S}$ be the space of sequences $\\{\psi_{i}\\}_{i>0}$ of
functions $\psi_{i}:S^{2}\to\mathbb{R}$ such that only finitely many are non-
zero 666In what follows we assume that the sequences of functions have this
property unless stated otherwise. Given a sequence $\Psi\in\mathcal{S}$, we
define the LGT associated to the sequence as

$\Lambda_{\Psi}:=\sum_{i>0}\Lambda^{i}_{\psi_{i}}$ (52)

where each $\Lambda^{i}_{\psi_{i}}$ is a pure $O(r^{i})$ LGT associated to
$\psi_{i}$, in the sense of section (2). Observe that the sum is finite for
every $\Psi\in\mathcal{S}$. We define the extended phase space as the
following set,

$\mathcal{F}_{\infty}=\mathcal{F}_{0}\times\mathcal{S},$ (53)

with the extended electromagnetic potential and scalar field are defined as

$\hat{\mathcal{A}}_{\mu}=\mathcal{A}_{\mu}+\partial_{\mu}\Lambda_{\Psi},\quad\hat{\phi}=e^{-ie\Lambda_{\Psi}}\phi,$
(54)

where $\mathcal{A}_{\mu}$ and $\phi$ are the vector potential and the scalar
field generated by the free data $A_{a}$ and $\varphi$ from the space
$\mathcal{F}_{0}$, respectively.

The admissible variations $\delta$ of this phase space are such that when
acting on the degrees of freedom parametrized by $\Psi$, it satisfies
$\delta\Psi\in\mathcal{S}$. This property is not restrictive regarding the
variations, as shown below.

Given a sequence
$\varepsilon=\\{\epsilon_{0},\epsilon_{1},...,\epsilon_{i},...\\}$ of free
$S^{2}$ functions, such that $\\{\epsilon_{i}\\}_{i>0}\in\mathcal{S}$,
consider the LGT associated to it,
$\Lambda_{\varepsilon}=\Lambda^{0}_{\epsilon_{0}}+\sum_{i>0}\Lambda^{i}_{\epsilon_{i}}$.
The variation generated by this LGT acts on $\mathcal{F}_{\infty}$ by acting
in $\mathcal{A}_{\mu}$ with its $O(r^{0})$ free function and by acting on
$\alpha$ on each sequence term,

$\delta_{\Lambda_{\varepsilon}}A_{A}=\partial_{A}\epsilon_{0},\quad\delta_{\Lambda_{\varepsilon}}\varphi=-ie\epsilon_{0}\varphi,\quad\delta_{\Lambda_{\varepsilon}}\Psi=\\{\epsilon_{i}\\}_{i>0}$
(55)

This structure is the same as in the previous section, extended to contain any
order in the $r-$expansion.

We can write the full symplectic potential, equation (41), and proceed in the
same way as in the previous section, obtaining the expression (45). In this
case, $\Lambda_{\Psi}$ is in place of $\Lambda_{\psi}^{1}$ and the split of
the symplectic potential in the radiative phase space contribution and the
extended part is given by

$\theta^{t}_{\infty}(\delta)=\sqrt{q}(\partial_{r}-\partial_{u})(r^{2}F_{ru}\delta\Lambda_{\Psi}),$
(56)

where the $\infty$ stands for the extension to all orders in $r$.

Given $\delta$ and $\Lambda_{\Psi}$, let us calculate the symplectic potential
evaluated at $\delta$. Consider the integral,

$\Theta_{t,\infty}(\delta)=\int_{\Sigma_{t}}\sqrt{q}(\partial_{r}-\partial_{u})(r^{2}F_{ru}\delta\Lambda_{\Psi})dx^{2}du,$
(57)

and observe that the term inside the integral is divergent in the limit
$t\rightarrow+\infty$ with the same order as the highest power of $r$ in
$\delta\Lambda_{\alpha}$. Our aim in this section is to understand better this
integral. For brevity let us call,

$\rho_{k}(\delta)=\sum_{i=k}^{+\infty}F_{ru}^{(-2+k-i)}\delta\Lambda_{\Psi}^{(i)},$
(58)

where $\delta\Lambda_{\Psi}^{(i)}$ is the coefficient corresponding to $r^{i}$
in the $r-$expansion of $\delta\Lambda_{\Psi}$. $\rho_{k}(\delta)$ is thus the
$O(r^{k})$ coefficient in the expansion of the term inside the brackets. Upon
direct computation, we have,

$\Theta_{t,\infty}(\delta)=\int_{\Sigma_{t}}\sqrt{q}\sum_{k=1}^{\infty}\left(kr^{k-1}\rho_{k}(\delta)-r^{k}\partial_{u}\rho_{k}(\delta)\right)dx^{2}du,$
(59)

which after we substitute $r=t-u$, gives,

$\Theta_{t,\infty}(\delta)=\sum_{j=0}^{\infty}t^{j}\int_{\Sigma_{t}}\theta_{j}^{t}(\delta)dx^{2}du,$
(60)

for some $t-$independent functions $\theta_{j}^{t}(\delta)$. This gives us a
$t-$expansion of the symplectic potential. In the next subsection, we show
that these divergences can be renormalized by adding total variations and
total derivatives (corner) terms to the symplectic potential.

Assuming such a procedure can be done, we are left with the $O(t^{0})$ term,
which satisfies the identity,

$\displaystyle\Theta^{\mathcal{I}}_{\infty}(\delta)$ $\displaystyle:=$
$\displaystyle\lim_{t\rightarrow+\infty}\Theta_{t,\infty}(\delta)=\int_{\mathcal{I}}\sqrt{q}\sum_{k=1}^{\infty}\left(k(-u)^{k-1}\rho_{k}(\delta)-(-u)^{k}\partial_{u}\rho_{k}(\delta)\right)dx^{2}du$
(61) $\displaystyle=$
$\displaystyle-\int_{\mathcal{I}}\partial_{u}\left(\sqrt{q}\sum_{k=1}^{\infty}(-u)^{k}\rho_{k}(\delta)\right)dx^{2}du,$

which gives us a boundary term. The charges associated with higher order LGT
can be directly computed using the identity $\delta
Q_{\Lambda_{\varepsilon}}=\Omega^{\mathcal{I}}_{\infty}(\delta,\delta_{\Lambda_{\varepsilon}})$,

$Q_{\epsilon}=\int_{\mathcal{I}}\partial_{u}\left(\sum_{k=1}^{\infty}(-u)^{k}\rho_{k}(\delta_{\Lambda_{\varepsilon}})\right)dud^{2}x.$
(62)

When evaluating the term in the brackets in the last line of (61) at
$u=+\infty$, we use the hypothesis that $F_{ru}=0$ at $\mathcal{I}^{+}_{+}$.
When evaluating at $\mathcal{I}^{+}_{-}$, we run into divergences. Since the
general behavior of $\rho_{k}(\delta)$ admitted by (13), (15) and (17)
777Remember that the fall off (14) is a consequence of these equations. near
spatial infinity is polynomial in $u$ plus a $O(1/|u|^{\infty})$ remainder, we
have that the above expression for $\Theta_{t}(\delta)$ is not well defined.
By the renormalization procedure of the next subsection, we will be able to
regularize the above expression, keeping only the $O(u^{0})$ in
$\rho_{0}(\delta)$,

$\Theta^{\mathcal{I}}_{\infty}(\delta)=\int_{S^{2}}\sqrt{q}\sum_{i=1}^{\infty}F_{ru}^{(-2-i,0)}\delta\psi_{i}dx^{2}du,$
(63)

where $F_{ru}^{(-2-i,0)}$ are the $O(u^{0})$ of $F_{ru}^{(-2-i)}$.

The renormalization procedure of the next subsection has to address the
previous two divergences: the $t$ divergence from the limit to $\mathcal{I}$,
and the $u$ divergences in the integrals over $\mathcal{I}$. We leave it as a
future work to understand the physical meaning of the boundary and corner
terms in the context of covariant phase space quantities.

We end this subsection with some remarks regarding previous works. The idea in
Campiglia:2016hvg  is to relate the divergent terms to the conserved
quantities, obtaining a "projected out" charge equal to the $t^{0}$ term,
while the discarded terms are proportional to lower order charges. While this
is the case for the $O(r)$ subleading charge (the $O(t^{1})$ part in the
charge is proportional to $Q_{\epsilon_{0}}$), there is, however, a remaining
divergent term in the $O(r^{2})$ that leads to an unresolved tension, in
particular the $O(t^{1})$ term is not proportional to any lower order charge.
This tension is solved once we renormalize the symplectic potential.

Regarding the concrete expressions of the charges, the order $O(u^{0})$ of
$Q_{\Lambda_{\varepsilon}}$ in equation (62) is exactly what was presented in
Campiglia:2018dyi . This is equivalent to prove that the $O(u^{0})$
coefficient of $Q_{\Lambda_{\varepsilon}}$ is
$\sum_{k=1}^{+\infty}\int_{S^{2}}\epsilon_{k}F_{ru}^{-2-k,0}d^{2}x$ (we are
considering $\epsilon_{0}=0$, only higher order charges).

As it stands, (62) diverges, due to the orders of $u^{n}$ that are involved in
the integral. If we want to write the charge as a corner integral on the
sphere at $u\rightarrow-\infty$, we should inspect the $O(u^{0})$ term,
corresponding to the finite limit term.

Here, we take the $u-$decay in the remainder functions $r_{i}$ in equation
(19) as faster than any polynomial decay. Therefore, inspecting the
expressions for $\Lambda_{\varepsilon}^{(k)}$ and $F_{ru}^{-2+k-i}$, we see
that each $\rho_{k}(\delta_{\Lambda_{\varepsilon}})$ has at least order
$u^{0}$, therefore the term in the sum contributes with at least $u^{k}$. The
only term with a possible $u^{0}$ order is thus
$\rho_{0}(\delta_{\Lambda_{\varepsilon}})$,

$\rho_{0}(\delta_{\Lambda_{\varepsilon}})=\sum_{i=1}^{\infty}\Lambda_{\varepsilon}^{(k)}F_{ru}^{(-2+k)}.$
(64)

Again, a close inspection in the $u-$ expansion of the functions shows that
the order $u^{0}$ is given by the sum of the products
$\epsilon_{k}F_{ru}^{(-2-k,0)}$.

###  4.2 Regularization procedure

In this subsection, following Freidel:2019ohg , we will renormalize the
symplectic potential for QED in the extended phase space to eliminate the
divergences. The idea is to write the higher order terms in the $t$ component
of the symplectic potential as boundary plus corner terms and to subtract them
from the original expression, thus obtaining a finite result in the limit
$t\rightarrow\infty$.

From the first variation of the Lagrangian (2), we have,

$\delta\mathcal{L}=E^{\mu}\delta\hat{\mathcal{A}}_{\mu}+E\delta\hat{\phi}+\partial_{\mu}\theta^{\mu}(\delta),$
(65)

where $E^{\mu}$ and $E$ are the field equations for $\hat{\mathcal{A}}_{\mu}$
and the massless scalar, respectively. By taking the retarded coordinates
${u,t,x^{1},x^{2}}$ on Minkowski spacetime, we write the previous equation on-
shell and obtain an equation for $\partial_{t}\theta^{t}(\delta)$,

$\partial_{t}\theta^{t}(\delta)=\delta\mathcal{L}-\partial_{u}\theta^{u}(\delta)-D_{a}\theta^{a}(\delta).$
(66)

We will assume that all the functions have $t$ and $u$ expansions around
$t=+\infty$ and $u=\pm\infty$, as is the case for $F_{ru}^{(2)}$, $A_{a}$ and
$\varphi$ (cf. equations (14), (12) and (10)).

Consider the derivation of the divergent part of the symplectic potential done
in the previous section but now applied to our extended phase space, i.e.,
starting from,

$\theta^{\mu}(\delta)=\sqrt{q}r^{2}\left(F^{\mu\nu}\delta\hat{\mathcal{A}}_{\nu})+\overline{\hat{\mathcal{D}}^{\mu}}\hat{\phi}\delta\hat{\phi}+c.c.\right),$
(67)

the general form for the symplectic potential on Cauchy slices at constant $t$
is the following, 888In the following equations we write the explicit
dependence of the functions on variations and coordinates.

$\theta^{t}(\delta)=Y_{0}(\delta)(u,t,x^{a})+\sum_{i=1}^{\infty}t^{i}Y_{i}(\delta)(u,x^{a}).$
(68)

where $Y_{0}(\delta)(u,t,x^{a})$ is such that
$\lim_{t\rightarrow+\infty}Y_{0}(\delta)(u,t,x^{a})=Y_{0}(\delta)(u,x^{a})$ is
a well defined function on $\mathcal{I}^{+}$. We introduce the renormalized
symplectic potential as $\theta_{ren}^{t}:=\theta^{t}-H_{ren}$, where
$H_{ren}$ satisfies the following equation,

$\partial_{t}\theta^{t}(\delta)-\partial_{t}H_{ren}(\delta)=K(\delta)(u,t,x^{a}),$
(69)

where $K$ is such that its limit when $t\rightarrow+\infty$ vanishes. In
general, $K$ and $H_{ren}$ are not uniquely determined by the previous
equation. The natural prescription for $H_{ren}$ to resolve the divergences is
the following,

$H_{ren}(\delta)=\sum_{i=1}^{+\infty}t^{i}Y_{i}(\delta)(u,x^{a})+C(\delta)(u,x^{a}),$
(70)

where $C(u,x^{a})$ is a function to be determined. Observe that $H_{ren}$ has
the same order as $\theta^{t}$ in the $t-$expansion, and that the divergences
in the $t$ parameter are canceled, so $\theta^{t}_{ren}$ converges in the
limit $t\to+\infty$. The coefficients $Y_{i}$ are obtained from the
integration of the terms in the variation of the lagrangian and the total
derivative of the symplectic potential in equation (66), on $\\{t=cnt\\}$
surfaces, directly by the $t$ expansion.

Therefore, we can prescribe

$Y_{i}(\delta)=\text{Finite
part}\left(\lim_{t\to+\infty}\frac{1}{t^{i}}\left(\delta\mathcal{L}-\partial_{u}\theta^{u}(\delta)-D_{a}\theta^{a}(\delta)\right)\right),$
(71)

for each $i$. Observe that in (71) each $Y_{i}$ can be written as a total
derivative plus a total variation.

By taking the free function $C$ as a total derivative,
$C=\partial_{u}X^{u}+D_{a}X^{a}$, we can add the last term in (71) to obtain a
new total derivative term. Then, the renormalized symplectic potential has the
form

$\theta^{t}_{ren}(\delta):=\theta^{t}(\delta)+\partial_{\nu}\Upsilon^{t\nu}(\delta)+\delta\Xi^{t}=P(\delta)(u,t,x^{a})$
(72)

where $\Upsilon$ and $\Xi$ are calculated from $Y_{i}$, $X^{u}_{i}$ and
$X^{a}_{i}$ directly, and $P$ is at most $O(t^{0})$ in the $t-$expansion. This
symplectic potential does not contain divergences in the $t\rightarrow\infty$
limit. The general form of the symplectic potential will be changing the upper
index $t$ by a 4d index $\mu$. We have that
$\Upsilon^{\mu\nu}=-\Upsilon^{\nu\mu}$, by definition of “corner terms” (see
Freidel:2019ohg ). Without any loss of generality, we can define
$\Upsilon^{jl}=0$, for $j,l$ running in the set $\\{u,x^{a}\\}$, since these
terms are not uniquely defined and do not affect the renormalization of
$\theta^{t}$. Therefore, we have a well-defined limit

$\theta^{\mathcal{I}}_{ren}(\delta)(u,x^{a}):=\lim_{t\rightarrow+\infty}\theta^{t}_{ren}(\delta)(t,u,x^{a})=Y_{0}(\delta)(u,x^{a})-C(\delta)(u,x^{a})$
(73)

We still have at our disposal the function $C(u,x^{1},x^{2})$ (the only
condition we imposed so far is that it is a total derivative), which can be
determined by imposing a finite limit when $u\rightarrow-\infty$ for the
symplectic potential, as we will show below.

Under a general LGT, the $O(t^{0})$ of the symplectic potential has $O(u^{N})$
terms. Therefore, $\theta_{ren}^{t}$ have an expansion in powers of $u$,
starting in some $u^{N}$ (corresponding to the highest power in $\delta$ or
$\alpha$), the coefficients of the expansion depending in general on which
limit we are computing, $u\rightarrow\pm\infty$. We consider the following
$u$-expansion for $Y_{0}(\delta)$ near $u=\pm\infty$,

$Y_{0}(\delta)(u,x^{a})\overset{u\rightarrow\pm\infty}{=}R_{Y_{0}}(\delta)(u,x^{a})+\sum_{k=1}^{\infty}u^{k}Y^{\pm}_{0,k}(\delta)(x^{a}),$
(74)

where $\partial_{u}R_{Y_{0}}(u,x^{a})=O(1/|u|^{\infty})$. This condition comes
from the tree-level assumption on the soft theorems and implies in particular
that the limits when $u\rightarrow\pm\infty$ are in principle different,

$R^{\pm}_{Y_{0}}(\delta)(x^{a}):=\lim_{u\rightarrow\pm\infty}R_{Y_{0}}(\delta)(u,x^{a}).$
(75)

By inserting (74) in (73), we have

$\theta^{\mathcal{I}}_{ren}(\delta)=R_{Y_{0}}(\delta)(u,x^{a})+\sum_{k=1}^{\infty}u^{k}Y^{\pm}_{0,k}(\delta)(x^{a})-\partial_{u}X^{u}(\delta)(u,x^{a})-D_{a}X^{a}(\delta)(u,x^{a}).$
(76)

Observe that we can find functions $X^{u},X^{a}$ such that their expansions
around $u=\pm\infty$ renormalize the limits of the symplectic potential. For
$X^{u}$ we find,

$X^{u}_{\pm}(\delta)(u,x^{a})=\sum_{k=1}^{\infty}\frac{1}{k+1}u^{k+1}Y^{\pm}_{0,k}(\delta)(x^{a}),$
(77)

Observe that the coefficient for $u^{0}$ vanishes to avoid ambiguities. For
$X^{a}$ we have,

$D_{a}X^{a}(\delta)(u,x^{a})=\left\\{\begin{array}[]{cc}R^{-}_{Y_{0}}(\delta)(x^{a})+O(1/|u|^{\infty})&\text{when
}u\rightarrow-\infty\\\
R^{+}_{Y_{0}}(\delta)(x^{a})+O(1/|u|^{\infty})&\text{when
}u\rightarrow+\infty\end{array}\right.$ (78)

Finally, the symplectic potential density gives a finite result upon
integration on $\mathcal{I}$ due to the fall-offs of $R_{Y_{0}}$.

Remark: It is a well-known fact that the equation,

$D_{a}V^{a}=0,$ (79)

for a certain vector field $V^{a}$ on the sphere, has infinite solutions. In
fact, it has as many solutions as there are scalar functions on the sphere
(since it is a simply connected manifold). On the one hand, we have an
ambiguity when solving $X^{a}$ in (78). Since $D_{a}X^{a}$ is the object that
enters in the definition of $C$, such ambiguity resolves trivially in the
renormalized symplectic potential, not affecting the outcome.

On the other hand, the contribution from $C$ to the symplectic potential is
computed only at the boundary. Thus, it is important only for the limit value
$u\rightarrow\pm\infty$ for $D_{a}X^{a}$. Then, any ambiguities in the
prescription of $X_{a}$ are “washed away” by the limiting process. More on
this detail in the next subsection.

As mentioned in the introduction, this renormalization is minimal because it
cancels all the divergent terms while keeping unchanged the finite ones. The
linearity of the theory played a central role in the previous derivation, and
also in the definition of the extended phase space. We leave for future works
to delve into such a renormalization procedure in the case of non-abelian
theories.

###  4.3 Electric-like charge algebra

The previous renormalization procedure adjusts exactly all the divergences
while maintaining the same convergent terms discussed in subsection 4.1. The
expression for the renormalized symplectic potential is therefore:

$\Theta_{ren}(\delta)=\int_{\mathcal{I}^{+}}\theta_{0}(\delta)dudx^{2}+\int_{S^{2}}\sum_{i=1}^{\infty}F_{ru}^{(-2-i,0)}\delta
a_{i}dx^{2}$ (80)

where $\theta_{0}$ is the usual symplectic potential in $\mathcal{F}_{0}$. The
symplectic form is the exterior derivative (in the extended phase space) of
the symplectic potential:

$\Omega_{ren}(\delta,\delta^{\prime})=\int_{\mathcal{I}}\omega_{0}(\delta,\delta^{\prime})dudx^{2}+\int_{S^{2}}\sum_{i=1}^{\infty}\delta
F_{ru}^{(-2-i,0)}\wedge\delta^{\prime}a_{i}dx^{2}$ (81)

All three ingredients in the charge calculation are well defined and finite:
the limit $t\to+\infty$, the integration on $\mathcal{I}$ and the series.

We can now show the full hierarchy of charges for arbitrary $O(r^{n})$ LGT in
QED. The electric charges associated with an LGT $\Lambda_{\varepsilon}$ can
be calculated from (81), substituting the sequence coordinates
$\\{\epsilon_{i}\\}$. By the equation

$\delta Q_{\varepsilon}=\Omega_{ren}(\delta,\delta_{\Lambda_{\varepsilon}}),$
(82)

we have

$Q_{\varepsilon}=\sum_{j=0}^{\infty}\int_{S^{2}}\sqrt{q}\epsilon_{j}F_{ru}^{-2-j,0}dx^{2}$
(83)

where we are using that $F_{\nu\mu}$ is invariant under
$\delta_{\Lambda_{\varepsilon}}$. This expression is the same as the one
obtained in Campiglia:2018dyi .

Observe that the full algebra of charges is abelian:

$\\{Q_{\varepsilon_{1}},Q_{\varepsilon_{2}}\\}=0,\quad\forall\varepsilon_{1},\varepsilon_{2}$
(84)

Remark: The expression we found for the renormalized symplectic potential
gives a symplectic form from which the electric charges $Q_{\varepsilon}$ can
be obtained. Given the procedure shown in this section, some ambiguities can,
in principle, spoil the conservation of the charges from $\mathcal{I}^{-}$ to
$\mathcal{I}^{+}$ in a scattering process. Nevertheless, observe that the
charges we obtain here are exactly the ones given in Campiglia:2018dyi  for
$\mathcal{I}^{+}$, and it can also be shown, via the same arguments and
prescriptions, that the charges for $\mathcal{I}^{-}$ are,

$Q_{\varepsilon}^{-}=\sum_{j=0}^{\infty}\int_{S^{2}}\sqrt{q}\epsilon_{j}F_{rv}^{-2-j,0}dx^{2},$
(85)

where $v$ is now the advanced coordinate (suitable to describe
$\mathcal{I}^{-}$), with respect to which the fall off’s (14), (10), etc., are
written in analogy. Since the conservation of the charges in the classical
theory was proven by Campiglia and Laddha (cf. Campiglia:2018dyi ), then any
possible ambiguities in the renormalization do not affect the scattering
processes between $\mathcal{I}^{-}$ and $\mathcal{I}^{+}$. It would be
interesting to study if there is any impact of these ambiguities beyond the
tree-level, in the non-abelian case, or in the presence of other fields.

##  5 Duality extension of tower of asymptotic charges

In the previous sections, we treated only the electric part of Maxwell theory,
renormalizing the symplectic potential in the extended phase space to contain
the subn-leading charges in a natural framework. In this section, we extend
the phase space (again) in order to include the magnetic freedom, á la
Freidel-Pranzetti, as in Freidel:2018fsk . This type of extension has been
thoroughly studied in recent years in several contexts: electromagnetic
duality (e.g., Hosseinzadeh:2018dkh ; Donnay:2020guq ), BF theories
(Geiller:2021gdk ) and under more general structures (Ciambelli:2021nmv ).
Throughout this section, we use differential form notation without explicitly
writing the indexes to ease the notation. Also, we are considering no extra
fields.

Electromagnetism possesses a duality symmetry, which can be characterized as
follows: the Lagrangian for the theory is

$\mathcal{L}[F]=\frac{1}{2}*F\wedge F,$ (86)

where $\wedge$ is the wedge product in the space of $p$-forms on Minkowski
space $M$ and $*$ is the Hodge dual operator,
$*:\Omega^{p}(M)\rightarrow\Omega^{4-p}(M)$, in $M$. This operator satisfies

$**\alpha=(-1)^{p(4-p)+1}\alpha,\quad\alpha\in\Omega^{p}(M),$ (87)

where the extra $+1$ in the exponent comes from the signature of the metric in
Minkowski space. Therefore, taking $p=2$ and applying $*$ to $F$ in (86), we
have

$\mathcal{L}[*F]=\frac{1}{2}*F\wedge F,$ (88)

which shows the duality symmetry.

The first step in including duality symmetry is to consider the duality
extension in the standard radiative phase space. On each $\Sigma_{t}$, we have
the Freidel-Pranzetti extension for the symplectic form, Freidel:2018fsk ,

$\Omega(\delta,\delta^{\prime})=\int_{\Sigma_{t}}\delta
A\wedge\delta^{\prime}\star F+\int_{S^{2}}\delta
a_{0}\wedge\delta^{\prime}B_{0},$ (89)

where $\star$ is the Hodge dual in the hypersurface,
$a_{0}\overset{S^{2}}{=}A+d\omega_{0}$ is the electric boundary gauge field,
and $B_{0}$ is the magnetic boundary gauge field. $\omega_{0}$ is the edge
mode, which extends the phase space, $(A,a_{0})$, which now contains this
boundary field. The symplectic form now contains a corner term living in
$\partial\Sigma_{t}$.

To make the connection with our past sections definition for $A$, we have

$A_{new}+d\omega=A_{old},$ (90)

where $old$ refers to the $A$ used in the previous sections, and $new$ is the
one in the present section. In particular, the expressions for curvature
tensor and the charges are still valid. Observe that $\omega$ can be thought
of as a zero-order extension, using the same idea as the previous sections:
extending the vector potential with a large gauge symmetry.

We distinguish between symmetries that leave fixed the bulk variable $A$ and
symmetries that act only on the boundary. In the previous sections, we use
this difference when defining the extension to higher order LGT, where
$\delta_{\Lambda_{\varepsilon}}$ only acts on $A_{a}$ through the first
component. In the present section, as it was done in (Freidel:2018fsk, ), we
are isolating the bulk from the boundary action on the $\epsilon_{0}$
variation in order to have a well-defined canonical action that includes the
duality symmetry, and such that the symplectic potential is invariant under
the gauge transformation of the fields.

We are working in $\mathcal{I}^{+}$, so in (89), we take
$t\rightarrow+\infty$. The “bulk” part now is $A$ along $\mathcal{I}$, while
the boundary is $\mathcal{I}^{+}_{-}$, with topology $S^{2}$. The values at
the boundary are not independent since the boundary symmetries act
simultaneously on both $\mathcal{I}^{+}_{\pm}$ (i.e., they are independent of
$u$). Under a gauge transformation generated by $G$, both the bulk and the
corner fields transform,

$\delta_{G}(A,a_{0},B_{0})=(dG,-dG,0),$ (91)

so the variation $\delta_{G}$ is indeed gauge, in the sense that it has a
vanishing charge $\Omega(\delta,\delta_{G})=0$, on-shell. The electric
(magnetic) symmetry $\delta_{\epsilon_{0}}(\delta_{\lambda_{0}})$ acts only on
the electric (magnetic) boundary field,

$\delta_{\epsilon_{0}}(A,a_{0},B_{0})=(0,d\epsilon_{0},0),\quad\delta_{\lambda_{0}}(A,a_{0},B_{0})=(0,0,d\lambda_{0}),$
(92)

where $d\lambda_{0}$ is locally but not globally exact (such as in the
standard examples of a charge in the $z-$axis, see section V in
Freidel:2018fsk ). Observe that on-shell, upon acting with $G$, we obtain the
identity

$dB_{0}=\star F,$ (93)

which on $\mathcal{I}^{+}$ gives us $dB_{0}=F_{ru}^{(-2,0)}$.

Our extended phase space of section 4.1 adapts well to the construction given
above to the duality extension. The gauge transformation $\Lambda_{\alpha}$ is
the “bulk” potential, generated by the boundary fields in the sequence
$\alpha$, in a hierarchy graded by the correspondent power of $r$. Therefore,
we can extend directly as,

$\Omega_{ren}(\delta,\delta^{\prime})=\int_{\mathcal{I}}\left[\delta
A\wedge\delta^{\prime}\star F\right]_{ren}+\int_{S^{2}}\delta
a_{0}\wedge\delta^{\prime}B_{0}+\int_{S^{2}}\sum_{k=1}^{\infty}\delta
a_{k}\delta^{\prime}dB_{k},$ (94)

where $a_{k}$ are functions on the sphere, $B_{k}$ are 1-forms in the sphere
locally (but not necessarily globally) exact, $a_{0}$ is a $1-$form 999$a_{0}$
is not generally a gradient., and $ren$ indicates that is the renormalized
term, given by (81). We define the action of a gauge transformations $G$ (of
order $r^{n}$ arbitrary) as

$\delta_{G}A=dG,\quad\delta_{G}a_{0}=dG_{0},\quad\delta_{G}a_{j}=G_{j},\quad\delta_{G}B_{j}=0,j\geq
1.$ (95)

Evaluating the symplectic form in $\delta_{G}$,

$\Omega_{ren}(\delta,\delta_{G})=-\delta\left(\int_{\mathcal{I}}\left[dG\wedge\star
F\right]_{ren}+\int_{S^{2}}\delta
dG_{0}\wedge\delta^{\prime}B_{0}+\int_{S^{2}}\sum_{k=0}^{\infty}G_{k}\delta
dB_{k}\right),$ (96)

which on-shell and after integrating by parts, we obtain (after the
renormalization, allowing variations $\delta$ such that $\delta A$ has order
higher than $r^{0}$ before taking the limit $t\rightarrow+\infty$)

$dB_{k}=F_{ru}^{(-2-k,0)},\quad k\geq 0.$ (97)

This equality establishes the value of the magnetic boundary gauge field as
the field strength functions.

Finally, we will denote the magnetic variations acting on $B_{k}$’s as
$\lambda=\\{\lambda_{i}\\}_{i\geq 0}$, in the same fashion as we define the
LGT generators. Electric (magnetic) variations act as follows on the extended
phase space variables,

$\delta_{\epsilon_{k}}A=0,\quad\delta_{\epsilon_{k}}a_{0}=\delta_{0k}d\epsilon_{k},\quad\delta_{\epsilon_{k}}a_{j}=\delta_{kj}\epsilon_{k},\quad\delta_{\epsilon_{k}}B_{j}=0,\quad
k\geq 0,j\geq 1$ (98)
$\delta_{\lambda_{k}}A=0,\quad\delta_{\lambda_{k}}a_{j}=0,\quad\delta_{\lambda_{k}}B_{j}=\delta_{kj}d\lambda_{k},\quad
k,j\geq 0,$ (99)

where $d\lambda_{k}$ is locally but not globally defined, and $\delta_{ij}$ is
Kroenecker delta.

###  5.1 Charges and dual charges and their algebra

By computing $\Omega_{ren}(\delta,\delta_{\Lambda_{\varepsilon}})$ and
$\Omega_{ren}(\delta,\delta_{\Lambda_{\lambda}})$, we obtain the electric
(denoted as $Q$) and magnetic (denoted as $\tilde{Q}$) charges,

$\displaystyle Q_{\varepsilon}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\int_{S^{2}}\epsilon_{k}dB_{k}$ (100)
$\displaystyle\tilde{Q}_{\lambda}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\int_{S^{2}}a_{k}d^{2}\lambda_{k},$ (101)

where the first integral gives directly 83, thanks to 97, and the last
integral does not vanish due to the failure of $d\lambda$ to be globally
exact.

Finally, we have can compute the charge algebra. As the electric charges, the
magnetic charges $\tilde{Q}_{\lambda}$ are abelian,

$\\{\tilde{Q}_{\lambda},\tilde{Q}_{\lambda^{\prime}}\\}=\delta_{\lambda}\sum_{k=0}^{\infty}\int_{S^{2}}a_{k}d^{2}\lambda^{\prime}_{k}=0.$
(102)

The mixed Poisson bracket gives a non-trivial component of the algebra,

$\\{Q_{\varepsilon},\tilde{Q}_{\lambda}\\}=\delta_{\varepsilon}\sum_{k=0}^{\infty}\int_{S^{2}}a_{k}d^{2}\lambda_{k}=\sum_{k=0}^{\infty}\int_{S^{2}}\epsilon_{k}d^{2}\lambda_{k}=:c_{k}$
(103)

This term shows that the boundary duality symmetry algebra possesses a
hierarchy of central charges, $\\{c_{k}\\}_{k\geq}$. We leave it to future
works to analyze in detail this fact in the context of soft theorems and Ward
identities.

##  6 Outlook

In this work, we obtain a well-defined symplectic form on $\mathcal{I}^{+}$
for the extended phase space of classical QED through a renormalization
procedure from the original symplectic form, giving a derivation from first
principles. With this symplectic form, the higher order LGT can be associated
with the subn-leading electric charges acting canonically on the phase space.
The expressions of the charges associated with the $O(r^{n})$ LGT are then
obtained, in agreement with the expressions previously proposed in
Campiglia:2018dyi  by means of the tree-level subn-leading formulas. We
compute the full electromagnetic charge algebra using the duality symmetry
extension, showing a hierarchy of central extensions. Several future
directions are possible in the framework of our work.

First, within the abelian theory, it would be interesting to extend the
analysis to include loop corrections to the soft photon factorization formulas
(Bern:2014oka ; He:2014bga ). This could lead to some new structure within the
charge hierarchy and between electric and magnetic charges. One of the main
difficulties in this line of work is the appearance of infrared divergences.
New advances in celestial CFT methods (see Pasterski:2021rjz ; Guevara:2021abz
; Albayrak:2020saa ; Gonzalez:2020tpi ; Raclariu:2021zjz , and references
therein) seem to be well suited for the incorporation of these effects. It
would also be interesting, given the recent developments in the study of
electromagnetic asymptotic charges at spatial infinity, such as consistently
accommodating $\ln(r)$ terms (Fuentealba:2023huv ) and the study in higher
dimensions (Fuentealba:2023rvf ), to establish a connection between the
symplectic structure at null infinity with that at spatial infinity.

Second, the extension to non-abelian gauge theories. In Yang-Mills theory,
extending the renormalization procedure would allow us to construct a well-
defined symplectic structure on an extended phase space to compute the
subleading charges and their algebra. As shown in Campiglia:2021oqz , the
first step towards this is to consider a linearized extension of the phase
space and restrict the charges up to $O(r)$ terms. Some progress is being made
in this direction, Nagy:2022xxs .

Finally, it would be interesting to study possible extensions of this
renormalization procedure in the context of gravity. As recent works suggest,
the study of higher-order diffeomorphisms seems to be a key ingredient in the
extensions of the phase spaces for gravity. In Compere:2017wrj , higher-order
multipole moments generated via specific diffeomorphisms were studied and
showed that they are Noether charges. In the null infinity sector it has been
proposed in Campiglia:2016jdj ; Campiglia:2016efb , and worked out more
recently in Freidel:2021dfs ; Horn:2022acq , that asymptotic diffeomorphisms
generated by certain $O(r)$ sphere-vector fields are behind the sub-subleading
soft graviton factorization Cachazo:2014fwa . A similar idea as the one
presented here could be used to identify an extended space supporting these
singular transformations.

## Aknowledgements

I would like to thank Miguel Campiglia for his comments, discussions, and
feedback on the process and the final manuscript, and also to the participants
of the Workshop on Celestial Symmetries, held in Montevideo in March 2022,
with whom the contents of this paper were discussed, in particular Laurent
Freidel, Marc Geiller, Alok Laddha, Silvia Nagy, Daniele Pranzetti and Céline
Zwickel. I would also like to thank Ali Seraj and Oscar Fuentealba for their
comments on the first version of the paper. I also thank the two referees for
useful comments and corrections. This work was partly supported by a CAP Ph.D.
fellowship, the ANII project FCE 2019-155865, and PEDECIBA.

##  Appendix A Recursive formula for $\epsilon^{ab}F_{ab}$

In this appendix we prove eq. (21). By writing the Bianchi identities

$\displaystyle\partial_{r}F_{ab}+\partial_{a}F_{br}+\partial_{b}F_{ra}$
$\displaystyle=$ $\displaystyle 0,$ (104)
$\displaystyle\partial_{u}F_{ab}+\partial_{a}F_{bu}+\partial_{b}F_{ua}$
$\displaystyle=$ $\displaystyle 0,$ (105) $\displaystyle
D_{c}F_{ab}+D_{a}F_{bc}+D_{b}F_{ca}$ $\displaystyle=$ $\displaystyle 0,$ (106)

and taking the $u$ and $r$ derivative of the first equation, the $r$
derivative of the second one, the $D_{d}$ derivative of the third one and
contracting with $\epsilon^{ab}$, we obtain,

$\displaystyle\partial_{u}\partial_{r}\epsilon^{ab}F_{ab}$ $\displaystyle=$
$\displaystyle 2\partial_{u}\epsilon^{ab}D_{a}F_{rb}$ (107)
$\displaystyle\partial_{r}\partial_{r}\epsilon^{ab}F_{ab}$ $\displaystyle=$
$\displaystyle 2\partial_{r}\epsilon^{ab}D_{a}F_{rb}$ (108)
$\displaystyle\partial_{u}\partial_{r}\epsilon^{ab}F_{ab}$ $\displaystyle=$
$\displaystyle 2\partial_{r}\epsilon^{ab}D_{a}F_{ub}$ (109) $\displaystyle
D_{d}\epsilon^{ab}D_{c}F_{ab}$ $\displaystyle=$
$\displaystyle-2\epsilon^{ab}D_{d}D_{a}F_{bc}.$ (110)

In equations (107), (108) and (109) we substituted $\partial_{i}$ by $D_{i}$
for every $i=a,b$, since we are contracting with $\epsilon^{ab}$.

Using identities

$D_{a}D_{b}F_{cd}=D_{b}D_{a}F_{cd}-q^{ef}R_{ecab}F_{fd}-q^{ef}R_{edab}F_{cf},\quad
R_{abcd}=\frac{R}{2}(q_{ac}q_{bd}-q_{ad}q_{bc}),$ (111)

we have

$D^{d}D_{a}F_{bd}=D_{a}D^{d}F_{bd}+RF_{ab}.$ (112)

By contracting (110) with $q^{cd}$ and u, and the previous equation,

$\Delta\epsilon^{ab}F_{ab}=-2\epsilon^{ab}D_{a}D^{d}F_{bd}-2R\epsilon^{ab}F_{ab}.$
(113)

Next, consider Maxwell equation (7), and take de $D_{d}$ derivative and
contract with $\epsilon^{da}$,

$\epsilon^{da}D_{d}j_{a}=-\partial_{r}\epsilon^{da}D_{d}(F_{ua}-F_{ra})+\partial_{u}\epsilon^{da}D_{d}F_{ra}+\frac{1}{r^{2}}\epsilon^{da}D_{d}D^{b}F_{ab}.$
(114)

Substituting the previous equations, we arrive at

$2\epsilon^{ab}D_{a}j_{b}=2\partial_{u}\partial_{r}\epsilon^{ab}F_{ab}-\partial_{r}\partial_{r}\epsilon^{ab}F_{ab}-\frac{1}{r^{2}}(\Delta\epsilon^{ab}F_{ab}+2R\epsilon^{ab}F_{ab}).$
(115)

where $R=2$, is the scalar curvature of $q_{ab}$.

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