# Asymptotic symmetries of gravity in the gauge PDE approach

Maxim Grigorieva,b and Mikhail Markova,b,c a Lebedev Physical Institute,
Leninsky Ave. 53, 119991 Moscow, Russia
b Institute for Theoretical and Mathematical Physics,
Lomonosov Moscow State University, 119991 Moscow, Russia
c Steklov Mathematical Institute of Russian Academy of Sciences,
Gubkina str. 8, 119991 Moscow, Russia

###### Abstract

We propose a framework to study local gauge theories on manifolds with
boundaries and asymptotic symmetries, which is based on representing them as
so-called gauge PDEs. These objects extend the conventional BV-AKSZ sigma-
models to the case of not necessarily topological and diffeomorphism invariant
systems and are known to behave well with respect to restrictions to
submanifolds and boundaries. We introduce the notion of gauge PDE with
boundaries, which takes into account generic boundary conditions, and apply
the framework to asymptotically flat gravity. In so doing we start with a
suitable representation of gravity as a gauge PDE with boundaries which
implements the Penrose’s description of asymptotically simple spacetimes. We
then derive the minimal model of the gauge PDE induced on the boundary and
observe that it provides the Cartan (frame-like) description of a (curved)
conformal Carollian structure on the boundary. Furthermore, imposing a
suitable version of the familiar boundary conditions in the induced boundary
gauge PDE immediately leads to the conventional BMS algebra of asymptotic
symmetries. Finally, we briefly sketch the construction in the case of
asymptotically (A)dS gravity.

###### Contents

  1. 1 Introduction
  2. 2 Gauge PDEs with boundaries and their symmetries
    1. 2.1 Gauge PDEs
    2. 2.2 Gauge PDEs with boundaries
    3. 2.3 (Asymptotic) symmetries in gPDE terms
    4. 2.4 Asymptotic symmetries in the presymplectic gPDE framework
  3. 3 GR as a gauge PDE
    1. 3.1 Off-shell GR as a gauge PDE
    2. 3.2 Conformal-like on-shell GR
    3. 3.3 Pre-minimal model for conformal-like on-shell GR
  4. 4 Boundary systems and asymptotic symmetries
    1. 4.1 Asymptotically simple GR as a gPDE with boundaries
    2. 4.2 Minimal model for the boundary gPDE of asymptotically simple GR
    3. 4.3 Boundary conditions and BMS symmetries
    4. 4.4 Field-theoretical interpretation of the minimal model
    5. 4.5 Asymptotically (A)dS spaces
  5. A Symmetries

##  1 Introduction

Asymptotic symmetries play a prominent role in modern QFT and gravity [1, 2,
3, 4, 5, 6, 7, 8, 9, 10, 11]. They originate from gauge transformations that
preserve the boundary conditions imposed on fields but, at the same time, are
not to be considered as genuine gauge transformations and hence define new
physical symmetries. In particular, asymptotic symmetries critically depend on
the choice of the boundary conditions for fields and gauge parameters.

Historically, the first and very influential example of asymptotic symmetries
is the Bondi-Metzner-Sachs (BMS) symmetries of asymptotically flat gravity [1,
2]. In contrast to naive expectations, the simplest natural choice of boundary
conditions employed in [1, 2] results in the enhancement of the Poincaré group
by the so-called supertranslations, giving the infinite-dimensional symmetry
group which is now known as the BMS group. The latter group, or its
generalizations associated with different choices of boundary conditions, is
now considered to be a proper symmetry group of the gravitational S-matrix and
has been related [10, 12] to the celebrated soft graviton theorem and the
gravitational memory effect.

The proper geometrical setup for studying asymptotic symmetries of gravity was
proposed by Roger Penrose, who introduced the notion of asymptotically simple
spacetime [13]. More specifically, this is a spacetime
$(\widetilde{M},\widetilde{g})$ that is diffeomorphic to the interior of the
spacetime $(M,g)$ with boundary such that in the interior
$g=\Omega^{2}\widetilde{g}$ for some smooth function $\Omega$ satisfying
$\Omega>0$ and $\Omega|_{\mathcal{J}}=0,d\Omega|_{\mathcal{J}}\neq 0$, where
$\mathcal{J}=\partial M$ is the boundary. In other words, the idea is to
realize the boundary at infinity as the usual boundary of the auxiliary
spacetime. More details can be found in e.g. [14, 15, 16].

Recent decades have shown an increasing interest in asymptotic symmetries, not
only in the context of gravity but also in general gauge theories, including
Yang-Mills, topological systems, and higher-spin gauge theories [17, 18, 19,
20, 21, 22, 23, 24]. From this perspective, asymptotic symmetries are to be
considered as a general feature of gauge theories on manifolds with
(asymptotic) boundaries. This calls for a proper gauge-theoretical
understanding of asymptotic symmetries. Various approaches are available in
the literature. In particular, the first principle understanding of asymptotic
symmetries is provided within the Hamiltonian approach [25, 4], see also [17],
at the price of manifest covariance. A covariant generalization can be
achieved with one or another version of the covariant phase space approach
[26, 27, 28], see also [29] and references therein.

A powerful and systematic framework for (quantum) gauge theories is provided
by the Batalin-Vilkovisky (BV) formalism [30, 31] or, more precisely, its
modern enhancements, such as the jet-bundle BV approach to local gauge
theories, see e.g. [32, 26]. Of special attention in the present work is the
so called BV-AKSZ framework [33], initially proposed in the context of
topological models. An interesting feature observed in [34, 35, 36, 37, 38]
(see also [39, 40, 40]) is that a BV-AKSZ system naturally induces a shifted
AKSZ system on any space-time submanifold. For instance, an AKSZ version of
the Hamiltonian BFV formulation is induced on a space-like submanifold of
spacetime. By combining this observation with the construction of [39], see
also [40, 38], which allows one to reformulate a general local gauge system as
an AKSZ-like model, one arrives at the framework to analyze boundary values
and symmetries of generic local gauge systems. This approach has been
successfully employed in [41, 42, 43] in the study of boundary values of
generic gauge fields on AdS space, see also [44] for a related approach, and
in the reconstruction of bulk theories from the boundary data [45, 46]. Note
that it does not employ the symplectic structure of BV formalism and is
applicable to non-Lagrangian systems or systems whose Lagrangian is not
specified. Let us also mention a somewhat related approach of [37] to
Lagrangian gauge systems on manifolds with boundaries, see also [47, 48, 49].

In this work we develop an approach to gauge theories on manifolds with
boundaries and their asymptotic symmetries, which is based on representing a
given local gauge theory as a so-called gauge PDE. Gauge PDE (gPDE) is a
generalization of the non-Lagrangian BV-AKSZ formulation to the case of
general gauge theories. Although the term gauge PDE and its geometrical
definition was introduced only in [50], the framework was originally put
forward already in [39], see also [51], under the name parent formulation.
Just like AKSZ systems, gPDEs behave well with respect to the restriction to
space-time submanifolds and hence provide a natural framework to study gauge
theories on manifolds with boundaries. More precisely, a gPDE is a bundle over
spacetime and its pullback to a submanifold is again a gPDE, see e.g. [52] and
references therein. Gauge PDEs can be also considered as a BV-BRST extension
and generalization of the so-called unfolded formalism developed in the
context of higher spin theories [53, 54].

We propose the notion of gPDEs with boundaries, which takes into account
boundary conditions on fields and gauge parameters. More precisely, the
boundary conditions are described by a sub-gPDE of the induced boundary gPDE
which is, by definition, the initial gPDE pulled back to the boundary. In
these terms asymptotic symmetries can be defined in a rather general and
purely geometrical way, giving a systematic description of such systems and
their symmetries in terms of differential graded geometry.

The approach is applied to asymptotically flat gravity and is shown to
reproduce celebrated BMS symmetries once a gauge PDE version of the well-known
boundary conditions is taken. A crucial point of the construction is the gauge
theoretical implementation of the Penrose asymptotically simple spacetime.
This is achieved by introducing a Weyl compensator field $\Omega$. In so doing
the metric sector of the system can be considered as that describing conformal
geometry, which allows to employ the equivalent reduction [55] (see also [56])
known in the context of conformal gravity. This later step leads to a
remarkably simple boundary system which also resembles the conformal-geometry
approach [57, 58, 59] to BMS symmetries and more general boundary calculus
[60, 61].

The paper is organized as follows: in Section 2, we briefly recall the gauge
PDE approach to local gauge theories and propose its extension to theories on
manifolds with boundaries and generally nontrivial boundary conditions. We
then define asymptotic symmetries in this setup. In Section 3, we present a
reformulation of gPDE for general relativity in a form convenient for studying
the asymptotic structure of this theory. The form is inspired by the Penrose’s
notion of an asymptotically simple spacetime. In Section 4 we derive the
induced boundary system and analyze boundary conditions and asymptotic
symmetries in the asymptotically flat case. This involves derivation of a
concise minimal model of the induced boundary system, which makes manifest the
Carrollian geometry structure. Finally, we sketch the construction in the the
case of nonzero cosmological constant and present the respective minimal
model.

##  2 Gauge PDEs with boundaries and their symmetries

###  2.1 Gauge PDEs

In the approach we employ in this work, local gauge theories, considered at
the level of equations of motion, are encoded in the geometrical objects
called gauge PDEs (or gPDE for short). A gPDE can be seen as a generalization
of the jet-bundle non-Lagrangian BV formalism, i.e. a jet-bundle of a graded
fiber bundle, equipped with the BV-BRST differential, which underlies a
conventional BV formalism for local gauge theories, see e.g. [32]. The non-
Lagrangian version was suggested in [51], see also [62, 63, 64].

We first need to briefly recall the necessary prerequisites. More detailed
exposition can be found in [50].

######  Definition 2.1.

A $Q$-manifold (also called dg-manifold) is a $\mathbb{Z}$-graded
supermanifold equipped with a homological vector field $Q$, i.e. a vector
field of degree 1 satisfying $Q^{2}=\dfrac{1}{2}[Q,Q]=0$, $\mathrm{gh}(Q)=1$,
$|Q|=1$, where $\mathrm{gh}(\cdot)$ denotes $\mathbb{Z}$-degree (often called
ghost degree), and $|\cdot|$ denotes Grassmann parity.

In this work we only deal with bosonic systems and hence one can simply assume
$|f|=\mathrm{gh}(f)=\,\rm mod\,{2}$ for any homogeneous functions, form,
vector field, etc. Of course, the framework extends to systems with fermions
in a standard way.

The standard simplest example of a $Q$ manifold is a shifted tangent bundle
$T[1]X$ over a smooth manifold $X$. Its algebra of functions is just the
algebra of differential forms on $X$. Under this identification the de Rham
differential corresponds to a homological vector field $\mathrm{d_{X}}$ on
$T[1]X$. If $x^{\mu}$ are local coordinates on $X$ and $\theta^{\mu}$ the
associated coordinates on the fibers,
$\mathrm{d_{X}}\equiv\theta^{\mu}\frac{\partial}{\partial x^{\mu}}$.

Let us also recall the definition of a $Q$-bundle, i.e. a fiber bundle in the
category of $Q$-manifolds:

######  Definition 2.2.

[65] 1\. $Q$-bundle $\pi:(M,Q)\rightarrow(N,q)$ is a locally trivial bundle of
graded manifolds $M$ and $N$ such that $\pi^{*}\circ q=Q\circ\pi^{*}$.

2\. A section $\sigma:N\rightarrow M$ is called a $Q$-section if
$q\circ\sigma^{*}=\sigma^{*}\circ Q$.

3\. A $Q$-bundle $\pi:(M,Q)\rightarrow(N,q)$ is called locally trivial (as a
$Q$-bundle) if it’s locally isomorphic to a direct product of the base $(N,q)$
and the fiber $(F,q^{\prime})$ in such a way that $Q=q+q^{\prime}$, i.e. $Q$
is locally the direct product $Q$-structure.

There is a natural notion of equivalence for $Q$ manifolds, which, roughly
speaking, corresponds to elimination of contractible pairs. From gauge-
theoretical viewpoint, such contractible coordinates correspond to auxiliary
fields, pure gauge variables and their associated ghosts/antifields. More
precisely:

######  Definition 2.3.

1\. A contractible $Q$-manifold is a $Q$-manifold of the form $(T[1]W,d_{W})$,
where $W$ is a graded vector space considered as a graded manifold, and
$d_{W}$ is the de Rham differential on $T[1]W$.

2\. A $Q$-manifold $(N,q)$ is called an equivalent reduction of $(M,Q)$ if
$(M,Q)$ is a locally trivial $Q$-bundle over $(N,q)$ admitting a global
$Q$-section, and the fibers of this bundle are contractible $Q$-manifolds.

Equivalent reduction generates the notion of equivalence. In particular,
cohomology of natural complexes, e.g. $Q$-cohomology in differential forms on
$E$, multivector fields, etc. on equivalent $Q$-manifolds are isomorphic.

Locally, the statement that $(N,q)$ is an equivalent reduction of $(M,Q)$
implies that, seen as a $Q$-manifold, $(M,Q)$ is a direct product of $(N,q)$
and a contractible $Q$-manifold. A useful way [51] to identify an equivalent
reduction in practice is to find independent functions $w^{a}$ such that
$Qw^{a}$ are independent functions as well. It then follows that at least
locally a submanifold defined by $Qw^{a}=0$ and $w^{a}=0$ is an equivalent
reduction of the initial $Q$-manifold. It follows one can find functions
$\phi^{i}$ such that $w^{a},v^{a}=Qw^{a},\phi^{i}$ form a local coordinate
system and $Q\phi^{i}=Q^{i}(\phi)$. In this case, $w^{a}$ and $v^{a}$ are
standard contractible pairs known in the context of local BRST cohomology, see
e.g. [66] and reference therein.

The above notions of equivalent reduction and of equivalence extend to
$Q$-bundles over the same base:

######  Definition 2.4.

Let $(M^{\prime},Q^{\prime})$ and $(M,Q)$ are $Q$-bundles over the same base
$(N,q)$. $(M,Q)$ is called an equivalent reduction of
$(M^{\prime},Q^{\prime})$ if $(M,Q)$ is a locally trivial $Q$-bundle over
$(M,Q)$ such that the projection and the local trivializations maps are
compatible with projections to $(N,q)$ (i.e. it is a bundle in the category of
bundles over $(N,q)$), $(M^{\prime},Q^{\prime})$ admits a global $Q$-section,
and, moreover, the fiber is a contractible $Q$-manifold.

This generates an equivalence relation for $Q$-bundles. Again, a practical way
to identify an equivalent reduction is to find functions $w^{a}$ such that
$w^{a},Qw^{a}$ are independent functions that remain independent when
restricted to a fiber (i.e. they can be taken as a part of a fiber coordinate
system). It follows that at least locally the subbundle of
$(M^{\prime},Q^{\prime})$ singled out by $w^{a}=0$ and $Qw^{a}=0$ is an
equivalent reduction. 111Strictly speaking, in the infinite-dimensional case
one should also require the existence of complementary fiber coordinates
$\phi^{i}$ such that $Q\phi^{i}=Q^{i}(\phi)$. See [39] for more details.

Finally, we are ready to formulate the definition of Gauge PDEs, which we
abbreviate gPDE in what follows.

######  Definition 2.5.

1\. Gauge PDE $(E,Q,T[1]X)$ is a $Q$ bundle
$\pi:(E,Q)\rightarrow(T[1]X,\mathrm{d_{X}})$, where $X$ is a real manifold
(independent variables). In addition it is assumed that $(E,Q,T[1]X)$ is
locally equivalent to nonnegatively graded $Q$-bundle. Moreover, it should be
equivalent to a jet-bundle BV-formulation seen as $Q$-bundle over $T[1]X$ with
$Q=\mathrm{d_{h}}+s$, where $s$ is the BV-BRST differential and
$\mathrm{d_{h}}$ the horizontal differential.
2\. Two gauge PDEs over $T[1]X$ are considered equivalent if they are
equivalent as $Q$-bundles.

Gauge PDEs encode local gauge theories. In particular, field configurations
are identified with their sections while equations of motion arise as
differential conditions on sections. More precisely, section
$\sigma:T[1]X\rightarrow E$ is a solution to $(E,Q,T[1]X)$ if

$\displaystyle\mathrm{d_{X}}\circ\sigma^{*}=\sigma^{*}\circ Q$ (2.1)

Infinitesimal gauge transformations of the section $\sigma$ are defined as

$\displaystyle\delta\sigma^{*}=\mathrm{d_{X}}\circ\chi_{\sigma}^{*}+\chi_{\sigma}^{*}\circ
Q,$ (2.2)

where
$\chi_{\sigma}^{*}:\mathcal{C}^{\infty}(E)\rightarrow\mathcal{C}^{\infty}(T[1]X)$
is of degree $-1$, satisfies

$\displaystyle\chi_{\sigma}^{*}(fg)=\chi_{\sigma}^{*}(f)\sigma^{*}(g)+(-1)^{|f|}\sigma^{*}(f)\chi_{\sigma}^{*}(g)\,,\qquad\forall
f,g\in\mathcal{C}^{\infty}(E)\,,$ (2.3)

and $\chi_{\sigma}^{*}(\pi^{*}(h))=0$ for all
$h\in\mathcal{C}^{\infty}(T[1]X)$. The map $\chi_{\sigma}^{*}$ is interpreted
as a gauge parameter. It is easy to check that the above gauge transformation
is an infinitesimal symmetry of the equations of motion (2.1). In a similar
way one defines gauge for gauge symmetries.

It is often convenient to parameterize $\chi_{\sigma}^{*}$ in terms of a
vertical vector field $Y$ on $E$ of degree $-1$:
$\chi_{\sigma}^{*}=\sigma^{*}\circ Y$. It is easy to check that for this
choice $\chi_{\sigma}^{*}$ (2.3) is indeed satisfied. Using this
representation the gauge transformation of $\sigma$ can be written as
$\delta\sigma^{*}=\sigma^{*}\circ[Q{,}\,Y]$. Note that a vector field
$V\equiv[Q,Y]$ is an infinitesimal symmetry of $(M,Q,T[1]X)$ because it
preserves $Q$, the degree, and the bundle structure.

In the case of diffeomorphism-invariant systems, for instance gravity, their
gPDE description usually requires the additional condition on the allowed
class of sections. More precisely, the fiber coordinates typically involve a
subset of ”diffeomorphism ghosts” $\xi^{a}$, $a=0,\ldots\dim{X}-1$,
$\mathrm{gh}(\xi^{a})=1$ and sections are restricted by the condition that
$e^{a}_{\mu}(x)$ defined via $\sigma^{*}(\xi^{a})=e^{a}_{\mu}(x)\theta^{\mu}$,
are invertible. This is of course a gPDE counterpart of the familiar condition
in the frame-like formulation of gravity. All the systems considered in this
work are of this type and the nondegeneracy condition on sections is assumed
in what follows.

To complete the discussion of gPDEs let us note that gPDE automatically
determine a nonlagrangian jet-bundle formulation of the underlying gauge
system. This is induced on the bundle of super-jets of $E$ and its BV-BRST
differential is the vertical part of the prolongation of $Q$ to the super-jet
bundle. More details can be found in [50, 52], see also [39] for the original
construction and local proof.

###  2.2 Gauge PDEs with boundaries

Let $X$ be a space-time manifold but now we assume that $X$ has a nontrivial
boundary $\Sigma=\partial X$ and let $i:\Sigma\to X$ denotes the embedding of
the boundary. Suppose we are given with a gPDE $(E,Q,T[1]X)$ on $X$. This
induces a new gPDE $i^{*}E$ on $\Sigma$ given by a pullback of $E$ to
$T[1]\Sigma\subset T[1]X$ (here by a slight abuse of notation $i$ also denotes
an induced pushforward $T[1]\Sigma\to T[1]X$). It is easy to check that this
is again a gPDE (e.g. by regarding it as a $Q$-submanifold of $E$), which we
call induced boundary gPDE. $i^{*}E$ can be considered as a gPDE describing a
gauge theory of unconstrained boundary values of the fields encoded in
$(E,Q,T[1]X)$, see [52] for more details and [41, 42] for the earlier and less
general construction and applications in the context of higher spin
holography.

Now we are interested in the gPDE description of systems with possibly
nontrivial boundary conditions. We have the following:

######  Definition 2.6.

By a gauge PDE with boundares we mean the following data:
$(E,Q,T[1]X,E_{\Sigma},T[1]\Sigma)$, where gPDE $(E,Q,T[1]X)$ is a gPDE on $X$
and $(E_{\Sigma},Q_{\Sigma},T[1]\Sigma)$ is a gPDE on the boundary
$\Sigma=\partial X$, which is a sub-gPDE of $i^{*}E$. In particular,
$Q_{\Sigma}$ is a restriction of $Q$ to $E_{\Sigma}\subset i^{*}E\subset E$.

Gauge PDE $(E_{\Sigma},Q_{\Sigma},T[1]\Sigma)$, which is a part of the above
definition, can be regarded as a gPDE of boundary conditions. For instance, if
$E_{\Sigma}=i^{*}E$ this means that no boundary conditions are imposed.
General situations are described by nontrivial $Q$-subbundles of $i^{*}E$. It
is important to stress that in general, $E_{\Sigma}$ restricts not only the
boundary values of fields but also the boundary values of gauge parameters and
parameters of gauge-for-gauge symmetries.

######  Remark 2.1.

Even if $E_{\Sigma}$ doesn’t coincide with $i^{*}E$ it doesn’t necessarily
mean that we are dealing with nontrivial boundary conditions. This happens if
$E_{\Sigma}$ is an equivalent reduction of $i^{*}E$ in which case $E_{\Sigma}$
implements elimination of auxiliary fields and pure gauge variables.

Although the above definition is quite general, in the context of asymptotic
symmetries we need to allow $(E,Q,T[1]X)$ to be slightly locally nontrivial.
Namely, $E$ restricted to the interior of $X$ and $i^{*}E$ are still required
to be locally trivial while the typical fiber of $i^{*}E$ can differ from the
fiber over the interior by a subset of measure zero. For our present purposes
it is enough to allow the fiber of $i^{*}E$ to be a manifold with boundary
whose interior coincides with the typical fiber over the interior. In this
case the total space of $E$ restricted to the interior of $X$ is itself the
interior of a manifold with corners. Restricting it to $\partial X$ gives a
total space of $i^{*}E$ whose fiber is a manifold with boundary. As we are
going to see, at more practical level we actually work with $i^{*}E$ and its
sub-gPDE $E_{\Sigma}$ which are locally trivial. Note that a gPDE with
boundary could be defined in terms of a single locally-nontrivial bundle
$E^{\prime}\to T[1]X$ such that its fibers over the boundary shrinks to those
of $E_{\Sigma}$ but we prefer to keep boundary conditions explicit.

The field theoretical interpretation of the above definition becomes clear
with the help of:

######  Definition 2.7.

  1. 1.

A solution of $(E,Q,T[1]X,E_{\Sigma},T[1]\Sigma)$ is a section
$\sigma:T[1]X\to E$ satisfying $\sigma^{*}\circ
Q=\mathrm{d_{X}}\circ\sigma^{*}$ and such that its restriction to $T[1]\Sigma$
belongs to $E_{\Sigma}$, i.e. is a solution to
$(E_{\Sigma},Q_{\Sigma},T[1]\Sigma)$.

  2. 2.

A gauge parameter is a vertical vector field $Y$ on $E$ such that
$\mathrm{gh}(Y)=-1$ and its restriction to $i^{*}E$ is tangent to
$E_{\Sigma}\subset i^{*}E$. In other words, gauge parameters should satisfy
the boundary conditions encoded in $E_{\Sigma}\subset i^{*}E$.

  3. 3.

A gauge transformation of section $\sigma$ is defined as

$\delta_{Y}\sigma^{*}=\mathrm{d_{X}}\circ\sigma^{*}\circ Y+\sigma^{*}\circ
Y\circ Q$ (2.4)

The following comments are in order: it is easy to check that if $\sigma$ is a
solution then $\sigma+\delta_{Y}\sigma$ with $\delta_{Y}\sigma$ determined by
(2.4), is again a solution (to first order in $Y$). Moreover, in this case the
gauge transformation (2.4) can be rewritten as:

$\delta_{Y}\sigma^{*}=\sigma^{*}\circ[Q{,}\,Y]\,.$ (2.5)

Restricting (2.4) to $T[1]\Sigma$ one finds a standard gauge transformation
for $(E_{\Sigma},Q_{\Sigma},T[1]\Sigma)$ whose parameter is $Y$ restricted to
$E_{\Sigma}\subset i^{*}E\subset E$ (recall that $Y$ is vertical and hence is
tangent to $i^{*}E$ and the above definition requires $Y$ to be tangent to
$E_{\Sigma}$). In a similar way one can define gauge-for-gauge
transformations. In particular, parameters of the gauge-for-gauge
transformations of stage 1 are vertical vector fields of degree $-2$ and
tangent to $E_{\Sigma}$.

All the above definitions can be generalised to the case where $\Sigma$ is a
generic submanifold. This version can be relevant in describing theories with
defects. Generalization to the case of manifolds with corners or higher
codimension strata is also possible.

###  2.3 (Asymptotic) symmetries in gPDE terms

Let us now turn to the discussion of symmetries. Given a gauge PDE, an
infinitesimal symmetry is by definition a vector field $W$ which preserves all
the structures, i.e. $[Q{,}\,W]=0$, $W$ is vertical, and $\mathrm{gh}(W)=0$
(though symmetries of nonvanishing ghost number are also of interest). The
infinitesimal transformation of a solution $\sigma$ under a symmetry
transformation determined by $W$ is defined to be:

$\delta_{W}\sigma^{*}=\sigma^{*}\circ W\,.$ (2.6)

It is easy to check that it defines an infinitesimal symmetry transformation
that takes solutions to solutions. Gauge symmetries are the ones where
$W=[Q{,}\,Y]$, where $Y$ is a gauge parameter. In particular, in this case the
above transformation coincides with (2.5) so that it is natural to regard
symmetries of the form $W=[Q{,}\,Y]$ as trivial. As we only consider
infinitesimal symmetries, in what follows we systematically omit
”infinitesimal”.

Inequivalent symmetries (also known as global or physical) can be defined as
the respective quotient of all symmetries modulo the ideal of the gauge ones
and hence are given by $Q$-cohomology in vertical vector fields. One can check
that in the case of usual jet-bundle BV formulation of a local gauge theory,
this definition reproduces the standard one, at least locally. Details can be
found in Appendix A.

As far as gauge PDEs with boundaries are concerned a natural definition of
symmetry is that it is a vertical vector field $W$ such that $[Q{,}\,W]=0$ and
$W$ restricted to $i^{*}E$ is tangent to $E_{\Sigma}\subset i^{*}E$. At the
same time, genuine gauge symmetries in this case are those symmetries of the
form $W=[Q{,}\,Y]$ whose parameters $Y$ are tangent to $E_{\Sigma}$. All the
above discussion applies to symmetries of arbitrary definite ghost degree.

Let now $(E,Q,T[1]X,E_{\Sigma},T[1]\Sigma)$ be a gPDE with boundaries. A
common lore is that asymptotic symmetries are gauge symmetries of the system
(defined as if there were no boundary conditions) that preserve boundary
conditions while those whose parameters satisfy boundary conditions for gauge
parameters are genuine gauge symmetries and should be considered trivial. In
the case of Lagrangian systems the extra requirements are to be imposed. In
the gPDE approach this can be formalised as follows:

######  Definition 2.8.

Asymptotic symmetries of $(E,Q,T[1]X,E_{\Sigma},T[1]\Sigma)$ are symmetries of
$(E_{\Sigma},Q_{\Sigma},T[1]\Sigma)$, which are restrictions to $E_{\Sigma}$
of those gauge symmetries of $i^{*}E$ that preserve $E_{\Sigma}\subset
i^{*}E$. Gauge symmetries of $(E_{\Sigma},Q_{\Sigma},T[1]\Sigma)$, i.e. vector
fields on $E_{\Sigma}$ of the form $[Q{,}\,Y]|_{E_{\Sigma}}$ with $Y$ tangent
to $E_{\Sigma}$, are considered trivial asymptotic symmetries.

More explicitly, asymptotic symmetries are vertical vector fields that have
the form $[Q{,}\,Y]$ with $Y$ vertical and are tangent to $E_{\Sigma}$. These
are considered modulo vector fields vanishing on $E_{\Sigma}$. Vector fields
$[Q{,}\,Y]$ with $Y$ tangent to $E_{\Sigma}$ are trivial (genuine gauge
symmetries). Asymptotic symmetries form a subalgebra of all symmetries of
$(E_{\Sigma},Q_{\Sigma},T[1]\Sigma)$.

An alternative would be to define asymptotic symmetries to be all symmetries
of $(E_{\Sigma},Q_{\Sigma},T[1]\Sigma)$ modulo its own gauge symmetries.
Another alternative is to require asymptotic symmetries to arise as
restrictions to $E_{\Sigma}$ of symmetries of $E$, but this does not seem to
make any difference if one restricts to local analysis as we do in this work.

Later on we also need a slightly more general framework applicable to the case
where $E_{\Sigma}$ is not a regular submanifold of $i^{*}E$ while its
prolongation to the bundle of jets of its sections is. This occurs in
applications because in the minimal formulations the frame fields arise as
components of $\sigma^{*}(\xi^{a})$, where $\sigma:T[1]\Sigma\to i^{*}E$ and
the condition that the frame field is invertible can not be implemented in
terms of the fiber geometry and is imposed as a condition on sections.

As we are going to see such formulations arise in practice if one is after a
concise formulation of the boundary gPDE. To cover this situation one allows
$Y$ to be a generalized vector field, i.e. its coefficients are allowed to
depend on jets of sections. If $\mathcal{I}_{\Sigma}$ is the ideal of
$E_{\Sigma}$ then the condition that $[Q{,}\,Y]$ is tangent to $E_{\Sigma}$ is
replaced by

$\mathrm{d_{X}}\sigma^{*}(Yf)+\sigma^{*}(YQf)=0\quad\forall
f\in\mathcal{I}_{\Sigma}\,.$ (2.7)

This should hold for all sections of the gPDE of boundary condition, i.e.
sections of $i^{*}E$ such that $\sigma^{*}(\mathcal{I}_{\Sigma})=0$. This is
precisely the condition that the symmetry transformation satisfying this
condition sends solutions to $E_{\Sigma}$ to themselves. Note that if one
doesn’t want to employ such a generalization it is always possible to work in
terms of the associated parent gPDE whose underlying bundle is the super-jet
bundle of $i^{*}E$ and hence the prolongation of $E_{\Sigma}$ is a smooth
submanifold.

###  2.4 Asymptotic symmetries in the presymplectic gPDE framework

In the case of Lagrangian systems, asymptotic symmetries can be defined as
gauge transformations whose associated charges become nontrivial due to
boundary conditions. Althogh in this work we restricted ourselves to the
analysis at the level of equations of motion, let us briefly comment on how
such an approach can be implemented in the gPDE framework.

A Lagrangian system can be described by a gPDE $(E,Q,T[1]X)$ equipped with the
compatible presymplectic structure $\omega$ of degree $n-1$, $n=\dim{X}$, such
that:

$d\omega=0\,,\qquad L_{Q}\omega\in\mathcal{I}\,,\qquad
i_{Q}L_{Q}\omega\in\mathcal{I}\,,$ (2.8)

where $\mathcal{I}$ denotes the ideal of forms on $E$ generated by the forms
of positive degree, pulled back from the base, i.e. by
$dx^{\mu},d\theta^{\mu}$ in standard coordinates. We also fix a presymplectic
potential $\chi$ such that $\omega=d\chi$. Note that for $n>1$ it exists
globally as the respective de Rham cohomology is empty. More details on
presymplectic gPDEs can be found in [52, 56, 67], see also [68, 69, 70] for
earlier relevant works.

We say that boundary gPDE $E_{\Sigma}$ and symplectic potential $\chi$ are
compatible if the pullback of $\chi$ to $E_{\Sigma}$ vanishes. It turns out
that this is a sufficient condition for the respective action to be
differentiable. Indeed, the above data defines a presymplectic AKSZ-like
action (also known as intrinsic action):

$S[\sigma]=\int_{T[1]X}\sigma^{*}(\chi)(\mathrm{d_{X}})+\sigma^{*}(H)\,,$
(2.9)

where the “covariant Hamiltonian/BRST charge” $H$ is defined through
$i_{Q}\omega+dH\in\mathcal{I}$. Note that picking an equivalent
$\chi=\chi+d\alpha$ doesn’t affect equations of motion but adds a boundary
term $\int\mathrm{d_{X}}\sigma^{*}(\alpha)$ to the above action. Consider a
variation of $S[\sigma]$ under $\sigma\to\sigma+\delta\sigma$. Representing
$\delta\sigma$ as $\sigma^{*}\circ V$, where $V$ is a vertical vector field on
$E$, one finds that the boundary term has the form:

$\delta S=\int\text{``EOM''}\,\,\delta\sigma+\int\sigma^{*}(i_{V}\chi)$ (2.10)

Because $\delta\sigma$ should preserve boundary conditions, $V$ is tangent to
$E_{\Sigma}$ so that boundary contribution vanishes provided $\chi$ and
$E_{\Sigma}$ are compatible as we assume in what follows.

Let us now turn to conserved currents (conservation laws) associated to
symmetries. A symmetry $W$ is called compatible with presymplectic structure
if

$L_{W}\omega+L_{Q}d\alpha\in\mathcal{I}\,,$ (2.11)

for some $1$-form $\alpha$ of ghost degree
$\mathrm{gh}(\alpha)=\mathrm{gh}(W)+n-2$. Given a compatible symmetry one can
define an associated generalised conserved current, which is a degree
$\mathrm{gh}(W)+n-1$ function defined through: 222The discussion of global
symmetries and conserved currents in the presymplectic BV-BRST approach can be
found in [64].

$i_{W}\omega-(-1)^{|W|}L_{Q}\alpha-dH_{W}\in\mathcal{I}\,,$ (2.12)

The consistency condition $d(i_{W}\omega-(-1)^{|W|}L_{Q}\alpha)\in\mathcal{I}$
holds thanks to (2.11). If $\alpha$ is fixed $H_{W}$ is defined modulo
functions of the form $\pi^{*}(f),f\in\mathcal{C}^{\infty}(T[1]X)$. Moreover,
it follows $d(QH_{W})\in\mathcal{I}$ and hence $QH_{W}=\pi^{*}(h)$ for some
function $h$ on $T[1]X$ so that by adding to $H_{W}$ a function of the form
$\pi^{*}(f)$ one can achieve $QH_{W}=0$. This defines a map from compatible
symmetries to conserved currents.

Given a conserved current, i.e. a $Q$-closed function $H_{W}$ on $E$, one can
define the respective charge as

$\mathbf{H_{W}}[\sigma]=\int_{T[1]C}\sigma^{*}(H_{W})\,,$ (2.13)

where $\sigma$ is a solution ($Q$-section) of $E$ restricted to a shifted
tangent bundle of submanifold $C\subset X$ of codimension $1-\mathrm{gh}(W)$.
Note that adding a $Q$-exact piece to $H_{W}$ results in the addition of a
$\mathrm{d_{X}}$-exact term in the integrand and hence this only contributes
to the boundary term (is $C$ has a nontrivial boundary). The above charge
doesn’t change under deformations of $C$ provided $\partial C$ is kept
undeformed, because $\mathrm{d_{X}}\sigma^{*}(H_{W})=\sigma^{*}(QH_{W})=0$ if
$\sigma$ is a solution.

If $W$ is a gauge symmetry, i.e. $W=[Q{,}\,Z]$ with $Z$ vertical it is
automatically compatible with $\omega$ because
$L_{[Q{,}\,Z]}\omega=-(-1)^{|Z|}L_{Q}dL_{Z}\chi+\mathcal{I}$. The associated
conserved current determined by the above map is $Q$-exact and can be taken in
the form $H_{W}=Q(i_{Z}\chi)$. In particular, in the case of a gPDE with
boundary the currents associated to gauge symmetries (i.e. with $Z$ tangent to
$E_{\Sigma}$) necessarily vanish on the boundary provided $\chi$ and
$E_{\Sigma}$ are compatible. In other words charges associated to genuine
gauge symmetries vanish while those associated to asymptotic symmetries are
generally nontrivial.

##  3 GR as a gauge PDE

###  3.1 Off-shell GR as a gauge PDE

We start by reformulating Riemannian geometry as a local gauge theory or, more
precisely, a gauge PDE. This system can also be seen as an off-shell gravity,
i.e. a gauge theory whose fields are components of the metric and gauge
transformations act as diffeomorphisms.

In the gPDE language the underlying bundle is given by:

$\mathcal{E}\to X\,,\qquad\mathcal{E}=(T^{*}X\vee T^{*}X)_{\mathrm{nd}}\oplus
T[1]X\,.$ (3.1)

Sections of the first summand are metrics $\widetilde{g}_{ab}(x)$ while
coordinates on the fibers of the second summand are diffeomorphism ghosts
$\xi^{a}$. The desired gPDE $\widetilde{E}\to T[1]X$ can be taken to be
$J^{\infty}(\mathcal{E})\to X$, pulled back to $T[1]X$ by the canonical
projection $T[1]X\to X$. In plain words, the fiber coordinates are
$D_{(a)}\widetilde{g}_{bc},D_{(a)}\xi^{b}$, where $D_{a}$ denote canonical
total derivative in $J^{\infty}(\mathcal{E})$, $(a)$ denotes a symmetric
multi-index, and the degree is assigned in a standard way:
$\mathrm{gh}(D_{(a)}\widetilde{g}_{bc})=0,\mathrm{gh}(D_{(a)}\xi^{b})=1$. In
terms of local coordinates the $Q$ structure is determined by

$\displaystyle Qx^{\mu}=\theta^{\mu},\qquad
Q\widetilde{g}_{bc}=\xi^{a}D_{a}\widetilde{g}_{bc}+\widetilde{g}_{ac}D_{b}\xi^{a}+\widetilde{g}_{ba}D_{c}\xi^{a},\qquad
Q\xi^{b}=\xi^{a}D_{a}\xi^{b},$ (3.2)

and $[Q,D_{a}]=0$. Note that $\widetilde{g}_{bc}=\widetilde{g}_{cb}$ and
$\widetilde{g}_{bc}$ is invertible. Note also that here we use generic
coordinates $x^{\mu}$ on the base $X$. To be more specific, once the jet
bundle and the action of $Q$ on the fiber is defined in terms of a fixed
coordinate system $x^{a}$ we have a freedom of using any coordinate system
$x^{\mu}$ on the base. This happens because $Q$ is locally a product
$Q$-structure of $\mathrm{d_{X}}$ and the $Q$-structure of the typical fiber
or, in other words, the underlying bundle is locally trivial as a $Q$-bundle.

In what follows we often refer to $\widetilde{g}_{ab}$ as metric. Similarly,
we refer to the Cristoffel symbols, Riemann curvature, etc. seen as respective
functions in $D_{(a)}\widetilde{g}_{bc}$ as just Christoffel symbols, Riemann
curvature, etc. This is natural because such local functions coincide with the
respective objects if one evaluates them on the prolongation of a section
$\sigma_{0}\,:X\,\to\mathcal{E}$. However, from the gPDE point of view this
identification only happens in a particular gauge (3.3).

######  Remark 3.1.

The above gPDE is not exactly the standard BV-BRST jet-bundle equipped with
the horizontal differential $\mathrm{d_{h}}$ and the BRST differential
$\gamma$, see e.g. [71, 72]. Although the action of $Q$ on fiber variables
coincides with that of the standard BV-BRST differential it actually
corresponds to the total BRST differential $\mathrm{d_{h}}+\gamma$. More
precisely, thanks to the diffeomorphism invariance one can bring
$\mathrm{d_{h}}+\gamma$ to the form (3.2) by a change of fiber coordinates,
see e.g. [39] for more details.

A local proof of the equivalence of the above gPDE and the standard jet-bundle
BV-BRST formulation of off-shell GR can be found in [39]. In any case, it is
not difficult to explicitly consider solutions and gauge transformations and
check that we are indeed dealing with the off-shell GR. Probably the simplest
way to see the equivalence is to observe that the gauge condition

$\sigma^{*}(\xi^{a})=\theta^{a},\qquad\sigma^{*}(D_{(b\ldots)}\xi^{a})=0$
(3.3)

is reachable locally. In this gauge the remaining equations of motion simply
tell us that $\widetilde{g}_{ab}$ is unconstrained and that
$\sigma^{*}(D_{(a)}\widetilde{g}_{bc})=\partial_{(a)}\sigma^{*}(\widetilde{g}_{bc})$.
Moreover, in this gauge the residual gauge parameters
$\chi^{*}(D_{(a)}\xi^{b})$ are all determined by
$\epsilon^{a}(x)=\chi^{*}(\xi^{a})$ and the residual gauge transformation of
$\sigma^{*}(\widetilde{g}_{ab})$ is given by
$L_{\epsilon}\sigma^{*}(\widetilde{g}_{ab})$ so that indeed we are dealing
with off-shell gravity. More detailed discussion of the analogous gauges in a
more general context can be found in [73].

The above system can be equivalently extended to provide a gauge-theoretical
implementation of the Penrose description of asymptotically simple spaces [74,
3]. More specifically, we extend the fiber of $\mathcal{E}$ with extra
coordinates $\Omega$, $\Omega>0$ and $\lambda$, with
$\mathrm{gh}(\Omega)=0,\mathrm{gh}(\lambda)=1$ and extend $Q$ as follows:

$\displaystyle Q\Omega=\xi^{a}D_{a}\Omega+\lambda\Omega,\qquad
Q\lambda=\xi^{a}D_{a}\lambda,\qquad[D_{a}{,}\,Q]=0\,.$ (3.4)

Condition $\Omega>0$ is crucial in ensuring the equivalence of the initial and
the extended system in the sense of 2.5. Thanks to this condition we can
introduce new coordinates $g_{bc}\equiv\Omega^{2}\widetilde{g}_{bc}$. In these
coordinates the action of $Q$ is given by:

$\displaystyle Qx^{\mu}=\theta^{\mu},\quad
Qg_{bc}=\xi^{a}D_{a}g_{bc}+g_{ac}D_{b}\xi^{a}+g_{ba}D_{c}\xi^{a}+2\lambda
g_{bc},\quad Q\xi^{b}=\xi^{a}D_{a}\xi^{b}\,.$ (3.5)

######  Definition 3.1.

The extended system with $\Omega>0$ and the $Q$ structure determined by
(3.5),(3.4) is called conformal-like off-shell gravity.

Note that the gauge transformations of $\sigma^{*}(g_{ab})$ can be identified
(for instance by employing partial gauge condition (3.3) along with
$\sigma^{*}(\lambda_{(a)})=0$) with the action of diffeomorphisms and Weyl
transformations whose parameters are associated to ghosts $\xi^{a}$ and
$\lambda$. In particular, the sub-gPDE determined by
$\Omega=1,\,D_{(a\ldots)}\Omega=0,D_{(a)}\lambda=0$ is the gPDE reformulation
of the conformal geometry, known in the literature in one or another version
[55, 75, 56].

###  3.2 Conformal-like on-shell GR

From the field theory perspective the systems presented above are off-shell
gauge theories, i.e. theories equivalent to a set of unconstrained fields
subject to gauge transformations. We are mostly interested in gravity-like
theories, where fields are subject to nontrivial differential equations. The
respective gPDE description can be obtained by considering a $Q$-subbundle of
the initial off-shell system. In the case of off-shell GR (3.2) the
$Q$-subbundle is defined as an infinite prolongation of the Einstein equations

$\displaystyle
D_{(a)}(\widetilde{R}_{bc}-\dfrac{\widetilde{g}_{bc}}{d}\widetilde{R})=0,\quad\widetilde{R}=\dfrac{2d}{d-2}\Lambda,$
(3.6)

where $\widetilde{R}_{bc}$,
$\widetilde{R}\equiv\widetilde{g}^{bc}\widetilde{R}_{bc}$ are local functions
in $D_{(a)}\widetilde{g}_{bc}$ corresponding to Ricci tensor and the scalar
curvature respectively. It is easy to see that $Q$ restricts to the
submanifold and hence this indeed defines a gPDE, to which we refer in what
follows as on-shell GR.

In what follows we often encounter gPDEs defined as subbundles of other gPDEs
i.e. jet-bundles. A convenient way to describe (coordinate) functions on such
a subbundle is to regard them as the equivalence classes of functions modulo
those vanishing on the subbundle. Alternatively, the restrictions of the
ambient coordinates to the subbundle can be regarded as an overcomplete
coordinate system therein.

Our aim now is to equivalently reformulate on-shell GR as a sub-gPDE of the
conformal-like off-shell GR defined in 3.1. To this end consider a subbundle
singled out by the following constraints:

$\displaystyle D_{(a)}F_{bc}=0,\qquad\Omega\rho+\frac{g^{ab}}{2}D_{a}\Omega
D_{b}\Omega=-\frac{\Lambda}{(d-1)(d-2)}\,,$ (3.7)

where

$\displaystyle F_{bc}\equiv D_{b}D_{c}\Omega-\Gamma_{bc}^{d}D_{d}\Omega+\Omega
P_{bc}+\rho g_{bc},$ (3.8)
$\displaystyle\rho\equiv-\dfrac{1}{d}g^{bc}(D_{b}D_{c}\Omega-\Gamma_{bc}^{d}D_{d}\Omega+P_{bc}\Omega)$
(3.9)

and $\Gamma_{bc}^{d}$, $P_{bc}$ are respectively the Christoffel symbols and
the Schouten tensor seen as functions in the jets of the metric.

Equations (3.7) are known as almost Einstein equation. However, usually they
are interpreted as equations on $\Omega$ while metric $g_{ab}$ is considered
fixed, see [76, 77] for more details. Now we treat (3.7) as equations
restricting both $g_{ab}$ and $\Omega$.

######  Definition 3.2.

The sub-gPDE of the conformal-like off-shell GR 3.1, which is determined by
constraints (3.7), is called conformal-like on-shell GR.

The name is justified by the following:

######  Proposition 3.3.

For $d\,{\geqslant}\,3$ conformal-like on-shell GR is equivalent to on-shell
GR (3.6).

###### Proof.

First of all recall that $\Omega>0$. In terms of
$\widetilde{g}_{ab}=\Omega^{-2}g_{ab}$ Einstein equations have the standard
form (3.6) while (the derivatives of) $\Omega$ and $\lambda$ form contractible
pairs and can be eliminated. There remains to show that the Einstein equations
rewritten in terms of $g_{ab}$ are equivalent to (3.7). This can be checked
using the well-known, see e.g. [76], transformation rules of the Schouten
tensor under Weyl transformations. ∎

It is important to stress that the above equivalence crucially relies on the
condition $\Omega>0$. At the same time, the conformal-like on-shell GR is
perfectly well-defined without this condition and is going to be very
instrumental in studying the boundary behaviour. Note also that the above
system with no restrictions on $\Omega$ provides a gPDE description of tractor
geometry though, to keep the exposition concise, we refrain from giving
details here.

###  3.3 Pre-minimal model for conformal-like on-shell GR

In what follows we need a certain equivalent reduction (in the sense of
definition 2.5) of conformal-like on-shell GR. This can be done in two steps.
The first step is to concentrate on the sector of $g_{ab},\xi^{a},\lambda$ and
their jets. This sector is precisely the one that gives a gPDE description of
conformal geometry (which can also be regarded as the off-shell conformal
gravity) so that one can eliminate contractible pairs as explained in [55]
(see also [56] for the discussion in simlar language). Namely,

$\displaystyle Q\Gamma_{ab}^{c}=\dots+D_{a}D_{b}\xi^{c},\qquad
QP_{ab}=\dots+D_{a}D_{b}\lambda$ (3.10)

allows one to eliminate $\Gamma_{ab}^{c}$, $D_{a}D_{b}\xi^{c}$, $P_{ab}$,
$D_{a}D_{b}\lambda$ as well as all their symmetrized total derivatives. This
reduction is quite different for the cases $d=3$ and $d\,{\geqslant}\,4$, so
in what follows we assume $d\,{\geqslant}\,4$. However, generalization to
$d=3$ is possible. The remaining jets of the metric have the meaning of the
Weyl tensor $\mathrm{W}^{b}{}_{cde;(a)}$ and its covariant total derivatives.
All the symmetries of the usual Weyl tensor are preserved: antisymmetry over
pairs of indices, Bianchi identities, tracelessness, and so on. It is always
possible to choose components of these tensors in such a way that they form a
part of the coordinate system. However, for our purposes it is more convenient
to keep all the components and use them as overcomplete coordinate system. We
introduce the notation
$\mathrm{C}_{dab}=-\dfrac{1}{(d-3)}\mathrm{W}^{c}{}_{dab;c}$.

Then the action of $Q$ on some coordinates in this sector is given by

$\displaystyle\begin{split}&Qg_{bc}=C_{b}{}^{a}g_{ac}+C_{c}{}^{a}g_{ba}+2\lambda
g_{bc},\quad Q\xi^{b}=\xi^{a}C_{a}{}^{b},\\\
&Q\lambda=\xi^{a}\lambda_{a},\quad
Q\lambda^{b}=C^{b}{}_{a}\lambda^{a}+\dfrac{1}{2}\xi^{a}\xi^{d}\mathrm{C}^{b}{}_{ad},\\\
&QC_{b}{}^{c}=C_{b}{}^{a}C_{a}{}^{c}+\lambda_{b}\xi^{c}-\lambda^{c}\xi_{b}+\delta_{b}^{c}\lambda_{a}\xi^{a}+\dfrac{1}{2}\xi^{a}\xi^{d}\mathrm{W}^{c}{}_{bad},\end{split}$
(3.11)

where the standard convention for raising and lowering indexes is used, for
example $\lambda^{a}\equiv g^{ab}\lambda_{b}$, and $C_{a}{}^{b}\equiv
D_{a}\xi^{b}$. The action of $Q$ on $\mathrm{W}^{b}{}_{cde;(a)}$ can also be
obtained by a straightforward (but tedious) calculation.

In the second step we analyze the sector of $\Omega$. If $\Omega_{(a)}$
denotes the restriction of $D_{(a)}\Omega$ to the system obtained in the 1st
step, one finds that equations $D_{(a)}F_{bc}=0$ from (3.7) restricted to the
surface defined by this reduction, takes the following form:

$\displaystyle\begin{split}&\Omega_{ab}+g_{ab}\rho=0,\\\ &\Omega_{a_{1}\dots
a_{n}}=0,\quad n\,{\geqslant}\,3\,\\\
&\nabla_{a_{1}}\dots\nabla_{a_{n-3}}(\mathrm{W}^{d}{}_{a_{n}a_{n-2}a_{n-1}}\nabla_{d}\Omega+\mathrm{C}_{a_{n}a_{n-2}a_{n-1}}\Omega)=0.\end{split}$
(3.12)

where $\rho=-\frac{1}{d}g^{ab}\Omega_{ab}$ and for any degree $0$ coordinate
$\varphi$, $\nabla_{a}\varphi$ is defined through
$Q\varphi=\xi^{a}\nabla_{a}\varphi+\dots$. The first equation is just
$F_{bc}=0$ restricted to the subbundle defined by previous reduction, the
second is the totally-symmetric component of
$D_{(a_{1}\ldots)}F_{a_{n-1}a_{n}}=0$, and the third one is the remaining
irreducible component in $D_{(a_{1}\ldots)}F_{a_{n-1}a_{n}}=0$. These
equations fix all the jets of $\Omega$ except for $\Omega$, $\Omega_{a}$ and
$\rho$. For $n=3$ the third equation in (3.12) takes the form

$\displaystyle\mathrm{W}^{d}{}_{a_{n}a_{n-2}a_{n-1}}\nabla_{d}\Omega+\mathrm{C}_{a_{n}a_{n-2}a_{n-1}}\Omega=0$
(3.13)

and is known as a part of the Fridrich equations, see e.g. [78]. After taking
into account the above equations and introducing $n^{a}\equiv
g^{ab}\Omega_{b}$ in place of $\Omega_{a}$, the action of $Q$ on
$\Omega_{a},n^{a},\rho$ takes the form:

$\displaystyle\begin{split}&Q\Omega=\xi^{a}g_{ab}n^{b}+\lambda\Omega,\quad
Qn^{b}=-\xi^{b}\rho-n^{a}C_{a}{}^{b}+\lambda^{b}\Omega-\lambda n^{b},\quad
Q\rho=-\lambda\rho-\lambda_{a}n^{a}.\end{split}$ (3.14)

The results of this subsection can be summarized in the following proposition:

######  Proposition 3.4.

For $d\,{\geqslant}\,4$ the gPDE defined in 3.2 is equivalent to its sub-gPDE
$(E,Q,T[1]X)$ with the following overcomplete set of fiber coordinates
$\\{g_{bc},\Omega,n^{b},\rho,\xi^{b},C_{b}{}^{c},\lambda,\lambda^{b},\mathrm{W}^{b}{}_{cde;(a)},|a|\,{\geqslant}\,0\\}$
which are understood modulo the ideal generated by the following constraints:

$\displaystyle\Omega\rho+\frac{1}{2}g_{ab}n^{a}n^{b}=-\frac{\Lambda}{(d-1)(d-2)},$
(3.15)
$\displaystyle\nabla_{a_{1}}\cdots\nabla_{a_{n}}(\mathrm{W}^{b_{3}}{}_{cb_{1}b_{2}}n^{c}-\mathrm{C}^{b_{3}}{}_{b_{1}b_{2}}\Omega)=0,\quad
n\,{\geqslant}\,0.$

The action of $Q$ on all coordinates except curvatures
$\mathrm{W}^{b}{}_{cde;(a)}$ is given by (3.11), (3.14).

##  4 Boundary systems and asymptotic symmetries

###  4.1 Asymptotically simple GR as a gPDE with boundaries

Having obtained a description of gravity in the bulk as a gPDE one can
immediately construct the induced gPDE $(i^{*}E,Q,T[1]\mathcal{J})$ on the
boundary $\mathcal{J}$. More specifically, we start with the gPDE defined in
the Proposition 3.4, which encodes the conformal-like on-shell GR in the bulk.
A slight but important modification is that fibers of $E$ over the boundary
are extended by their own boundary by allowing $\Omega$ to take value $0$
(recall that in the bulk $\Omega>0$). In what follows we restrict ourselves to
the local analysis and hence do not discuss global geometry of the space-time
and its boundary. More specifically, we restrict to a neighbourhood of the
boundary of topology $S^{d-2}\times\mathbb{R}$.

Now we identify a gPDE with boundaries which describes asymptotically simple
GR. To this end we impose additional conditions which implements Penrose’s
definition of asymptotically simple spacetime in the gPDE terms. More
specifically, we take $E_{B}\subset i^{*}E$ to be a sub gPDE of $i^{*}E$
determined by

$\displaystyle\Omega=0\,,\qquad Q\Omega=0\,.\qquad D_{a}\Omega\neq 0\,.$ (4.1)

This gives a gPDE with boundaries $(E,Q,T[1]X,E_{B},T[1]\mathcal{J})$ which we
refer to as asymptotically simple GR. Here we keep using $Q$ to denote the
homological vector field on $i^{*}E$ as well as on $E_{B}$ as these are
restrictions of the initial $Q$ on $E$ to the respective submanifolds.
Analogous systems for asymptotically-simple spacetimes are obtained by not
imposing the Einstein equations.

It is important to stress that if $D_{a}\Omega$ were nonvanishing everywhere
in $i^{*}E$, functions $\Omega$ and $Q\Omega$ would be independent on $i^{*}E$
so that setting $\Omega=0$ and $Q\Omega=0$ can be understood as an equivalent
reduction. However, $D_{a}\Omega\neq 0$ is imposed at $\Omega=0$ only so that
it is better to regard (4.1) as the boundary conditions determining
asymptotically simple GR. In any case, (4.1) effectively implements only minor
restrictions on the moduli of solutions, which can be thought of as partial
gauge conditions. Another remark is that, as we discussed in Section 2, the
total space $E$ can be extended to a manifold with corners by allowing
$\Omega\,{\geqslant}\,0$ everywhere. From this perspective $E_{B}$ can be
identified with the respective corner provided one also excludes points where
$D_{a}\Omega\neq 0$.

As an (overcomplete) coordinate system on $i^{*}(E)$ we use coordinates on $E$
restricted to $i^{*}E$ seen as a submanifold in $E$. In particular, $\Omega$
in (4.1) is, strictly speaking, a restriction of the initial coordinate
$\Omega$ to $i^{*}(E)$. In what follows we restrict ourselves to the local
analysis and hence do not discuss global geometry of the space-time and its
boundary.

Taking into account constraints (4.1) in $(i^{*}E,Q,T[1]\mathcal{J})$ results
in the boundary gPDE $(E_{B},Q,T[1]\mathcal{J})$. The overcomplete set of
fiber coordinates can be obtained by restricting the coordinates from
Proposition 3.4 to $E_{B}$ and is given by

$\\{g_{bc},n^{b},\rho,\xi^{b},C_{b}{}^{c},\lambda,\lambda^{b},\mathrm{W}^{b}{}_{cde;(a)},|a|\,{\geqslant}\,0\\}\,.$
(4.2)

The action of $Q$ on some of the coordinates is easily obtained by restricting
(3.11), (3.14):

$\displaystyle\begin{split}&Qg_{bc}=C_{b}{}^{a}g_{ac}+C_{c}{}^{a}g_{ba}+2\lambda
g_{bc},\quad Q\xi^{b}=\xi^{a}C_{a}{}^{b},\\\
&Qn^{b}=-\xi^{b}\rho-n^{a}C_{a}{}^{b}-\lambda n^{b},\quad
Q\lambda^{b}=C^{b}{}_{a}\lambda^{a}+\dfrac{1}{2}\xi^{a}\xi^{c}\mathrm{C}^{b}{}_{ac},\\\
&Q\rho=-\lambda\rho-\lambda_{a}n^{a},\quad Q\lambda=\xi^{a}\lambda_{a},\\\
&QC_{b}{}^{c}=C_{b}{}^{a}C_{a}{}^{c}+\lambda_{b}\xi^{c}-\lambda^{c}\xi_{b}+\delta_{b}^{c}\lambda_{a}\xi^{a}+\dfrac{1}{2}\xi^{a}\xi^{d}\mathrm{W}^{c}{}_{bab}\,.\end{split}$
(4.3)

At the same time constraints (LABEL:constraints-mb) take the form:

$\displaystyle\begin{split}&g_{ab}n^{a}n^{b}=-\frac{2}{(d-1)(d-2)}\Lambda,\\\
&\left(\nabla_{a_{1}}\cdots\nabla_{a_{n}}(\mathrm{W}^{b_{3}}{}_{cb_{1}b_{2}}n^{c}-\mathrm{C}^{b_{3}}{}_{b_{1}b_{2}}\Omega)\right)\big{|}_{\Omega=0}=0,\quad
n\,{\geqslant}\,0,\,,\\\ \end{split}$ (4.4)

and, finally, the last constraint is given by $\xi^{a}g_{ab}n^{b}=0$ and
originates from $Q\Omega=0$.

######  Definition 4.1.

The above gPDE $(E_{B},Q,T[1]\mathcal{J})$ is refereed to as the boundary gPDE
for asymptotically simple GR.

As we are going to see it is very convenient to work in terms of the minimal
model of the system 4.1. However, the respective minimal gPDE crucially
depends on the value of the cosmological constant. As we are mostly interested
in the null-infinity we assume $\Lambda=0$ unless otherwise specified.

###  4.2 Minimal model for the boundary gPDE of asymptotically simple GR

We now take $\Lambda=0$ and find a minimal model of the boundary gPDE obtained
in the previous Section. We have the following:

######  Proposition 4.2.

In the case $\Lambda=0$ and $g_{ab}$ of Lorentz signature, gPDE
$(E_{B},Q,T[1]\mathcal{J})$ defined in 4.1 is equivalent to its subbundle
determined by the following conditions:

$\displaystyle g_{ab}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&-\delta_{AB}\end{pmatrix},\quad n^{a}=\begin{pmatrix}0\\\ 1\\\
0\end{pmatrix},$ (4.5) $\displaystyle
C_{a}{}^{b}=\begin{pmatrix}-\lambda&0&C_{A}\\\ 0&-\lambda&0\\\
0&-C_{A}&\rho_{A}{}^{B}-\lambda\delta_{A}^{B}\end{pmatrix},$ (4.6)
$\displaystyle\rho=0,\quad\xi^{\Omega}=0,\lambda^{\Omega}=0,$ (4.7)

where we used the adapted basis so that $\\{a\\}=\\{\Omega,u,A\\}$,
$A=1,\ldots,d-2$ and introduced the following new coordinates:
$\rho_{AB}\equiv C_{[AB]}$, $C_{A}\equiv C_{\Omega\,A}$. Among the constraints
(4.4) on the degree-zero variables there only remain:

$\displaystyle\nabla_{a_{1}}\cdots\nabla_{a_{n}}\mathrm{W}_{b_{3}ub_{1}b_{2}}-\sum_{i=1}^{n}g_{ua_{i}}\nabla_{a_{1}}\cdots\widehat{\nabla}_{a_{i}}\cdots\nabla_{a_{n}}\mathrm{C}_{b_{3}b_{1}b_{2}}=0,\quad
n\,{\geqslant}\,0.$ (4.8)

Note that $g_{ua_{i}}=\delta_{a_{i}\Omega}$ on the subbundle.

We denote the minimal model introduced in the above Proposition by
$(E_{B}^{\text{min}},Q,T[1]\mathcal{J})$. This gPDE is explicitly defined as a
sub-gPDE of $(E_{B},Q,T[1]\mathcal{J})$ which, in its turn, is a sub-gPDE of
$i^{*}E$.

###### Proof.

As usual, the proof is based on identification of the contractible pairs.
Using

$\displaystyle Qn^{b}=-\xi^{b}\rho-n^{a}C_{a}{}^{b}-\lambda n^{b}$ (4.9)

and taking into account $n^{b}\neq 0$ one can set:

$\displaystyle\begin{split}n^{\Omega}=0,\quad
C_{u}{}^{\Omega}=-\xi^{\Omega}\rho,\qquad n^{u}=1,\quad
C_{u}{}^{u}=-\xi^{u}\rho-\lambda,\qquad n^{A}=0,\quad
C_{u}{}^{A}=-\xi^{A}\rho\,,\end{split}$ (4.10)

which also gives $g_{uu}=0$ thanks to the first constraint in (4.4). Using
then

$\displaystyle Qg_{ua}=-\xi^{\Omega}\rho g_{\Omega a}-\xi^{u}\rho
g_{ua}-\xi^{B}\rho g_{Ba}+C_{a}{}^{b}g_{ub}+\lambda g_{ua}\,,$ (4.11)

we can eliminate $g_{ua}$ as well as $C_{a}{}^{b}g_{ub}$. Note that
$g_{ub}\neq 0$ because of $det(g_{ab})\neq 0$. More precisely, we set

$\displaystyle\begin{split}&g_{u\Omega}=1,\quad
C_{\Omega}{}^{\Omega}=\xi^{u}\rho-\lambda,\\\ &g_{uA}=0,\quad
C_{A}{}^{\Omega}=\xi^{\Omega}\rho g_{\Omega A}+\xi^{B}\rho g_{BA}.\end{split}$
(4.12)

The second constraint in (4.4) then gives $\xi^{\Omega}=0$. Using

$\displaystyle Q\rho=-\lambda\rho-\lambda^{\Omega},$ (4.13)

allows us to set $\rho=0$ and $\lambda^{\Omega}=0$. Furthermore, using

$\displaystyle
Qg_{\Omega\Omega}=2C_{\Omega}{}^{u}+2C_{\Omega}{}^{B}g_{B\Omega}+2\lambda
g_{\Omega\Omega}$ (4.14)

we can set $g_{\Omega\Omega}=0$ and
$C_{\Omega}{}^{u}=-C_{\Omega}{}^{B}g_{B\Omega}$. Similarly,

$\displaystyle Qg_{\Omega A}=\lambda g_{\Omega
A}+C_{\Omega}{}^{B}g_{BA}+C_{A}{}^{u}$ (4.15)

allows us to set $g_{\Omega A}=0$ and $C_{A}{}^{u}=-C_{\Omega}{}^{B}g_{BA}$.
Finally, eliminating the remaining components $g_{AB}$ of the metric we set

$\displaystyle g_{AB}=-\delta_{AB},\quad C_{(AB)}=\lambda\delta_{AB}.$ (4.16)

∎

To summarize, we explicitly found a minimal model of the boundary gPDE for
asymptotically simple GR. Its overcomplete fiber coordinates are
$\\{\xi^{u},C^{A},\xi^{A},\rho_{A}{}^{B},\lambda,\lambda^{u},\lambda^{A},\mathrm{W}^{b}{}_{cde;(a)},|a|\,{\geqslant}\,0\\}$.
The action of $Q$ on the degree $1$-coordinates
$\xi^{A},\lambda,\lambda_{A},\rho_{A}{}^{B}$ is given by:

$\displaystyle\begin{split}&Q\xi^{A}=\xi^{B}\rho_{B}{}^{A}-\xi^{A}\lambda,\\\
&Q\rho_{A}{}^{B}=\rho_{A}{}^{C}\rho_{C}{}^{B}+\lambda_{A}\xi^{B}-\lambda^{B}\xi_{A}+\frac{1}{2}\xi^{C}\xi^{D}\mathrm{W}^{B}{}_{ACD},\\\
&Q\lambda=\xi^{A}\lambda_{A},\\\
&Q\lambda^{A}=\rho^{A}{}_{B}\lambda^{B}-\lambda\lambda^{A}+\dfrac{1}{2}\xi^{C}\xi^{D}\mathrm{C}^{A}{}_{CD}+\xi^{u}\xi^{D}\mathrm{C}^{A}{}_{uD}.\end{split}$
(4.17)

Setting the curvatures (these enter the right hand sides multiplied by
$\xi^{a}\xi^{b}$) to zero gives the Chevalley-Eilenberg differential of the
$so(d-1,1)$ subalgebra of $iso(d-1,1)$ algebra. This subalgebra can be
identified with the conformal algebra associated to $d-2$-dimensional
submanifold of the boundary. Moreover, setting to zero only the components
$\mathrm{C}^{A}{}_{uD}$ gives the respective sector of the minimal model of
the conformal geometry in $d-2$ dimensions. Note, however, that the entire
system differs form that of conformal geometry. In particular, extra
curvatures are present and the action of $Q$ on the curvatures is different.

The action of $Q$ on the remaining degree-1 coordinates reads as:

$\displaystyle\begin{split}&Q\xi^{u}=-\xi^{u}\lambda-\xi^{A}C_{A},\\\
&QC^{A}=C^{B}\rho_{B}{}^{A}+\lambda^{u}\xi^{A}-\lambda^{A}\xi^{u}+\dfrac{1}{2}\xi^{C}\xi^{D}\mathrm{W}^{A}{}_{\Omega
CD},\\\
&Q\lambda^{u}=C^{A}\lambda_{A}-\lambda\lambda^{u}+\dfrac{1}{2}\xi^{C}\xi^{D}\mathrm{C}_{\Omega
CD}+\xi^{u}\xi^{D}\mathrm{C}_{\Omega uD}.\end{split}$ (4.18)

With all the curvatures set to zero, the actions of $Q$ is that of the
Chevalley-Eilenberg differential of $iso(d-1,1)$, where (4.17) corresponds to
$so(d-1,1)$ while (4.18) to the $iso(d-1,1)$ translations.

Fields parameterizing solutions of the above minimal model are the
$iso(d-1,1)$ connection on the boundary along with the bunch of the degree
zero fields (curvatures) some of which are expressed in term of the connection
through the equations of motion (and generally impose some differential
equations on the connection) while the remaining ones are independent fields.
It is of course natural that the boundary system can be formulated in terms of
the Poincarè connection because the minimal model of the bulk gravity has an
analogous formulation. Similar but not identical formulations of the
asymptotically simple GR were considered in [79],[59]. More details on the
field theory encoded in the above minimal model are given in Section 4.4.

In the next two sections, in order to agree with the standard conventions for
connections and curvatures, we redefine all fiber coordinates $\varphi$ such
that $\mathrm{gh}(\varphi)=1$ as $\varphi\rightarrow-\varphi$. In particular,
this affects the explicit formulas for the action of $Q$ on fiber coordinates
(an alternative way is to reverse the sign at the vertical part of $Q$).

###  4.3 Boundary conditions and BMS symmetries

Now we plan to identify a proper counterpart of the BMS boundary conditions in
this setup. Strictly speaking the minimal model constructed in the previous
section is too “minimal” to incorporate a sub-gPDE of boundary conditions as a
regular submanifold. Nevertheless it is not difficult to identify, generally
non-regular, constraints which do the job, giving a rather concise description
of the boundary conditions and asymptotic symmetries.

As explained in Section 2.3 in this setup we are forced to allow $Y$ to depend
on jets of sections of $E^{min}_{B}$ (recall that we treat the gPDE of
boundary conditions as a subbundle in $E^{min}_{B}$). Here we use
$D^{\theta}_{\mu}$ to denote the total derivative in $\theta^{\mu}$ direction
(it’s a total derivative in the super-jet bundle of $E^{min}_{B}$ and should
not be confused with the total derivative in the initial jet-bundle from which
$E^{min}_{B}$ has been constructed). For instance, if $\sigma$ is a section
and $\sigma^{*}(\lambda_{A})=\lambda_{A\mu}(x)\theta^{\mu}$ then
$\sigma^{*}(D^{\theta}_{\mu}\lambda_{A})=\lambda_{A\mu}(x)$. Note that
$\sigma^{*}(D^{\theta}_{\mu}D^{\theta}_{\nu}\lambda_{A})=0$ by the degree
reasoning.

We define $E_{\mathcal{J}}\subset E_{B}^{min}$ as a zero locus of the
constraints defined on $E^{min}_{B}$. We first introduce constraints which set
the frame field encoded in $\xi^{a}$ to be a fixed frame:

$\xi^{A}-e^{A}\sim 0,\qquad\xi^{u}-\theta^{u}\sim 0\,,$ (4.19)

where we use adapted coordinates $y^{\alpha}$ and $u$ on the boundary
$\mathcal{J}$ and assumed for simplicity that
$e^{A}=e^{A}{}_{\alpha}(u,y)\theta^{\alpha}$. It is easy to see that $Q$ is
not tangent to the surface and hence extra boundary conditions are necessary.
Consider the following extra constraints:

$\lambda\sim 0\,,\quad\lambda_{A}\xi^{A}\sim 0\,,\quad C_{A}\xi^{A}\sim
0\,,\quad\mathrm{d_{\mathcal{J}}}e^{A}+\xi^{B}\rho_{B}{}^{A}\sim 0\,,$ (4.20)

where the last three ones coincide with $Q\lambda$, $Q(\xi^{u}-\theta^{u})$,
and $Q(\xi^{A}-e^{A})$ modulo terms proportional to $\lambda$. This can be
easily seen using (4.17) and (4.18) as well as the following representation:

$C_{A}\xi^{A}=Q\xi^{u}-\xi^{u}\lambda\,,\qquad\rho^{A}{}_{B}\xi^{B}=Q\xi^{A}-\xi^{A}\lambda\,.$
(4.21)

Constraints (4.20) and (4.19) define an ideal $\mathcal{I}_{\mathcal{J}}$ in
the algebra of functions on $E_{B}^{min}$ introduced in Proposition 4.2. It is
easy to see that $Q$ is well defined on the quotient as
$\mathcal{I}_{\mathcal{J}}$ is $Q$-invariant. Because some of the constraints
are quadratic, the quotient is not an algebra of functions on a regular
subbundle. However, we can still think of it as determining a $Q$-subbundle
$E_{\mathcal{J}}$ which is defined in the algebraic sense only. This does not
really lead to problems because its prolongation to jets of supersections is a
regular submanifold, provided we restrict to sections such that the frame
field is invertible. In this sense working in terms of $E_{\mathcal{J}}$ only
gives an economical framework to analyse asymptotic symmetries. All the steps
can be repeated in terms of its jet-prolongation which is a genuine subbundle
of the jet-bundle. Disregarding the above subtlety, constraints (4.20) and
(4.19) define a gauge PDE with boundary in the sense of Definition 2.6.
Indeed, $E_{\mathcal{J}}$ is a sub-gPDE of the $E_{B}^{min}$ which, in turn,
is defined as a sub-gPDE of $i^{*}E$.

Now we are ready to study gauge symmetries that preserve the gPDE of boundary
conditions. Consider a gauge parameter vector field

$Y=\epsilon^{u}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\xi^{u}}$}\kern
1.0pt}{\frac{\partial}{\partial\xi^{u}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\xi^{u}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\xi^{u}}$}\kern
1.0pt}+\epsilon^{A}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\xi^{A}}$}\kern
1.0pt}{\frac{\partial}{\partial\xi^{A}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\xi^{A}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\xi^{A}}$}\kern
1.0pt}+\bar{\lambda}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda}$}\kern
1.0pt}{\frac{\partial}{\partial\lambda}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda}$}\kern
1.0pt}+\bar{\lambda}_{A}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda_{A}}$}\kern
1.0pt}{\frac{\partial}{\partial\lambda_{A}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda_{A}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda_{A}}$}\kern
1.0pt}+\bar{\rho}^{AB}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\rho^{AB}}$}\kern
1.0pt}{\frac{\partial}{\partial\rho^{AB}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\rho^{AB}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\rho^{AB}}$}\kern
1.0pt}+\bar{C}^{A}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
C^{A}}$}\kern 1.0pt}{\frac{\partial}{\partial
C^{A}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
C^{A}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
C^{A}}$}\kern
1.0pt}+\bar{\lambda}{}^{u}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda^{u}}$}\kern
1.0pt}{\frac{\partial}{\partial\lambda^{u}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda^{u}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda^{u}}$}\kern
1.0pt}\,,$ (4.22)

where $\epsilon^{u},\epsilon^{A},\bar{\lambda},\ldots$ are functions in $x$
while $\bar{\lambda}_{A},\bar{C}^{A},\bar{\rho}^{AB}$ are also allowed to
depend on the $\theta$-jets of $\lambda_{A},C^{A},\rho^{AB}$. Note that the
component
$\bar{\lambda}{}^{u}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda^{u}}$}\kern
1.0pt}{\frac{\partial}{\partial\lambda^{u}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda^{u}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda^{u}}$}\kern
1.0pt}$ clearly preserves the constraints and hence correspond to trivial
asymptotic symmetries.

We are interested in $Y$ such that the respective symmetry transformations
preserves the ideal and hence induces a symmetry transformation that take
solutions of $E_{\mathcal{J}}$ to solutions. We have:

$\mathrm{d_{\mathcal{J}}}\sigma^{*}(Yf)+\sigma^{*}(YQf)=0\quad\forall
f\in\mathcal{I}$ (4.23)

This should hold for all section of the gPDE of boundary condition, i.e.
sections of $E_{B}^{min}$ such that $\sigma^{*}(\text{``constraints''})=0$.

Taking $f=\lambda$ gives

$\sigma^{*}(\mathrm{d_{\mathcal{J}}}\bar{\lambda}-\epsilon^{A}\lambda_{A}+\xi^{A}\bar{\lambda}_{A})=0$
(4.24)

This implies $\partial_{u}\bar{\lambda}=0$ because we assumed $\theta^{u}$
unconstrained and because $\sigma^{*}(\lambda_{A})=\lambda_{AB}(x)e^{B}$ for
some $\lambda_{AB}(x)=\lambda_{BA}(x)$ thanks to
$\sigma^{*}(\xi^{A}\lambda_{A})=0$. Furthermore, (4.24) also implies:

$e_{B}{}^{\alpha}\partial_{\alpha}\bar{\lambda}-\epsilon^{A}\lambda_{AB}+\sigma^{*}(\bar{\lambda}_{B})=0\,.$
(4.25)

This can be solved for $\bar{\lambda}_{B}$ by e.g.
$\bar{\lambda}_{B}=-e_{B}{}^{\alpha}\partial_{\alpha}\bar{\lambda}+e_{B}{}^{\alpha}\epsilon^{A}D^{\theta}_{\alpha}\lambda_{A}$.
Indeed,
$\sigma^{*}(D^{\theta}_{\alpha}\lambda_{A})=\lambda_{AB}(x)e^{B}{}_{\alpha}$
and hence imposes no restrictions on $\partial_{\alpha}\bar{\lambda}$.

Taking $f=\xi^{u}-\theta^{u}$ in (4.23) one finds

$\mathrm{d_{\mathcal{J}}}\epsilon^{u}+\sigma^{*}(\epsilon^{u}\lambda-\xi^{u}\bar{\lambda}+\epsilon^{A}C_{A}-\xi^{A}\bar{C}_{A})=0$
(4.26)

This implies $\epsilon^{u}=u\bar{\lambda}(y)+T(y)$, with $T$ unconstrained.
Moreover, the remaining equation can be satisfied by taking
$\bar{C}_{A}=e_{A}{}^{\alpha}(\partial_{\alpha}\epsilon^{u}+\epsilon^{B}D^{\theta}_{\alpha}C_{B})$.

Taking $f=\xi^{A}-e^{A}$ one gets

$\mathrm{d_{\mathcal{J}}}\epsilon^{A}+\sigma^{*}(-\epsilon^{B}\rho_{B}{}^{A}+\xi^{B}\bar{\rho}_{B}{}^{A}+\epsilon^{B}\lambda-\xi^{B}\bar{\lambda})=0$
(4.27)

Thanks to $\sigma^{*}(\mathrm{d_{\mathcal{J}}}e^{A}+\rho^{A}{}_{B}\xi^{B})=0$
one finds that $\sigma^{*}(\rho^{A}{}_{B})=\omega^{A}{}_{B\mu}\theta^{\mu}$,
where $\omega^{A}{}_{B\alpha}(u,y)$ can be expressed in terms of
$e^{A}=e^{A}{}_{\alpha}(u,y)\theta^{\alpha}$ through standard formulas for
Levi-Civita connection and
$\omega_{B}{}^{A}{}_{u}=\sigma^{*}(e^{\alpha}{}_{B}\partial_{u}e^{A}{}_{\alpha})$.
Then, in terms of $\epsilon^{\alpha}\equiv e^{\alpha}{}_{A}\epsilon^{A}$, the
equation (4.27) implies

$\partial_{u}\epsilon^{\alpha}=0,\quad\partial_{\alpha}\epsilon_{\beta}-\Gamma^{\gamma}_{\alpha\beta}(e)\epsilon_{\gamma}+e^{A}{}_{\alpha}e^{B}{}_{\beta}\sigma^{*}(\bar{\rho}_{AB})-g_{\alpha\beta}\bar{\lambda}=0.$
(4.28)

Here, similarly to the standard formulas, $g_{\alpha\beta}\equiv
e_{A\alpha}e^{A}{}_{\beta}$ and $\Gamma^{\gamma}{}_{\alpha\beta}\equiv
e^{\gamma}{}_{A}(\partial_{\alpha}e^{A}{}_{\beta}-e^{B}{}_{\beta}\omega_{B}{}^{A}{}_{\alpha})$.
The antisymmetric part of the second equation in (4.28) can be solved for
$\bar{\rho}_{AB}$, and the symmetric part is nothing but a conformal Killing
equation. In particular this fixes $\bar{\lambda}$ in terms of $\epsilon^{A}$.

Finally, there remains to check (4.23) for the last three constraints from
(4.20). However, these three are all of the form $Qg$, modulo terms
proportional to $\lambda$, with $g$ being $\lambda$ or $\xi^{u}-\theta^{u}$ or
$\xi^{A}-e^{A}$. It follows (4.23) always holds because

$\mathrm{d_{\mathcal{J}}}\sigma^{*}(YQg)+\sigma^{*}(YQQg)=\mathrm{d_{\mathcal{J}}}\sigma^{*}(YQg)=-\mathrm{d_{\mathcal{J}}}(\mathrm{d_{\mathcal{J}}}\sigma^{*}(Yg))=0\,,$
(4.29)

where in the last equality we made use of (4.23), with $f$ replaced by $g$,
and the fact that $Y$ was chosen in such a way that (4.23) holds for $f$ being
$\lambda$ or $\xi^{u}-\theta^{u}$ or $\xi^{A}-e^{A}$.

In this way we are left with $Y$ parameterized by $u$-independent $T$ and
$\epsilon^{\alpha}$. Interpreting $\epsilon^{u}(u,y),\epsilon^{\alpha}(y)$ as
components of a vector field on the boundary it is easy to check that this is
precisely BMS vector field on the boundary, which encodes conformal isometries
of $d-2$-dimensional space and supertranslations. More specifically, the BMS
vector field on $E_{\mathcal{J}}$ reads as

$\epsilon^{BMS}=(u\bar{\lambda}+T(y))\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
u}$}\kern 1.0pt}{\frac{\partial}{\partial
u}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
u}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
u}$}\kern
1.0pt}+\epsilon^{\alpha}(y)\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
y^{\alpha}}$}\kern 1.0pt}{\frac{\partial}{\partial
y^{\alpha}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
y^{\alpha}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
y^{\alpha}}$}\kern 1.0pt}\,,$ (4.30)

where we use adapted coordinates $u,y^{\alpha}$ on $\mathcal{J}$ and where
$T(y)$ is a generic function in $y^{\alpha}$, $\epsilon^{\alpha}(y)$ are
components of a conformal Killing vector in $d-2$ dimensions, and
$\bar{\lambda}$ is determined by (4.28). This is precisely how the
infinitesimal BMS transformations act as symmetries of the conformal
Carrollian geometry, see e.g. [80] for more details.

To make sure we are dealing with nontrivial asymptotic symmetries one should,
strictly speaking, show that these symmetries are not equivalent to trivial.
I.e. that $[Q{,}\,Y]|_{E_{\mathcal{J}}}$ can not be represented as
$[Q|_{E_{\mathcal{J}}}{,}\,Y^{\prime}]$ for some vertical vector field
$Y^{\prime}$ on ${E_{\mathcal{J}}}$. Considering $Y^{\prime}$ as a
representative of an equivalence class of vertical vector fields tangent to
$E_{\mathcal{J}}$ modulo those vanishing on $E_{\mathcal{J}}$ one can assume
that $Y^{\prime}\lambda=Y^{\prime}\xi^{A}=Y^{\prime}\xi^{u}=0$. Repeating the
analysis of this section for such $Y^{\prime}$ one concludes that
$\sigma^{*}(\bar{\lambda}_{A})=\sigma^{*}(\bar{C}_{A})=\sigma^{*}(\bar{\rho}^{A}_{B})=0$.
Considering, for instance, $\sigma^{*}([Q{,}\,Y^{\prime}]C_{A})$ one finds
that
$(\delta_{Y^{\prime}}\sigma)^{*}(C_{A})=\sigma^{*}(e^{A}\bar{\lambda}^{u})$.
Then introducing components $C_{AB}$ as $\sigma^{*}(C_{A})=e^{B}C_{BA}(x)$,
the transformation takes the form:

$\delta_{Y^{\prime}}C_{AB}=\eta_{AB}\bar{\lambda}{}^{u}$ (4.31)

so that it cannot affect the trace-free components of coordinates $(C_{AB})$.
As will be shown in the next section, these components parameterize the
asymptotic shear. At the same time transformations with nontrivial
$\epsilon^{u},\epsilon^{\alpha}$ do affect the asymptotic shear.

###  4.4 Field-theoretical interpretation of the minimal model

As we have seen the minimal model $(E^{min}_{B},Q,T[1]\mathcal{J})$ of the
boundary gPDE for asymptotically simple GR, defined in Section 4.2, plays a
crucial role in our approach to asymptotic symmetries. In this section we
study its solutions and gauge symmetries and explain how the BMS symmetries
can be derived in these terms.

We now study the space of solutions, i.e. sections of
$(E^{min}_{B},Q,T[1]\mathcal{J})$ satisfying
$\mathrm{d_{\mathcal{J}}}\circ\sigma^{*}=\sigma^{*}\circ Q$. By some abuse of
notation we introduce the following parameterization of sections:

$\displaystyle\sigma^{*}\rho_{A}{}^{B}=\omega_{A}{}^{B},\quad\sigma^{*}\xi^{A}=e^{A},\quad\sigma^{*}\xi^{u}=l\,,$
(4.32)

where all the new functions are linear in $\theta^{\mu}$, i.e. can be seen as
1-forms on $X$, by the degree reasoning. For the remaining fiber coordinates
we take $\sigma^{*}\phi=\phi(x,\theta)$. The equations of motion in the sector
of degree-1 fiber coordinates read as:

$\begin{gathered}\mathrm{d_{\mathcal{J}}}e^{A}+\omega^{A}{}_{B}e^{B}+\lambda
e^{A}=0,\qquad\mathrm{d_{\mathcal{J}}}\lambda+e^{A}\lambda_{A}=0,\qquad\mathrm{d_{\mathcal{J}}}l+\lambda
l-e^{A}C_{A}=0\,,\\\
\mathrm{d_{\mathcal{J}}}\omega_{A}{}^{B}+\omega_{A}{}^{C}\omega_{C}{}^{B}+\lambda_{A}e^{B}-\lambda^{B}e_{A}=\frac{1}{2}e^{C}e^{D}\mathrm{W}_{A}{}^{B}{}_{CD},\\\
\mathrm{d_{\mathcal{J}}}\lambda^{A}+\omega^{A}{}_{B}\lambda^{B}-\lambda\lambda^{A}=-le^{D}\mathrm{C}^{A}{}_{uD}-\dfrac{1}{2}e^{C}e^{D}\mathrm{C}^{A}{}_{CD},\\\
\mathrm{d_{\mathcal{J}}}C^{A}+\omega^{A}{}_{B}C^{B}+\lambda^{u}e^{A}-\lambda^{A}l=\dfrac{1}{2}e^{C}e^{D}\mathrm{W}_{\Omega}{}^{A}{}_{CD},\\\
\mathrm{d_{\mathcal{J}}}\lambda^{u}+C_{A}\lambda^{A}-\lambda\lambda^{u}=-le^{D}\mathrm{C}_{\Omega
uD}-\dfrac{1}{2}e^{C}e^{D}\mathrm{C}_{\Omega CD}\,.\end{gathered}$ (4.33)

These are Cartan structure equations for the $iso(1,d-1)$ connection written
in the special basis, where the $so(1,d-1)$ subalgebra is made explicit as the
conformal algebra in $d-2$-dimensions and in contrast to the usual Cartan
description of Riemannian or Einstein geometry these equations are defined in
$d-1$-dimensional space rather than $d$-dimensional one. Moreover, the
curvatures appearing in the right hand sides of the above equations are
subject to specific constraints. For instance, components of the curvature in
the sector of varibales $e^{A},l$ and $\lambda$ vanish.

In the case of $d=4$ the curvature of this connection contains 5 independent
components, namely $\mathrm{C}_{\Omega cd}$ and $\mathrm{C}_{Bcd}$ (other
components vanish in $d=4$), which can be identified with Newman-Penrose
coefficients $\Psi_{4},\Psi_{3},Im\Psi_{2}$ encoding the gravitational
radiation. At the same time, fields $\mathrm{W}_{A\Omega\Omega B}$,
$\mathrm{W}_{\Omega uA\Omega}$, and $\mathrm{W}_{\Omega uu\Omega}$ also
contain 5 independent components which correspond to the remaining Newman-
Penrose coefficients $\Psi_{0},\Psi_{1}$ and $Re(\Psi_{2})$, see e.g. [81].
Let us stress that in contrast to the former, the latter 5 components do not
enter the Cartan structure equations and hence can not be interpreted as
components of the curvature of the $iso(1,d-1)$-connection on the boundary.
These are known to capture the longitudinal information and indeed are not
described by the curvature [82], see also [57, 59] for more details.333They
are analogous to the components of subleading modes appearing in the near-
boundary analysis of critical fields in the AdS/CFT context, see e.g. [83,
84]. For generic fields, these modes were described in [41, 42] within a
version of gPDE approach.

Let us introduce the components of the dual frame according to
$e^{A}=\sigma^{*}(\xi^{A})=e^{A}{}_{\mu}\theta^{\mu}$ and
$l=\sigma^{*}(\xi^{u})=l_{\mu}\theta^{\mu}$ and restrict to sections with
invertible frame. Components $(e^{\mu}{}_{A},n^{\mu})$ of the frame are
introduced via

$\displaystyle n^{\mu}l_{\mu}=1,\quad n^{\mu}e^{A}{}_{\mu}=0,\quad
l_{\mu}e^{\mu}{}_{A}=0,\quad e^{A}{}_{\mu}e^{\mu}{}_{B}=\delta^{A}_{B}.$
(4.34)

Taking into account the constraints on the curvature one can check that as
independent components of the connection one can take
$\\{l_{\mu},e^{A}{}_{\mu},e^{\nu}{}_{A}\lambda_{\nu},C_{(AB)}\\}$ because the
remaining components can be expressed through them.

The parameterization of the space of solutions to (4.33) can be described more
efficiently if one makes use of the gauge freedom (2.2). Introducing gauge
parameter vector field $Y$ as in (4.22) and assuming coefficients to depend on
$x$ only the gauge transformation for $\lambda_{\mu}$ reads as

$\displaystyle\lambda_{\mu}\rightarrow\lambda_{\mu}+\partial_{\mu}\bar{\lambda}-\epsilon^{A}\lambda_{A\mu}+e^{A}{}_{\mu}\bar{\lambda}_{A}\,,$
(4.35)

so that the following gauge condition can be imposed:

$\displaystyle e^{\nu}{}_{A}\lambda_{\nu}=0\,.$ (4.36)

In this gauge the components of the gauge parameter satisfy
$\bar{\lambda}_{A}=-e^{\mu}{}_{A}(\partial_{\mu}\bar{\lambda}-\epsilon^{B}\lambda_{B\mu})$.
In a similar way, we can achieve $C^{A}{}_{A}=0$, leading to further relations
between gauge parameters:

$\displaystyle\bar{\lambda}^{u}=\dfrac{1}{d-2}e^{\mu}{}_{A}(\partial_{\mu}\bar{C}^{A}-\bar{C}^{B}\omega_{B}{}^{A}{}_{\mu}+C^{B}{}_{\mu}\bar{\rho}_{B}{}^{A}+\lambda^{u}{}_{\mu}\epsilon^{A}-\lambda^{A}{}_{\mu}\epsilon^{u}).$
(4.37)

Furthermore, using

$\displaystyle\delta
l_{\mu}=\partial_{\mu}\epsilon^{u}+\epsilon^{u}\lambda_{\mu}-l\bar{\lambda}+\epsilon^{A}C_{A\mu}-e^{A}{}_{\mu}\bar{C}_{A}\,,$
(4.38)

the following gauge can be reached $l_{\mu}=\partial_{\mu}u$, where $u$ is a
function of $x^{\mu}$ satisfying $n^{\mu}\partial_{\mu}u=1$. Function $u$ is
often employed in the literature on BMS symmetries and it is convenient to
take it as one of the coordinate functions
$\\{x^{\mu}\\}\rightarrow\\{u,y^{\alpha}\\}$, $\alpha=1,\ldots,d-2$. Let us
also list the constraints on gauge parameters, which ensure preservation of
$l_{\mu}=\partial_{\mu}u$:

$\displaystyle\bar{C}_{B}=e^{\mu}{}_{B}(\partial_{\mu}\epsilon^{u}+\epsilon^{u}\lambda_{\mu}+\epsilon^{A}C_{A\mu}),\qquad\bar{\lambda}=n^{\mu}(\partial_{\mu}\epsilon^{u}+\epsilon^{u}\lambda_{\mu}+\epsilon^{A}C_{A\mu})\,.$
(4.39)

To summarize: by imposing gauge condition as explained above one can
parameterize the connection in terms of algebraically independent components
$\\{e^{A}{}_{\mu},C_{(AB)}|_{tf}\\}$ (of course there can be nontrivial
differential constraints following from the constraints on the curvature). In
so doing ${e^{A}}{}_{\mu}$ encodes the degenerate metric $g_{\mu\nu}\equiv
e^{A}{}_{\nu}g_{AB}e^{B}{}_{\nu}$ whose kernel is generated by
$n=\frac{\partial}{\partial u}$, while $-\frac{1}{2}C_{(AB)|_{tf}}$ is the so-
called asymptotic shear, see e.g.[85], which parameterize torsion-free and
metric-compatible affine connections on the boundary. Recall that such a
connection is not unique if metric is degenerate. The geometry determined by
$g_{\mu\nu}$ and $n^{\mu}$ defined up to an overall Weyl-like rescalings is
often refereed to as conformal Carroll geometry.

The setup of this section gives an alternative framework to study asymptotic
symmetries, which in contrast to the more algebraic approach of Section 4.3,
is somewhat analogous to the standard analysis, see e.g. [79] for the
derivation in the first-order formalism. Let us sketch how this can be done.
First of all one imposes boundary condition on sections of $E_{B}^{min}$ and
then, in order to simplify the system, one imposes partial gauge conditions,
e.g. the one discussed above. In the next step one studies gauge
transformations that preserve this boundary conditions. For instance, to
arrive at BMS symmetries in the present framework it is enough to fix a
concrete frame $e^{A}=e^{A}{}_{\mu}(x)\theta^{\mu}$ and set $\lambda=0$. BMS
symmetries are then obtained as the residual symmetries preserving these
boundary conditions. Note that these boundary conditions correspond to only a
subset of the conditions (4.19) and (4.20) of Section 4.3. The remaining
conditions correspond to solving some of the equations of motion and imposing
partial gauge conditions, cf. Remark 2.1.

###  4.5 Asymptotically (A)dS spaces

In the above analysis we concentrated on asymptotically flat spacetimes. It
turns out that the boundary system $(E_{B},Q,T[1]\mathcal{J})$ defined in 4.1
equally well works for asymptotically (A)dS spacetimes444In our analysis
$\Lambda>0$ for asymptotically AdS spacetimes or $\Lambda<0$ for
asymptotically dS spacetimes, since we work in the signature $(+,-,\dots,-)$.
See e.g. [78]. In this case the metric induced on the boundary is
nondegenerate and hence the respective minimal model differs substantially
from the case of $\Lambda=0$. More precisely, we have

######  Proposition 4.3.

For $\Lambda\neq 0$ Gauge PDE (4.1) is equivalent to its sub-gPDE defined as
follows:

$\displaystyle\begin{split}&g_{ab}=\begin{pmatrix}-\widetilde{\Lambda}&0\\\
0&\eta^{\varepsilon}_{AB}\end{pmatrix},\quad n^{a}=\begin{pmatrix}1\\\ 0\\\
\end{pmatrix},\quad C_{a}{}^{b}=\begin{pmatrix}-\lambda&0\\\
0&\rho_{A}{}^{B}-\lambda\delta_{A}^{B},\end{pmatrix},\\\
&\rho=0,\quad\xi^{\Omega}=0,\quad\lambda^{\Omega}=0,\end{split}$ (4.40)

where the adapted partition of indexes, e.g. $\\{a\\}=\\{\Omega,A\\}$,
$A=0,\ldots,d-1$ has been employed and

$\displaystyle\eta^{\varepsilon}_{AB}\equiv(\varepsilon,-1,\dots,-1),\quad\rho_{AB}\equiv
C_{[AB]},\quad\widetilde{\Lambda}\equiv\frac{2}{(d-1)(d-2)}\Lambda,\quad\varepsilon\equiv\operatorname{sign}\Lambda\,.$
(4.41)

Among the constraints on the degree-zero variables (4.4) there only remain:

$\displaystyle\nabla_{a_{1}}\cdots\nabla_{a_{n}}\mathrm{W}_{b_{3}\Omega
b_{1}b_{2}}-\sum_{i=1}^{n}g_{\Omega
a_{i}}\nabla_{a_{1}}\cdots\widehat{\nabla}_{a_{i}}\cdots\nabla_{a_{n}}\mathrm{C}_{b_{3}b_{1}b_{2}}=0,\quad
n\,{\geqslant}\,0,$ (4.42)

where the hatted symbols are assumed omitted and $g_{\Omega
a_{i}}=-\widetilde{\Lambda}\delta_{\Omega a_{i}}$.

The proof is fully analogous to that of 4.2. As an overcomplete coordinate
system on the above sub-gPDE we can take the restrictions of :
$\\{\xi^{A},\rho_{A}{}^{B},\lambda,\lambda^{A},\mathrm{W}^{m}{}_{nkp;(a)},|a|\,{\geqslant}\,0\\}$.
In these coordinates the action of $Q$ on the degree $1$ coordinates reads as:

$\displaystyle\begin{split}&Q\xi^{A}=\xi^{B}\rho_{B}{}^{A}-\xi^{A}\lambda,\\\
&Q\rho_{A}{}^{B}=\rho_{A}{}^{C}\rho_{C}{}^{B}+\lambda_{A}\xi^{B}-\lambda^{B}\xi_{A}+\dfrac{1}{2}\xi^{C}\xi^{D}\mathrm{W}^{B}{}_{ACD},\\\
&Q\lambda=\xi^{A}\lambda_{A},\\\
&Q\lambda^{A}=\rho^{A}{}_{C}\lambda^{C}-\lambda\lambda^{A}+\dfrac{1}{2}\xi^{B}\xi^{C}\mathrm{C}^{A}{}_{BC}\,.\end{split}$
(4.43)

It is easy to see that this coincides with the definition of CE differential
of $o(d-1,1)$ for dS and $o(d-1,2)$ for AdS respectively, written in the
conformal-like basis. This of course signals that in the case at hand the
boundary is naturally equipped with the conformal structure. More precisely,
solutions to the above sub-gPDE in the sector of degree $1$ coordinates define
a Cartan connection of the respective conformal geometry. However, the gauge
theory encoded in this sub-gPDE is not generally equivalent to conformal
geometry. For instance, in the case of $d=5$ the respective conformal geometry
is Bach-flat. The Bach flatness condition is encoded in the equations on
curvatures arising in the sector of degree $0$ variables. This is the
realization in our approach of the well-known Fefferman-Graham analysis [86,
83] (see also [41, 42, 43, 45] for the analogous considerations for generic
gauge fields within a version of gPDE framework).

As for asymptotic symmetries, one can consider an analogous boundary condition
$\sigma^{*}(\xi^{A})=e^{A}$, where $e^{A}$ is a fixed frame on the boundary.
The analysis of Section 4.4 can be easily repeated in the case at hand, giving
the conformal Killings of $g_{\mu\nu}e_{\mu}^{A}e_{\nu}^{B}$ as the basis in
the algebra of asymptotic symmetries. Of course, one can equally well repeat
the analysis of Section 4.3 in which case together with $\xi^{A}-e^{A}\sim 0$
and $Q(\xi^{A}-e^{A})\sim 0$ one should also impose additional boundary
conditions $\lambda\sim 0$ and $Q\lambda\sim 0$.

## Acknowledgments

We wish to thank I. Dneprov and Th. Popelensky for fruitful discussions. M.G.
is also grateful to G. Barnich, X. Bekaert, M. Henneaux, and J. Herfray for
useful exchanges. The work of M. M. was supported by the Russian Science
Foundation grant No 22-72-10122 (https://rscf.ru/en/project/22-72-10122/).
Part of this work was done when MG participated in the thematic program
”Emergent Geometries from Strings and Quantum Fields” at the Galileo Galilei
Institute for Theoretical Physics, Florence, Italy.

##  Appendix A Symmetries

Let us restrict to local analysis. Because locally gauge PDE can be
equivalently represented by the nonlagrangian local BV system it is enough to
give a proof in this setup. In this case $E$ is a $J^{\infty}(\mathcal{E})\to
X$ pulled back to $T[1]X$. In particular, functions on $E$ can be identified
with horizontal forms on $J^{\infty}(\mathcal{E})$. $E$ is equipped with the
evolutionary homological vector filed $s$ of ghost degree $1$ and the
homological vector field $\mathrm{d_{h}}=\theta^{a}D_{a}$ of
$\theta$-homogeneity $1$. We have the following:

######  Proposition A.1.

Locally, cohomology of $[\mathrm{d_{h}}{,}\,\cdot]$ in the space of vertical
vector fields is trivial in positive form degree (homogeneity in $\theta$). In
the space of vanishing $\theta$-homogeneity vertical vector fields, it is
given by evolutionary vector fields on $E$.

###### Proof.

To give an idea of the proof let us work in local coordinates
$x^{a},\theta^{b},\phi^{i}_{(a)}$ and conisider a $\theta$-homogeneity 1
vector field to begin with. It reads as

$V=\theta^{a}V_{a}^{i}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\phi^{i}}$}\kern
1.0pt}{\frac{\partial}{\partial\phi^{i}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\phi^{i}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\phi^{i}}$}\kern
1.0pt}+\theta^{a}V_{a|b}^{i}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\phi^{i}_{b}}$}\kern
1.0pt}{\frac{\partial}{\partial\phi^{i}_{b}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\phi^{i}_{b}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\phi^{i}_{b}}$}\kern
1.0pt}+\ldots$ (A.1)

The cocycle condition reads as $[\mathrm{d_{h}}{,}\,V]=0$ and implies, in
particular,

$[\mathrm{d_{h}}{,}\,V]\phi^{i}=\theta^{a}\theta^{b}(D_{b}V^{i}_{a}-V^{i}_{a|b})=0$
(A.2)

At the same time the coboundary $[\mathrm{d_{h}}{,}\,W]$ acting on 0-th jets
has the following structure:

$[\mathrm{d_{h}}{,}\,W]\phi^{i}=\theta^{a}(D_{a}W^{i}-W^{i}_{a})$ (A.3)

It follows, by adding a coboundary one can always set the coefficient
$V^{i}_{a}=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta^{a}}$}\kern
1.0pt}{\frac{\partial}{\partial\theta^{a}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta^{a}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta^{a}}$}\kern
1.0pt}(V\phi^{i})$ to zero. Then the cocycle condition implies that
$V^{i}_{a|b}-V^{i}_{b|a}=0$ so that $V^{i}_{b|a}$ can be also set to zero by
adding $[\mathrm{d_{h}}{,}\,W]$ such that the only nonvanishing coefficient is
$W^{i}_{ab}=W\phi^{i}_{(ab)}=-V^{i}_{(a|b)}$. The proof can be completed by
induction. The analysis for higher $\theta$-homogeneity vector field is
analogous. ∎

Let us now turn to the cohomology of $[Q{,}\,\cdot]$, $Q=\mathrm{d_{h}}+s$ in
the space of vertical vector fields. Expanding the cocycle condition in the
$\theta$-homogeneity one gets

$\begin{gathered}[s{,}\,V_{0}]=0\,,\quad[s{,}\,V_{1}]+[\mathrm{d_{h}}{,}\,V_{0}]=0\,,\quad\ldots\\\
[s{,}\,V_{1}]+[\mathrm{d_{h}}{,}\,V_{0}]=0\,,\quad[\mathrm{d_{h}}{,}\,V_{k}]=0\end{gathered}$
(A.4)

where we assumed that $V_{l}=0$ for all $l>k$, with $0<k\,{\leqslant}\,n$.
Applying the above Proposition we conclude that
$V_{l}=[\mathrm{d_{h}}{,}\,W_{l-1}]$. Subtracting the trivial cocycle
$[Q{,}\,W_{l-1}]$ we arrive at a new representative $V^{\prime}$ for which
$V^{\prime}_{l}=0$ for all $l>k-1$. Applying the same procedure again we
arrive at an equivalent representative for which $V_{l}=0$ for $l>0$. The
cocycle condition then implies that $[s{,}\,V_{0}]=0$ and
$[\mathrm{d_{h}}{,}\,V_{0}]=0$. In other words we have arrived at the standard
representative of a global symmetry. Note that in the above we did not assume
$\mathrm{gh}(V)=0$ so it applies to generalized symmetries as well.

## References

  * [1] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, “Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc. Lond. A 269 (1962) 21–52.
  * [2] R. Sachs, “Asymptotic symmetries in gravitational theory,” Phys. Rev. 128 (1962) 2851–2864.
  * [3] R. Penrose, “Conformal treatment of infinity,”.
  * [4] J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math. Phys. 104 (1986) 207–226.
  * [5] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, “Large N field theories, string theory and gravity,” Phys.Rept. 323 (2000) 183–386, arXiv:hep-th/9905111 [hep-th].
  * [6] A. Bagchi and R. Gopakumar, “Galilean Conformal Algebras and AdS/CFT,” JHEP 07 (2009) 037, arXiv:0902.1385 [hep-th].
  * [7] G. Barnich and C. Troessaert, “Aspects of the BMS/CFT correspondence,” JHEP 05 (2010) 062, arXiv:1001.1541 [hep-th].
  * [8] A. Bagchi, S. Detournay, and D. Grumiller, “Flat-Space Chiral Gravity,” Phys. Rev. Lett. 109 (2012) 151301, arXiv:1208.1658 [hep-th].
  * [9] A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 07 152, arXiv:1312.2229 [hep-th].
  * [10] A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP 01 (2016) 086, arXiv:1411.5745 [hep-th].
  * [11] A. B. Prema, G. Compère, L. Pipolo de Gioia, I. Mol, and B. Swidler, “Celestial holography: Lectures on asymptotic symmetries,” SciPost Phys. Lect. Notes 47 (2022) 1, arXiv:2109.00997 [hep-th].
  * [12] T. He, V. Lysov, P. Mitra, and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP 05 (2015) 151, arXiv:1401.7026 [hep-th].
  * [13] R. Penrose, “Zero rest mass fields including gravitation: Asymptotic behavior,” Proc. Roy. Soc. Lond. A284 (1965) 159.
  * [14] R. Penrose, “Relativistic symmetry groups,” in NATO Advanced Study Institute – Group theory in non-linear problems.  1972\.
  * [15] R. P. Geroch, “Asymptotic structure of space-time,”.
  * [16] F. Alessio and G. Esposito, “On the structure and applications of the Bondi–Metzner–Sachs group,” Int. J. Geom. Meth. Mod. Phys. 15 no. 02, (2018) 1830002, arXiv:1709.05134 [gr-qc].
  * [17] M. Banados, “Three-dimensional quantum geometry and black holes,” arXiv:hep-th/9901148.
  * [18] M. Henneaux and S.-J. Rey, “Nonlinear $W_{infinity}$ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity,” JHEP 12 (2010) 007, arXiv:1008.4579 [hep-th].
  * [19] A. Campoleoni, S. Fredenhagen, S. Pfenninger, and S. Theisen, “Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields,” JHEP 11 (2010) 007, arXiv:1008.4744 [hep-th].
  * [20] M. R. Gaberdiel and R. Gopakumar, “An ${A}d{S}_{3}$ Dual for Minimal Model CFTs,” arXiv:1011.2986 [hep-th].
  * [21] H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller, and J. Rosseel, “Spin-3 Gravity in Three-Dimensional Flat Space,” Phys. Rev. Lett. 111 no. 12, (2013) 121603, arXiv:1307.4768 [hep-th].
  * [22] A. Strominger, “Asymptotic Symmetries of Yang-Mills Theory,” JHEP 07 (2014) 151, arXiv:1308.0589 [hep-th].
  * [23] A. Campoleoni, D. Francia, and C. Heissenberg, “On higher-spin supertranslations and superrotations,” JHEP 05 (2017) 120, arXiv:1703.01351 [hep-th].
  * [24] X. Bekaert and B. Oblak, “Massless scalars and higher-spin BMS in any dimension,” JHEP 11 (2022) 022, arXiv:2209.02253 [hep-th].
  * [25] J. D. Brown and M. Henneaux, “On the Poisson Brackets of Differentiable Generators in Classical Field Theory,” J. Math. Phys. 27 (1986) 489–491.
  * [26] G. Barnich and F. Brandt, “Covariant theory of asymptotic symmetries, conservation laws and central charges,” Nucl. Phys. B633 (2002) 3–82, hep-th/0111246.
  * [27] L. Freidel, R. Oliveri, D. Pranzetti, and S. Speziale, “The Weyl BMS group and Einstein’s equations,” JHEP 07 (2021) 170, arXiv:2104.05793 [hep-th].
  * [28] L. Freidel, R. Oliveri, D. Pranzetti, and S. Speziale, “Extended corner symmetry, charge bracket and Einstein’s equations,” JHEP 09 (2021) 083, arXiv:2104.12881 [hep-th].
  * [29] G. Compère and A. Fiorucci, “Advanced Lectures on General Relativity,” arXiv:1801.07064 [hep-th].
  * [30] I. Batalin and G. Vilkovisky, “Gauge Algebra and Quantization,” Phys.Lett. B102 (1981) 27–31.
  * [31] I. Batalin and G. Vilkovisky, “Feynman Rules For Reducible Gauge Theories,” Phys.Lett. B120 (1983) 166–170.
  * [32] G. Barnich, F. Brandt, and M. Henneaux, “Local BRST cohomology in gauge theories,” Phys.Rept. 338 (2000) 439–569, hep-th/0002245.
  * [33] M. Alexandrov, M. Kontsevich, A. Schwartz, and O. Zaboronsky, “The Geometry of the master equation and topological quantum field theory,” Int.J.Mod.Phys. A12 (1997) 1405–1430, hep-th/9502010.
  * [34] M. A. Grigoriev and P. H. Damgaard, “Superfield BRST charge and the master action,” Phys. Lett. B474 (2000) 323–330, arXiv:hep-th/9911092 [hep-th].
  * [35] G. Barnich and M. Grigoriev, “Hamiltonian BRST and Batalin-Vilkovisky formalisms for second quantization of gauge theories,” Commun.Math.Phys. 254 (2005) 581–601, arXiv:hep-th/0310083 [hep-th].
  * [36] G. Barnich and M. Grigoriev, “BRST extension of the non-linear unfolded formalism,” in International School / Seminar on Quantum Field Theory, Supersymmetry, High Spin Fields, Gravity Tomsk, Russia, March 20-26, 2005.  2005\.  arXiv:hep-th/0504119 [hep-th].
  * [37] A. Cattaneo, P. Mnev, and N. Reshetikhin, “Classical BV theories on manifolds with boundary,” Commun. Math. Phys. 332 (2014) 535–603, arXiv:1201.0290 [math-ph].
  * [38] M. Grigoriev, “Parent formulations, frame-like Lagrangians, and generalized auxiliary fields,” JHEP 1212 (2012) 048, arXiv:1204.1793 [hep-th].
  * [39] G. Barnich and M. Grigoriev, “First order parent formulation for generic gauge field theories,” JHEP 01 (2011) 122, arXiv:1009.0190 [hep-th].
  * [40] M. Grigoriev, “Parent formulation at the Lagrangian level,” JHEP 07 (2011) 061, arXiv:1012.1903 [hep-th].
  * [41] X. Bekaert and M. Grigoriev, “Notes on the ambient approach to boundary values of AdS gauge fields,” J. Phys. A46 (2013) 214008, arXiv:1207.3439 [hep-th].
  * [42] X. Bekaert and M. Grigoriev, “Higher order singletons, partially massless fields and their boundary values in the ambient approach,” Nucl.Phys. B876 (2013) 667–714, arXiv:1305.0162 [hep-th].
  * [43] A. Chekmenev and M. Grigoriev, “Boundary values of mixed-symmetry massless fields in AdS space,” Nucl. Phys. B913 (2016) 769–791, arXiv:1512.06443 [hep-th].
  * [44] M. A. Vasiliev, “Holography, Unfolding and Higher-Spin Theory,” J.Phys. A46 (2013) 214013, arXiv:1203.5554 [hep-th].
  * [45] X. Bekaert, M. Grigoriev, and E. D. Skvortsov, “Higher Spin Extension of Fefferman-Graham Construction,” Universe 4 no. 2, (2018) 17, arXiv:1710.11463 [hep-th].
  * [46] M. Grigoriev and E. D. Skvortsov, “Type-B Formal Higher Spin Gravity,” JHEP 05 (2018) 138, arXiv:1804.03196 [hep-th].
  * [47] P. Mnev, M. Schiavina, and K. Wernli, “Towards holography in the BV-BFV setting,” Annales Henri Poincare 21 no. 3, (2019) 993–1044, arXiv:1905.00952 [math-ph].
  * [48] K. Rejzner and M. Schiavina, “Asymptotic symmetries in the BV-BFV formalism,” arXiv:2002.09957 [math-ph].
  * [49] A. Riello and M. Schiavina, “Null Hamiltonian Yang-Mills theory. Soft symmetries and memory as superselection,” arXiv:2303.03531 [math-ph].
  * [50] M. Grigoriev and A. Kotov, “Gauge PDE and AKSZ-type Sigma Models,” Fortsch. Phys. (2019) , arXiv:1903.02820 [hep-th].
  * [51] G. Barnich, M. Grigoriev, A. Semikhatov, and I. Tipunin, “Parent field theory and unfolding in BRST first-quantized terms,” Commun.Math.Phys. 260 (2005) 147–181, arXiv:hep-th/0406192 [hep-th].
  * [52] M. Grigoriev, “Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles,” arXiv:2212.11350 [math-ph].
  * [53] M. A. Vasiliev, “Equations of Motion of Interacting Massless Fields of All Spins as a Free Differential Algebra,” Phys. Lett. B209 (1988) 491–497.
  * [54] M. A. Vasiliev, “Actions, charges and off-shell fields in the unfolded dynamics approach,” Int. J. Geom. Meth. Mod. Phys. 3 (2006) 37–80, hep-th/0504090.
  * [55] N. Boulanger, “A Weyl-covariant tensor calculus,” J. Math. Phys. 46 (2005) 053508, arXiv:hep-th/0412314.
  * [56] I. Dneprov and M. Grigoriev, “Presymplectic BV-AKSZ formulation of conformal gravity,” Eur. Phys. J. C 83 no. 1, (2023) 6, arXiv:2208.02933 [hep-th].
  * [57] Y. Herfray, “Asymptotic shear and the intrinsic conformal geometry of null-infinity,” J. Math. Phys. 61 no. 7, (2020) 072502, arXiv:2001.01281 [gr-qc].
  * [58] Y. Herfray, “Tractor Geometry of Asymptotically Flat Spacetimes,” Annales Henri Poincare 23 no. 9, (2022) 3265–3310, arXiv:2103.10405 [gr-qc].
  * [59] Y. Herfray, “Carrollian manifolds and null infinity: a view from Cartan geometry,” Class. Quant. Grav. 39 no. 21, (2022) 215005, arXiv:2112.09048 [gr-qc].
  * [60] A. R. Gover and A. Waldron, “Boundary calculus for conformally compact manifolds,” Indiana Univ. Math. J. 63 (2014) 119–163, arXiv:1104.2991 [math.DG].
  * [61] A. Rod Gover, E. Latini, and A. Waldron, “Poincare-Einstein Holography for Forms via Conformal Geometry in the Bulk,” arXiv:1205.3489 [math.DG].
  * [62] G. Barnich and M. Grigoriev, “A Poincare lemma for sigma models of AKSZ type,” J.Geom.Phys. 61 (2011) 663–674, arXiv:0905.0547 [math-ph].
  * [63] D. S. Kaparulin, S. L. Lyakhovich, and A. A. Sharapov, “Local BRST cohomology in (non-)Lagrangian field theory,” JHEP 09 (2011) 006, arXiv:1106.4252 [hep-th].
  * [64] A. A. Sharapov, “Variational Tricomplex, Global Symmetries and Conservation Laws of Gauge Systems,” SIGMA 12 (2016) 098, arXiv:1607.01626 [math-ph].
  * [65] A. Kotov and T. Strobl, “Characteristic classes associated to Q-bundles,” Int. J. Geom. Meth. Mod. Phys. 12 no. 01, (2014) 1550006, arXiv:0711.4106 [math.DG].
  * [66] F. Brandt, “Local BRST Cohomology and Covariance,” Commun. Math. Phys. 190 (1997) 459–489, hep-th/9604025.
  * [67] M. Grigoriev and A. Kotov, “Presymplectic AKSZ formulation of Einstein gravity,” JHEP 09 (2021) 181, arXiv:2008.11690 [hep-th].
  * [68] M. Grigoriev, “Presymplectic structures and intrinsic Lagrangians,” arXiv:1606.07532 [hep-th].
  * [69] M. Grigoriev and V. Gritzaenko, “Presymplectic structures and intrinsic Lagrangians for massive fields,” Nucl. Phys. B 975 (2022) 115686, arXiv:2109.05596 [hep-th].
  * [70] K. B. Alkalaev and M. Grigoriev, “Frame-like Lagrangians and presymplectic AKSZ-type sigma models,” Int. J. Mod. Phys. A29 no. 18, (2014) 1450103, arXiv:1312.5296 [hep-th].
  * [71] G. Barnich, F. Brandt, and M. Henneaux, “Local BRST cohomology in Einstein Yang-Mills theory,” Nucl. Phys. B455 (1995) 357–408, hep-th/9505173.
  * [72] G. Barnich, F. Brandt, and M. Henneaux, “Local BRST cohomology in the antifield formalism. I. General theorems,” Commun. Math. Phys. 174 (1995) 57–92, hep-th/9405109.
  * [73] T. Basile, M. Grigoriev, and E. Skvortsov, “Covariant action for conformal higher spin gravity,” arXiv:2212.10336 [hep-th].
  * [74] R. Penrose, “Asymptotic properties of fields and space-times,” Phys. Rev. Lett. 10 (1963) 66–68.
  * [75] E. Joung, M.-g. Kim, and Y. Kim, “Unfolding conformal geometry,” JHEP 12 (2021) 092, arXiv:2108.05535 [hep-th].
  * [76] T. Bailey, M. Eastwood, and A. Gover, “ Thomas’s structure bundle for conformal, projective and related structures,” Rocky Mountain J. Math. 24 (1994) 1191–1217.
  * [77] S. Curry and A. R. Gover, “An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity,” arXiv:1412.7559 [math.DG].
  * [78] J. A. V. Kroon, Conformal Methods in General Relativity.  Cambridge University Press, Cambridge, 2016.
  * [79] K. Nguyen and J. Salzer, “The effective action of superrotation modes,” JHEP 02 (2021) 108, arXiv:2008.03321 [hep-th].
  * [80] C. Duval, G. W. Gibbons, and P. A. Horvathy, “Conformal Carroll groups and BMS symmetry,” Class. Quant. Grav. 31 (2014) 092001, arXiv:1402.5894 [gr-qc].
  * [81] E. Newman and R. Penrose, “An Approach to gravitational radiation by a method of spin coefficients,” J. Math. Phys. 3 (1962) 566–578.
  * [82] A. Ashtekar, “Radiative Degrees of Freedom of the Gravitational Field in Exact General Relativity,” J. Math. Phys. 22 (1981) 2885–2895.
  * [83] C. Fefferman and C. R. Graham, “The ambient metric,” arXiv:0710.0919 [math.DG].
  * [84] K. Skenderis, “Lecture notes on holographic renormalization,” Class. Quant. Grav. 19 (2002) 5849–5876, arXiv:hep-th/0209067.
  * [85] A. Ashtekar, “Geometry and Physics of Null Infinity,” arXiv:1409.1800 [gr-qc].
  * [86] C. Fefferman and C. R. Graham, “Conformal invariants,” Astérisque no. Numéro Hors Série, (1985) 95–116. The mathematical heritage of Élie Cartan (Lyon, 1984).