aainstitutetext: Université Libre de Bruxelles and International Solvay
Institutes, ULB-Campus Plaine CP231, B-1050 Brussels, Belgiumbbinstitutetext:
Collège de France, 11 place Marcelin Berthelot, 75005 Paris,
Franceccinstitutetext: Department of Theoretical Physics, Wroclaw University
of Science and Technology, 50-370 Wroclaw, Poland

# Asymptotic structure of Carrollian limits of Einstein-Yang-Mills theory in
four spacetime dimensions

Oscar Fuentealba a,b Marc Henneaux c Patricio Salgado-Rebolledo a and Jakob
Salzer<EMAIL_ADDRESS><EMAIL_ADDRESS>patricio.salgado-
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, three things are done. First, we study from an algebraic point
of view the infinite-dimensional BMS-like extensions of the Carroll algebra
relevant to the asymptotic structure of the electric and magnetic Carrollian
limits of Einstein gravity. In the course of this study we exhibit by
“Carroll-Galileo duality” a new infinite-dimensional BMS-like extension of the
Galilean algebra and of its centrally extended Bargmann algebra. Second, we
consider the electric Carrollian limit of the pure Einstein theory and
indicate that more flexible boundary conditions than the ones that follow from
just taking the limit of the Einsteinian boundary conditions are actually
consistent. These boundary conditions lead to a bigger asymptotic symmetry
algebra that involves spatial supertranslations depending on three functions
of the angles (instead of one). Third, we turn to the Carrollian limit of the
coupled Einstein-Yang-Mills system. An infinite-dimensional color enhancement
of the gauge algebra is found in the electric Carrollian limit of the Yang-
Mills field, which allows angle-dependent Yang-Mills transformations at
spatial infinity, not available in the Einstein-Yang-Mills case prior to
taking the Carrollian electric limit. This enhancement does not occur in the
magnetic limit.

††preprint: ULB-MathPhys-22/05

##  1 Introduction

The Carroll algebra $\mathcal{C}$ is one of the contractions of the Poincaré
algebra, obtained by sending the speed of light $c$ to zero
(“ultrarelativistic limit”) LevyLeblond:1965 ; Bacry:1968zf . Its commutation
relations are

$\displaystyle[M_{i},M_{j}]=\epsilon_{ijk}M_{k},\quad[M_{i},B_{j}]=\epsilon_{ijk}B_{k},\quad[B_{i},B_{j}]=0,$
$\displaystyle[M_{i},P_{j}]=\epsilon_{ijk}P_{k},\quad[P_{i},B_{j}]=\delta_{ij}E,\quad[M_{i},E]=0\quad[E,B_{i}]=0$
(1) $\displaystyle[P_{i},P_{j}]=0,\quad[P_{i},E]=0.$

where $P_{i}$, $M_{i}$, $E$ and $B_{i}$ are respectively the generators of
spatial translations, spatial rotations, time translations and Carroll boosts.

The Carrollian limit of Einstein gravity arose initially as the “strong
coupling limit” Isham:1975ur  or “zero signature limit” Teitelboim:1978wv ;
Henneaux:1981su  of general relativity. It is relevant to the description of
the generic behaviour of the gravitational field in the vicinity of a
spacelike singularity Belinsky:1970ew ; Belinsky:1982pk  and even more so when
$p$-forms are included Damour:2000wm ; Damour:2002et  (see the review
Belinski:2017fas ). The geometry of the Carrollian limit was constructed in
Henneaux:1979vn . Since then, many applications and properties of Carrollian
structures have been studied, in particular in the context of the geometry of
null surfaces and the BMS symmetry Dautcourt:1997hb ; Duval:2014uoa ;
Duval:2014uva  (see the recent articles Donnay:2022aba ; Figueroa-
OFarrill:2021sxz  for an updated, comprehensive list of references and
Bergshoeff:2022eog  for a review of non-lorentzian theories).

It was shown recently that there are at least two different Carrollian limits
of general relativity, an “electric” one and a “magnetic” one Henneaux:2021yzg
(see also deBoer:2021jej ; Figueroa-OFarrill:2022mcy ; Hansen:2021fxi  for a
different perspective on Carrollian gravity theories). The electric Carrollian
action corresponds to the zero signature limit of Einstein gravity, whereas
the magnetic limit is equivalent to the Carrollian gravity theory found in
Bergshoeff:2017btm  by gauging the Carroll algebra toappear .

In a very interesting paper Perez:2021abf , the asymptotic structures of the
electric and magnetic limits of the Einstein theory were determined (see
Perez:2022jpr  for the analysis in the presence of negative cosmological
constant). It was shown that the magnetic and electric limits have distinct
asymptotic symmetry groups. While the asymptotic symmetries in the magnetic
case are simply the $c\rightarrow 0$ contractions of those of Einstein’s
theory with the same number of improper Benguria:1976in  gauge
transformations, there is an effective disappearance in the electric case of
all the dynamical improper gauge symmetries (boosts and time
(super)translations), because these become in the limit proper gauge
transformations acting trivially on the physical states.

The boundary conditions studied in Perez:2021abf  were naturally taken to be
the Carrollian contractions of the various boundary conditions considered at
spatial infinity for Einstein’s theory (Regge:1974zd ; Henneaux:2018cst ;
Henneaux:2018hdj ; Henneaux:2019yax ). It turns out that in the Carrollian
electric limit, a less strict set of boundary conditions can also be
consistently imposed. These are not obtainable through the limiting process
and lead to an infinite-dimensional enhancement of the Carroll spatial
supertranslations.

One purpose of this paper is to analyse these more flexible boundary
conditions for the electric Carrollian limit of Einstein gravity. We
explicitly show that while the boosts and the time supertranslations remain
pure gauge, as in Perez:2021abf , new space supertranslations introducing
further functions of the angles indeed appear. These involve three functions
of the angles and have no analog in the investigations of Regge:1974zd ;
Henneaux:2018cst ; Henneaux:2018hdj ; Henneaux:2019yax  of the Einstein case.

In order to shed light on the algebras that can occur asymptotically, we study
prior to the asymptotic considerations the Carroll-like contractions of the
BMS4 algebra Bondi:1962px ; Sachs:1962wk  from a purely algebraic viewpoint.
This analysis enables one to exhibit, by “Carroll-Galileo duality”, a new
infinite-dimensional BMS-like extension of the Galilean algebra and of its
centrally extended Bargmann algebra. We also study the BMS-like extensions of
some ideals of the Carroll algebra relevant to the asymptotic symmetries.

Another purpose of our paper is to include the coupling to the Yang-Mills
theory. This system is of interest in the asymptotic analysis context since it
was shown in Tanzi:2020fmt ; Tanzi:2021prq  that there were obstructions to
finding at spatial infinity the angle-dependent color transformations
exhibited at null infinity Strominger:2013lka ; Barnich:2013sxa ;
Banerjee:2021uxe , except in the abelian case studied in Henneaux:2018gfi . We
start by considering the Yang-Mills theory on a flat Carroll background. We
show that while the magnetic limit presents the same difficulties as its
Lorentzian counterpart, the electric limit allows again for a greater
flexibility. We present boundary conditions that are compatible with the
Carroll symmetry and invariant under an angle-dependent color group.

We then extend these results to the different Carrollian limits of the coupled
Einstein-Yang-Mills system, which is a rather direct task once the flat
Carroll case has been understood.

Our paper is organized as follows. Sections 2 and 3 study the algebraic
structure of the Carroll algebra and its BMS-like extensions. A BMS-like
extension of the Carroll algebra is by definition a semi-direct sum of the
homogeneous Carroll algebra and an infinite-dimensional abelian algebra
parametrized by functions on the sphere (the “supertranslations”) that
contains the ordinary spacetime translations. This is just the analog of the
BMS extension of the Poincaré group Bondi:1962px ; Sachs:1962wk . We use the
3+1 parametrization of the symmetry introduced in Troessaert:2017jcm  (see
also Henneaux:2018cst ). We show that there exist at least three inequivalent
BMS-like extensions. By applying the same methods to the non-relativistic
limit, we exhibit similarly a third, and to our knowledge new, BMS-like
extension of the Galilean algebra and of its centrally extended version, the
Bargmann algebra, in addition to the two extensions already constructed in
Batlle:2017yuz . We study next the asymptotic analysis of the electric
Carrollian limit of pure Einstein gravity, after a brief survey of the
Minkowskian results (Sections 4 and 5). We then turn the electric Carrollian
limit of the Yang-Mills theory in flat Carroll space (Section 6) and finally
to the different Carrollian limits of the combined Einstein-Yang-Mills system
(Section 7). Section 8 contains concluding comments.

##  2 Ideals of the Carroll algebra

Among the Carroll transformations, $P_{i}$ are $M_{i}$ are kinematical
transformations defined within equal time hypersurfaces, while $E$ and $B_{i}$
are dynamical transformations involving time evolution. The kinematical
transformations form a subalgebra isomorphic to the algebra of Euclidean
displacements $iso(3)$.

The Carroll algebra is not simple and possesses many ideals. For instance, the
dynamical transformations $E$ and $B_{i}$ form an abelian ideal $\mathcal{D}$.
The quotient of the Carroll algebra by the ideal $\mathcal{D}$ is isomorphic
to $iso(3)$,

$\frac{\mathcal{C}}{\mathcal{D}}\simeq iso(3).$ (2)

The energy $E$ (time translations) by itself also generates a one-dimensional
(abelian) ideal $\mathcal{I}$. The quotient of the Carroll algebra by the
ideal $\mathcal{I}$ is isomorphic to the semi-direct sum of $iso(3)$ and a
three-dimensional abelian algebra $t_{3}$ transforming in the vector
representation of $iso(3)$, denoted $d_{3}$

$\frac{\mathcal{C}}{\mathcal{I}}\simeq iso(3)\oplus_{\sigma}t_{3}\equiv d_{3}$
(3)

(of which $t_{3}$ is an ideal).

The subalgebra of homogeneous Carroll transformations is spanned by the
spatial rotations and the boosts. It is six-dimensional and isomorphic to
$so(3)\oplus_{\sigma}t_{3}$. These transformations grow up at infinity
linearly with $r$, while the spatial translations and the time translations
tend to constants.

##  3 Supertranslations

###  3.1 Carroll-BMS ($\mathcal{C}$-BMS) algebra

Each of the previous quotient algebras, including the Carroll algebra itself,
admits infinite-dimensional extensions by supertranslations. We now describe
extensions for each of them, even though only the extensions of $\mathcal{C}$
and $iso(3)$ are realized through the asymptotic conditions displayed below.

Before turning to these infinite-dimensional extensions, it is helpful to
briefly review the Lorentzian case. As is well-known, both at null
Bondi:1962px ; Sachs:1962wk  and spatial infinity Henneaux:2018cst ;
Henneaux:2018hdj , the Poincaré algebra is extended by supertranslations that
are labelled by an arbitrary function on the 2-sphere. These transform as an
infinite-dimensional representation of the homogeneous part of the Poincaré
algebra, i.e., the Lorentz algebra. In a Hamiltonian description around
spatial infinity Henneaux:2018cst ; Henneaux:2018hdj , one naturally finds a
parametrization in terms of one even $T(\theta,\varphi)$ and one odd function
$W(\theta,\varphi)$ on the 2-sphere that transform as

$\hat{T}=Y^{A}\partial_{A}T-3bW-\partial_{A}b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W-b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W,\qquad\hat{W}=Y^{A}\partial_{A}W-bT.$ (4)

In these expressions, $b=b_{i}n^{i}$ parametrizes the boosts and $Y^{A}$ the
rotations, where the $n_{i}$’s are the components of the unit normal vector to
the 2-spheres – we refer the reader to the original works for our notation. An
analysis in hyperbolic coordinates in the neighbourhood of spatial infinity
Troessaert:2017jcm ; Henneaux:2018cst  allows one to match this Hamiltonian
description involving an even and an odd function on the 2-sphere to the
description in terms of the single, unrestricted function on the 2-sphere that
one finds at null infinity.

Let us now turn to the Carroll algebra $\mathcal{C}$. In a similar way to the
Poincaré algebra, the Carroll algebra can be extended by Carroll
supertranslations, which form an infinite-dimensional representation of the
homogeneous Carroll subalgebra and commute among themselves. One expedient
manner to obtain the extension is to take the $c\rightarrow 0$ contraction of
the BMS4 algebra in a way compatible with the Carrollian contraction of the
Poincaré algebra.

In a formulation adapted to the Hamiltonian description, Carroll
supertranslations are parametrized by one even function $T(\theta,\varphi)$
and one odd function $W(\theta,\varphi)$ on the $2$-sphere. The transformation
properties of the parameters of supertranslations under boosts can be obtained
from (4) by the scaling limit $T\rightarrow cT,b\rightarrow cb,W\rightarrow
W$, yielding the transformation law

$\hat{T}=Y^{A}\partial_{A}T-3bW-\partial_{A}b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W-b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W,\qquad\hat{W}=Y^{A}\partial_{A}W.$ (5)

Here, $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}$ stands for the covariant derivative associated to the metric
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}_{AB}$ on the 2-sphere. This contraction reproduces in particular the
commutation relations (1) of the energy and the linear momentum with the
Carroll boosts, i.e., $[P_{i},B_{j}]=\delta_{ij}E$ and $[E,B_{i}]=0$.

The symmetry algebra defined by (5), together with the commutators
$[M_{i},M_{j}]=\epsilon_{ijk}M_{k}$, $[M_{i},B_{j}]=\epsilon_{ijk}B_{k}$ and
$[B_{i},B_{j}]=0$ of the homogeneous Carroll generators, is denoted by
$\mathcal{C}$-BMS. It arises in the magnetic Carrollian limit of Einstein
gravity Perez:2021abf . The charge associated with the $\mathcal{C}$-BMS
transformations were found to take the form

$Q^{M}_{\xi}=b_{i}B^{i}+\frac{1}{2}b_{ij}M^{ij}+\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,T\,\mathcal{T}+\oint d^{2}xW\,\mathcal{W}\,,$ (6)

($M^{ij}=\epsilon^{ijk}M_{k}$) where the Poisson-Dirac brackets of the Carroll
boosts and the spatial rotations with the Carroll supertranslations are given
by

$\displaystyle\\{B_{i},\mathcal{T}(\theta,\phi)\\}$ $\displaystyle=$
$\displaystyle 0\,,$ (7) $\displaystyle\\{B_{i},\mathcal{W}(\theta,\phi)\\}$
$\displaystyle=$
$\displaystyle-3n_{i}\mathcal{T}(\theta,\phi)-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}n_{i}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\mathcal{T}(\theta,\phi)-n_{i}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\mathcal{T}(\theta,\phi)\,,$ (8)
$\displaystyle\\{M^{ij},\mathcal{T}(\theta,\phi)\\}$ $\displaystyle=$
$\displaystyle-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{T}(\theta,\phi))\,,$ (9)
$\displaystyle\\{M^{ij},\mathcal{W}(\theta,\phi)\\}$ $\displaystyle=$
$\displaystyle-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{W}(\theta,\phi))\,,$ (10)

in agreement with (5).

Interestingly, the transformation laws

$\hat{T}=Y^{A}\partial_{A}T,\qquad\hat{W}=Y^{A}\partial_{A}W-bT$ (11)

arising from the opposite scaling limit $T\rightarrow
T,b\rightarrow\frac{1}{c}b$, $W\rightarrow\frac{1}{c}W$ with
$c\rightarrow\infty$ define one BMS extension of the Galilean algebra
(reproducing in particular the Galilean relation $[P_{i},B_{j}]=0$ and
$[E,B_{i}]=P_{i}$). The comparison of this BMS extension with the extensions
constructed in Batlle:2017yuz  is discussed in the next subsection.

###  3.2 $d_{3}$-BMS algebra

The ideals of the Carroll algebra $\mathcal{C}$ listed above can be extented
to include supertranslations. One can for instance enlarge the ideal generated
by the time translation generator $E$ to include all time supertranslations.

If one takes the quotient of the algebra $\mathcal{C}$-BMS by this ideal, one
gets an extension of $d_{3}$ by space supertranslations, i.e., the algebra
generated by $B_{i}$, $M_{ij}$ and $\mathcal{W}$ with (8) replaced by

$\\{B_{i},\mathcal{W}(\theta,\phi)\\}=0.$ (12)

We call this algebra the $d_{3}$-BMS algebra. It is not realized through the
asymptotic conditions given below.

###  3.3 $iso(3)$-BMS algebra

Similarly, one can enlarge the ideal generated by the time translation
generator $E$ and the boosts $B_{i}$ to include all time supertranslations.
This time, the quotient of the Carroll-BMS algebra by this ideal is spanned by
$M_{ij}$ and the spatial supertranslations $\mathcal{W}$, with the above
commutation relations.

This algebra is called the $iso(3)$-BMS algebra and arises in the electric
Carrollian limit of Einstein gravity Perez:2021abf .

###  3.4 Extended $iso(3)$-BMS algebra

Another infinite-dimensional extension of $iso(3)$ relevant to the asymptotic
analysis of the electric limit of Einstein theory is obtained by adding more
spatial supertranslations, which we denote $\mathcal{W}_{A}(\theta,\phi)$
(parametrized by $I^{A}$) and which commute with the spatial supertranslations
parametrized by $W$. While the $W$’s are odd functions on the sphere, the
$I^{A}$ are even functions.

The transformation of the functions $I^{A}$ under rotations is

$\hat{I}^{A}=Y^{B}\partial_{B}I^{A}-I^{B}\partial_{B}Y^{A}\,,$ (13)

leading to the corresponding brackets for the generators $\mathcal{W}_{A}$,

$\displaystyle\\{M^{ij},\mathcal{W}_{A}(\theta,\phi)\\}$
$\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern
0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{W}_{A}(\theta,\phi))-x^{[i}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}e^{j]B}\mathcal{W}_{B}(\theta,\phi)\,.$ (14)

This algebra is called the extended $iso(3)$-BMS algebra. It cannot be
obtained by contraction of the BMS4 algebra since it contains more spatial
supertranslations. These are parametrized by one odd function ($W$) and two
even functions ($I^{A}$) on the sphere.

###  3.5 Spherical harmonics presentation

####  3.5.1 BMS4 algebra

The supertranslation generators $\mathcal{T}$ and $\mathcal{W}$ can be
decomposed in terms of spherical harmonics,

$\mathcal{T}=\sum_{l\geq
0,\hbox{\scriptsize{even}}}\;\sum_{m=-l}^{m=l}P_{lm}Y^{l}_{m}\,,\qquad\mathcal{W}=\sum_{l\geq
1,\hbox{\scriptsize{odd}}}\;\sum_{m=-l}^{m=l}P_{lm}Y^{l}_{m}\,,$ (15)

The BMS4 commutation rules take then the form Sachs:1962zza

$[B_{i},P_{lm}]=\sum_{l^{\prime}}\sum_{m^{\prime}}\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}P_{l^{\prime}m^{\prime}}$
(16)

in addition to the commutation relations of the homogeneous Lorentz group
$[M_{i},M_{j}]=\epsilon_{ijk}M_{k}$, $[M_{i},B_{j}]=\epsilon_{ijk}B_{k}$,
$[B_{i},B_{j}]=-\epsilon_{ijk}M_{k}$ and the commutation relations
$[M_{i},P_{lm}]$ that express that the spherical harmonics $\\{Y^{l}_{m}\\}$
($m=-l,\cdots,l$) form a basis of the spin-$l$ representation of the rotation
group, so that $\mathcal{T}$ and $\mathcal{W}$ transform as scalar functions
on the sphere.

The detailed form of the structure constants
$\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}$ will not be needed here. It
can be found in Sachs:1962zza  (after making the appropriate normalization of
the $P_{lm}$’s explained in Troessaert:2017jcm  and Henneaux:2018cst ). The
only property of the $\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}$’s that
we will need is

$\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}=0\hbox{ unless
$l=l^{\prime}\pm 1$},$ (17)

so that (16) involves only two contributions,

$[B_{i},P_{lm}]=\sum_{m^{\prime}}\left(C_{i}\right)_{lm}^{l-1,m^{\prime}}P_{l-1,m^{\prime}}+\sum_{m^{\prime}}\left(C_{i}\right)_{lm}^{l+1,m^{\prime}}P_{l+1,m^{\prime}}$
(18)

Furthermore $|m^{\prime}-m|$ must be at most equal to one.

Irreducible representations of the Lorentz algebra ($so(3,1)$) have been
systematically investigated in Naimark62 ; Gel'fand63 ; HarishChandra47 . They
are characterized by two numbers. In the work of Naimark62 ; Gel'fand63 ,
which we follow, these two numbers are denoted $l_{0}$ and $l_{1}$, and the
corresponding representation is denoted $(l_{0},l_{1})$. The first number
$l_{0}$ is a non-negative integer or half-integer and is the minimum
$so(3)$-spin occurring in the decomposition of the representation of the
Lorentz algebra according to its $so(3)$-subalgebra. The second number $l_{1}$
is an arbitrary complex number. When $l_{1}-l_{0}$ is a strictly positive
integer, the representation is finite-dimensional and $l_{1}$ is equal to the
maximum $so(3)$-spin plus one. There exist then another representation of
$so(3,1)$ characterized by the “dual” values $l^{\prime}_{0}=l_{1}$ and
$l^{\prime}_{1}=l_{0}$. This representation $(l_{1},l_{0})$ is called the
“tail” of the finite-dimensional representation $(l_{0},l_{1})$. It is
infinite-dimensional since $l^{\prime}_{1}-l^{\prime}_{0}=-(l_{1}-l_{0})$ is
strictly negative.

The two numbers $l_{0}$ and $l_{1}$ determine the structure constants
$\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}$ and the two $so(3,1)$
Casimirs Naimark62 ; Gel'fand63 ,

$C_{1}\equiv
J^{2}\equiv\frac{1}{2}M^{\alpha\beta}M_{\alpha\beta}={\mathbf{M}}^{2}-{\mathbf{B}}^{2}=-(l_{0}^{2}+l_{1}^{2})+1$
(19)

and

$C_{2}\equiv-\frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}M_{\alpha\beta}M_{\gamma\delta}=-{\mathbf{M}}\cdot{\mathbf{B}}=-il_{0}l_{1}$
(20)

where $M_{\alpha\beta}$ are the Lorentz generators in covariant form
($M_{12}=M_{3}$ etc). This formula shows that the Casimirs are invariant under
the exchange of $l_{0}$ with $l_{1}$, $(l_{0},l_{1})\rightarrow(l_{1},l_{0})$,
and thus, cannot distinguish the corresponding (in general distinct)
irreducible representations. In particular, it cannot distinguish between a
finite-dimensional representation and its tail, which have the same Casimirs.

The representation of the Lorentz group given by the supertranslations is not
irreducible, but is indecomposable Sachs:1962zza . There is a four-dimensional
invariant subspace, characterized by the values $l_{0}=0$ and $l_{1}=2$. This
is just the standard vector representation, spanned by ordinary spacetime
translations with maximum $so(3)$-spin equal to $1$. The Casimirs are
$C_{1}=-3$, $C_{2}=0$. The quotient representation of the supertranslations by
the translations is infinite-dimensional and isomorphic to the irreducible
representation with dual values $l_{0}=2$ and $l_{1}=0$. This infinite-
dimensional representation is thus the “tail” of the finite-dimensional vector
representation.

We close this brief survey of the irreducible representations of the Lorentz
algebra by giving explicitly the action of $B_{3}$ in the irreducible
representation $(l_{0},l_{1})$. To write this action, it is convenient to
decompose the representation in terms of irreducible representations of the
compact $so(3)$ subalgebra. The representations that occur have $so(3)$-spin
equal to $l_{0}$, $l_{0}+1$, $l_{0}+2$, etc and this never stops unless
$l_{1}-l_{0}$ is a positive integer. In a standard spin-basis
$\\{\xi_{lm}\\}$, the action of $B_{3}$ reads

$\displaystyle
iB_{3}\xi_{lm}=c_{l}\sqrt{l^{2}-m^{2}}\xi_{l-1,m}-a_{l}m\xi_{l,m}-c_{l+1}\sqrt{(l+1)^{2}-m^{2}}\xi_{l+1,m}$
(21) $\displaystyle l=l_{0},l_{0}+1,\cdots,\qquad m=-l,-l+1,\cdots,l-1,l,$
(22)

where

$a_{l}=\frac{il_{0}l_{1}}{l(l+1)},\qquad
c_{l}=\frac{i}{l}\sqrt{\frac{(l^{2}-l_{0}^{2})(l^{2}-l_{1}^{2})}{4l^{2}-1}}.$
(23)

The relation (21) involves a specific choice of relative $l$-dependent
normalization of the $\\{\xi_{lm}\\}$, made such that it is the same set of
coefficients $c_{l}$ that characterizes the component of $B_{3}\xi_{lm}$ along
$\xi_{l-1,m}$and $\xi_{l+1,m}$. Note that $a_{l}$ is equal to zero whenever
$l_{0}$ or $l_{1}$ vanishes.

One can bring the commutation relations (16) with $i=3$, $l=0$ and $l=1$ to
the form (21) with $l_{0}=0$, $l_{1}=2$. Similarly, modulo terms involving
$P_{1m}$ when $l=2$, one can bring the commutation relations (16) with $i=3$
and $l\geq 2$ to the form (21) with $l_{0}=2$, $l_{1}=0$.

####  3.5.2 $\mathcal{C}$-BMS algebra

With the separation of the spherical harmonics into even components
(containing the time translations) and odd components (containing the space
translations), the Carroll contraction of the BMS4 algebra is direct. Indeed,
to get from the Poincaré algebra to the Carroll algebra, one must rescale
differently the time translations ($l=0$) and the space translations ($l=1$),

$E\rightarrow cE,\qquad P_{i}\rightarrow P_{i}$ (24)

together with $B_{i}\rightarrow cB_{i}$ ($c\rightarrow 0$). This can
consistently be extended to all the supertranslations as follows,

$P_{lm}\rightarrow cP_{lm}\quad\hbox{for $l$ even, $l\not=0$},\qquad
P_{lm}\rightarrow P_{lm}\quad\hbox{for $l$ odd}$ (25)

leading to the $\mathcal{C}$-BMS algebra of the rescaled generators in the
limit $c\rightarrow 0$ (rescaled generators kept fixed),

$[\tilde{B}_{i},\tilde{P}_{lm}]=\sum_{l^{\prime}}\sum_{m^{\prime}}\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}\tilde{P}_{l^{\prime}m^{\prime}}\quad\hbox{for
$l$ odd},\qquad[\tilde{B}_{i},\tilde{P}_{lm}]=0\quad\hbox{for $l$ even}$ (26)

with $\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}$ unchanged for $l$ odd,
and being equal to zero for $l$ even. Here

$\tilde{B}_{i}=cB_{i}\,,\qquad\tilde{P}_{lm}=cP_{lm}\quad\hbox{for $l$ even,
$l\not=0$},\qquad\tilde{P}_{lm}=P_{lm}\quad\hbox{for $l$ odd.}$ (27)

From now on, we shall drop the tildes and keep the same symbol for the
rescaled generators since no confusion should arise. So we rewrite the above
brackets as

$[B_{i},P_{lm}]=\sum_{l^{\prime}}\sum_{m^{\prime}}\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}P_{l^{\prime}m^{\prime}}\quad\hbox{for
$l$ odd},\qquad[B_{i},P_{lm}]=0\quad\hbox{for $l$ even}$ (28)

The same convention will be adopted below when we apply other rescalings to
get the other Carrollian (and Galilean) contractions.

The splitting of the supertranslations into even and odd parts, natural from
the Hamiltonian description, is particularly well adapted to the Carroll
contraction.

The homogeneous Carroll algebra spanned by $M_{i},B_{i}$, being isomorphic to
$\mathfrak{iso}(3)$, has the two Casimirs

$C_{1}=B_{i}B_{i}\qquad C_{2}=B_{i}M_{i}\,.$ (29)

The supertranslation representation (5) ($\Leftrightarrow$ (28)) has
$C_{1}=C_{2}=0$.

One can imagine different rescalings of the supertranslations, where the power
of $c$ depends on $l$. The only restriction is that these should reproduce the
rescalings $E\rightarrow cE$, $P_{i}\rightarrow P_{i}$ and yield a well-
defined limit consistent with $B_{i}\rightarrow cB_{i}$ when $c\rightarrow 0$.
One possibility is to take

$P_{lm}\rightarrow c^{1-l}P_{lm}.$ (30)

In that case, one gets

$[B_{i},P_{lm}]=\sum_{m^{\prime}}\left(C_{i}\right)_{lm}^{l-1,m^{\prime}}P_{l-1,m^{\prime}}$
(31)

with unchanged $\left(C_{i}\right)_{lm}^{l-1,m^{\prime}}$ but with
$\left(C_{i}\right)_{lm}^{l+1,m^{\prime}}=0$. This representation of the
homogeneous Carroll group has also its Casimirs both equal to zero, but is
inequivalent to the previous one. Both representations are not irreducible,
but indecomposable. In the case of $\mathcal{C}$-BMS, each subspace with even
$\mathfrak{so}(3)$-spin $2k$ ($k$ arbitrary integer) is invariant and form the
finite-dimensional spin-$2k$ representation of $\mathfrak{iso}(3)$. Once one
takes the quotient by the adjacent even spins, the subspaces with definite odd
$\mathfrak{so}(3)$-spin are also invariant and provide the finite-dimensional
representations of $\mathfrak{iso}(3)$ with odd spin. In the case when the
supertranslations transform as in (31), one arrives at the same content in
terms of irreducible representations, but the quotients are nested, in the
sense that the representation with spin $j$ is obtained by taking the quotient
of the subspace with $\mathfrak{so}(3)$-spin $j$ by the subspaces with lower
$\mathfrak{so}(3)$-spin $j^{\prime}<j$.

The infinite-dimensional algebras $d_{3}$-BMS and $iso(3)$-BMS are easy to
describe in terms of a spherical harmonic decomposition of the supertranlation
generators, restricted in that case to the odd generator
$\mathcal{W}(\theta,\phi)$, since only the rotation subgroup $so(3)$ acts non
trivially. Similarly, the extended $iso(3)$-BMS has additional
supertranslantions transforming as vector fields on the sphere, which are most
conveniently expanded in vector spherical harmonics.

####  3.5.3 A note on the Galilean contractions

The Hamiltonian description of the BMS4 algebra is also adapted to the study
of the Galilean contraction

$B_{i}\rightarrow\frac{1}{c}B_{i},\qquad E\rightarrow E,\qquad
P_{i}\rightarrow\frac{1}{c}P_{i}$ (32)

($c\rightarrow\infty$). Again, there exist various possibilities. One is the
direct analog of the $\mathcal{C}$-BMS algebra and corresponds to (11). It
reads

$P_{lm}\rightarrow P_{lm}\quad\hbox{for $l$ even},\qquad
P_{lm}\rightarrow\frac{1}{c}P_{lm}\quad\hbox{for $l$ odd}$ (33)

leading to the Galilean-BMS algebra

$[B_{i},P_{lm}]=\sum_{l^{\prime}}\sum_{m^{\prime}}\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}P_{l^{\prime}m^{\prime}}\quad\hbox{for
$l$ even},\qquad[B_{i},P_{lm}]=0\quad\hbox{for $l$ odd}$ (34)

with $\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}$ unchanged for $l$ even,
and being equal to zero for $l$ odd.

This contracted algebra differs from the algebras $\mathfrak{nrbms}^{\pm}$
considered in Batlle:2017yuz  and is thus new. The algebra
$\mathfrak{nrbms}^{+}$ corresponds to a rescaling analogous to (30),

$P_{lm}\rightarrow c^{-l}P_{lm},$ (35)

leading to

$[B_{i},P_{lm}]=\sum_{m^{\prime}}\left(C_{i}\right)_{lm}^{l+1,m^{\prime}}P_{l+1,m^{\prime}}$
(36)

with unchanged $\left(C_{i}\right)_{lm}^{l+1,m^{\prime}}$ but with
$\left(C_{i}\right)_{lm}^{l-1,m^{\prime}}=0$. As to the algebra
$\mathfrak{nrbms}^{-}$, it corresponds to the rescaling

$P_{00}\rightarrow c^{2}P_{00},\qquad P_{lm}\rightarrow
c^{l}P_{lm}\quad(l\not=0)$ (37)

leading to

$[B_{i},P_{1m}]=0,\quad[B_{i},P_{lm}]=\sum_{m^{\prime}}\left(C_{i}\right)_{lm}^{l-1,m^{\prime}}P_{l-1,m^{\prime}}\quad(l\not=1)$
(38)

with unchanged $\left(C_{i}\right)_{lm}^{l-1,m^{\prime}}$ ($l\not=1$) but with
$\left(C_{i}\right)_{lm}^{l+1,m^{\prime}}=0$. [The Galilean algebra can also
be obtained from the Poincaré algebra through the rescalings
$B_{i}\rightarrow\frac{1}{c}B_{i}$, $E\rightarrow c^{2}E$, $P_{i}\rightarrow
cP_{i}$.]

In fact, the paper Batlle:2017yuz , to which we refer for the details, studied
the extensions of the Bargmann algebra, which is the central extension of the
Galilean algebra. They considered scalings equivalent to the ones considered
here, adapted to the central extension. This raises the question as to whether
the Galilean-BMS algebra (34) admits a central extension that would make it
another infinite-dimensional BMS-extension of the Bargmann algebra.

The answer is affirmative, as can be seen by taking the appropriate limit of
the direct sum BMS${}_{4}\oplus u(1)$ of the BMS4 algebra with the abelian
algebra $u(1)$, the generator of which is denoted by $C$. One thus has prior
to contraction

$[B_{i},C]=0,\qquad[P_{lm},C]=0,\qquad[M_{i},C]=0$ (39)

We then “twist” the zero mode sector spanned by the generators $(P_{00},C)$
that commute with $M_{i}$ through the redefinitions

$E=P_{00}+C,\quad Z=\frac{1}{2}(P_{00}-C),\quad\Leftrightarrow\quad
P_{00}=\frac{1}{2}E+Z,\quad C=\frac{1}{2}E-Z$ (40)

Finally, we perform the standard rescalings
$B_{i}\rightarrow\tilde{B}_{i}=\frac{1}{c}B_{i}$,
$M_{i}\rightarrow\tilde{M}_{i}=M_{i}$ and (33) for $l>0$, together with the
zero-mode rescaling

$E\rightarrow\tilde{E}=E,\qquad Z\rightarrow\tilde{Z}=\frac{1}{c^{2}}Z\,.$
(41)

This brings the algebra of the rescaled generators with the boosts to the form
(dropping the tildes)

$\displaystyle[B_{i},E]=\sum_{m}\left(C_{i}\right)_{00}^{1m}P_{1m},\qquad[B_{i},Z]=0$
(42)
$\displaystyle[B_{i},P_{lm}]=\sum_{l^{\prime}}\sum_{m^{\prime}}\left(C_{i}\right)_{lm}^{l^{\prime}m^{\prime}}P_{l^{\prime}m^{\prime}}\>\;\hbox{for
$l$ even $>0$},$ (43)
$\displaystyle[B_{i},P_{1m}]=\left(C_{i}\right)_{1m}^{00}Z$ (44)
$\displaystyle[B_{i},P_{lm}]=0\;\;\hbox{for $l$ odd $>1$}$ (45)

and provides indeed a BMS-like extension of the Bargmann algebra different
from those of Batlle:2017yuz .

##  4 Brief overview of boundary conditions in Einstein gravity

After these algebraic preliminaries, we now turn to the asymptotic analysis of
Carrollian gravities, starting with a brief overview of Einstein gravity.

The Hamiltonian action of Einstein gravity in four spacetime dimensions reads

$S[g_{ij},\pi^{ij},N,N^{i}]=\int dt\left[\int
d^{3}x\left(\pi^{ij}\dot{g}_{ij}-N\mathcal{H}-N^{i}\mathcal{H}_{i}\right)-B_{\infty}\right]\,.$
(46)

Here, $N$ and $N^{i}$ stand for the lapse and shift functions, respectively.
The variation of the Hamiltonian action with respect to these functions
imposes the following constraints on the momentum $\pi^{ij}$ and the
3-dimensional metric $g_{ij}$

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{g}}\left(\pi^{ij}\pi_{ij}-\frac{1}{2}\pi^{2}\right)-\sqrt{g}R\approx
0\,,$ (47) $\displaystyle\mathcal{H}_{i}$ $\displaystyle=$
$\displaystyle-2\nabla^{j}\pi_{ij}\approx 0\,.$ (48)

The boundary term at spatial infinity $B_{\infty}$ depends on the boundary
conditions and it turns out to be the standard ADM energy when the lapse and
shift functions behave asymptotically as $N\rightarrow 1$ and
$N^{i}\rightarrow 0$, respectively Regge:1974zd . The action of an arbitrary
diffeomorphism $\xi^{\mu}=(\xi^{\perp}\equiv\xi,\xi^{i})$ on the dynamical
fields yields the following infinitesimal transformation laws

$\displaystyle\delta_{\xi,\xi^{i}}g_{ij}$ $\displaystyle=$
$\displaystyle\frac{2\xi}{\sqrt{g}}\left(\pi_{ij}-\frac{1}{2}g_{ij}\pi\right)+\mathcal{L}_{\xi}g_{ij}\,,$
(49) $\displaystyle\delta_{\xi,\xi^{i}}\pi^{ij}$ $\displaystyle=$
$\displaystyle-\frac{\xi}{\sqrt{g}}\left(R^{ij}-\frac{1}{2}g^{ij}R\right)+\frac{\xi
g^{ij}}{2\sqrt{g}}\left(\pi^{mn}\pi_{mn}-\frac{\pi^{2}}{2}\right)$ (50)
$\displaystyle-\frac{2\xi}{\sqrt{g}}\left(\pi^{im}\pi^{j}_{m}-\frac{1}{2}\pi^{ij}\pi\right)+\sqrt{g}\left(\nabla^{i}\nabla^{j}\xi-g^{ij}\nabla_{m}\nabla^{m}\xi\right)+\mathcal{L}_{\xi}\pi^{ij}\,,$

where $\mathcal{L}_{\xi}$ denotes the spatial Lie derivative, which acts on
the fields as

$\displaystyle\mathcal{L}_{\xi}g_{ij}$ $\displaystyle=$
$\displaystyle\xi^{k}\partial_{k}g_{ij}+\partial_{i}\xi^{k}g_{kj}+\partial_{j}\xi^{k}g_{ki}\,,$
(51) $\displaystyle\mathcal{L}_{\xi}\pi^{ij}$ $\displaystyle=$
$\displaystyle\partial_{k}(\xi^{k}\pi^{ij})-\partial_{k}\xi^{i}\pi^{jk}-\partial_{k}\xi^{j}\pi^{ik}\,.$
(52)

There exist different sets of boundary conditions at spatial infinity that
have been proposed in the literature Regge:1974zd ; Henneaux:2018cst ;
Henneaux:2018hdj . For all of them, the fall-off of the dynamical fields in
spherical polar coordinates is given by

$\displaystyle g_{rr}$ $\displaystyle=1+\frac{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{rr}}{r}+\frac{h^{(2)}_{rr}}{r^{2}}+\mathcal{O}\left(r^{-3}\right)\,,$
(53) $\displaystyle g_{rA}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}_{A}+\frac{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern
0.0pt}}}_{rA}}{r}+\frac{h^{(2)}_{rA}}{r^{2}}+\mathcal{O}\left(r^{-3}\right)\,,$
(54) $\displaystyle g_{AB}$ $\displaystyle=r^{2}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}_{AB}+r\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}_{AB}+h^{(2)}_{AB}+\mathcal{O}\left(r^{-1}\right)\,,$
(55) $\displaystyle\pi^{rr}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rr}+\frac{\pi^{(2)}_{rr}}{r}+\mathcal{O}\left(r^{-2}\right)\,,$ (56)
$\displaystyle\pi^{rA}$ $\displaystyle=\frac{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rA}}{r}+\frac{\pi^{(2)rA}}{r^{2}}+\mathcal{O}\left(r^{-3}\right)\,,$
(57) $\displaystyle\pi^{AB}$ $\displaystyle=\frac{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{AB}}{r^{2}}+\frac{\pi^{(2)AB}}{r^{3}}+\mathcal{O}\left(r^{-4}\right)\,.$
(58)

What distinguishes the different sets of boundary conditions are the
conditions imposed on the leading orders $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{ij}$, $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{ij}$ in the
expansion of the fields, which must be imposed for the symplectic structure to
be finite. The Carroll contractions of the boundary conditions proposed in
Regge:1974zd ; Henneaux:2018cst  have been analysed in detail in Perez:2021abf
, with the following conclusions: (i) the Carrollian limit of the boundary
conditions of Regge:1974zd  leads to the finite-dimensional asymptotic
symmetry algebras $\mathcal{C}$ in the magnetic case, and $iso(3)$ in the
electric one; (ii) the Carrollian limit of the boundary conditions of
Henneaux:2018cst  leads to the infinite-dimensional asymptotic symmetry
algebras $\mathcal{C}$-BMS in the magnetic case, and $iso(3)$-BMS in the
electric one. Since the Carroll contractions of the boundary conditions of
Regge:1974zd ; Henneaux:2018cst  are fully understood, we shall focus here on
the boundary conditions proposed in Henneaux:2018hdj .

###  4.1 Boundary conditions of Henneaux:2018hdj

The third set of boundary conditions at spatial infinity compatible with
finiteness of the symplectic structure that has been proposed in the
literature is also BMS invariant, as are the ones of Henneaux:2018cst . It was
introduced in Henneaux:2018hdj  (see also Henneaux:2019yax  for more
information). The corresponding boundary conditions differ from the boundary
conditions of Regge:1974zd  (which do not have the BMS group as asymptotic
symmetry group) by an improper gauge transformation and for that reason, are
sometimes called diffeomorphism-twisted parity conditions.

In spherical coordinates, the boundary conditions read Henneaux:2018hdj ;
Henneaux:2019yax

$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}_{rr}$ $\displaystyle=\text{even}\,,$ (59)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern 0.0pt}}}_{A}$ $\displaystyle=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}_{A})^{\text{odd}}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\zeta_{r}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\zeta$\kern 0.0pt}}}_{A}\,,$ (60)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}_{AB}$ $\displaystyle=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{AB})^{\text{even}}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\zeta$\kern
0.0pt}}}_{B}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\zeta$\kern 0.0pt}}}_{A}+2\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}_{AB}\zeta_{r}\,,$ (61) $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rr}$
$\displaystyle=(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr})^{\text{odd}}-\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\triangle$\kern 0.0pt}}}V\,,$ (62) $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rA}$
$\displaystyle=(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rA})^{\text{even}}-\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}V\,,$ (63) $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{AB}$
$\displaystyle=(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{AB})^{\text{odd}}+\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{B}V-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}^{AB}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\triangle$\kern 0.0pt}}}V)\,,$ (64)

where $\zeta_{i}dx^{i}=\zeta_{r}(\theta,\phi)dr+r\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\zeta$\kern
0.0pt}}}_{A}(\theta,\phi)dx^{A}$, $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}$ is the covariant derivative
on the 2-sphere, and  $\triangle$ is the Laplace operator on the 2-sphere. The
function $\zeta_{r}$ is odd, while the angular component $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\zeta$\kern 0.0pt}}}_{A}$ and the
function $V$ are even under the antipodal map.

The condition $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern 0.0pt}}}_{A}=0$ (which implies $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\zeta$\kern
0.0pt}}}_{A}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\zeta_{r}$) is also imposed to insure integrability
of the boost charges, but since it can be relaxed in the Carroll electric
limit, we keep this term here.

###  4.2 Asymptotic symmetries and BMS4 algebra

This set of asymptotic conditions is preserved by the following surface
deformation parameters

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle
br+T+\text{``more''}+\mathcal{O}\left(r^{-2}\right)\,,$ (65)
$\displaystyle\xi^{r}$ $\displaystyle=$ $\displaystyle
W+\mathcal{O}\left(r^{-1}\right)\,,$ (66) $\displaystyle\xi^{A}$
$\displaystyle=$ $\displaystyle Y^{A}+\frac{1}{r}(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W+\text{``more''})+\mathcal{O}\left(r^{-2}\right)\,.$ (67)

Here:

  * •

Lorentz boosts are generated by $b=b_{i}n^{i}$ (with $b_{i}$ arbitrary
constants). Being pure vector spherical harmonic, the function $b$ obeys the
equation $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{B}b+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}b=0$.

  * •

Spatial rotations are generated by the vectors
$Y^{A}=\frac{1}{2}b_{ij}x^{i}e^{jA}$ (with $b_{ij}=-b_{ji}$ constant and
$e^{jA}$ vectors tangent to the unit sphere), which are Killing vectors of the
round 2-sphere, i.e., $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}Y_{B}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{B}Y_{A}=0$.

  * •

The parameters $T$ and $W$ are respectively arbitrary even and odd functions
on the 2-sphere under the antipodal map $\theta\rightarrow\pi-\theta$ and
$\phi\rightarrow\phi+\pi$. They generate all the supertranslations.

  * •

The terms “more” correspond to correcting improper diffeomorphisms that
preserve the condition $g_{rA}=\mathcal{O}(r^{-1})$, or equivalently
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}_{A}=0$, which makes the Lorentz boosts canonical transformations.

The even parameter $T$ and odd parameter $W$ do not obey any additional
condition, and possess non-vanishing generators. The supertranslations are
thus non-trivially realized. It follows that the asymptotic symmetry algebra
at spatial infinity is given by the BMS algebra. We refer to Henneaux:2018hdj
; Henneaux:2019yax  for the details.

##  5 Electric Carrollian limit of Einstein gravity with twisted parity
conditions

###  5.1 Magnetic limit with diffeomorphism-twisted parity conditions

The magnetic Carroll contraction of the boundary conditions given above takes
exactly the same form, because the leading orders of the transformation of the
fields under surface deformations coincide with the Einstein case. Indeed, one
now has

$\displaystyle\delta_{\xi,\xi^{i}}g_{ij}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\xi}g_{ij}\,,$ (68)
$\displaystyle\delta_{\xi,\xi^{i}}\pi^{ij}$ $\displaystyle=$
$\displaystyle-\frac{\xi}{\sqrt{g}}\left(R^{ij}-\frac{1}{2}g^{ij}R\right)+\sqrt{g}\left(\nabla^{i}\nabla^{j}\xi-g^{ij}\nabla_{m}\nabla^{m}\xi\right)+\mathcal{L}_{\xi}\pi^{ij}\,,$
(69)

and one thus sees that for $\xi$ and $\xi^{k}$ of order one, the leading terms
of (68)-(69) and (49)-(50) coincide. The conditions on the leading terms of
the metric and the conjugate momentum at spatial infinity are thus naturally
taken to be exactly the same as in (59)-(64).

Because the terms being dropped in the Hamiltonian constraints are algebraic,
the analysis of the surface terms in the canonical generators proceeeds as in
the Einstein theory. One can in particular easily verify that Carroll boosts
are integrable when $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern 0.0pt}}}_{A}=0$ and that the asymptotic symmetry algebra
is $\mathcal{C}$-BMS, as found in Perez:2021abf  for the boundary conditions
of Henneaux:2018cst .

The twisted parity conditions produce therefore the expected result in the
magnetic contraction. The electric Carrollian limit, on the other hand, opens
new possibilities and we focus on it in the rest of this section.

###  5.2 Electric limit: Action principle and transformation laws

The Hamiltonian action principle of the electric Carrollian theory of gravity
in four dimensions is given by Henneaux:1979vn , Henneaux:2021yzg

$S^{E}[g_{ij},\pi^{ij},N,N^{i}]=\int dt\left[\int
d^{3}x\left(\pi^{ij}\dot{g}_{ij}-N\mathcal{H}^{E}-N^{i}\mathcal{H}^{E}_{i}\right)-B^{E}_{\infty}\right]\,.$
(70)

Variation with respect to the lapse $N$ and shift $N^{i}$ functions enforces
again the Hamiltonian and momentum constraints, which reads in the electric
contraction,

$\displaystyle\mathcal{H}^{E}$
$\displaystyle=\frac{1}{\sqrt{g}}\left(\pi^{ij}\pi_{ij}-\frac{\pi^{2}}{2}\right)\approx
0\,,$ (71) $\displaystyle\mathcal{H}^{E}_{i}$
$\displaystyle=-2\nabla^{j}\pi_{ij}\approx 0\,.$ (72)

These constraints obey the “zero-signature” deformation algebra
Teitelboim:1978wv , Henneaux:1979vn

$\displaystyle\\{\mathcal{H}^{E}(x),\mathcal{H}^{E}(x^{\prime})\\}$
$\displaystyle=$ $\displaystyle 0\,,$ (73)
$\displaystyle\\{\mathcal{H}^{E}(x),\mathcal{H}^{E}_{i}(x^{\prime})\\}$
$\displaystyle=$
$\displaystyle\mathcal{H}^{E}(x)\partial_{i}\delta(x,x^{\prime})\,,$ (74)
$\displaystyle\\{\mathcal{H}^{E}_{i}(x),\mathcal{H}^{E}_{j}(x^{\prime})\\}$
$\displaystyle=$
$\displaystyle\mathcal{H}^{E}_{i}(x^{\prime})\partial_{j}\delta(x,x^{\prime})+\mathcal{H}^{E}_{j}(x)\partial_{i}\delta(x,x^{\prime})\,.$
(75)

Variation of the action with respect to the momentum $\pi^{ij}$ and the
3-dimensional metric $g_{ij}$ gives the following equations of motion

$\displaystyle\dot{g}_{ij}$
$\displaystyle=\frac{2N}{\sqrt{g}}\left(\pi_{ij}-\frac{1}{2}g_{ij}\pi\right)+N_{i|j}+N_{j|i}\,,$
(76) $\displaystyle\dot{\pi}^{ij}$
$\displaystyle=\frac{N}{2\sqrt{g}}g^{ij}\left(\pi^{mn}\pi_{mn}-\frac{\pi^{2}}{2}\right)-\frac{2N}{\sqrt{g}}\left(\pi^{im}\pi_{m}^{j}-\frac{1}{2}\pi^{ij}\pi\right)+\mathcal{L}_{N}\pi^{ij}\,.$
(77)

The infinitesimal transformation laws of the canonical pair
$(g_{ij},\pi^{ij})$ under diffeomorphisms generated by
$\left(\xi^{\perp}\equiv\xi,\xi^{i}\right)$ are now

$\displaystyle\delta_{\xi,\xi^{i}}g_{ij}$
$\displaystyle=\frac{2\xi}{\sqrt{g}}\left(\pi_{ij}-\frac{1}{2}g_{ij}\pi\right)+\xi_{i|j}+\xi_{j|i}\,,$
(78) $\displaystyle\delta_{\xi,\xi^{i}}\pi^{ij}$
$\displaystyle=\frac{\xi}{2\sqrt{g}}g^{ij}\left(\pi^{mn}\pi_{mn}-\frac{\pi^{2}}{2}\right)-\frac{2\xi}{\sqrt{g}}\left(\pi^{im}\pi_{m}^{j}-\frac{1}{2}\pi^{ij}\pi\right)+\mathcal{L}_{\xi}\pi^{ij}\,.$
(79)

One should note that in (77) and (79), the terms involving $\xi$ decay at
least as $r^{-3}$ at infinity, even for parameters $\xi$ that blow up as $r$,
so that the leading order $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{ij}$ is invariant under normal
hypersurface deformations.

###  5.3 Boundary conditions in the electric Carroll limit

In view of the remark just made, the Carroll contraction in the electric case
of the twisted boundary conditions yield

$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}_{rr}$ $\displaystyle=\text{even}\,,$ (80)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern 0.0pt}}}_{A}$ $\displaystyle=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}_{A})^{\text{odd}}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\zeta_{r}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\zeta$\kern 0.0pt}}}_{A}\,,$ (81)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}_{AB}$ $\displaystyle=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{AB})^{\text{even}}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\zeta$\kern
0.0pt}}}_{B}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\zeta$\kern 0.0pt}}}_{A}+2\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}_{AB}\zeta_{r}\,,$ (82) $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rr}$
$\displaystyle=(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr})^{\text{odd}}\,,$ (83)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rA}$ $\displaystyle=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rA})^{\text{even}}\,,$ (84) $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{AB}$
$\displaystyle=(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{AB})^{\text{odd}}\,,$ (85)

where the leading order of the conjugate momentum is now strictly even or odd
since the corresponding improper gauge terms are of lower order.

The contraction of the twisted boundary conditions also implies
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}_{A}=0$, but this condition turns out not to be necessary in the
limit. This condition can be consistently avoided, leading to a bigger
symmetry group.

###  5.4 Asymptotic symmetries of the electric Carroll limit

We now give the transformation laws of the leading orders of the fields under
Carrollian diffeomorphisms with asymptotic form

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle
br+T+\mathcal{O}\left(r^{-2}\right)\,,$ (86) $\displaystyle\xi^{r}$
$\displaystyle=$ $\displaystyle W+\mathcal{O}\left(r^{-1}\right)\,,$ (87)
$\displaystyle\xi^{A}$ $\displaystyle=$ $\displaystyle
Y^{A}+\frac{1}{r}I^{A}+\mathcal{O}\left(r^{-2}\right)\,,$ (88)

where $b$ describes Carrollian boosts, $Y^{A}$ spatial rotations, and $T$ and
$W$ stand for Carrollian supertranslations. The parameter $I^{A}$ also
describes spatial supertranslations and its precise role will depend on
whether we impose $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern 0.0pt}}}_{A}=0$ or not (see below).

We find that the transformations of the leading orders under space-like
diffeomorphisms read

$\displaystyle\delta_{\xi^{i}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{rr}$
$\displaystyle=Y^{A}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{rr}\,,$ (89)
$\displaystyle\delta_{\xi^{i}}\overline{\lambda}_{A}$
$\displaystyle=\mathcal{L}_{Y}\overline{\lambda}_{A}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}W-I_{A}\,,$
(90) $\displaystyle\delta_{\xi^{i}}\overline{h}_{AB}$
$\displaystyle=\mathcal{L}_{Y}\overline{h}_{AB}+2\left(\overline{D}_{(A}I_{B)}+\overline{g}_{AB}W\right)\,.$
(91)

We also find that the leading orders of the momentum transform as

$\displaystyle\delta_{\xi^{i}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rr}$
$\displaystyle=\partial_{A}\left(Y^{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rr}\right)\,,$ (92)
$\displaystyle\delta_{\xi^{i}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rA}$
$\displaystyle=\partial_{B}\left(Y^{B}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rA}\right)-\partial_{B}Y^{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rB}\,,$ (93)
$\displaystyle\delta_{\xi^{i}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{AB}$
$\displaystyle=\partial_{C}\left(Y^{C}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{AB}\right)-\partial_{C}Y^{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{CB}-\partial_{C}Y^{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{CA}\,,$ (94)

while the subleading order $\pi^{(2)rA}$, which appears in the expression of
the charges as in the Einstein theory, transforms as

$\displaystyle\delta_{\xi^{i}}\pi^{(2)rA}$
$\displaystyle=\partial_{B}\left(Y^{B}\pi^{(2)rA}\right)-\partial_{B}Y^{A}\pi^{(2)rB}$
$\displaystyle\quad+\partial_{B}\left(I^{B}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rA}\right)-\partial_{B}I^{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rB}+I^{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rr}-\partial_{B}W\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{AB}-W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rA}\,.$ (95)

Similarly, we find that the transformation law under time-like deformations
are

$\displaystyle\delta_{\xi}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{rr}$
$\displaystyle=\frac{b}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\Big{(}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rr}-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{A}_{A}\Big{)}\,,$ (96)
$\displaystyle\delta_{\xi}\overline{\lambda}_{A}$
$\displaystyle=\frac{2b}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{A}\,,$
(97) $\displaystyle\delta_{\xi}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{AB}$
$\displaystyle=\frac{2b}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\Big{[}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}_{AB}-\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rr}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{A}_{A})\Big{]}\,.$ (98)

The leading orders of the momentum do not transform under time-like
deformations,

$\delta_{\xi}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr}=\delta_{\xi^{i}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rA}=\delta_{\xi^{i}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{AB}=0\,,$ (99)

but one finds for $\pi^{(2)rA}$,

$\delta_{\xi}\pi^{(2)rA}=-\frac{2b}{3\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}\Big{(}2\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rA}+3\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}_{B}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{AB}-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rA}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{B}_{B}\Big{)}\,.$ (100)

####  5.4.1 Canonical generator

Because the symplectic structure takes the standard form $\int
d^{3}x\,d_{V}\pi^{ij}\wedge d_{V}g_{ij}$, the canonical generator of the
asymptotic symmetries is simply obtained through the approach of Regge:1974zd
. The surface integral in this case reads

$\delta Q^{E}_{\xi}[g_{ij},\pi^{ij}]=\oint
d^{2}x\Big{(}2\xi^{i}\delta\pi^{r}_{i}-\xi^{r}\delta g_{jk}\pi^{jk}\Big{)}\,.$
(101)

Replacing the asymptotic conditions in (101), we obtain

$\displaystyle\delta Q^{E}_{\xi}[g_{ij},\pi^{ij}]$ $\displaystyle=r\oint
d^{2}x\Big{(}2Y^{A}\delta\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{B}\Big{)}$ (102)
$\displaystyle\quad+\oint
d^{2}x\Big{[}2Y^{A}\delta\left(\pi^{(2)r}_{A}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{AB}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rB}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rr}\right)+2I^{A}\delta\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}_{A}+2W\delta\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rr}\Big{]}\,.$ (103)

The variation of the canonical generator has a linear divergent term. However,
this is equal to zero by virtue of the asymptotic constraint
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{AB}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rA}=0$, and using the fact that
$Y^{A}$ is the Killing vector of the 2-sphere at infinity. Then, the variation
of the charge is given by

$\displaystyle\delta Q^{E}_{\xi}[g_{ij},\pi^{ij}]$ $\displaystyle=\oint
d^{2}x\Big{[}2Y^{A}\delta\left(\pi^{(2)r}_{A}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{AB}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rB}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rr}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}_{A}\right)+2I^{A}\delta\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}_{A}+2W\delta\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rr}\Big{]}\,.$ (104)

To proceed further, we need to distinguish two cases, according to whether the
additional asymptotic condition $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}_{A}=0$ is imposed as in the
Einstein theory, or is not imposed, as it turns out to be possible.

####  5.4.2 Asymptotic symmetry algebra - Case with $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}_{A}=0$:
$iso_{3}$-BMS algebra

When $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern 0.0pt}}}_{A}=0$, the parameters $I_{A}$ are not
independent. Indeed, preservation of the condition $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}_{A}=0$
requires one to add the following correcting gauge transformation when one
performs a Carroll boost and a spatial supertranslation parametrized by $W$,

$\xi^{r}=\mathcal{O}\left(r^{-2}\right)\,,\qquad\xi^{A}=\frac{1}{r}(I^{A}_{(b)}+I^{A}_{(W)})+\mathcal{O}\left(r^{-2}\right)\,,$
(105)

with

$I^{A}_{(b)}=\frac{2b}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rA}\qquad
I^{A}_{(W)}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}W\,,$ (106)

so that the $I_{A}$’s are entirely determined by $b$ and $W$.

Substituting this expression for $I_{A}$ in $\delta
Q^{E}_{\xi}[g_{ij},\pi^{ij}]$, one finds after integration in phase space that
the canonical generator is equal to

$Q^{E}_{\xi}=\frac{1}{2}b_{ij}M^{ij}+Q_{W}\,,$ (107)

where

$\displaystyle M^{ij}$ $\displaystyle=$ $\displaystyle\oint
d^{2}x\,2x^{[i}e^{j]A}\left(\pi^{(2)r}_{A}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{AB}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rB}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rr}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}_{A}\right)\,,$ (108) $\displaystyle Q_{W}$ $\displaystyle=$
$\displaystyle\oint d^{2}x\,W\mathcal{W}\,,$ (109)

with $\mathcal{W}=2(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr}-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}_{A})$. The surface term
accompanying the boost is given by

$\oint d^{2}x\,\frac{2n^{i}}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rA}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}_{rA}\,,$ (110)

but this integral is equal to zero due to the parity conditions. It follows
that the boosts are also proper gauge transformations for the diffeomorphism-
twisted parity conditions, as in Perez:2021abf . Furthermore, it can be
checked that only the odd part of $W$ is an improper gauge transformation.

The parameters obey the following transformation laws

$\displaystyle\hat{Y}^{A}$ $\displaystyle=$ $\displaystyle
Y_{1}^{B}\partial_{B}Y_{2}^{A}-(1\leftrightarrow 2)\,,$ (111)
$\displaystyle\hat{W}$ $\displaystyle=$ $\displaystyle
Y_{1}^{B}\partial_{B}W_{2}-(1\leftrightarrow 2)\,,$ (112)

leading to the brackets

$\displaystyle\\{M^{ij},M^{kl}\\}$
$\displaystyle=\frac{1}{2}\Big{(}\delta^{jk}M^{il}-\delta^{ik}M^{jl}+\delta^{li}M^{jk}-\delta^{lj}M^{ik}\Big{)}\,,$
(113) $\displaystyle\\{M^{ij},\mathcal{W}(\theta,\phi)\\}$
$\displaystyle=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{W}(\theta,\phi))\,.$ (114)

This is the $iso(3)$-BMS algebra.

####  5.4.3 Asymptotic symmetry algebra - Case with $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}_{A}\neq
0$: Extended $iso(3)$-BMS algebra

Unlike what happens in the magnetic Carrollian theory, the expression (104) is
integrable for arbitrary $Y^{A}$’s, $I^{A}$’s and $W$’s. Thus, there is no
need to impose the restriction $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}_{A}=0$ which was necessary
in that case (and also in Einstein gravity) to ensure integrability. We thus
lift this restriction and allow configurations with $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}_{A}\neq
0$.

The electric theory is then invariant under the semi-direct product of spatial
rotations and the Abelian set of “generalized” spatial Carrollian
supertranslations parametrized by $W$ and $I^{A}$, with canonical generator
that can be written as

$Q^{E}_{\xi}=\frac{1}{2}b_{ij}M^{ij}+Q_{W}+Q_{I}\,,$ (115)

where

$Q_{W}=\oint d^{2}x\,W\mathcal{W}\,,\qquad Q_{I}=\oint
d^{2}x\,I^{A}\mathcal{W}_{A}\,.$ (116)

Here, $\mathcal{W}=2\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr}$ and $\mathcal{W}_{A}=2\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{A}$.
Because $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr}$ is odd and $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{A}$ even,
only odd parameters $W$ and even parameters $I^{A}$, which are otherwise
arbitrary functions and vectors on the 2-sphere, define improper gauge
transformations.

The transformation laws of the parameters are given by

$\displaystyle\hat{Y}^{A}$ $\displaystyle=$ $\displaystyle
Y_{1}^{B}\partial_{B}Y_{2}^{A}-(1\leftrightarrow 2)\,,$ (117)
$\displaystyle\hat{W}$ $\displaystyle=$ $\displaystyle
Y_{1}^{B}\partial_{B}W_{2}-(1\leftrightarrow 2)\,,$ (118)
$\displaystyle\hat{I}^{A}$ $\displaystyle=$ $\displaystyle
Y_{1}^{B}\partial_{B}I^{A}_{2}-I^{B}_{2}\partial_{B}Y_{1}^{A}-(1\leftrightarrow
2)\,,$ (119)

and hence, the nonvanishing brackets of the generators read

$\displaystyle\\{M^{ij},M^{kl}\\}$
$\displaystyle=\frac{1}{2}\Big{(}\delta^{jk}M^{il}-\delta^{ik}M^{jl}+\delta^{li}M^{jk}-\delta^{lj}M^{ik}\Big{)}\,,$
(120) $\displaystyle\\{M^{ij},\mathcal{W}(\theta,\phi)\\}$
$\displaystyle=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{W}(\theta,\phi))\,,$ (121)
$\displaystyle\\{M^{ij},\mathcal{W}_{A}(\theta,\phi)\\}$
$\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern
0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{W}_{A}(\theta,\phi))-x^{[i}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}e^{j]B}\mathcal{W}_{B}(\theta,\phi)\,.$ (122)

This is the extended $iso(3)$-BMS algebra.

This enlargment of the $iso(3)$-BMS algebra cannot be obtained as a
contraction of the BMS4 algebra since it contains more spatial
supertranslations. It is somewhat reminiscent of the Spi group of
Ashtekar:1978zz , but the parity of the supertranslations are restricted here.

##  6 Asymptotic symmetries of the Yang-Mills field in flat Carroll spacetime

Before turning to the coupled Einstein-Yang-Mills system in the Carroll limit,
we first consider the asymptotic symmetries of the Yang-Mills field on a flat
Carroll background. We start with a brief overview of the asymptotic structure
of the Yang-Mills theory in Minkowski space. We then proceed to the analysis
of the electric and magnetic Carrollian limits.

###  6.1 Brief overview of boundary conditions in Yang-Mills theory

The asymptotic structure of Yang-Mills theory has been rigorously studied in
Tanzi:2020fmt . Here, we give a brief summary of their results.

The Hamiltonian action for the Yang-Mills theory on a Minkowski background in
four spacetime dimensions reads

$S_{\text{YM}}[A_{i},\pi^{i},A_{0}]=\int
dtd^{3}x\left(\pi^{i}_{a}\dot{A}^{a}_{i}-\mathcal{H}+A^{a}_{0}D_{i}\pi^{i}_{a}\right)\,,$
(123)

where $\pi^{i}_{a}$ is the conjugate momentum to the non-Abelian gauge field
$A_{i}=A^{a}_{i}T_{a}$. This canonical pair takes values on some compact semi-
simple Lie algebra $[T_{a},T_{b}]=f^{c}_{\;\;ab}T_{c}$. By using the inverse
of the invariant metric, one can raise the internal index $a$ of $\pi^{i}_{a}$
to form an object $\pi^{i}=\pi^{ia}T_{a}$ that transforms as the vector
potential in the adjoint representation.111Here, we follow the conventions of
Tanzi:2020fmt . The inner product is given by $S_{ab}=-\text{tr}(T_{a}T_{b})$,
where $S^{ab}S_{bc}=\delta^{a}_{c}$. Latin indices $a,b,c,\dots$ refer to the
internal Lie algebra and are lowered and raised with $S_{ab}$ and its inverse
$S^{bc}$, respectively. The variation of the action with respect to the
Lagrange multiplier $A^{a}_{0}$ enforces the Gauss constraint

$\mathcal{G}_{a}=D_{i}\pi^{i}_{a}\approx 0\,,$ (124)

where the covariant derivative is defined as
$D_{i}X^{a}=\partial_{i}X^{a}+\alpha\,f^{a}_{\;\;bc}A^{b}_{i}X^{c}$, with
$\alpha$ the Yang-Mills coupling constant. The Hamiltonian density is given by

$\mathcal{H}=\frac{1}{2}\pi^{a}_{i}\pi_{a}^{i}+\frac{1}{4}F^{a}_{ij}F_{a}^{ij}\,,$
(125)

while the momentum density reads

$\mathcal{H}_{i}=\pi^{j}_{a}\partial_{i}A^{a}_{j}-\partial_{j}(\pi^{j}_{a}A^{a}_{i})\,.$
(126)

Infinitesimal transformation laws of the fields under Poincaré and non-Abelian
gauge symmetries, generated by $\xi^{\mu}=(\xi,\xi^{i})$ and
$\varepsilon=\varepsilon^{a}T_{a}$, read

$\displaystyle\delta_{\xi,\xi^{i},\varepsilon}A_{i}^{a}$
$\displaystyle=\xi\pi_{i}^{a}+\xi^{j}\partial_{i}A_{j}^{a}+\partial_{i}\xi^{j}A_{j}^{a}-D_{i}\varepsilon^{a}\,,$
(127) $\displaystyle\delta_{\xi,\xi^{i},\varepsilon}\pi^{i}_{a}$
$\displaystyle=-D_{j}(\xi
F^{ij}_{a})+\partial_{j}\left(\xi^{j}\pi^{i}_{a}\right)-\partial_{j}\xi^{i}\pi^{j}_{a}+\alpha
f^{c}_{\;\;ab}\pi^{i}_{c}\varepsilon^{b}\,.$ (128)

In spherical coordinates, the vector fields defining the infinitesimal
Poincaré transformations take the form

$\xi=br+T_{0}\,,\quad\xi^{r}=W_{0}\,,\quad\xi^{A}=Y^{A}+\frac{1}{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}W_{0}\,,$
(129)

where $T_{0}$ stand for time translations $\partial_{A}T_{0}=0$ and $W_{0}$,
which corresponds to the spatial translations, is a combination of the $l=1$
spherical harmonics and satisfies therefore the conditions $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}W_{0}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}W_{0}=0$. The decay of the
gauge parameter reads

$\varepsilon=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\varepsilon$\kern 0.0pt}}}+\mathcal{O}(r^{-1})\,.$ (130)

The fall-off of the gauge field and its conjugate momentum is given by

$\displaystyle A_{r}$ $\displaystyle=\frac{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}}{r}+\mathcal{O}\left(r^{-2}\right)\,,\quad
A_{A}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{A}+\mathcal{O}\left(r^{-1}\right)\,,$ (131) $\displaystyle\pi^{r}$
$\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern
0.0pt}}}^{r}+\mathcal{O}\left(r^{-1}\right)\,\quad\pi^{A}=\frac{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}}{r}+\mathcal{O}\left(r^{-2}\right)\,.$ (132)

As in the case of electromagnetism Henneaux:2018gfi  and Einstein gravity
Henneaux:2018hdj ; Henneaux:2019yax , the symplectic structure is
logarithmically divergent. This can be solved by imposing parity conditions.
The authors of Tanzi:2020fmt  first propose the following set of strict parity
conditions

$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\sim\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{A}=\text{even}\,,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{A}\sim\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}=\text{odd}\,,$ (133)

which ensures the finiteness of the symplectic structure. These parity
conditions differ from those usually imposed in electromagnetism where
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{i}$ is odd instead of being even, but possess two good features: (i)
to leading order, the vector potential $A_{i}$ and the derivative operator
$\partial_{i}$ possess same odd parity, so that $D_{i}=\partial_{i}-A_{i}$ has
a definite parity (namely, also odd parity); and (ii) the $SU(2)$ Wu-Yang
monopole solution of the pure Yang-Mills theory Wu:1967vp ,

$A_{i}^{a}=\delta^{ab}\epsilon_{ibc}\frac{x^{c}}{r^{2}}$ (134)

fulfills these parity conditions.

Furthermore, these boundary conditions are Poincaré invariant and make the
Lorentz boosts canonical. They are also preserved by gauge transformations
with gauge parameter that asymptotically tends to an even function on the
2-sphere, i.e., $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\varepsilon$\kern 0.0pt}}}=\text{even}$. However, the boundary term of
the canonical generator of gauge transformations

$G[\varepsilon]=\int d^{3}x\,\varepsilon^{a}\,\mathcal{G}_{a}-\oint
d^{2}S_{i}\varepsilon^{a}\pi^{i}_{a}\,,$ (135)

is then equal to zero since $\varepsilon^{a}$ and $\pi^{i}_{a}$ have opposite
parity. There is no improper gauge symmetry at all, contrary to what happens
for electromagnetism. The only symmetries are the Poincaré transformations,
and these are rigid symmetries with a non-vanishing bulk integral.

The authors of Tanzi:2020fmt  relaxed then the boundary conditions by twisting
the parity conditions as in Henneaux:2018gfi ; Henneaux:2018hdj ;
Henneaux:2019yax . They imposed accordingly strict parity conditions up to an
improper gauge transformation. Specifically, they proposed that the Yang-Mills
field should behave asymptotically as

$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$U$\kern
0.0pt}}}^{-1}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}_{r}^{\text{even}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$U$\kern 0.0pt}}}\,,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{A}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$U$\kern 0.0pt}}}^{-1}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}_{A}^{\text{odd}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$U$\kern
0.0pt}}}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$U$\kern 0.0pt}}}^{-1}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$U$\kern 0.0pt}}}\,,$ (136)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}$ $\displaystyle=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$U$\kern
0.0pt}}}^{-1}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}_{\text{odd}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$U$\kern 0.0pt}}}\,,\qquad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$U$\kern 0.0pt}}}^{-1}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{A}_{\text{even}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$U$\kern 0.0pt}}}\,,$ (137)

where $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$U$\kern
0.0pt}}}=\exp(-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\phi$\kern 0.0pt}}}^{a}\,T_{a})$, where $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\phi$\kern 0.0pt}}}^{a}$ depends
on the angles. They verified that these boundary conditions lead to a finite
symplectic structure (by imposing a faster fall-off of the Gauss constraint
$\mathcal{G}_{a}\sim r^{-4}$). By construction, these conditions are preserved
by angle-dependent gauge symmetries with no definite parity and hence non-zero
charges. However, the authors of Tanzi:2020fmt  also showed that there was a
clash with Poincaré invariance, in the sense that the symplectic structure is
not invariant under Lorentz boosts, which transforms into a non-trivial
surface term. While this problem can be cured in electromagnetism by adding
boundary terms in the symplectic form Henneaux:2018gfi , the same method does
not work in the non-abelian case Tanzi:2020fmt . The difficulty is that the
interaction term $A_{i}A_{j}$ in the curvature $F_{ij}$ is of the same order
as the free term $\partial_{i}A_{j}$ and hence, the interactions cannot be
neglected asymptotically.

The conclusion is therefore that there is no known set of asymptotic
conditions at spatial infinity simultaneously consistent with Lorentz
invariance and accommodating improper color gauge symmetries in the non-
Abelian case.

###  6.2 Electric Carrollian limit of Yang-Mills

####  6.2.1 Boundary conditions and asymptotic transformations

We now show that the aforementioned difficulties do not appear in the electric
Carrollian limit of Yang-Mills theory, which possesses an infinite-dimensional
color symmetry group consistent with Carroll invariance. The situation in the
magnetic Carrollian limit, on the other hand, is similar to the Lorentzian
case, i.e., one easily verifies that it is not possible to have improper gauge
transformations consistent with Carroll invariance. Thus, the asymptotic
symmetry algebra is given in that case by the Carroll algebra only. For that
reason, from now on we focus on the electric limit, which has a richer
structure.

The Hamiltonian action for the electric Carrollian limit of Yang-Mills theory
reads Henneaux:2021yzg

$S^{E}_{\text{YM}}[A_{i},\pi^{i},A_{0}]=\int
dtd^{3}x\left(\pi^{i}_{a}\dot{A}^{a}_{i}-\mathcal{H}^{E}+A^{a}_{0}D_{i}\pi^{i}_{a}\right)\,,$
(138)

The Hamiltonian density is given by

$\mathcal{H}^{E}=\frac{1}{2}\pi^{a}_{i}\pi_{a}^{i}\,,$ (139)

and satisfies the relation
$\\{\mathcal{H}^{E}(x),\mathcal{H}^{E}(x^{\prime})\\}=0$ characteristic of
Carroll-invariant theories. The momentum density and the Gauss constraint are
unchanged. The variation of the action with respect to the fields yields the
following equations of motion

$\displaystyle\dot{A}_{i}^{a}$ $\displaystyle=\pi_{i}^{a}-D_{i}A^{a}_{0}\,,$
(140) $\displaystyle\dot{\pi}^{i}_{a}$ $\displaystyle=0\,.$ (141)

Carroll transformations are generated by the same surface-deformation vector
fields as in the Lorentzian case, i.e.,

$\xi=br+T_{0}\,,\quad\xi^{r}=W_{0}\,,\quad\xi^{A}=Y^{A}+\frac{1}{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}W_{0}\,,$
(142)

where $b$ is the Carrollian boost parameter, $Y^{A}$ is the Killing vector of
the 2-sphere at infinity, the constant $T_{0}$ stands for time translations
and $W_{0}$ corresponds to the spatial translations (satisfying the property
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}W_{0}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}W_{0}=0$). The action on the
fields is however different because the Hamiltonian density is different,
leading to variations of the fields under Carroll and gauge transformations
that read

$\displaystyle\delta_{\xi,\xi^{i},\varepsilon}A_{i}^{a}$
$\displaystyle=\xi\pi_{i}^{a}+\xi^{j}\partial_{i}A_{j}^{a}+\partial_{i}\xi^{j}A_{j}^{a}-D_{i}\varepsilon^{a}\,,$
(143) $\displaystyle\delta_{\xi,\xi^{i},\varepsilon}\pi^{i}_{a}$
$\displaystyle=\partial_{j}\left(\xi^{j}\pi^{i}_{a}\right)-\partial_{j}\xi^{i}\pi^{j}_{a}+\alpha
f^{c}_{\;\;ab}\pi^{i}_{c}\varepsilon^{b}\,.$ (144)

The notable difference lies in the transformation rule of the momenta
$\pi^{i}_{a}$ under timelike deformations (boosts and time translations),
which is now $\delta_{\xi}\pi^{i}_{a}=0$.

We adopt the same fall-off of the fields as in the previous section, i.e.,
(131), (132) and the set of twisted parity conditions (136), (137). We also
take the same asymptotic behaviour (130) of the gauge parameter $\varepsilon$.
As we recalled, these boundary conditions make the symplectic form finite
under the additional requirement that the Gauss constraint should decay as
$D_{i}\pi^{i}_{a}\sim r^{-4}$. It is direct to show that this set of boundary
conditions is preserved by both Carroll and non-Abelian gauge symmetries
behaving as in (130).

The transformation laws of the leading orders of the fields in the asymptotic
expansion ($r\rightarrow\infty$) are given in spherical coordinates by

$\displaystyle\delta_{\xi,\varepsilon}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{a}_{r}$ $\displaystyle=$
$\displaystyle\frac{b\,\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{ar}}{\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}+Y^{A}\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{a}_{r}+\alpha
f^{a}_{\,\,\,bc}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\varepsilon$\kern 0.0pt}}}^{b}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{c}_{r}\,,$ (145)
$\displaystyle\delta_{\xi,\varepsilon}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{a}_{A}$ $\displaystyle=$
$\displaystyle\frac{b\,\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{a}_{A}}{\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}}+Y^{B}\partial_{B}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}^{a}_{A}+\partial_{B}Y^{B}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}^{a}_{A}-D_{A}\bar{\varepsilon}^{a}\,,$ (146)
$\displaystyle\delta_{\xi,\varepsilon}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{a}$ $\displaystyle=$
$\displaystyle\partial_{A}\left(Y^{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{a}\right)+\alpha
f_{abc}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\varepsilon$\kern 0.0pt}}}^{b}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{cr}\,,$ (147)
$\displaystyle\delta_{\xi,\varepsilon}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}_{a}$ $\displaystyle=$
$\displaystyle\partial_{B}Y^{B}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}_{a}^{A}-\partial_{B}Y^{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}_{a}^{B}+\alpha
f_{abc}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\varepsilon$\kern 0.0pt}}}^{b}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{c\,A}\,.$ (148)

####  6.2.2 Boosts are canonical transformations - Improper gauge
transformations

The great simplification that occurs in the electric Carroll contraction of
Yang-Mills theory is that the boosts are now canonical transformations, even
with the parity conditions twisted by a gauge transformation. This is because
there are no spatial derivatives in the energy density (139). The twisted
parity conditions are therefore compatible with Carroll invariance, while they
were not (and could not be improved to become so) in the Lorentzian case.

As we have also explained above, the twisted parity conditions (136), (137)
are invariant under angle-dependent $\mathcal{O}(1)$ color gauge
transformations, which are improper when they are odd (in a sense to make
precise in the subsection below).

The electric Carroll contraction of Yang-Mills theory accommodates
consequently an infinite-dimensional angle-dependent color group without
conflict with Carroll covariance.

####  6.2.3 Asymptotic symmetry algebra: infinite-dimensional color group

By applying the standard canonical methods, one then finds that the canonical
generator of the asymptotic symmetries is given by

$C^{E}_{\xi,\varepsilon}[A_{i},\pi^{i}]=\int
d^{3}x\left(\xi\mathcal{H}^{E}+\xi^{i}\mathcal{H}_{i}+\varepsilon^{a}\mathcal{G}_{a}\right)+Q^{E}_{\xi,\epsilon}[A_{i},\pi^{i}]\,,$
(149)

where the surface term reads

$Q^{E}_{\xi,\varepsilon}[A_{i},\pi^{i}]=\oint d^{2}x\left(-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\varepsilon$\kern
0.0pt}}}^{a}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}_{a}+Y^{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{a}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}_{a}\right)\,.$ (150)

The generator of spatial rotations is the sum of both a non-vanishing bulk
term $\int d^{4}x\xi^{i}\mathcal{H}_{i}$ (recall that $\mathcal{H}^{E}$ and $\
\mathcal{H}_{i}$ are not constrained to vanish when Carroll gravity is not
included) and a non-vanishing surface integral $\oint
d^{2}xY^{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}^{a}_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{a}$

A direct computation shows that the Poisson brackets of the canonical
generators are given by

$\Big{\\{}C^{E}_{\xi_{1},\varepsilon_{1}}[A_{i},\pi^{i}],C^{E}_{\xi_{2},,\varepsilon_{2}}[A_{i},\pi^{i}]\Big{\\}}=C^{E}_{\hat{\xi},\hat{\varepsilon}}[A_{i},\pi^{i}]\,,$
(151)

where

$\displaystyle\hat{\xi}$ $\displaystyle=$
$\displaystyle\xi^{i}_{1}\partial_{i}\xi_{2}-(1\leftrightarrow 2)\,,$ (152)
$\displaystyle\hat{\xi}^{i}$ $\displaystyle=$
$\displaystyle\xi^{j}_{1}\partial_{j}\xi^{i}_{2}-(1\leftrightarrow 2)\,,$
(153) $\displaystyle\hat{\varepsilon}^{a}$ $\displaystyle=$
$\displaystyle\xi^{i}_{1}\partial_{i}\varepsilon_{2}^{a}+\frac{1}{2}f^{a}_{\,\,\,bc}\varepsilon^{b}_{1}\varepsilon^{c}_{2}-(1\leftrightarrow
2)\,.$ (154)

The symmetry algebra is the semi-direct sum of the Carroll algebra and an
infinite-dimensional set of non-Abelian angle-dependent color charges. In
order to write the algebra in a more explicit way, we can recast the canonical
generator $C^{E}_{\xi,\varepsilon}[A_{i},\pi^{i}]$ as

$C^{E}_{\xi,\varepsilon}=b_{i}B^{i}+a_{0}E+\frac{1}{2}b_{ij}M^{ij}+a_{i}P^{i}+Q^{\text{YM}}_{\varepsilon}\,,$
(155)

where

$\displaystyle B^{i}$ $\displaystyle=$ $\displaystyle\int
d^{3}xx^{i}\mathcal{H}^{E}\,,$ (156) $\displaystyle E$ $\displaystyle=$
$\displaystyle\int d^{3}x\,\mathcal{H}^{E}\,,$ (157) $\displaystyle M^{ij}$
$\displaystyle=$ $\displaystyle\int d^{3}x\,2x^{[i}\mathcal{H}^{j]}+\oint
d^{2}x\,x^{[i}e^{j]A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$A$\kern 0.0pt}}}^{a}_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{a}\,,$ (158) $\displaystyle
P^{i}$ $\displaystyle=$ $\displaystyle\int d^{3}x\,\mathcal{H}^{i}\,,$ (159)
$\displaystyle Q^{\text{YM}}_{\epsilon}$ $\displaystyle=$ $\displaystyle\oint
d^{2}x\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}^{a}\mathcal{T}_{a}\,,$ (160)

with $\mathcal{T}_{a}=-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{a}$. Then, the non-
vanishing brackets of the asymptotic symmetry algebra are explicitly given by

$\displaystyle\\{P^{i},B^{j}\\}$ $\displaystyle=\delta^{ij}E\,,$ (161)
$\displaystyle\\{B^{i},M^{jk}\\}$
$\displaystyle=\frac{1}{2}\Big{(}\delta^{ik}B^{j}-\delta^{ij}B^{k}\Big{)}\,,$
(162) $\displaystyle\\{P^{i},M^{jk}\\}$
$\displaystyle=\frac{1}{2}\Big{(}\delta^{ik}P^{j}-\delta^{ij}P^{k}\Big{)}\,,$
(163) $\displaystyle\\{M^{ij},M^{kl}\\}$
$\displaystyle=\frac{1}{2}\Big{(}\delta^{jk}M^{il}-\delta^{ik}M^{jl}+\delta^{li}M^{jk}-\delta^{lj}M^{ik}\Big{)}\,,$
(164) $\displaystyle\\{M^{ij},\mathcal{T}_{a}(x^{A})\\}$
$\displaystyle=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{T}_{a}(x^{A}))\,,$ (165)
$\displaystyle\\{\mathcal{T}_{a}(x^{A}_{1}),\mathcal{T}_{b}(x^{A}_{2})\\}$
$\displaystyle=f^{c}_{\,\,\,ab}\mathcal{T}_{c}(x^{A}_{1})\delta^{(2)}(x^{A}_{1}-x^{A}_{2})\,.$
(166)

Note that the generators $\mathcal{T}_{a}=-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{a}$ of
the improper color gauge transformations, which form a centerless Kac-Moody
algebra, obey a twisted parity condition and thus they are not independent
functions on the sphere.

##  7 Carrollian limits of Einstein-Yang-Mills system

In this section, we proceed to combine our previous results, which can be done
easily. One can take different limits in the Einstein and Yang-Mills sectors
of the coupled theory, which would lead to different consistent Carrollian
limits of the Einstein-Yang-Mills system. We will consider first, and somewhat
in detail, the theory that results from taking the magnetic Carrollian limit
for gravity and the electric Carrollian limit for Yang-Mills, since it is the
one that admits the most interesting asymptotic structure. We will then make a
summary of the asymptotic symmetry algebras associated to the remaining
Carrollian limits.

###  7.1 Magnetic Carrollian gravity coupled to the electric limit of the
Yang-Mills field

The Hamiltonian action for this Carrollian limit of the Einstein-Yang-Mills
system reads

$S=\int dt\left[\int
d^{3}x\left(\pi^{ij}\dot{g}_{ij}+\pi^{i}_{a}\dot{A}^{a}_{i}-N\mathcal{H}-N^{i}\mathcal{H}_{i}+A^{a}_{0}\mathcal{G}_{a}\right)-B_{\infty}\right]\,.$
(167)

The variation of the action with respect to the Lagrange multipliers $N$,
$N^{i}$ and $A^{a}_{0}$ yields the following constraints

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle-\sqrt{g}R+\frac{1}{2\sqrt{g}}\pi^{a}_{i}\pi_{a}^{i}\approx
0\,,$ (168) $\displaystyle\mathcal{H}_{i}$ $\displaystyle=$
$\displaystyle-2\nabla^{j}\pi_{ij}+\pi^{j}_{a}\partial_{i}A^{a}_{j}-\partial_{j}(\pi^{j}_{a}A^{a}_{i})\approx
0\,,$ (169) $\displaystyle\mathcal{G}_{a}$ $\displaystyle=$ $\displaystyle
D_{i}\pi^{i}_{a}\approx 0\,,$ (170)

where the covariant derivative $D_{i}$ is the complete spacetime
($\nabla_{i}$) and gauge covariant derivative,
$D_{i}X^{a}=\nabla_{i}X^{a}+\alpha\,f^{a}_{\;\;bc}A^{b}_{i}X^{c}$.

The constraints are all first-class and satisfy the following algebra

$\displaystyle\\{\mathcal{H}(x),\mathcal{H}(x^{\prime})\\}$ $\displaystyle=$
$\displaystyle 0\,,$ (171)
$\displaystyle\\{\mathcal{H}(x),\mathcal{H}_{i}(x^{\prime})\\}$
$\displaystyle=$
$\displaystyle\mathcal{H}(x)\partial_{i}\delta(x,x^{\prime})\,,$ (172)
$\displaystyle\\{\mathcal{H}_{i}(x),\mathcal{H}_{j}(x^{\prime})\\}$
$\displaystyle=$
$\displaystyle\mathcal{H}_{i}(x^{\prime})\partial_{j}\delta(x,x^{\prime})+\mathcal{H}_{j}(x)\partial_{i}\delta(x,x^{\prime})\,,$
(173) $\displaystyle\\{\mathcal{G}_{a}(x),\mathcal{H}(x^{\prime})\\}$
$\displaystyle=$ $\displaystyle 0\,,$ (174)
$\displaystyle\\{\mathcal{G}_{a}(x),\mathcal{H}_{i}(x^{\prime})\\}$
$\displaystyle=$
$\displaystyle\mathcal{G}_{a}(x)\partial_{i}\delta(x,x^{\prime})\,,$ (175)
$\displaystyle\\{\mathcal{G}_{a}(x),\mathcal{G}_{b}(x^{\prime})\\}$
$\displaystyle=$ $\displaystyle
f^{c}_{\,\,\,ab}\mathcal{G}_{c}(x)\delta(x,x^{\prime})\,.$ (176)

The infinitesimal transformation laws of the fields generated by Carrollian
diffeomorphisms $\left(\xi^{\perp}\equiv\xi,\xi^{i}\right)$ and non-Abelian
gauge transformation with parameter $\varepsilon^{a}$ read

$\displaystyle\delta_{\xi,\xi^{i}}g_{ij}$
$\displaystyle=\mathcal{L}_{\xi}g_{ij}\,,$ (177)
$\displaystyle\delta_{\xi,\xi^{i}}\pi^{ij}$
$\displaystyle=-\xi\sqrt{g}\left(R^{ij}-\frac{1}{2}g^{ij}R\right)+\frac{\xi}{2\sqrt{g}}\left(\pi^{ai}\pi^{j}_{a}-\frac{1}{2}g^{ij}\pi^{a}_{k}\pi_{a}^{k}\right)$
(178)
$\displaystyle\quad+\sqrt{g}\left(\xi^{|ij}-g^{ij}\xi_{\quad|m}^{|m}\right)+\mathcal{L}_{\xi}\pi^{ij}\,,$
(179) $\displaystyle\delta_{\xi,\xi^{i},\varepsilon}A_{i}^{a}$
$\displaystyle=\frac{\xi}{\sqrt{g}}\pi_{i}^{a}+\xi^{j}\partial_{i}A_{j}^{a}+\partial_{i}\xi^{j}A_{j}^{a}-D_{i}\varepsilon^{a}\,,$
(180) $\displaystyle\delta_{\xi,\xi^{i},\varepsilon}\pi^{i}_{a}$
$\displaystyle=\partial_{j}\left(\xi^{j}\pi^{i}_{a}\right)-\partial_{j}\xi^{i}\pi^{j}_{a}+\alpha
f^{c}_{\;\;ab}\pi^{i}_{c}\varepsilon^{b}\,.$ (181)

We take as asymptotic conditions:

  * •

for gravity, the twisted parity conditions (59)-(64), with the additional
condition $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern 0.0pt}}}_{A}=0$ (which implies $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\zeta$\kern
0.0pt}}}_{A}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{A}\zeta_{r}$);

  * •

for the Yang-Mills field, the twisted conditions (131), (132), (136) and
(137).

Additionally, we also assume the faster fall-off for the constraints

$\mathcal{H}=\mathcal{O}(r^{-2})\,,\quad\mathcal{H}_{r}=\mathcal{O}(r^{-2})\,,\quad\mathcal{H}_{A}=\mathcal{O}(r^{-1})\,,\quad\mathcal{G}_{a}=\mathcal{O}(r^{-4})\,.$
(182)

The asymptotic Killing vectors and the non-Abelian gauge parameters that
preserve the fall-off of the fields read

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle
br+T-\frac{1}{2}b\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}+\mathcal{O}\left(r^{-2}\right)\,,$ (183)
$\displaystyle\xi^{r}$ $\displaystyle=$ $\displaystyle
W+\mathcal{O}\left(r^{-1}\right)\,,$ (184) $\displaystyle\xi^{A}$
$\displaystyle=$ $\displaystyle Y^{A}+\frac{1}{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W+\mathcal{O}\left(r^{-2}\right)\,,$ (185)
$\displaystyle\varepsilon$ $\displaystyle=$ $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\varepsilon$\kern
0.0pt}}}+\mathcal{O}(r^{-1})\,,$ (186)

where $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}b+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}b=0$, $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}Y_{B}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}Y_{A}=0$ and where the boost-dependent
$\mathcal{O}(1)$ compensating term $-\frac{1}{2}b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}$ is included in
order to make the boost charges integrable, as in Einstein gravity (the
details of this derivation can be found in Henneaux:2018cst ; Henneaux:2019yax
; Perez:2021abf ).

These transformations are canonical and hence define asymptotic symmetries.
Their canonical generators are again obtained through the standard canonical
procedure, and explicitly given by

$G_{\xi,\epsilon}[g_{ij},\pi^{ij};A_{i},\pi^{i}]=\int
d^{3}x\left(\xi\mathcal{H}+\xi^{i}\mathcal{H}_{i}+\epsilon^{a}\mathcal{G}_{a}\right)+Q_{\xi,\epsilon}[g_{ij},\pi^{ij};A_{i},\pi^{i}]\,$
(187)

where the boundary terms read

$\displaystyle Q_{\xi,\epsilon}$ $\displaystyle=\oint
d^{2}x\Big{[}Y^{A}(2\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{(2)r}_{A}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{a}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}_{a})+2\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,T\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{rr}+2W(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr}-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}_{A})$
$\displaystyle\quad\qquad\qquad+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,b\Big{(}2k^{(2)}+\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{rr}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}+\frac{1}{4}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}^{2}+\frac{1}{4}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}^{A}_{B}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}^{B}_{A}\Big{)}-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\varepsilon$\kern 0.0pt}}}^{a}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}_{a}\Big{]}\,.$ (188)

with

$k$ $\displaystyle=\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{rr}\,,$ (189)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$k$\kern 0.0pt}}}^{(2)}$
$\displaystyle=h_{rr}^{(2)}+h^{(2)}-\frac{1}{4}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{rr}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}-\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{B}^{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{A}^{B}-\frac{3}{4}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{rr}^{2}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}h_{r}^{(2)A}\,,$ $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}_{A}^{(2)r}$
$\displaystyle=\pi_{A}^{(2)r}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{AB}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rB}\,.$ (190)

The brackets of the canonical generators are given by

$\Big{\\{}G_{\xi_{1},\varepsilon_{1}}[g_{ij},\pi^{ij};A_{i},\pi^{i}],G_{\xi_{2},\varepsilon_{2}}[g_{ij},\pi^{ij};A_{i},\pi^{i}]\Big{\\}}=G_{\hat{\xi},\hat{\varepsilon}}[g_{ij},\pi^{ij};A_{i},\pi^{i}]\,,$
(191)

where the hatted parameters of the commutator transformation
$(\hat{\xi},\hat{\varepsilon})$ are

$\displaystyle\hat{Y}^{A}$ $\displaystyle=$ $\displaystyle
Y_{1}^{B}\partial_{B}Y_{2}^{A}-(1\leftrightarrow 2)\,,$ (192)
$\displaystyle\hat{b}$ $\displaystyle=$ $\displaystyle
Y_{1}^{B}\partial_{B}b_{2}-(1\leftrightarrow 2)\,,$ (193)
$\displaystyle\hat{T}$ $\displaystyle=$ $\displaystyle
Y_{1}^{B}\partial_{B}T_{2}-3b_{1}W_{2}-\partial_{A}b_{1}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W_{2}-b_{1}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W_{2}-(1\leftrightarrow 2)\,,$ (194) $\displaystyle\hat{W}$
$\displaystyle=$ $\displaystyle Y_{1}^{B}\partial_{B}W_{2}-(1\leftrightarrow
2)\,,$ (195) $\displaystyle\hat{\varepsilon}^{a}$ $\displaystyle=$
$\displaystyle\xi^{i}_{1}\partial_{i}\varepsilon_{2}^{a}+\frac{1}{2}f^{a}_{\,\,\,bc}\varepsilon^{b}_{1}\varepsilon^{c}_{2}-(1\leftrightarrow
2)\,.$ (196)

These transformations define the Carroll-BMS algebra $\mathcal{C}$-BMS endowed
with an infinite set of color charges.

Writing the canonical charges as

$Q_{\xi,\epsilon}=b_{i}B^{i}+\frac{1}{2}b_{ij}M^{ij}+Q_{W}+Q_{T}+Q^{\text{YM}}_{\epsilon}\,,$
(197)

where

$\displaystyle B^{i}=\oint d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,n^{i}\Big{(}2k^{(2)}+\frac{1}{4}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}^{2}+\frac{1}{4}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}^{A}_{B}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}^{B}_{A}\Big{)}\,,$ (198) $\displaystyle M^{ij}=\oint
d^{2}xx^{[i}e^{j]A}\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{(2)r}_{A}+\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{a}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}_{a}\Big{)}\,,$ (199) $\displaystyle Q_{T}=\oint
d^{2}x\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,T\,\mathcal{T}\,,\quad Q_{W}=\oint
d^{2}x\,W\,\mathcal{W}\,,$ (200) $\displaystyle Q^{\text{YM}}_{\epsilon}=\oint
d^{2}x\,\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}^{a}\mathcal{T}_{a}\,,$ (201)

with

$\mathcal{T}=2\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}_{rr}\,,\qquad\mathcal{W}=2(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rr}-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{A}_{A})\,,\qquad\mathcal{T}_{a}=-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{a}\,,$
(202)

we then find that the brackets of the Carroll boosts and spatial rotations are
given by

$\displaystyle\\{B^{i},B^{j}\\}$ $\displaystyle=0\,,$ (203)
$\displaystyle\\{B^{i},M^{jk}\\}$
$\displaystyle=\frac{1}{2}\Big{(}\delta^{ik}B^{j}-\delta^{ij}B^{k}\Big{)}\,,$
(204) $\displaystyle\\{M^{ij},M^{kl}\\}$
$\displaystyle=\frac{1}{2}\Big{(}\delta^{ik}M^{jl}-\delta^{il}M^{jk}-\delta^{jk}M^{il}+\delta^{jl}M^{ik}\Big{)}\,.$
(205)

The brackets of the time and spatial Carrollian supertranslations and the non-
Abelian charges with Carroll boosts and spatial rotations read

$\displaystyle\\{B^{i},\mathcal{W}(x^{A})\\}$
$\displaystyle=-3n^{i}\mathcal{T}-\partial_{A}n^{i}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\mathcal{T}-n^{i}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}}\mathcal{T}\,,$ (206)
$\displaystyle\\{B^{i},\mathcal{T}(x^{A})\\}$ $\displaystyle=0\,,$ (207)
$\displaystyle\\{B^{i},\mathcal{T}_{a}(x^{A})\\}$ $\displaystyle=0\,,$ (208)
$\displaystyle\\{M^{ij},\mathcal{W}(x^{A})\\}$
$\displaystyle=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{W}(x^{A}))\,,$ (209)
$\displaystyle\\{M^{ij},\mathcal{T}(x^{A})\\}$
$\displaystyle=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{T}(x^{A}))\,,$ (210)
$\displaystyle\\{M^{ij},\mathcal{T}_{a}(x^{A})\\}$
$\displaystyle=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{T}_{a}(x^{A}))\,.$ (211)

Finally, the bracket between the non-Abelian charges is given by

$\displaystyle\\{\mathcal{T}_{a}(x^{A}_{1}),\mathcal{T}_{b}(x^{A}_{2})\\}$
$\displaystyle=f^{c}_{\,\,\,ab}\mathcal{T}_{c}(x^{A}_{1})\delta^{(2)}(x^{A}_{1}-x^{A}_{2})\,.$
(212)

###  7.2 Magnetic Carrollian gravity coupled to the magnetic limit of the
Yang-Mills field

In this case, the color charges vanish because of the strict set of parity
conditions (of Tanzi:2020fmt ),

$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{r}\sim\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{A}=\text{even}\,,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$A$\kern
0.0pt}}}_{A}\sim\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{r}=\text{odd}\,,$ (213)

that one must impose on the Yang-Mills field in order to render Carroll boosts
canonical transformations. Thus, the asymptotic symmetry algebra is only given
by the Carroll-BMS algebra.

###  7.3 Electric Carrollian gravity coupled to the electric limit of the
Yang-Mills field

In this case, the theory is invariant under the semi-direct sum of $so(3)$
with the direct sum of the infinite-dimensional sets of generalized Carrollian
supertranslations (extended $iso(3)$-Carroll algebra) and color charges.

The asymptotic Killing vector and the non-Abelian gauge parameter that
preserve the fall-off of the fields read

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle
br+T+\mathcal{O}\left(r^{-2}\right)\,,$ (214) $\displaystyle\xi^{r}$
$\displaystyle=$ $\displaystyle W+\mathcal{O}\left(r^{-1}\right)\,,$ (215)
$\displaystyle\xi^{A}$ $\displaystyle=$ $\displaystyle
Y^{A}+\frac{1}{r}\left(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}W+\frac{2b}{\sqrt{g}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rA}\right)+\mathcal{O}\left(r^{-2}\right)\,,$ (216)
$\displaystyle\varepsilon$ $\displaystyle=$ $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\varepsilon$\kern
0.0pt}}}+\mathcal{O}(r^{-1})\,,$ (217)

where $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}b+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}b=0$ and $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}Y_{B}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}Y_{A}=0$.

The boundary term of the canonical generator is given by

$\displaystyle Q_{\xi,\epsilon}$ $\displaystyle=\oint
d^{2}x\Big{[}2Y^{A}\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{(2)r}_{A}+\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{a}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}_{a}\Big{)}+2I^{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{A}+2W\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rr}-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\epsilon$\kern 0.0pt}}}^{a}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{a}\Big{]}\,.$ (218)

The latter can be re-written as

$Q_{\xi,\epsilon}=\frac{1}{2}b_{ij}M^{ij}+Q_{I}+Q_{W}+Q^{\text{YM}}_{\epsilon}\,,$
(219)

where

$\displaystyle M^{ij}=\oint d^{2}xx^{[i}e^{j]A}\Big{(}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{(2)r}_{A}+\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$A$\kern 0.0pt}}}^{a}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{r}_{a}\Big{)}\,,$ (220) $\displaystyle Q_{I}=\oint
d^{2}x\,I^{A}\mathcal{W}_{A}\,\qquad Q_{W}=\oint d^{2}x\,W\,\mathcal{W}\,,$
(221) $\displaystyle Q^{\text{YM}}_{\epsilon}=\oint d^{2}x\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}^{a}\mathcal{T}_{a}\,,$ (222)

with $\mathcal{W}_{A}=2\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{A}$,
$\mathcal{W}=2\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr}$ and $\mathcal{T}_{a}=-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{r}_{a}$.

The brackets of the charges are then given by

$\displaystyle\\{M^{ij},M^{kl}\\}$
$\displaystyle=\frac{1}{2}\Big{(}\delta^{ik}M^{jl}-\delta^{il}M^{jk}-\delta^{jk}M^{il}+\delta^{jl}M^{ik}\Big{)}\,,$
(223) $\displaystyle\\{M^{ij},\mathcal{W}(x^{A})\\}$
$\displaystyle=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{W}(x^{A}))\,,$ (224)
$\displaystyle\\{M^{ij},\mathcal{W}_{A}(x^{A})\\}$
$\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern
0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{W}_{A}(x^{A}))-x^{[i}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}e^{j]B}\mathcal{W}_{B}(x^{A})\,,$ (225)
$\displaystyle\\{M^{ij},\mathcal{T}_{a}(x^{A})\\}$
$\displaystyle=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{T}_{a}(x^{A}))\,,$ (226)
$\displaystyle\\{\mathcal{T}_{a}(x^{A}_{1}),\mathcal{T}_{b}(x^{A}_{2})\\}$
$\displaystyle=f^{c}_{\,\,\,ab}\mathcal{T}_{c}(x^{A}_{1})\delta^{(2)}(x^{A}_{1}-x^{A}_{2})\,.$
(227)

###  7.4 Electric Carrollian gravity coupled to the magnetic limit of the
Yang-Mills field

The color charges in this case vanish because of the strict set of parity
conditions on the Yang-Mills field. The asymptotic symmetry algebra is
accordingly only given by the semi-direct sum of $so(3)$ with the infinite-
dimensional sets of generalized Carrollian supertranslations. The latter
explicitly reads

$\displaystyle\\{M^{ij},M^{kl}\\}$
$\displaystyle=\frac{1}{2}\Big{(}\delta^{ik}M^{jl}-\delta^{il}M^{jk}-\delta^{jk}M^{il}+\delta^{jl}M^{ik}\Big{)}\,,$
(228) $\displaystyle\\{M^{ij},\mathcal{W}(x^{A})\\}$
$\displaystyle=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{W}(x^{A}))\,,$ (229)
$\displaystyle\\{M^{ij},\mathcal{W}_{A}(x^{A})\\}$
$\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern
0.0pt}}}_{B}(x^{[i}e^{j]B}\mathcal{W}_{A}(x^{A}))-x^{[i}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}e^{j]B}\mathcal{W}_{B}(x^{A})\,.$ (230)

##  8 Conclusions

In this paper, we have analyzed the asymptotic structure of the electric and
magnetic Carrollian limits of the Enstein-Yang-Mills theory. Our results can
be summarized as follows:

  * •

We have first constructed BMS-like extensions of the Carroll group on purely
algebraic grounds, without reference to the dynamics. We used the 3+1
description of the symmetry, particularly well suited to rescalings of the
energy and the linear momentum involving different powers of $c$. The same
techniques applied to the more familiar non-relativistic case have led us to a
new BMS-like extension of the Galilean and Bargmann algebras.

  * •

We have then considered the Carrollian limit of the pure Einstein theory,
where we have shown that the electric contraction allows for a larger set of
supertranslations involving three functions on the sphere (with definite
parity conditions). Technically, this is because the condition
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{rA}=0$ on the leading order of the mixed radial-angular component of
the metric, which reduces the number of symmetry generators in the Einstein
case, is not needed any more in the electric contraction.

  * •

We studied next the asymptotic symmetries of Carrollian Yang-Mills theories
and have shown that contrary to its Minkowskian parent Tanzi:2020fmt , the
electric Carroll contraction admits at spatial infinity an infinite set of
angle-dependent color gauge symmetries fully compatible with Carroll spacetime
covariance, including the Carroll boosts. The result extends directly to the
combined Einstein-Yang-Mills system.

Our results can be extended to higher dimensions by following the methods of
Fuentealba:2021yvo ; Fuentealba:2022yqt , as well as to the coupled
electromagnetic-massless scalar field system for which the asymptotic analysis
has been performed in Tanzi:2021xva .

Another possible future direction is to explore the possibility of realizing
the Galilean-BMS algebra (34) as an asymptotic symmetry by imposing suitable
boundary conditions and parity conditions in Newton-Cartan gravity
Andringa:2010it .

It is rather interesting that the electric Carroll limit of the Yang-Mills
theory admits the infinite-dimensional color symmetry group found at null
infinity (but not at spatial infinity) prior to taking the limit. This
property is in line with the general expectation that the Carroll limit
describes well the dynamics at null infinity but definitely deserves further
study.

## Acknowledgments

Discussions with Andrés Gomberoff, Joaquim Gomis, Alfredo Pérez and Ricardo
Troncoso are gratefully acknowledged. This work was partially supported by
FNRS-Belgium (conventions FRFC PDRT.1025.14 and IISN 4.4503.15), as well as by
funds from the Solvay Family. P.S-R. has received funding from the Norwegian
Financial Mechanism 2014-2021 via the National Science Centre (NCN) POLS grant
2020/37/K/ST3/03390. JS was supported by a Marina Solvay fellowship.

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