††institutetext: Institute of Mathematical Sciences,
IV Cross Road, CIT Campus,
Taramani, Chennai, India 600 113
††institutetext: Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar,
Mumbai, India 400 085, India

# The Radiative Phase Space for the Dynamical Celestial Metric

Adarsh Sudhakar  Amit Suthar<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Generalized BMS (gBMS) is the Lie group of the asymptotic symmetries at null
infinity, and is proposed to be a symmetry of the quantum S-matrix. Despite
much progress in understanding the symplectic structure at null infinity
consistent with the gBMS symmetries, the construction of a radiative phase
space where all the physical soft modes and their conjugate partners are
identified remains elusive. We construct just such a radiative phase space for
linearized gravity by a systematic constraint analysis. In addition, we
highlight the difficulties that arise in extending this analysis to the non-
linear case. In order to analyze the difficulties we face in extending these
ideas to the non-linear setting, we consider a toy model in which we gauge the
action of the Weyl scaling in the Weyl BMS group. We find that
supertranslations are no longer well-defined symmetries on the reduced phase
space of the gauged Weyl, as Weyl scalings do not commute with
supertranslations. In this restricted case we obtain the symplectic form and
derive the reduced phase space.

##  1 Introduction

Over the last decade or so, there has been a renewed interest in understanding
physics at the boundaries of asymptotically flat space-times. The most well-
understood components of the boundary are the null infinities
$\mathcal{I}^{\pm}$. Their rich structure encompasses the beautiful discovery
of Bondi, Metzner, and Sachs BMS ; Sachs , that the symmetry group that
preserves asymptotic flatness at $\mathcal{I}^{\pm}$ is an infinite
dimensional group which is termed the BMS group. It is generated by the
_super-translations_ , and the Lorentz group on the celestial sphere.
Supertranslations are the angle-dependent translations in the null coordinates
of $\mathcal{I}^{\pm}$, and ordinary translations are a subgroup of it.

Starting from the seminal work of Barnich and Troessart bt ; bt2 , there have
been numerous enhancements of the BMS group obtained by relaxing the boundary
conditions on the space-time metric at null infinity. Allowing the celestial
metric to fluctuate, while keeping the determinant of the celestial metric
fixed, we arrive at the well known extended BMS (eBMS) and generalized BMS
(gBMS) groups. In all these extensions and generalizations of the $bms$
algebra, super-translations form an abelian ideal. The Lorentz algebra is
isomorphic to the algebra of global CKVs of the celestial sphere,
$sl(2,\mathbb{C})$. Hence the original $bms$ algebra is a semi direct sum of
super-translations and $sl(2,\mathbb{C})$. $ebms$ algebra extends the Lorentz
algebra to include all the local CKVs (meromorphic vector fields) on the
celestial sphere. The super-rotations ($ebms$ modulo super-translations) form
two copies of the Witt algebra. The other generalization, known as gBMS,
enhances the Lorentz group to include all the smooth diffeomorphisms on the
celestial sphere. In the case of $gbms$ algebra, super-rotations are generated
by smooth vector fields on the sphere. See strominger lectures ; compere
lectures  for a comprehensive review of these developments.

Even though the asymptotic symmetries of asymptotically flat spacetimes have
been studied since the 1960s, the main reason for their resurgence in the last
decade was the realization that the conservation law associated with the
super-translation symmetry is equivalent to the well known factorization of
scattering amplitudes in the soft limits , Weinberg soft graviton
theoremWeinberg soft . The eBMS and gBMS extensions were further consolidation
of this connection between the asymptotic symmetries and constraints on the
S-matrix, as their conservation law was proved to be equivalent to the sub-
leading soft graviton theorem discovered by Cachazo and Strominger sub ;
gBMS_Ward ; pks ; alok miguel 2014 . The new extensions also have an
interesting consequence in classical gravity, as the corresponding Noether
charges defined at null infinity have been shown to be associated with so-
called spin-memory effect pas .

Once we accept the paradigm of studying the group of all symmetries that
preserve asymptotic flatness, it leads to further enhancements of the BMS
group at null infinity. It was shown in f  that asymptotic flatness remains
preserved even after allowing the area element (celestial metric determinant)
to fluctuate. This enhancement of BMS group is known as the Weyl BMS (WBMS)
group. In this, an arbitrary Weyl scaling of the celestial metric is allowed
along with the superrotations generated by all smooth diffeomorphisms of
$S^{2}$. In this article, we mainly focus on the $gbms$ and $wbms$ algebras.

The space of solutions to Einstein equations for a fixed celestial metric is
parameterized by the shear tensor $\sigma_{ab}(u,\hat{x})$. As was shown by
Ashtekar and Streubel in the 1980s as , one can associate a phase space called
radiative phase space, parameterized by the shear tensor $(\sigma_{ab})$ to
this space of solutions. In order for the phase space to have a faithful
representation of the $bms$ algebra and its enhancements, the radiative phase
space was enhanced so that it is parameterized by the shear tensor
$\sigma_{ab}(u,\hat{x})$, and the boundary modes $C(\hat{x}),T_{ab}(\hat{x})$
that transforms inhomogeneously under supertranslations and superrotations
respectively. The symplectic structure for this setup splits into the hard
sector, parametrized by the shear and News tensors
($N_{ab}(u,\hat{x})=\partial_{u}\sigma_{ab}$) and the soft sector containing
the boundary modes and their conjugate partners. The soft sector is termed so,
because the quantized conjugate partners to the boundary modes can be
identified with the soft modes of the gravitational field.

The hard and soft sectors are not independent. The conjugate momenta of the
soft modes are related to the hard modes by constraints. Prior to imposing
these constraints, we call the phase space _kinematical_. The particular case
in which the celestial metric is fixed away from all but one point of the
celestial sphere, was analyzed in am . In that case, one could solve these
constraints and obtain a ‘physical radiative phase space’ in which hard and
independent soft modes are identified and which generate a Poisson algebra.
However, the situation is far more intricate in the presence of a dynamical
celestial metric. The conserved charges corresponding to all the gBMS
generators have been derived mp , such that their canonical action on the
shear field matches with the space-time action of gBMS symmetries. Despite
this, the radiative phase space that comprises the independent set of fields
obtained by relating the hard shear modes and the soft conjugate momenta to
the boundary modes has not yet been derived.

Our goal in this paper is to initiate a study of just such a radiative phase
space at future null infinity $\mathcal{I}^{+}$ in which the sphere metric is
smooth and dynamical. This would be a direct extension of the phase space
obtained by He, Lysov, Mitra, and Strominger (HLMS) which admits a faithful
action of the BMS group hlms .

In particular, we consider two scenarios. Firstly, we obtain the physical
radiative phase space for the smooth celestial metric, corresponding to gBMS
in the linearized setting. Secondly, we consider the case of HLMS phase space
augmented by arbitrary Weyl scalings of the celestial metric, and gauge the
Weyl scaling. The main results of the paper and its organization are
summarized below.

##### Summary and organization of the paper:

In Section 2, we begin with a review of supertranslations, gBMS, and Weyl-BMS
groups. In Section 3, we consider the radiative phase space for the $gbms$
algebra. In Section 3.1, we start with the symplectic structure proposed by
Campiglia-Peraza in mp , suitably adapted to linearized gravity. We then
obtain the physical radiative phase space by imposing the appropriate second
class constraints via the Dirac bracket analysisd . The main result of this
analysis is the identification of the physical mode conjugate to the
subleading soft News. To the best of our knowledge, this is the first model of
a radiative phase space that allows for a smooth and dynamical celestial
sphere metric. We then outline the difficulties and subtleties in extending
our results to full general relativity 111Note that the Ashtekar-Streubel, as
well as HLMS phase spaces obtained in linearized gravity, are isomorphic to
the ones obtained in the fully non-linear theory. However, in the cases
involving dynamical smooth celestial metric, the radiative phase space
structure is considerably simplified due to linearization..

In Section 4, we move on to the second case of interest. We consider the sub-
group of Weyl-BMS which is generated by super-translations and Weyl
transformations at $\mathcal{I}^{+}$. In particular we start with the expanded
radiative phase space comprising of the News tensors, super-translation soft
modes as well as the new modes corresponding to the dynamical area element.
The symplectic structure is determined by demanding that it is degenerate
along the Weyl orbit, rendering Weyl rescaling to be pure gauge. We then show
that the constraints can be solved unambiguously, leading to a radiative phase
space. The reduced phase space turns out to be different from the HLMS phase
space in the following manner: there is no notion of local hard News in the
reduced phase space. We also show explicitly that supertranslations are not
well defined on our reduced phase space. The reduced phase space is
parameterized by the usual soft modes and the $u$-integrals of arbitrary
quantities made out of the hard News. We note that they form a closed algebra.

### Notations and Conventions

  * •

$\nabla_{a}$ is the two dimensional covariant derivative compatible with the
celestial metric $q_{ab}$. $D_{a}$ is the Diff-$S^{2}$ covariant derivative as
introduced in mp . We also denote by $\mathcal{L}_{V}$ the Lie derivative
along vector field $V$.

  * •

The identity for the space of symmetric trace-free two tensors is constructed
out of the Kronecker delta and the flat metric $\eta$ on the sphere. It is
denoted by :

$\displaystyle\mathbb{I}^{ab}_{cd}=\delta^{(a}_{c}\delta^{b)}_{d}-\frac{1}{2}\eta^{ab}\eta^{cd}\
\ ,\qquad\mathbb{I}^{ab}_{cd}\,\mathbb{I}^{cd}_{mn}=\mathbb{I}^{ab}_{mn}\ \
,\qquad\mathbb{I}_{ab,cd}=q_{am}q_{bn}\,\mathbb{I}^{mn}_{cd}~{}.$ (1)

  * •

The symbols $d$ and $\delta$ denote the exterior derivative on spacetime and
field-space respectively.

  * •

We denote by $X_{A}$ the Hamiltonian vector field associated with arbitrary
phase space variable $A$. For a given symplectic form $\Omega$, it is defined
through:

$\displaystyle\Omega(Y,X_{A})$ $\displaystyle=\delta A[Y]\ ,$ (2)

where $Y$ is any vector field on phase space.

  * •

We use the following conventions for the Poisson Brackets endowed by a
symplectic form $\Omega$ :

$\displaystyle\\{f,g\\}=\Omega(X_{g},X_{f})$ $\displaystyle=\delta
f(X_{g})=X_{g}(f)\ .$ (3)

##  2 The Story So Far

We are interested in analyzing the null boundaries of the asymptotically flat
spacetimes. The null boundary $\mathcal{I}^{+}$ is parameterized by
$(u,\hat{x}^{a})$. Note that $\hat{x}^{a}$ are the coordinates on the two
sphere, transverse to the radial and temporal directions. We choose the
following gauge conditions:
$g_{rr}=g_{ra}=0=\partial_{r}\,\text{det}(g_{ab}/r^{2})$, referred to as Bondi
gauge. In this particular gauge choice, the metric at $\mathcal{I}^{+}$ takes
the following form :

$\displaystyle\mathrm{d}s^{2}=-2\mathrm{d}u^{2}-\mathrm{d}u\,\mathrm{d}r+\mathrm{d}\hat{x}^{a}\mathrm{d}\hat{x}^{b}\left(r^{2}\,q_{ab}+r\sigma_{ab}+\ldots\right)+\ldots$
(4)

Now, we review the phase spaces for supertranslations, gBMS and Weyl.

###  2.1 Review of the HLMS Phase Space

The symplectic potential at $\mathcal{I}^{+}$ for the radiative data (shear
and News tensor) was famously derived by Ashtekar and Struebel (AS) as , and
is given by:

$\Theta_{AS}=\int_{\mathcal{I}}\sqrt{q}\ \sigma^{ab}\delta N_{ab}\ ,$ (5)

For a well-defined symplectic structure, the News tensor is required to have
the following fall-off conditions:

$\displaystyle
N_{ab}\equiv\partial_{u}\sigma_{ab}\xrightarrow{u\to\pm\infty}\mathcal{O}(\rvert
u\rvert^{-1-\epsilon})\qquad(\epsilon\ >\ 0)\ .$ (6)

It leads to the following symplectic form and Poisson bracket:

$\displaystyle\Omega$ $\displaystyle=\int_{\mathcal{I}}\sqrt{q}\,\delta
N_{ab}\wedge\delta\sigma^{ab}~{},$ (7)
$\displaystyle\big{[}N_{ab}(u,\hat{x}),N_{cd}(u^{\prime},\hat{y})\big{]}$
$\displaystyle=\frac{1}{2}\partial_{u}\delta(u-u^{\prime})\,\mathbb{I}_{ab,cd}\,\frac{1}{\sqrt{q}}\delta^{2}(\hat{x}-\hat{y})~{},$
(8)

where $\mathbb{I}_{ab,cd}$ was introduced in (1).

Super-translation symmetries generate a Hamiltonian action on all the smooth
functions defined on the Ashtekar-Streubel phase space ($\Gamma_{AS}$). The
corresponding charge (or flux) for super-translation contains a term, linear
in soft News tensor ($\int\mathrm{d}u\,N_{ab}$). However, the soft News or its
expected conjugate mode does not exist in $\Gamma_{AS}$. He, Lysov, Mitra,
Strominger (HLMS) revisited the canonical derivation of the symplectic
structure at $\mathcal{I}$, and showed that there is an enhanced phase space,
which includes the constant shear as a boundary mode.222The HLMS phase space
can also be denoted as $\Gamma_{\text{HLMS}}$. The soft News tensor is its
conjugate. HLMS analysis involved extending the AS phase space by adding the
soft sector to it, and then imposing physical constraints that relate the soft
News as the zero mode of the News tensor strominger lectures ; hlms .

The constant shear mode is separated from the full shear tensor as follows:

$\displaystyle\sigma_{ab}(u,\hat{x})=\
\overset{o}{\sigma}_{ab}(u,\hat{x})+C_{ab}(\hat{x}).$ (9)

This split is unique if we demand

$\displaystyle\lim_{u\rightarrow\infty}\big{[}\overset{o}{\sigma}_{ab}(u,\hat{x})\
+\ \overset{o}{\sigma}_{ab}(-u,\hat{x})\big{]}\equiv\ \
\overset{o}{\sigma}{}^{+}_{ab}(\hat{x})\
+\overset{o}{\sigma}{}^{-}_{ab}(\hat{x})=0~{}.$ (10)

$C_{ab}$ is the boundary mode that HLMS introduced into the Ashtekar-Streubel
phase space structure. Making this mode explicit within the original Ashtekar-
Streubel symplectic form, one can obtain the following symplectic structure:

$\displaystyle\Omega$ $\displaystyle=\int_{\mathcal{I}}\sqrt{q}\,\delta
N_{ab}\wedge\delta\overset{o}{\sigma}{}^{ab}+\int_{S^{2}}\sqrt{q}\,\delta\left(\int\mathrm{d}uN_{ab}\right)\wedge\delta
C^{ab}$ (11) $\displaystyle\equiv\int_{\mathcal{I}}\sqrt{q}\,\delta
N_{ab}\wedge\delta\overset{o}{\sigma}{}^{ab}+\int_{S^{2}}\sqrt{q}\,\delta\overset{o}{\mathcal{N}}_{ab}\wedge\,\delta
C^{ab}~{},$ (12)

where $\overset{o}{\mathcal{N}}_{ab}$ is defined through:

$\overset{o}{\mathcal{N}}_{ab}\equiv\int\mathrm{d}u\ N_{ab}~{}.$ (13)

Let us treat the mode conjugate to the constant shear as independent of the
hard News, and denote it by $\overset{o}{N}_{ab}$, to allow for the soft
($C_{ab},\overset{o}{N}_{ab}$) and the hard ($N_{ab}$) factorization of the
phase space. The physical phase space is now a constraint surface inside
$\Gamma_{AS}\times\Gamma_{s}$. The following are the constraints:

$\displaystyle\overset{o}{N}_{ab}(\hat{x})\
-\overset{o}{\mathcal{N}}_{ab}(\hat{x})$ $\displaystyle=0~{},$ (14)
$\displaystyle\overset{o}{\sigma}{}^{+}_{ab}(\hat{x})\
+\overset{o}{\sigma}{}^{-}_{ab}(\hat{x})$ $\displaystyle=0\ .$ (15)

Before solving the second class constraints and computing the physical
brackets, there is a subtlety to be considered. Note that the components of
Weyl curvature tensor can be identified as electric and magnetic components,
in analogy with the gauge theories. For Christodolou Klainermann spacetimes
Christ  and their appropriate generalizations, the magnetic part of the Weyl
tensor vanishes at $\mathcal{I}$. This amounts to the following condition:

$\displaystyle\lim_{u\to\pm\infty}\nabla_{[a}\nabla^{c}\sigma_{cb]}\equiv\nabla_{[a}\nabla^{c}\sigma^{\pm}_{cb]}=0$
(16)

Since $\ \sigma_{cb}^{+}+\sigma_{cb}^{-}=2C_{cb}\ \text{and}\
\sigma_{cb}^{+}-\sigma_{cb}^{-}=\,\overset{o}{\mathcal{N}}_{cb}\ $, the
general solution to (16) is the following:

$\displaystyle
C_{ab}=-2\nabla_{a}\nabla_{b}C^{\text{TF}}\qquad;\qquad\overset{o}{\mathcal{N}}_{ab}=-2\nabla_{a}\nabla_{b}\overset{o}{\mathcal{N}}^{\text{TF}}\
.$ (17)

This reduces the degrees of freedom of the boundary modes by half.

Let us consider the symplectic form, (12), treat the soft News
($\overset{o}{N}_{ab}$) as independent from the hard News ($N_{ab}$), find the
HVFs and kinematical brackets and perform the Dirac analysis to obtain the
following physical brackets:

$\displaystyle\big{[}N_{ab}(u,\hat{x}),N_{cd}(u^{\prime},\hat{y})\big{]}$
$\displaystyle=\frac{1}{2}\partial_{u}\delta(u-u^{\prime})\,\mathbb{I}_{ab,cd}\,\frac{1}{\sqrt{q}}\delta^{2}(\hat{x}-\hat{y})$
(18) $\displaystyle\big{[}C(\hat{x}),\overset{o}{N}(\hat{y})\big{]}$
$\displaystyle=G(\hat{x},\hat{y})\ ,$ (19)

with $G(\hat{x},\hat{y})$ being the Green’s function for the operator
$4\nabla^{a}\nabla^{b}(\nabla_{a}\nabla_{b})^{\text{TF}}$.

###  2.2 gBMS Phase Space

Generalized BMS is one generalization of the BMS group, which generalizes the
Lorentz algebra to include all the smooth diffeomorphisms of the celestial
sphere as superrotations. The following is the action of $gbms$ on the
constant shear mode and the celestial metric:

$\delta_{V}C_{ab}(\hat{x})=\left(\mathcal{L}_{V}-\frac{1}{2}\nabla_{m}V^{m}\right)C_{ab}~{},$
(20a)
$\delta_{V}q_{ab}(\hat{x})=\left(\mathcal{L}_{V}-\nabla_{m}V^{m}\right)q_{ab}~{}.$
(20b)

Note that for arbitrary $V^{a}$, $\text{det }q_{ab}$ is fixed. Since the
action of the $gbms$ algebra deforms the celestial metric $q_{ab}$, the
associated phase space must include $q_{ab}$ and its conjugate mode $p^{ab}$.

The existence of superrotations requires relaxing the boundary conditions on
the metric components. For example, the shear tensor $(\sigma_{ab})$, upon the
action of superrotations picks up a linear in $u$ contribution:
$\sigma_{ab}(u,\hat{x})\to\sigma_{ab}(u,\hat{x})+uT_{ab}(\hat{x})$. The
coefficient of the linear in $u$ term is well known as the _Geroch tensor_
$(T_{ab})$. If we wish to separate the $T_{ab}$ mode from shear, then the
phase space is further expanded to include $T_{ab}$ and its conjugate
$\Pi^{ab}$. Note that both $p^{ab}$ and $\Pi^{ab}$ are some functionals of the
subleading $(\overset{1}{\mathcal{N}}_{ab})$ and the leading
$(\overset{o}{\mathcal{N}}_{ab})$ soft News tensors. The leading and
subleading soft News tensors are defined as follows:

$\displaystyle\overset{o}{\mathcal{N}}_{ab}=\int\mathrm{d}u\,N_{ab}\
,\qquad\overset{1}{\mathcal{N}}_{ab}=\int\mathrm{d}u\,uN_{ab}\ ,\qquad
N_{ab}(u,\hat{x})\xrightarrow{u\to\pm\infty}|u|^{-2-\delta}~{}.$ (21)

Since gBMS corresponds to the subleading soft graviton theorem, it requires
new subleading soft News modes in phase space, as shown in alok miguel 2014 ;
gBMS_Ward . One way to see how subleading soft News comes into the picture is
to see how the $\Gamma_{AS}$ changes by the introduction of the Geroch tensor.

$\displaystyle\int_{\mathcal{I}}\delta N_{ab}\wedge\delta\sigma^{ab}$
$\displaystyle\to\int_{\mathcal{I}}\left(\delta N_{ab}+\delta
T_{ab}\right)\wedge\left(\delta\overset{o}{\sigma}{}^{ab}+u\,\delta
T^{ab}\right)$ (22) $\displaystyle=\int_{\mathcal{I}}\delta
N_{ab}\wedge\delta\overset{o}{\sigma}{}^{ab}+\int_{S^{2}}\delta
T_{ab}\wedge\left(\int\mathrm{d}u\,\overset{o}{\sigma}{}^{ab}\right)+\int_{S^{2}}\delta\left(\int\mathrm{d}u\,uN_{ab}\right)\wedge\delta
T^{ab}$ $\displaystyle=\int_{\mathcal{I}}\delta
N_{ab}\wedge\delta\overset{o}{\sigma}{}^{ab}-\int_{S^{2}}\delta
T_{ab}\wedge\delta\overset{1}{\mathcal{N}}{}^{ab}+\int_{S^{2}}\delta\overset{1}{\mathcal{N}}_{ab}\wedge\,\delta
T^{ab}\ .$ (23)

Note that, for simplicity, we are not separating out $\sigma_{ab}$ into the
constant shear mode $(C_{ab})$, which would lead to
$\int\delta\overset{o}{\mathcal{N}}_{ab}\wedge\delta C^{ab}$ term.

With the relaxed fall-off conditions on the spacetime metric, and the metric
on the celestial sphere being dynamical, it is not straightforward to obtain a
symplectic structure from first principles. After the initial works by Laddha
and Campiglia alok miguel 2014 ; gBMS_Ward , Compere et al compere  improved
the understanding of gBMS phase space by writing the renormalized surface
charges (with controlled radial divergences). However, the charges in compere
are non-integrable $(\Omega(\delta,\delta_{V})$ is not an exact form.) The
integrable part of the charge can be separated out, and the associated Ward
identity turns out to be equivalent to the subleading soft graviton theorem.
Charges that differ by a boundary term in the soft sector are equivalent to
the correct subleading soft theorem ashokesen1 ; ashokesen2 , because the
quadratic boundary modes have trivial action on the scattering states. The
absence of a canonical hard charge is precisely the angular momentum ambiguity
in general relativity.

The charges (integrable part) in compere  do not close among themselves and
have a 2-cocycle extension. This sits as an obstruction if the superrotations
are to be symmetries of the quantum S-matrix. Campiglia and Peraza, in mp ,
obtained the charges exploiting the angular momentum ambiguity (adding
quadratic boundary modes), such that there was no 2-cocycle. Demanding the
charges close, they were able to write down the following symplectic form
compatible with these charges:333In mp , $C_{ab}(u,\hat{x})$ refers to the
shear tensor without separating the constant shear mode. Here, we refer to the
constant shear mode as $C_{ab}(\hat{x})$, and the rest of shear tensor as
$\overset{o}{\sigma}_{ab}$.

$\displaystyle\Omega=\int_{\mathcal{I}}$ $\displaystyle\delta
N_{ab}\wedge\delta\overset{o}{\sigma}{}^{ab}+\int_{S^{2}}\left(\delta\overset{o}{\mathcal{N}}_{ab}\wedge\,\delta
C^{ab}+\delta p^{ab}\wedge\delta q_{ab}+\delta\Pi^{ab}\wedge\delta
T_{ab}\right)~{},$ (24) $\displaystyle p^{ab}$
$\displaystyle=\nabla^{(a}\nabla_{c}\overset{1}{\mathcal{N}}{}^{b)c}-\frac{R}{2}\overset{1}{\mathcal{N}}{}^{ab}+\left(\text{Bilinear
in }C,\overset{o}{N}\right)~{},$ (25) $\displaystyle\Pi^{ab}$
$\displaystyle=2\overset{1}{\mathcal{N}}{}^{ab}+\left(\text{Bilinear in
}C,\overset{o}{N}\right)~{}.$ (26)

$R$ here refers to the Ricci scalar curvature corresponding to the celestial
metric $q_{ab}$. Note that even though we have the symplectic structure, since
there is no hard and soft factorization, we do not get a trivial reduced phase
space. This symplectic form shall be our starting point to obtain the reduced
phase space in Section 3.

###  2.3 Weyl BMS

Weyl BMS is a further generalization of the gBMS group, where a Weyl rescaling
of the metric on the celestial sphere is allowed, along with the smooth
diffeomorphisms. It was introduced in f , where, the authors separate the
scaling action due to Diff-$S^{2}$, and include it into the action of Weyl. A
general infinitesimal asymptotic symmetry transformation $\xi$ is parametrized
by a pair of functions $\mathcal{T}$, W and a vector field $V$ on $S^{2}$. The
associated Lie bracket can be written as :

$[\xi_{(\mathcal{T}_{1},V_{1},W_{1})},\xi_{(\mathcal{T}_{2},V_{2},W_{2})}]=\xi_{(\mathcal{T}_{12},V_{12},W_{12})}~{},$
(27)

where the parameters $\mathcal{T}_{12}$, $W_{12}$ and $V_{12}$ are given by :

$\displaystyle\mathcal{T}_{12}$
$\displaystyle=Y_{1}[\mathcal{T}_{2}]-Y_{2}[\mathcal{T}_{1}]+W_{2}\mathcal{T}_{1}-W_{1}\mathcal{T}_{2}~{},$
(28) $\displaystyle W_{12}$ $\displaystyle=Y_{1}[W_{2}]-Y_{2}[W_{1}]~{},$ (29)
$\displaystyle V_{12}$ $\displaystyle=[V_{1},V_{2}]~{}.$ (30)

The right hand side of the last equation being the Lie bracket of vector
fields.

The infinitesimal transformations $\xi_{(\mathcal{T},V=0,W=0)}$ form an
abelian ideal and are the usual super-translations. Similarly, the Weyl-part
of the Weyl BMS are the transformations $\xi_{(\mathcal{T}=0,V=0,W)}$ and gBMS
super-rotations are given by
$\xi_{(\mathcal{T}=0,V,W=\frac{1}{2}\nabla_{a}V^{a})}$. The form of the latter
is dictated by the property that these transformations preserve the
determinant of the celestial sphere metric, as can be seen from the variations
written below :

$\displaystyle\delta_{V}C_{ab}(\hat{x})=\mathcal{L}_{V}C_{ab}(\hat{x})\ \
,\qquad\delta_{w}C_{ab}(\hat{x})=-wC_{ab}(\hat{x})$ (31)
$\displaystyle\delta_{V}q_{ab}(\hat{x})=\mathcal{L}_{V}q_{ab}(\hat{x})\ \
,\qquad\delta_{w}q_{ab}(\hat{x})=-2wq_{ab}(\hat{x})$ (32)

The following are the finite Weyl transformation of the radiative modes:

$\displaystyle N_{ab}(u,\hat{x})$ $\displaystyle\to
N_{ab}(e^{w(\hat{x})}u,\hat{x})~{},$ (33) $\displaystyle\sqrt{q}(\hat{x})$
$\displaystyle\to e^{-2w(\hat{x})}\sqrt{q}(\hat{x})~{},$ (34) $\displaystyle
C(\hat{x})$ $\displaystyle\to e^{-w(\hat{x})}C(\hat{x})~{},$ (35)
$\displaystyle\overset{o}{N}(\hat{x})$ $\displaystyle\to
e^{-w(\hat{x})}\overset{o}{N}(\hat{x})~{}.$ (36)

In f , the authors have performed detailed covariant phase space analysis, and
obtained the symplectic structure and charges corresponding to the sphere
diffeomorphisms and the Weyl rescalings.

##  3 Generalized BMS Reduced Phase Space

Our interest lies in understanding the reduced phase space structure for the
gBMS, the Diff($S^{2}$) algebra. We wish to find the physical brackets between
the fundamental fields so that they can be promoted to quantum commutation
relations. We start from the Campiglia-Peraza symplectic structure mp  (See
(24)):

$\displaystyle\Omega=\int_{\mathcal{I}}$ $\displaystyle\delta
N_{ab}\wedge\delta\overset{o}{\sigma}{}^{ab}+\int_{S^{2}}\left(\delta\overset{o}{\mathcal{N}}_{ab}\wedge\,\delta
C^{ab}+\delta p^{ab}\wedge\delta q_{ab}+\delta\Pi^{ab}\wedge\delta
T_{ab}\right)~{},$ (37) $\displaystyle p^{ab}$
$\displaystyle=\nabla^{(a}\nabla_{c}\overset{1}{\mathcal{N}}{}^{b)c}-\frac{R}{2}\overset{1}{\mathcal{N}}{}^{ab}+\left(\text{Bilinear
in }C,\overset{o}{N}\right)~{},$ (38) $\displaystyle\Pi^{ab}$
$\displaystyle=2\overset{1}{\mathcal{N}}{}^{ab}+\left(\text{Bilinear in
}C,\overset{o}{N}\right)~{}.$ (39)

Note that this particular symplectic form was not derived from covariant phase
space methods Wald ; crnk witten ; speranza . Even though we have this
symplectic form, and the action of superrotations on all the fundamental
fields, we lack the reduced phase space analysis. The ‘kinematical’ phase
space is parameterized by the following conjugate pairs:

$\\{\overset{o}{\sigma}_{ab}\\}\cup\\{\\{\Pi^{ab},T_{ab}\\},\\{p^{ab},q_{ab}\\},\\{C,\overset{o}{N}\\}\\}~{}.$

There is no hard and soft factorization, and modes $p^{ab}$ and $\Pi^{ab}$ are
non-linearly related to the radiative soft modes. Also, $T_{ab}$ and $q_{ab}$
are not independent, but are related by the following constraint:

$\displaystyle\nabla^{b}T_{ab}+\frac{1}{2}\nabla_{a}R=0~{}.$ (40)

Thus, to obtain the reduced phase space for gBMS, we need to find the physical
brackets between the radiative modes and the Goldstone modes $(C,T_{ab})$,
starting from (24), with (25), (26) and (40) as constraints, in addition to
the HLMS constraints. However, such a constraint analysis turns out to be
difficult. We enumerate some of the obstacles in Section 3.2 and in Appendix
B. But first, we perform this Dirac analysis for the simplified case of
linearized gravity.

###  3.1 Linearized Gravity

The goal of this section is to study the symplectic structure corresponding to
the $gbms$ algebra in linearized gravity. Even though we have dynamical
$q_{ab}$, it differs only infinitesimally from the plane metric $\eta_{ab}$.
The radiative data and the soft News modes are also to be treated
perturbatively. We wish to find the physical brackets between the modes of the
HLMS phase space supplemented by the mode $q_{ab}$ corresponding to the
celestial metric. The angular part of the metric at $\mathcal{I}^{+}$ takes
the following form:

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=\ldots+\mathrm{d}x^{a}\mathrm{d}x^{b}\left[r^{2}\left(\eta_{ab}+h_{ab}(\hat{x})\right)+r\left(\sigma_{ab}(u,\hat{x})+C_{ab}(\hat{x})\right)+\ldots\right]$
(41)

$N_{ab}=\partial_{u}\sigma_{ab}$, the News tensor, is treated infinitesimally
and has the following falloff:

$\displaystyle N_{ab}(u,\hat{x})\xrightarrow{u\to\pm\infty}(\text{Constant
Mode})+|u|^{-2-\delta}~{}.$ (42)

The constant mode is the Geroch tensor $T_{ab}$. Note that $\sigma_{ab}$ in
this section differs from $\overset{o}{\sigma}_{ab}$ from earlier sections by
the $uT_{ab}$ contribution.

The leading and sub-leading soft News tensors can be defined as follows:

$\displaystyle\overset{o}{\mathcal{N}}_{ab}(\hat{x})$
$\displaystyle=\int\mathrm{d}u\,\left(N_{ab}(u,\hat{x})-\lim_{u^{\prime}\to\infty}N_{ab}(u^{\prime},\hat{x})\right)~{},$
(43) $\displaystyle\overset{1}{\mathcal{N}}_{ab}(\hat{x})$
$\displaystyle=\int\mathrm{d}u\,uN_{ab}(u,\hat{x})~{}.$ (44)

In the second equation, we are relying on the prescription
$\int\mathrm{d}u\,u=0$. Note that the $\sigma_{ab}$, $C,p^{ab},\Pi^{ab}$ all
are infinitesimal. The celestial metric is written as a perturbation $h_{ab}$
around the plane metric $\eta_{ab}$:

$\displaystyle q_{ab}=\eta_{ab}+h_{ab}\ ,\qquad\qquad
q^{ab}=\eta^{ab}-h^{ab}~{}.$ (45)

The condition that the $\text{det }q_{ab}=\text{det }\eta_{ab}=1$ translates
to the following condition for $h_{ab}$:

$\displaystyle 1=\text{det}(\eta+h)$
$\displaystyle=\exp\text{tr}\log(\eta+h)=1+\eta^{ab}h_{ab}+\ldots$ (46)
$\displaystyle\Rightarrow\qquad\eta^{ab}h_{ab}$ $\displaystyle=0~{}.$ (47)

Note that if $q^{ab}$ and $\Pi_{cd}$ are conjugates, then their traces are
conjugate modes to each other as well. Hence the conjugate mode $p_{ab}$ to
the dynamical metric is tracefree.

####  3.1.1 Setup: symplectic form and constraints

Since the indices are raised/lowered using the non-dynamical metric
$\eta_{ab}$ and all the quadratic terms in radiative data and soft modes are
omitted, the symplectic form (37) simplifies greatly. The constant shear mode
$C_{ab}$ and leading soft News tensor simplifies as follows:

$\displaystyle C_{ab}$
$\displaystyle=-2\partial_{a}\partial_{b}C^{\text{TF}}~{},\qquad\qquad\overset{o}{\mathcal{N}}_{ab}=-2\partial_{a}\partial_{b}\overset{o}{N}{}^{\text{TF}}~{}.$
(48)

Let us introduce
$\mathfrak{C}=4(\partial_{a}\partial_{b})^{\text{TF}}\partial_{a}\partial_{b}C$
for the linearized case. Since the differential operator relating
$\mathfrak{C}$ and $C$ is invertible hlms , we do not lose any information
treating $\mathfrak{C}$ as the fundamental field.444For the non-linear case,
the relation between $\mathfrak{C}$ and $C$ is rather complicated and contains
Geroch tensor. With $\mathfrak{C}$, $\overset{o}{N}$, $q_{ab}$ and $p^{ab}$
parameterizing the soft sector, the symplectic form (37) can be expressed as
follows:

$\Omega=\int_{\mathcal{I}}\partial_{u}\delta\sigma_{ab}\wedge\delta\sigma^{ab}+\int_{S^{2}}\left(\delta\overset{o}{N}\wedge\,\delta\mathfrak{C}\,+\delta
p^{ab}\wedge\delta q_{ab}\right)~{}.$ (49)

Note that for the linearized case, the conjugate mode to $T_{ab}$ is
$\Pi^{ab}=2\overset{1}{\mathcal{N}}{}^{ab}$, and we have absorbed the term
$\delta\Pi^{ab}\wedge\delta T_{ab}$ into the Ashtekar-Streubel term, as done
in (23).

Let us enumerate the constraints that define the physical phase space. The
relation between the conjugate mode to $q_{ab}$ and the subleading soft News
tensor is a constraint, which we denote $\mathcal{F}_{1}$. Another constraint,
relating the Geroch tensor to the dynamical celestial metric now manifests
itself as $\mathcal{F}_{2}$. The rest of the two constraints are the same as
those from the HLMS construction. We collect the constraints below:

$\displaystyle\mathcal{F}_{1}^{ab}$
$\displaystyle=p^{ab}-\left(\partial^{a}\partial_{c}\overset{1}{\mathcal{N}}{}^{bc}\right){}^{\text{STF}}~{},$
(50) $\displaystyle\mathcal{F}_{2a}$
$\displaystyle=\lim_{u\rightarrow\infty}\partial^{b}N_{ab}+\frac{1}{2}\partial_{a}\partial_{b}\partial_{c}h^{bc}~{},$
(51) $\displaystyle\mathcal{F}_{3ab}$
$\displaystyle=\sigma^{+}_{ab}+\sigma^{-}_{ab}~{},$ (52)
$\displaystyle\mathcal{F}_{4ab}$
$\displaystyle=\overset{o}{\mathcal{N}}_{ab}+2\partial_{a}\partial_{b}\overset{o}{N}~{}.$
(53)

####  3.1.2 Kinematical structure

The following kinematical brackets can be derived 555We have obtained these
brackets by calculating the appropriate Hamiltonian Vector Fields (HVFs). See
Appendix B for details. Strictly speaking, due to fall off properties of the
first term of (49) the HVF’s derived from it are well defined only up to the
regularization $\int_{-\infty}^{\infty}du\ u=0$ and
$\int_{-\infty}^{\infty}du\ \partial_{u}\left(u\sigma_{ab}\right)=0$, the
latter motivated by the falloffs on
$N_{ab}(u)-\lim_{u^{\prime}\rightarrow\infty}N_{ab}(u^{\prime})$ and the
constraint $\sigma^{+}_{ab}+\sigma^{-}_{ab}=0$ from (49) :

$\displaystyle\big{[}\sigma_{ab}(u,\hat{x})\,,\,N^{cd}(u^{\prime},\hat{y})\big{]}$
$\displaystyle=\frac{1}{2}\mathbb{I}^{cd}_{ab}\,\delta(u-u^{\prime})\delta^{2}(\hat{x}-\hat{y})~{},$
(54)
$\displaystyle\big{[}\sigma_{ab}(u,\hat{x})\,,\,\overset{o}{\mathcal{N}}{}^{cd}(\hat{y})\big{]}$
$\displaystyle=\mathbb{I}^{cd}_{ab}\,\delta^{2}(\hat{x}-\hat{y})~{},$ (55)
$\displaystyle\big{[}\sigma_{ab}(u,\hat{x})\,,\,\overset{1}{\mathcal{N}}{}^{cd}(\hat{y})\big{]}$
$\displaystyle=\frac{1}{2}u\,\mathbb{I}^{cd}_{ab}\
\delta^{2}(\hat{x}-\hat{y})~{},$ (56)
$\displaystyle\big{[}q_{cd}(\hat{x})\,,\,p^{ab}(\hat{y})\big{]}$
$\displaystyle=\mathbb{I}^{ab}_{cd}\ \delta^{2}(\hat{x}-\hat{y})~{},$ (57)
$\displaystyle\big{[}\mathfrak{C}(\hat{x})\,,\,\overset{o}{N}(\hat{y})\big{]}$
$\displaystyle=\delta^{2}(\hat{x}-\hat{y})~{},$ (58)

where $\mathbb{I}_{cd}^{ab}$, introduced in (1), ensures that the trace modes
are non dynamical. The rest of the brackets do not survive at the linearized
order. For example, all the brackets of $p^{ab}$, except for $[q,p]$ have a
radiative field on the RHS, and hence are irrelevant in the linearized
setting. For the detailed kinematical structure of the full non-linear gBMS,
please refer to Appendix B.

Thus, using these kinematical brackets, one can find the non-zero brackets
between the constraints:

$\displaystyle\big{[}\mathcal{F}_{1}^{ab}(\hat{x})\,,\,\mathcal{F}_{2m}(\hat{y})\big{]}$
$\displaystyle=\left(\frac{1}{2}\partial_{m}\partial^{a}\partial^{b}-\frac{1}{4}\partial^{2}\partial^{(a}\delta^{b)}_{m}-\frac{1}{8}\eta^{ab}\partial^{2}\partial_{m}\right)\delta^{2}(x-y)~{},$
(59)
$\displaystyle\big{[}\mathcal{F}_{3}^{ab}(\hat{x})\,,\,\mathcal{F}_{4cd}(\hat{y})\big{]}$
$\displaystyle=2\left(\delta^{(a}_{c}\delta^{b)}_{d}-\frac{1}{2}\eta^{ab}\eta_{cd}\right)\delta^{2}(\hat{x}-\hat{y})~{}.$
(60)

Here we wish to clarify two things: (i) The Hamiltonian vector fields for the
mode $\sigma_{ab}$ are not well defined and we require a prescription for the
$[\mathcal{F}_{3ab},\mathcal{F}_{3cd}]$ bracket. Following the reference am ,
we set this bracket to $0$. (ii) To evaluate
$[\mathcal{F}_{2a},\mathcal{F}_{3bc}]$ we need

$\displaystyle\left[\lim_{u^{\prime}\to\infty}N_{ab}(u^{\prime},\hat{x})\,,\,\sigma_{cd}^{+}(\hat{y})+\sigma^{-}_{cd}(\hat{y})\right]$
$\displaystyle\hskip
22.76228pt=\lim_{u^{\prime}\to\infty}\lim_{u\to\infty}\big{[}N_{ab}(u^{\prime},\hat{x})\,,\,\sigma_{cd}(u,\hat{y})+\sigma_{cd}(-u,\hat{y})\big{]}\sim\lim_{u^{\prime}\to\infty}\lim_{u\to\infty}\delta(u-u^{\prime})=0~{}.$
(61)

Another justification for the same is as follows:

$\displaystyle\lim_{u^{\prime}\to\infty}\lim_{u\to\infty}u\left[N_{ab}(u^{\prime})\,,\,\sigma^{cd}(u)+\sigma^{cd}(-u)\right]$
$\displaystyle=\lim_{u^{\prime}\rightarrow\infty}\bigg{[}N_{ab}(u^{\prime})\,,\int\mathrm{d}u\,\partial_{u}(u\sigma^{cd}(u))\bigg{]}$
$\displaystyle=\lim_{u^{\prime}\rightarrow\infty}\bigg{[}N_{ab}(u^{\prime})\,,\,\overset{1}{\mathcal{N}}{}^{cd}+\int\mathrm{d}u\
\sigma^{cd}(u)\bigg{]}$ $\displaystyle=\lim_{u^{\prime}\rightarrow\infty}\
\bigg{(}\frac{1}{2}\delta^{cd}_{ab}\delta^{2}(\hat{x}-\hat{y})$
$\displaystyle-\frac{1}{2}\frac{\delta}{\delta\sigma_{ab}(u^{\prime},\hat{x})}\left(\int\mathrm{d}u\
\sigma^{cd}(u,\hat{y})\right)\bigg{)}=0~{},$ (62)

where we explicitly used the Hamiltonian vector field corresponding to
$N_{ab}(u^{\prime},\hat{x})$.

####  3.1.3 Inverting the Dirac matrix

Getting on with the constraint analysis, we can see that the matrix of
brackets of constraints, i.e., the Dirac matrix, is block-diagonal. This
simplifies the inversion of the Dirac matrix, which we denote by the letter
$M$. The ‘inverse’ of the Dirac matrix will be denoted by $W$ and it is
required to satisfy the following equation(s) :

$\int\mathrm{d}^{2}\hat{y}\,M_{IJ}(\hat{x},\hat{y})W^{JK}(\hat{y},\hat{z})=\delta^{2}(\hat{x}-\hat{z})\delta^{K}_{I}~{},$
(63)

$\displaystyle\int\mathrm{d}^{2}\hat{z}\left(\begin{matrix}0&M_{12}{}^{ab}_{m}&0&0\\\
M_{21}{}^{ab}_{m}&0&0&0\\\ 0&0&0&M_{34}{}_{ab}^{cd}\\\
0&0&M_{43}{}_{ab}^{cd}&0\end{matrix}\right)(\hat{x},\hat{z})$
$\displaystyle\left(\begin{matrix}0&W_{12}{}_{ab}^{n}&0&0\\\
W_{21}{}_{cd}^{m}&0&0&0\\\ 0&0&0&W_{34}{}_{cd}^{kl}\\\
0&0&W_{43}{}_{cd}^{kl}&0\end{matrix}\right)(\hat{z},\hat{y})$
$\displaystyle\qquad\quad=\delta^{2}(\hat{x}-\hat{y})\left(\begin{matrix}\mathbb{I}^{ab}_{cd}&0&0&0\\\
0&\delta_{m}^{n}&0&0\\\ 0&0&\mathbb{I}_{ab}^{kl}&0\\\
0&0&0&\mathbb{I}_{ab}^{kl}\end{matrix}\right)~{}.$ (64)

Some explanation of the notation used above is in order: the capital letters
$I,J,K,$ range from 1-4. The sphere indices that each constraint carries are
understood. For instance:

$M_{IJ}\rvert_{I=1,J=2}=M_{12m}^{ab}=[\mathcal{F}_{1}^{ab},\mathcal{F}_{2a}]~{}.$
(65)

Let us write the equations that define the inverse for $M_{12}{}^{ab}_{m}$:

$\displaystyle\int\mathrm{d}^{2}z\,\big{[}\mathcal{F}_{1}{}^{ab}(\hat{x})\,,\,\mathcal{F}_{2m}(\hat{z})\big{]}\,W_{21}{}^{m}_{cd}(\hat{z},\hat{y})$
$\displaystyle=\delta^{ab}_{cd}\,\delta^{2}(\hat{x}-\hat{y})~{},$ (66)
$\displaystyle\int\mathrm{d}^{2}z\,\big{[}\mathcal{F}_{2m}(\hat{x})\,,\,\mathcal{F}_{1}{}^{ab}(\hat{z})\big{]}\,W_{12}{}_{ab}^{n}(\hat{z},\hat{y})$
$\displaystyle=\delta^{n}_{m}\,\delta^{2}(\hat{x}-\hat{y})~{}.$ (67)

We require the Green’s function for the following differential operator:

$\left(\frac{1}{2}\partial_{m}\partial^{a}\partial^{b}-\frac{1}{4}\partial^{2}\partial^{(a}\delta^{b)}_{m}-\frac{1}{8}\eta^{ab}\partial^{2}\partial_{m}\right)_{(x)}W_{21}{}^{m}_{cd}(\hat{x},\hat{y})=\mathcal{P}\
\mathbb{I}^{ab}_{cd}\ \delta^{2}(\hat{x}-\hat{y})$ (68a)
$\left(\frac{1}{2}\partial_{m}\partial^{a}\partial^{b}-\frac{1}{4}\partial^{2}\partial^{(a}\delta^{b)}_{m}-\frac{1}{8}\eta^{ab}\partial^{2}\partial_{m}\right)_{(x)}W_{21}{}^{n}_{ab}(\hat{y},\hat{x})=-\mathcal{P}\
\delta^{n}_{m}\delta^{2}(\hat{x}-\hat{y})$ (68b)

As these differential operators may have non-trivial kernels, we have included
formal operators $\mathcal{P}$ that project functions onto the subspace
wherein they are invertible. In terms of these abstract inverses, we can write
down all the non-zero entries of the inverse Dirac matrix:

$\displaystyle W_{21ab}^{m}(\hat{x},\hat{y})$
$\displaystyle=-W_{12ab}^{m}(\hat{y},\hat{x})~{},$ (69) $\displaystyle
W_{34ab}^{cd}(\hat{x},\hat{y})$
$\displaystyle=-W_{43ab}^{cd}(\hat{x},\hat{y})=\frac{1}{2}\delta^{cd}_{ab}\delta^{2}(\hat{x}-\hat{y})~{}.$
(70)

Before moving on, let us check that the inverse of the Dirac matrix is
tracefree $\eta^{ab}W_{21}{}_{ab}^{m}=0$.

$\left[\int\mathrm{d}^{2}x\,W_{21}{}^{p}_{ab}(\hat{w},\hat{x})\right]\int\mathrm{d}^{2}z\,M_{12}{}^{ab}_{m}(\hat{x},\hat{z})W_{21}{}_{cd}^{m}(\hat{z},\hat{y})=\bigg{[}\int\mathrm{d}^{2}x\,W_{21}{}^{p}_{ab}(\hat{w},\hat{x})\bigg{]}\mathcal{P}\,\mathbb{I}^{ab}_{cd}\,\delta^{2}(\hat{x}-\hat{y})~{},$
$\displaystyle\Rightarrow\int\mathrm{d}^{2}z\,W_{21}{}_{cd}^{m}(\hat{z},\hat{y})\left[\int\mathrm{d}^{2}x\,W_{21}{}^{p}_{ab}(\hat{w},\hat{x})M_{12}{}^{ab}_{m}(\hat{x},\hat{z})\right]$
$\displaystyle=W_{21}{}^{p}_{ab}(\hat{w},\hat{y})\mathcal{P}\mathbb{I}^{ab}_{cd}$
(71) $\displaystyle\Rightarrow\quad
W_{21}{}_{cd}^{p}(\hat{w},\hat{y})\mathcal{P}$
$\displaystyle=W_{21}{}^{p}_{ab}(\hat{w},\hat{y})\mathcal{P}\mathbb{I}^{ab}_{cd}$
(72) $\displaystyle\Rightarrow\qquad\eta^{cd}W_{21}{}_{cd}^{p}$
$\displaystyle=0~{}.$ (73)

####  3.1.4 Dirac brackets

We are now ready to find the physical brackets between the fundamental modes.
We identify the dynamical modes of interest that have kinematical brackets
with particular constraints. This is illustrated through the following cartoon
:

$\mathcal{F}_{1}$ $\mathcal{F}_{2}$
$q_{ab}$$N_{ab}$$\overset{1}{\mathcal{N}}_{ab}$$\overset{o}{\mathcal{N}}_{ab}$
$\mathcal{F}_{3}$ $\mathcal{F}_{4}$ $\mathfrak{C}$$\sigma_{ab}$(a)(b)

An edge between the nodes denoting the constraints indicates that the
corresponding entry from the inverse Dirac matrix is non-zero. The kinematical
bracket between quantities from the left and right ends of the graph gets
corrections from the Dirac analysis. As can be noted from the diagrams,
certain brackets are identical to those of the kinematical ones. The only non-
vanishing bracket of this kind is:

$\big{[}N_{ab}(u,\hat{x})\,,\,N^{cd}(u^{\prime},\hat{y})\big{]}_{*}=\frac{1}{2}\,\mathbb{I}^{cd}_{ab}\,\partial_{u}\delta(u-u^{\prime})\delta^{2}(\hat{x}-\hat{y}).$
(74)

Upon imposing $\mathcal{F}_{3}$ and $\mathcal{F}_{4}$, we find that all the
brackets of $\overset{o}{\mathcal{N}}_{ab}$ and $\overset{o}{N}_{ab}$ turn out
to be the same. So, from now on, we don’t differentiate between the two. The
only non-vanishing bracket involving $\overset{o}{\mathcal{N}}_{ab}$ is:

$\big{[}\overset{o}{\mathcal{N}}_{ab}(\hat{x})\,,\,\mathfrak{C}(\hat{y})\big{]}_{*}=2\partial_{a}\partial_{b}^{\text{TF}}\delta^{2}(\hat{x}-\hat{y}).$
(75)

This matches with the corresponding brackets from HLMS hlms . We next consider
:

$\displaystyle\big{[}\,h_{ab}(\hat{x})\,,\,\overset{1}{\mathcal{N}}{}^{cd}(\hat{y})\big{]}_{*}$
$\displaystyle=-\int\mathrm{d}^{2}\hat{z}\ \mathrm{d}^{2}\hat{z}^{\prime}\
\big{[}h_{ab}(\hat{x})\,,\,\mathcal{F}^{ij}_{1}(\hat{z})\big{]}\,W^{m}_{12ij}(\hat{z},\hat{z}^{\prime})\,\big{[}\mathcal{F}_{2m}(\hat{z}^{\prime})\,,\,\overset{1}{\mathcal{N}}{}^{cd}(\hat{y})\big{]}$
$\displaystyle=\frac{1}{2}\mathbb{I}^{ij}_{ab}\,\mathbb{I}^{cd}_{mn}\,\frac{\partial}{\partial
y_{n}}W^{m}_{12ij}(\hat{x},\hat{y})=\frac{1}{2}\mathbb{I}^{cd}_{mn}\,\frac{\partial}{\partial
y_{n}}W^{m}_{12ab}(\hat{x},\hat{y}).$ (76)

Note that in the last equality, we have used $\eta^{ij}W_{12}{}_{ij}^{m}=0$.

We note from diagram (a):

$\displaystyle\big{[}\,q_{ab}(\hat{x})\,,\,N^{cd}(u,\hat{y})\big{]}_{*}=0\qquad\qquad\forall\text{
finite }u.$ (77)

However, the two equations, (77) and (76) are not in contradiction with each
other, because, in general, $[a,\int b]\neq\int[a,b]$. The other non-vanishing
bracket is :

$\displaystyle\big{[}$ $\displaystyle
N_{ab}(u,\hat{x})\,,\,\overset{1}{\mathcal{N}}{}^{cd}(\hat{y})\big{]}_{*}$
$\displaystyle=\frac{1}{2}\mathbb{I}^{cd}_{ab}\,\delta^{2}(\hat{x}-\hat{y})-\int\mathrm{d}^{2}\hat{z}\,\mathrm{d}^{2}\hat{z}^{\prime}\,\big{[}N_{ab}(u,\hat{x})\,,\,\mathcal{F}^{ij}_{1}(\hat{z})]\,W_{12}{}_{ij}^{m}(\hat{z},\hat{z}^{\prime})\,\big{[}\mathcal{F}_{2m}(\hat{z}^{\prime})\,,\,\overset{1}{\mathcal{N}}{}^{cd}(\hat{y})\big{]}$
(78)
$\displaystyle=\frac{1}{2}\mathbb{I}^{cd}_{ab}\,\delta^{2}(\hat{x}-\hat{y})-\frac{1}{4}\mathbb{I}^{ki}_{ab}\,\frac{\partial}{\partial
y_{e}}\frac{\partial^{2}}{\partial x^{k}\partial
x_{j}}W_{12}{}_{ij}^{m}(\hat{x},\hat{y})\,\mathbb{I}^{cd}_{em}~{}.$ (79)

Note that the soft News sector does not decouple completely from the hard
News, as was the case in am .

Eliminating the inverse matrix element from (76) and (79), we obtain

$\displaystyle\bigg{[}\left(2N_{ab}(u,\hat{x})+\partial^{m}\partial_{(a}h_{b)m}-\frac{1}{2}\eta_{ab}\partial^{m}\partial^{n}h_{mn}\right)\,,\
\overset{1}{\mathcal{N}}{}^{cd}(\hat{y})\bigg{]}_{*}=\mathbb{I}^{cd}_{ab}\,\delta^{2}(\hat{x}-\hat{y})~{}.$
(80)

Thus at the level of linearized gravity, we can find the conjugate mode to
$\overset{1}{\mathcal{N}}{}^{ab}$. The striking thing about this result is
that the explicit form of Green’s function does not appear in this equation.
We only require its existence. Note that since (80) holds for all values of
$u$, only the constant mode in News, which is exactly the Geroch tensor, is
contributing to the Dirac bracket. Had $q_{ab}$ not been dynamical, we would
have gotten $2T_{ab}$ to be the conjugate to the subleading soft News tensor,
as expected.

$\displaystyle\big{[}N_{ab}(u,\hat{x})\,,\,N^{cd}(u^{\prime},\hat{y})\big{]}_{*}$
$\displaystyle=\frac{1}{2}\,\mathbb{I}^{cd}_{ab}\,\partial_{u}\delta(u-u^{\prime})\delta^{2}(\hat{x}-\hat{y})$
(81a)
$\displaystyle\bigg{[}\big{(}2N_{ab}(u,\hat{x})+\mathbb{I}_{ab}^{mn}\,\partial^{p}\partial_{m}q_{np}\big{)}\,,\
\overset{1}{\mathcal{N}}{}^{cd}(\hat{y})\bigg{]}_{*}$
$\displaystyle=\mathbb{I}^{cd}_{ab}\,\delta^{2}(\hat{x}-\hat{y})$ (81b)
$\displaystyle\big{[}\mathfrak{C}(\hat{x})\,,\,\overset{o}{N}(\hat{y})\big{]}_{*}$
$\displaystyle=\delta^{2}(\hat{x}-\hat{y})$ (81c)

##### Jacobi identities

At the linearized level, all the Dirac brackets are differential operators (or
constants) acting on the Dirac delta function, hence the Dirac brackets
satisfy Jacobi identities.

####  3.1.5 Celestial plane vs celestial sphere

In the analysis so far, we have expanded the metric $q_{ab}$ around the flat
$\eta_{ab}$ perturbatively. However, we could have done the same analysis with
$q_{ab}$ being perturbed around round sphere metric $\overset{o}{q}_{ab}$, for
which the Ricci scalar $\overset{o}{R}\,=2$. The significant difference lies
in the kinematical mode conjugate to $h_{ab}$ (38). The updated constraints
would be as follows:

$\displaystyle\mathcal{F}_{1}^{ab}$
$\displaystyle=p^{ab}+\mathbb{I}^{ab}_{mn}\left(\overset{1}{\mathcal{N}}{}^{mn}-\overset{o}{\nabla}{}^{m}\overset{o}{\nabla}_{c}\overset{1}{\mathcal{N}}{}^{nc}\right)=p^{ab}+\left(\overset{1}{\mathcal{N}}{}^{ab}-\overset{o}{\nabla}{}^{a}\overset{o}{\nabla}_{c}\overset{1}{\mathcal{N}}{}^{bc}\right)^{\text{STF}}~{},$
(82) $\displaystyle\mathcal{F}_{2a}$
$\displaystyle=\lim_{u\to\infty}\overset{o}{\nabla}{}^{b}N_{ab}+\frac{1}{2}\overset{o}{\nabla}_{a}\overset{o}{\nabla}{}^{b}\overset{o}{\nabla}{}^{c}h_{bc}~{},$
(83) $\displaystyle\mathcal{F}_{3ab}$
$\displaystyle=\sigma^{+}_{ab}+\sigma^{-}_{ab}~{},$ (84)
$\displaystyle\mathcal{F}_{4ab}$
$\displaystyle=\,\overset{o}{\mathcal{N}}_{ab}+2\overset{o}{\nabla}_{a}\overset{o}{\nabla}_{b}\overset{o}{N}~{}.$
(85)

Note that the covariant derivative $\overset{o}{\nabla}_{a}$ is compatible
with the round sphere metric $\overset{o}{q}_{ab}$. The non-trivial element of
the Dirac matrix is:

$\displaystyle\big{[}\mathcal{F}_{1}^{ab}$
$\displaystyle(\hat{x})\,,\,\mathcal{F}_{2m}(\hat{y})\big{]}$
$\displaystyle=\bigg{[}\,\frac{1}{2}\overset{o}{\nabla}_{m}\overset{o}{\nabla}{}^{a}\overset{o}{\nabla}{}^{b}-\frac{1}{4}\delta^{(a}_{m}\overset{o}{\nabla}{}^{b)}\overset{o}{\nabla}{}^{2}-\frac{1}{8}\overset{o}{q}{}^{ab}\overset{o}{\nabla}_{m}\,\overset{o}{\nabla}{}^{2}+\frac{1}{2}\big{(}\overset{o}{\nabla}{}^{a}\delta^{b}_{m}\big{)}^{\text{STF}}\bigg{]}\delta^{2}(\hat{x}-\hat{y})~{}.$
(86)

Proceeding as before, the final Dirac brackets are:

$\displaystyle\big{[}N_{ab}(u,\hat{x})\,,\,N^{cd}(u^{\prime},\hat{y})\big{]}_{*}$
$\displaystyle=\frac{1}{2}\,\mathbb{I}^{cd}_{ab}\,\partial_{u}\delta(u-u^{\prime})\delta^{2}(\hat{x}-\hat{y})$
(87a)
$\displaystyle\bigg{[}\bigg{(}2N_{ab}(u,\hat{x})-h_{ab}(\hat{x})+\mathbb{I}_{ab}^{ik}\,\overset{o}{\nabla}_{k}\overset{o}{\nabla}{}^{j}q_{ij}\bigg{)}\,,\
$
$\displaystyle\overset{1}{\mathcal{N}}{}^{cd}(\hat{y})\bigg{]}_{*}=\mathbb{I}^{cd}_{ab}\,\delta^{2}(\hat{x}-\hat{y})$
(87b)
$\displaystyle\big{[}\mathfrak{C}(\hat{x})\,,\,\overset{o}{N}(\hat{y})\big{]}_{*}$
$\displaystyle=\delta^{2}(\hat{x}-\hat{y})$ (87c)

We end this section with a few remarks.

  * •

We have separated $\overset{o}{N},\,C$ from the hard modes and the reduced
phase space admits Poisson brackets between these soft modes along with the
usual AS bracket. Hence $\Gamma_{\text{HLMS}}$ is a subspace of the phase
space defined in this section.

  * •

An important distinction of the present analysis from the earlier attempts mp
; am  to derive the symplectic structure on gBMS (eBMS) phase space is the
following: In am  the authors separated the fields in hard and soft sector. In
the gBMS case, the soft sector is parametrized by $q_{ab},T_{ab}$ which are
related by a constraint, namely the defining equation for the Geroch tensor.
As we have shown, at least in the linearized gravity, separating hard shear
tensor in $\overset{o}{\sigma}_{ab}$ and $T_{ab}$ is not necessary.
Parameterizing the sub-leading soft sector by sphere metric and its conjugate
is sufficient to obtain the reduced phase space in which all the functionally
independent conjugate partners can be identified. It is this phase space which
should be quantized and may lead us to a more refined understanding of soft
vacuua in quantum theory. However, we leave the quantization of this phase
space for future work.

###  3.2 Obstacles in gBMS Phase Space Analysis

In this section, we outline some difficulties we faced while undertaking the
constraint analysis for $gbms$ algebra. The following is the symplectic form
proposed in mp , out of which charges that faithfully represent the $gbms$
algebra can be computed:

$\begin{gathered}\Omega=\int_{\mathcal{I}}\sqrt{q}\ \delta
N_{ab}\wedge\delta\overset{o}{\sigma}{}^{ab}+\int_{S^{2}}\sqrt{q}\
\left(\delta\overset{o}{N}\wedge\
\delta\mathfrak{C}+\delta\Pi^{ab}\wedge\delta T_{ab}+\delta p^{ab}\wedge\delta
q_{ab}\right)\ ,\end{gathered}$ (88)

where
$\mathfrak{C}=\left(-2\nabla_{a}\nabla_{b}+q_{ab}\Delta+T_{ab}\right)\left(-2\nabla^{a}\nabla^{b}+T^{ab}\right)C$.
Furthermore, $p^{ab}$ and $\Pi^{ab}$, the modes conjugate to $q_{ab}$ and
$T_{ab}$ respectively, are functions of subleading and leading soft News
modes. We include their definitions in the set of constraints:

$\displaystyle\mathcal{F}^{ab}_{1}$
$\displaystyle=p^{ab}-\left[\nabla^{a}\nabla_{c}\overset{1}{\mathcal{N}}{}^{bc}-\
\frac{R}{2}\overset{1}{\mathcal{N}}{}^{ab}+\left(\text{bilinear in}\ C,\
\overset{o}{N}\right)\right]~{},$ (89) $\displaystyle\mathcal{F}^{ab}_{2}$
$\displaystyle=\Pi^{ab}-\left[2\overset{1}{\mathcal{N}}{}^{ab}+\left(\text{bilinear
in}\ C,\ \overset{o}{N}\right)\right]\ .$ (90)

$R$ is the Ricci scalar for the two dimensional celestial sphere. Since we
have included the definitions of $p^{ab}$ and $\Pi^{ab}$ as constraints, we
can treat them as independent from other soft or hard modes, at the
kinematical level.

Apart from these two, we have two constraints, (almost) the same as the HLMS
case. We refer to them as $\mathcal{F}_{3,4}$. The relation between $T_{ab}$
and $q_{ab}$ is referred to as $\mathcal{F}_{5}$. The same constraints appear
in am .

$\displaystyle\mathcal{F}_{3\,ab}$
$\displaystyle=\,\,\,\overset{o}{\mathcal{N}}_{ab}+2\,[D_{a}D_{b}\overset{o}{N}]^{\text{TF}}~{},$
(91) $\displaystyle\mathcal{F}_{4\,ab}$
$\displaystyle=\,\,\,\overset{o}{\sigma}{}^{+}_{ab}+\overset{o}{\sigma}{}^{-}_{ab}~{},$
(92) $\displaystyle\mathcal{F}_{5\,a}$
$\displaystyle=\nabla^{b}T_{ab}+\frac{1}{2}\nabla_{a}R\ .$ (93)

Here the symbol $D$ denotes the $\text{Diff}(S^{2})$ covariant derivative. We
have to find the kinematical brackets for this setup and perform the second
class constraint analysis to obtain the reduced phase space. However, this
problem proved rather difficult. The interested reader may consult Appendix B
for further details.

##  4 Reduced Phase Space for Gauged Weyl BMS

In this section, we analyze another example of the radiative phase space with
relaxed boundary conditions at null infinity. The boundary conditions are such
that the celestial metric is fixed up to a conformal factor. These boundary
conditions are ‘complementary’ to those which lead to gBMS symmetries, for
which the area form on the celestial sphere is fixed.

It was shown in f  that such a boundary condition preserves asymptotic
flatness. The resulting symmetry group which we denote as ${\cal W}$ is a
subgroup of the so-called Weyl-BMS group, discovered in f , which is a semi-
direct product of the BMS group and the Weyl scaling of the celestial sphere
metric. The Weyl-BMS group contains super-translations, celestial
diffeomorphisms, as well as Weyl scalings. We focus on the phase space which
admits an action of ${\cal W}$, that excludes the area-preserving (celestial)
diffeomorphisms.

Although ${\cal W}$ (as well as Weyl-BMS) generates an action on the solutions
of Einstein’s equations, the charge associated with Weyl rescaling does not
constrain classical scattering. Moreover, it can be argued that the
conservation law for flux $Q_{{\cal W}}$ at ${\cal I}^{+}$, generated by the
conformal scaling, is a consequence of the super-translation conservation
law.666We thank Daniele Pranzetti and Laurent Freidel for communicating this
result to us.

Motivated by this result, we consider the scenario where the asymptotic
structure at $\mathcal{I}$ is fixed up to rescaling freedom of the celestial
sphere metric but we then gauge the Weyl symmetry. This model for a radiative
phase space is not physical as in generic radiative space-time the Weyl flux
at $\mathcal{I}^{+}$ need not vanish. However, this example helps us in
elucidating subtleties in analyzing the phase space at $\mathcal{I}$ with a
dynamical celestial sphere metric which we did not encounter in the case of
linearized gravity in Section 3.1.

As we show below, the $Q_{w}=0$ hypersurface is symplectic and is a direct
product of the ‘soft-sector’ parametrized by the super-translation Goldstone
mode, its symplectic partner, and an additional pair of fields associated with
the shear field at $\mathcal{I}$. The soft sector turns out to be isomorphic
to the soft sector in $\Gamma_{\text{HLMS}}$ as the only soft modes are
$(C,\overset{o}{N})$. The symplectic structure on this hypersurface which is
induced by the symplectic structure on $\Gamma_{{\cal W}}$ nicely elucidates
the difficulties we face in separating hard and soft degrees of freedom once
the celestial metric is a dynamical mode.

###  4.1 Weyl Invariance of The CK Condition

There are two polarizations of the graviton and hence two soft theorems.
However, there is only one charge associated with each supertranslation
parameter $f(\hat{x})$. The apparent discrepancy is resolved by the
_Christodoulou-Klainerman (CK) condition_ Christ , which relates the positive
and negative helicity graviton insertions. CK conditions can be interpreted as
the vanishing of the magnetic charge at null infinity am2016 .777In analogy
with the gauge theories, certain components of the Weyl tensor for the
4-dimensional metric can be identified as electric fields and magnetic fields.
In the presence of a dynamical metric, the magnetic part of the
supertranslation charge is as follows:

$\displaystyle
Q_{\mathcal{I}}[\xi_{f}]=\int_{S^{2}}f\int\mathrm{d}u\left(\nabla_{[a}\nabla^{c}N_{cb]}-\frac{1}{2}T_{[a}{}^{c}N_{cb]}\right)$
(94)

Note the presence of an extra term containing Geroch tensor compared to the
HLMS setup.

For the charge (94) to vanish for arbitrary $f(z,\bar{z})$, the integrand
should vanish, hence:

$\displaystyle\nabla_{[a}\nabla^{c}\overset{o}{N}_{cb]}-\frac{1}{2}T_{[a}{}^{c}\overset{o}{N}_{cb]}=D_{[a}D^{c}\overset{o}{N}_{cb]}=0$
(95)

Here, the $D_{a}$ are the gBMS covariant derivative, as introduced in mp . The
general solution to the above constraint is
$\overset{o}{N}_{ab}=-2D_{a}D_{b}\overset{o}{N}{}^{TF}$ and hence:

$\displaystyle
D_{z}D_{z}\overset{o}{N}{}^{\bar{z}\bar{z}}=D_{\bar{z}}D_{\bar{z}}\overset{o}{N}{}^{zz}$
(96)

Since $\overset{o}{N}_{zz}$ and $\overset{o}{N}_{\bar{z}\bar{z}}$ create
leading soft gravitons of different helicities, hence the above equation
implies that the two polarizations of the leading soft graviton are not
independenthlms 888Superrotations corresponds to the subleading soft
insertions of gravitons, and the insertions of the two helicities are
independent. There are two charges corresponding to $V^{a}\quad a\in[1,2]$ and
the electric and magnetic parts for each vector field are identical, hence a
total of only 2 independent chargesam2016 ..

The counting for the leading soft gravitons remains the same even in the
presence of super-rotations, as the CK condition is invariant under super-
rotations:

$\displaystyle\delta_{V}\left(D_{[a}D^{c}\overset{o}{N}_{cb]}\right)=\left(\mathcal{L}_{V}+\frac{1}{2}\nabla_{a}V^{a}\right)\left(\nabla_{[a}\nabla^{c}\overset{o}{N}_{cb]}-\frac{1}{2}T_{[a}{}^{c}\overset{o}{N}_{cb]}\right)=0$
(97)

Hence, even in the presence of superrotations, the counting for leading soft
graviton insertion is still the same.

In order to compute the transformation properties of the magnetic charge under
Weyl BMS, we make use of the following identity:

$\displaystyle\delta_{w}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}$
$\displaystyle=k\,w\,\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}\quad\Rightarrow\qquad\delta_{w}\,D_{a}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}=k\,w\,D_{a}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}$
(98)

We make use of the crucial insight that the Weyl weights ($k$ in
$\delta_{w}\vartheta=k\,w\,\vartheta$) and the gBMS weights ($k$ in
$\delta_{V}\vartheta=(\mathcal{L}_{V}+k\alpha)\vartheta$) coincide for all the
quantities.999This follows from the fact that the $k\alpha$ term in gBMS
action: $\delta_{V}\vartheta=\left(\mathcal{L}_{V}+k\alpha\right)\vartheta$,
upon generalization, leads to the Weyl scalings. $\alpha$ becomes the
independent Weyl parameter $w$. This implies that the gBMS covariant
derivative introduced by Campiglia-Peraza in mp  is also Weyl
covariant.101010As an interesting exercise, had the Weyl weight and the gBMS
weight been different, we could have constructed a Weyl-Diff-covariant
derivative ($\bar{D}$) as follows:
$\displaystyle\delta_{V}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}$
$\displaystyle=\left(\mathcal{L}_{V}+k\alpha\right)\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}\
\
\Rightarrow\qquad\delta_{V}\bar{D}_{a}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}=\left(\mathcal{L}_{V}+k\alpha\right)\bar{D}_{a}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}~{},$
$\displaystyle\delta_{w}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}$
$\displaystyle=l\,w\,\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}\ \ \qquad\quad\
\Rightarrow\qquad\delta_{w}\,\bar{D}_{a}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}=l\,w\,\bar{D}_{a}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}$
This derivative is metric compatible and has the following form:
$\displaystyle\bar{D}_{a}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}$
$\displaystyle=D_{a}\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}+(k-l)A_{b}\,\vartheta_{a_{1}\ldots}{}^{b_{1}\ldots}~{},$
$\displaystyle\text{such that,}\qquad\delta_{w}A_{b}$
$\displaystyle=\nabla_{b}w\ \ ,\qquad\delta_{V}A_{b}=\mathcal{L}_{V}A_{b}$ As
a consequence, the CK condition is invariant under the Weyl scalings:

$\displaystyle\delta_{w}\bigg{(}D_{[a}D^{c}\overset{o}{N}_{cb]}\bigg{)}=0$
(99)

The usual counting of charges and soft theorems remains unchanged and we can
express $C_{ab}$ and $\overset{o}{N}_{ab}$ in terms of scalar functions:

$C_{ab}\equiv-2D_{a}D_{b}C^{\text{TF}},\quad\overset{o}{N}_{ab}\equiv-2D_{a}D_{b}\overset{o}{N}{}^{\text{TF}}$
(100)

###  4.2 Symplectic Structure and Kinematical Brackets

In this case, a general ansatz for the pre-symplectic form is a sum of the
super-translation sector from am  and a term corresponding to the mode
$\sqrt{q}$. Concretely :

$\Omega=\int_{\mathcal{I}}\delta
N_{ab}\wedge\delta\left(\sqrt{q}\,\overset{o}{\sigma}{}^{ab}\right)+\int_{S^{2}}\delta\overset{o}{N}\wedge\,\delta\big{(}\sqrt{q}\,\mathfrak{D}C\big{)}+\frac{1}{2}\int_{S^{2}}\delta\Pi\wedge\delta\sqrt{q}~{}.$
(101)

The first two terms are the usual terms from $\Gamma_{\text{HLMS}}$, with the
last term indicating that the $\sqrt{q}$ is dynamical.111111Note that (101)
reduces to the HLMS symplectic structure once $\sqrt{q}$ is non-dynamical.
Recall that $\mathfrak{D}=4D_{a}D_{b}(D^{a}D^{b})^{TF}$. The presence of the
dynamical celestial sphere area element leads to a non-zero Geroch tensor, and
hence we have Diff-$S^{2}$ covariant derivative $D_{a}$, rather than
$\nabla_{a}$. Also note that since $T_{ab}$ is a functional of $\sqrt{q}$, we
are not treating it as an independent boundary mode. The News tensor does not
have the Geroch tensor as the constant mode, and hence has the following fall-
offs:

$\displaystyle
N_{ab}(u,\hat{x})\xrightarrow{u\to\pm\infty}|u|^{-2-\delta}~{}.$ (102)

An expression for the mode $\Pi$ in terms of the other phase space variables
can be obtained from covariant phase space techniques f . Here we take a more
simplistic approach to fix $\Pi$: we compute it on the hypersurface
$\Gamma_{\mathcal{W}}$ within the phase space, on which we demand that the
Weyl flux $Q_{\mathcal{W}}=\Omega(\delta,\delta_{w})$ vanishes identically.
The rest of our analysis will be restricted to this hypersurface. The action
of Weyl BMS transformations on the modes in the phase space are as follows:

$\displaystyle\delta_{w}N_{ab}(u,\hat{x})$
$\displaystyle=w\,u\partial_{u}N_{ab}(u,\hat{x})$ (103)
$\displaystyle\delta_{w}\overset{o}{\sigma}_{ab}(u,\hat{x})$
$\displaystyle=w\left(-\overset{o}{\sigma}_{ab}+uN_{ab}\right)(u,\hat{x})$
(104) $\displaystyle\delta_{w}C_{ab}(\hat{x})$
$\displaystyle=-wC_{ab}(\hat{x})\qquad\Longleftrightarrow\qquad\delta_{w}C=-wC$
(105) $\displaystyle\delta_{w}\overset{o}{N}_{ab}(\hat{x})$
$\displaystyle=-w\overset{o}{N}_{ab}(\hat{x})\qquad\Longleftrightarrow\qquad\delta_{w}\overset{o}{N}=-w\overset{o}{N}$
(106) $\displaystyle\delta_{w}\sqrt{q}(\hat{x})$
$\displaystyle=-2w\sqrt{q}(\hat{x})\ .$ (107)

For more details, we refer to f .

Demanding that the Weyl flux vanishes puts the following constraint on the
phase space variables:

$\displaystyle\Omega(\delta,\delta_{w})=\int\mathrm{d}u\ \delta
N_{ab}\sqrt{q}\bigg{(}\overset{o}{\sigma}{}^{ab}$
$\displaystyle+uN_{ab}\bigg{)}-\int\mathrm{d}u\
u\,\partial_{u}N_{ab}\,\delta\left(\sqrt{q}\,\overset{o}{\sigma}{}^{ab}\right)$
$\displaystyle+\overset{o}{N}\ \delta\left(\sqrt{q}\,C^{ab}\right)$
$\displaystyle+\delta\overset{o}{N}\left(\sqrt{q}\,\mathfrak{D}C\right)-\
\delta\Pi\sqrt{q}-\frac{1}{2w}\delta_{w}\Pi\,\delta\sqrt{q}=0\ .$ (108)

Using the falloff conditions on $N_{ab}$, and the relation
$\overset{o}{\sigma}{}^{+}_{ab}=-\overset{o}{\sigma}{}^{-}_{ab}$, we obtain :

$\delta\left(\int\mathrm{d}u\
u\sqrt{q}N_{ab}N^{ab}+\sqrt{q}\overset{o}{N}\mathfrak{D}C\right)=\sqrt{q}\
\delta\Pi+\frac{1}{2w}\delta_{w}\Pi\delta\sqrt{q}\ .$ (109)

For the right-hand side to be a total variation, we must have the following:

$\delta_{w}\Pi=2w\left(g(z,\bar{z})f^{\prime}(\sqrt{q})+\Pi\right)\ .$ (110)

This implies that $\Pi$ has to be of the form :

$\Pi(\hat{x})=\int\mathrm{d}u\
uN_{ab}N^{ab}+\overset{o}{N}\mathfrak{D}C+g(\hat{x})f(\sqrt{q})\ .$ (111)

Substituting it back into the symplectic form (101), we see that the term
$g(\hat{x})f(\sqrt{q})$ does not contribute to the symplectic form. We,
therefore, omit this term altogether. We have the symplectic form (101), from
which kinematical Poisson brackets may be derived, and from which the reduced
phase space may be obtained by imposing on the kinematic phase space the
following constraints:

$\displaystyle\mathcal{F}_{1}$ $\displaystyle=\
\Pi-\overset{o}{N}\mathfrak{D}C-\int\mathrm{d}u\ uN_{ab}N^{ab}~{},$ (112)
$\displaystyle\mathcal{F}_{2}{}_{ab}$ $\displaystyle=\
\overset{o}{N}_{ab}-\overset{o}{\mathcal{N}}_{ab}~{},$ (113)
$\displaystyle\mathcal{F}_{3}{}_{ab}$ $\displaystyle=\
\overset{o}{\sigma}{}^{+}_{ab}\ +\ \overset{o}{\sigma}{}^{-}_{ab}\ .$ (114)

The symplectic form is degenerate by construction, i.e., $\exists
X:\Omega(Y,X)=0\ \ \forall\ Y$. The $X$ in question is the Hamiltonian vector
field for the constraint $\mathcal{F}_{1}$. This is because $\mathcal{F}_{1}$
is by construction the Noether charge that generates the Weyl transformations.

At the kinematical level, i.e., prior to the constraints being imposed, we
find the brackets using the kinematical HVFs derived from the symplectic form
given by (101). The second class constraints among these involve the modes
$\overset{o}{N}$, $\overset{o}{\mathcal{N}}{}^{ab}$,
$\overset{o}{\sigma}_{ab}$ and $\sqrt{q}$. The non-vanishing brackets amongst
these modes are listed below, and these will prove useful when we solve these
constraints via the Dirac procedure.

$\displaystyle\big{[}N^{ab}(u,\hat{x})\,,\,N_{cd}(u^{\prime},\hat{y})\big{]}$
$\displaystyle=\frac{1}{2}\,\partial_{u}\delta(u-u^{\prime})\,\mathbb{I}^{ab}_{cd}\
\frac{1}{\sqrt{q}}\delta^{2}(\hat{x}-\hat{y})~{},$ (115)
$\displaystyle\big{[}\overset{o}{\mathcal{N}}{}^{ab}(\hat{x})\,,\,\overset{o}{\sigma}_{cd}(u^{\prime},\hat{y})\big{]}$
$\displaystyle=-\mathbb{I}^{ab}_{cd}\
\frac{1}{\sqrt{q}}\delta^{2}(\hat{x}-\hat{y})~{},$ (116)
$\displaystyle\big{[}\overset{o}{N}(\hat{x})\,,\,C(\hat{y})]$
$\displaystyle=-\,\frac{1}{\sqrt{q}}G(\hat{x},\hat{y})~{},$ (117)
$\displaystyle\big{[}\overset{o}{\sigma}_{ab}(u,\hat{x})\,,\,\Pi(\hat{y})\big{]}$
$\displaystyle=\overset{o}{\sigma}_{ab}\,\frac{1}{\sqrt{q}}\delta^{2}(\hat{x}-\hat{y})~{},$
(118) $\displaystyle\big{[}\sqrt{q}(\hat{x})\,,\,\Pi(\hat{y})]$
$\displaystyle=2\delta^{2}(\hat{x}-\hat{y})~{}.$ (119)

###  4.3 Physical Phase Space

We have a dynamical system with Weyl scalings along $\mathcal{I}$ as the gauge
redundancy. Gauging the Weyl transformations implies that we consider the
constraint hypersurface $\mathcal{F}_{1}\sim 0$ and quotient it out by the
Weyl transformations. It can be checked that this is a consistent restriction
as the HVF generating the Weyl action is tangential to the
$\mathcal{F}_{1}\sim 0$ hypersurface. We present two ways to ‘solve’ such a
system, by gauge fixing, and by finding the explicit reduced phase space.

####  4.3.1 Gauge fixing

One way to ‘solve’ the first class constraint is by introducing another
constraint that has a non-zero bracket with the first class constraint,
rendering it second classGitman ; sundermeyer . To begin with, we have a total
of $5\times\infty$ constraints, out of which $1\times\infty$ are first class.
$\mathcal{F}_{1}$ is the first class constraint and is the generator of the
Weyl scalings. The new constraint is the gauge fixing constraint. One natural
gauge fixing constraint for our current context is $\sqrt{q}=1$121212Other
possibilities are $C$ or $\overset{o}{N}$ = $\\{-1,0,+1\\}$. Note that the
Weyl can not switch the sign, and if for instance $C=0$ in some region, Weyl
action will keep it zero. Hence such a gauge fixing will only be partial.. We
have thus the following second-class constraints to solve:

$\displaystyle\mathcal{F}_{1}$ $\displaystyle=\
\Pi-\overset{o}{N}\mathfrak{D}C-\int\mathrm{d}u\ uN_{ab}N^{ab}~{},$ (120)
$\displaystyle\mathcal{F}_{2}{}_{ab}$ $\displaystyle=\
-2[D_{a}D_{b}\overset{o}{N}]^{\text{TF}}-\overset{o}{\mathcal{N}}_{ab}~{},$
(121) $\displaystyle\mathcal{F}_{3}{}_{ab}$ $\displaystyle=\
\overset{o}{\sigma}{}^{+}_{ab}\ +\ \overset{o}{\sigma}{}^{-}_{ab}\ ,$ (122)
$\displaystyle\mathcal{F}_{4}$ $\displaystyle=\sqrt{q}-1\ .$ (123)

We can find the kinematical brackets among the constraints using the
kinematical brackets from earlier. Using the HVF:

$\displaystyle
X_{\int\mathrm{d}u\,uN_{ab}N^{ab}}=\frac{1}{\sqrt{q}}\int\mathrm{d}u\,uN_{ab}\frac{\delta}{\delta\overset{o}{\sigma}_{ab}}+2\int\mathrm{d}u\,uN_{ab}N^{ab}\frac{\delta}{\delta\Pi}~{},$
(124)

we can check that:

$\displaystyle\left[\int\mathrm{d}u\,uN_{ab}N^{ab}\,,\,\overset{o}{\mathcal{N}}_{mn}\right]$
$\displaystyle=0=\left[\int\mathrm{d}u\,uN_{ab}N^{ab}\,,\,\mathcal{F}_{3mn}\right]~{}.$
(125)

Hence, the Dirac matrix is as follows131313Note that the antisymmetry of
$\big{[}\mathcal{F}_{1}\,,\,\mathcal{F}_{1}\big{]}$ rules out terms
proportional to $\delta^{2}(x-y)$. One can check that the rest of the terms
vanish, and hence $\big{[}\mathcal{F}_{1}\,,\,\mathcal{F}_{1}\big{]}=0$. :

$\displaystyle\left(\begin{matrix}0&M_{12ab}(\hat{x},\hat{z})&0&-\delta^{2}(\hat{x}-\hat{z})\\\
M_{21ab}(\hat{x},\hat{z})&0&\frac{2}{\sqrt{q}}\mathbb{I}_{ab,cd}\delta^{2}(\hat{x}-\hat{z})&0\\\
0&\frac{2}{\sqrt{q}}\mathbb{I}_{ab,cd}\delta^{2}(\hat{x}-\hat{z})&0&0\\\
\delta^{2}(\hat{x}-\hat{z})&0&0&0\end{matrix}\right)~{}.$ (126)

The inverse of the constraint matrix is defined as follows: (Note that
$I,J,\ldots$ refer to the constraint number $[1,2,3,4]$ along with the sphere
indices.)

$\displaystyle\int\mathrm{d}^{2}\hat{z}\,M_{IJ}(\hat{x},\hat{z})W^{JK}(\hat{z},\hat{y})$
$\displaystyle=\delta^{2}(\hat{x}-\hat{y})\delta^{K}_{I}~{}.$ (127)

The explicit inverse of the Dirac matrix is as follows:

$\begin{gathered}\int\mathrm{d}^{2}\hat{z}\left(\begin{matrix}0&M_{12mn}&0&-\delta^{2}_{\hat{x},\hat{z}}\\\
M_{21ab}&0&\frac{2}{\sqrt{q}}\mathbb{I}_{ab,mn}\delta^{2}_{\hat{x},\hat{z}}&0\\\
0&-\frac{2}{\sqrt{q}}\mathbb{I}_{ab,mn}\delta^{2}_{\hat{x},\hat{z}}&0&0\\\
\delta^{2}_{\hat{x},\hat{z}}&0&0&0\end{matrix}\right)~{}.\hskip 142.26378pt\\\
\hskip 42.67912pt\left(\begin{matrix}0&0&0&\delta^{2}_{\hat{z},\hat{y}}\\\
0&0&-\frac{\sqrt{q}}{2}\mathbb{I}^{mn,cd}\delta^{2}_{\hat{z},\hat{y}}&0\\\
0&\frac{\sqrt{q}}{2}\mathbb{I}^{mn,cd}\delta^{2}_{\hat{z},\hat{y}}&0&-\frac{\sqrt{q}}{2}M_{21}{}^{mn}\\\
-\delta^{2}_{\hat{z},\hat{y}}&0&-\frac{\sqrt{q}}{2}M_{12}{}^{cd}&0\end{matrix}\right)=\left(\begin{matrix}1&0&0&0\\\
0&\mathbb{I}^{cd}_{ab}&0&0\\\ 0&0&\mathbb{I}^{cd}_{ab}&0\\\
0&0&0&1\end{matrix}\right)\delta^{2}_{\hat{x},\hat{y}}~{}.\end{gathered}$
(128)

Let us look at the corrected brackets.

$\displaystyle\big{[}\varphi,\alpha\big{]}_{*}$
$\displaystyle=\big{[}\varphi,\alpha\big{]}-\sum_{IJ}\int\mathrm{d}^{2}\hat{z}_{1}\mathrm{d}^{2}\hat{z}_{2}\,\big{[}\varphi,\mathcal{F}_{I}(\hat{z}_{1})\big{]}{W}_{IJ}(\hat{z}_{1},\hat{z}_{2})\big{[}\mathcal{F}_{J}(\hat{z}_{2}),\alpha\big{]}~{}.$
(129)

One immediate thing to notice is that141414Note that after gauge fixing
$\sqrt{q}=1$, $T_{ab}$ vanishes, and hence
$\overset{o}{N}_{ab}=-2\nabla_{a}\nabla_{b}\overset{o}{N}{}^{\text{TF}}$.

$\displaystyle\big{[}\mathcal{F}_{3ab}(\hat{x})\,,\,\alpha\big{]}_{*}=0\qquad\Rightarrow\qquad\big{[}\,\overset{o}{\mathcal{N}}_{ab}(\hat{x})\,,\,\alpha\big{]}_{*}=\big{[}-2\nabla_{a}\nabla_{b}\overset{o}{N}{}^{\text{TF}}\,,\,\alpha\big{]}_{*}~{}.$
(130)

The only independent modes after imposing the constraints are
$\\{\overset{o}{N},C,N_{ab}\\}$. One can notice that all of these modes have
vanishing brackets with $\mathcal{F}_{3}$ and $\mathcal{F}_{4}$. Hence, only
the $\mathcal{F}_{1}$, $\mathcal{F}_{2}$ block of the inverse matrix
contributes to the correction of brackets. This particular block (upper left
$2\times 2$ in (128)) is identically zero. Hence the brackets of these modes
remain uncorrected:

$\displaystyle\big{[}N_{ab}(u,\hat{x})\,,\,N_{cd}(u^{\prime},\hat{y})\big{]}_{*}$
$\displaystyle=\frac{1}{2}\,\partial_{u}\delta(u-u^{\prime})\,\mathbb{I}_{ab,cd}\
\delta^{2}(\hat{x}-\hat{y})$ (131)
$\displaystyle\big{[}C(\hat{x})\,,\,\overset{o}{N}(\hat{y})\big{]}_{*}$
$\displaystyle=G(\hat{x},\hat{y})$ (132)

These brackets will be useful in computing the Poisson algebra of the Weyl
invariants in the next section. The structure of the gauged fixed phase space
is not meant to imply that the reduced phase space is $\Gamma_{\text{HLMS}}$.
We show in the next section that the supertranslations, which are a genuine
symmetry of $\Gamma_{\text{HLMS}}$ are not well defined on the reduced phase
space of the gauged Weyl model.

####  4.3.2 Supertranslations in the presence of the gauged Weyl

In this section, we show that the action of supetranslations is not well-
defined on the reduced phase space.

Consider the action of supertranslations on the News tensor:

$N_{ab}(u,\hat{x})\rightarrow N_{ab}(u+f(\hat{x}),\hat{x})$ (133)

Since Weyl is a gauge redundancy, we may choose to act the same
supertranslation on another representative of the orbit of $N_{ab}(u,\hat{x})$
under the Weyl scaling, say $N_{ab}(e^{w}u,\hat{x})$ :

$N_{ab}(e^{w}u,\hat{x})\rightarrow N_{ab}(e^{w}u+f(\hat{x}),\hat{x})$ (134)

As the right hand sides of (133) and (134) are not related by a Weyl
transformation, we conclude that the action of supertranslations take
different points on the same gauge orbit to points that do not lie on the same
gauge orbit, and are thus ill-defined on the reduced phase space. This is
because the supertranslations and Weyl generating vector fields do not
commute, see Weyl algebra from Section 2.3 :

$\displaystyle\big{[}\delta_{(f,0,0)}\,,\,\delta_{(0,0,w)}\big{]}=\delta_{(wf,0,0)}~{}.$
(135)

Thus, if we gauge Weyl, supertranslations are no longer symmetries of the
dynamical system. Since supertranslations are true symmetries of the quantum
S-matrix, the gauged Weyl setup is unphysical. The absence of
supertranslations as a symmetry manifests itself in the reduced phase space
approach to gauging the Weyl action.

####  4.3.3 Reduced Phase Space

The Weyl action on $N_{ab}$ is induced by a scale transformation of the null
coordinate. The algebra of dilatation invariant functions on $\mathbb{R}$ is
isomorphic to the algebra of functions on a non-Hausdorff space consisting of
three points. This non-trivial topology makes it clear why supertranslations
no longer remain global symmetries in the reduced phase space.

Let us now isolate the ‘physical’ degrees of freedom without gauge fixing. To
begin with, we have the following coordinates for the kinematical phase space:

$\displaystyle\big{\\{}\,\sqrt{q}\,,\,C(\hat{x})\,,\,\overset{o}{N}(\hat{x})\,\big{\\}}\cup\big{\\{}N_{ab}(u,\hat{x})\big{\\}}~{}.$
(136)

The reduced phase space is the one constructed by endowing the space of Weyl-
orbits a symplectic form induced from the kinematical phase space Gitman ;
sundermeyer . Since each orbit may be parameterized by the values of the Weyl
invariant quantities, we may treat independent Weyl invariant quantities as
the coordinates on the reduced phase space. For instance, in the soft sector,
kinematically, $\\{C\,,\,\overset{o}{N}\,,\,\sqrt{q}\\}$ are the coordinates
in the phase space. We can choose the following coordinates for the soft
sector, in which case, the gauge mode and the physical modes are identifiable:

$\displaystyle\big{\\{}\,\sqrt{q}\,,\,\underbrace{\sqrt{q}^{-1/2}C\,,\,\sqrt{q}^{-1/2}\overset{o}{N}}_{\text{Dynamical
modes}}\,\big{\\}}~{}.$ (137)

In addition to the $\\{C,\overset{o}{N}\\}$ modes, we have the following
integrals151515Note that even though $u$ integrals,
$\int_{0}^{\pm}\mathrm{d}u$, can be made Weyl invariant, they are not a part
of the phase space since one needs distributional smearing functions to define
them. over $\mathcal{I}$ that are naturally Weyl-invariant:

$\displaystyle I^{\\{a_{n}\\}}_{\ldots}\equiv\sqrt{q}^{(\sum
na_{n}-1)/2}\int_{-\infty}^{\infty}\mathrm{d}u\prod_{n=0}^{\infty}\big{(}\partial_{u}^{n}N_{\cdot\cdot}(u)\big{)}^{a_{n}}\,,\qquad
a_{n}\in\mathbb{Z}_{\geq 0}~{}.$ (138)

The $\ldots$ in $I^{\\{a_{n}\\}}_{\ldots}$ denotes the arbitrary sphere
indices161616One can verify the Weyl invariance of $I$ by using
$\quad\partial_{u}^{n}N_{ab}(u)\xrightarrow{w}e^{nw}\partial_{u^{\prime}}^{n}N_{ab}(u^{\prime})\big{|}_{u^{\prime}=e^{w}u}$..
Note that we require only a finite number of $a_{n}$s to be non-zero for $I$
to be well defined. In holomorphic coordinates, these integrals take the
following form:

$\displaystyle I^{\\{a_{n}\\},\\{b_{n}\\}}\equiv\sqrt{q}^{(\sum
n(a_{n}+b_{n})-1)/2}\int_{-\infty}^{\infty}\mathrm{d}u\prod_{n=0}^{\infty}\big{(}\partial_{u}^{n}N_{zz}(u)\big{)}^{a_{n}}\big{(}\partial_{u}^{n}N_{\bar{z}\bar{z}}(u)\big{)}^{b_{n}}\,,\qquad
a_{n},b_{n}\in\mathbb{Z}_{\geq 0}~{}.$ (139)

Not all of the $I^{\\{a_{n}\\}}$ are independent - while some of them
trivially vanish, others have relations amongst them arising out of
integration by parts. However, such relations keep the rank of the tensor,
$\sum na_{n}$ and $\sum a_{n}$ fixed. Examples include:

$q\int\mathrm{d}u\ N_{ab}N^{ab}\partial_{u}N_{cd}+2q\int\mathrm{d}u\
N_{ab}\partial_{u}N^{ab}N_{cd}=0~{}.$ (140)

We can use the kinematical brackets to evaluate the physical Poisson brackets
amongst Weyl invariant quantities. One can see clearly that the $I^{a_{n}}$
form a closed algebra: this is because, by virtue of (131), the Poisson
bracket of any two of these invariants is a sum over terms with one higher
power of $\sqrt{q}$, one extra $u$ derivative. This ensures that the power of
the prefactor $\sqrt{q}$ is always appropriate to make the Weyl weight of the
overall quantity zero. Now we give a few examples of what the brackets look
like. First, we note that the following set of observables forms an abelian
subalgebra:

$\displaystyle\left[\sqrt{q}^{(4n-1)/2}\int\mathrm{d}u\left(N_{ab}N^{ab}\right)^{n}\,,\,\sqrt{q}^{(4m-1)/2}\int\mathrm{d}u^{\prime}\left(N_{cd}N^{cd}\right)^{m}\right]=0~{}.$
(141)

For illustration, we write another Poisson bracket:

$\displaystyle\left[\sqrt{q}^{3/2}\int\mathrm{d}u\,N_{ab}\left(N_{mn}N^{mn}\right)(\hat{x})\,,\,\sqrt{q}^{2m-1/2}\int\mathrm{d}u^{\prime}\left(N_{cd}N^{cd}\right)^{m}(\hat{y})\right]$
$\displaystyle\hskip
113.81102pt=2(1-m)\sqrt{q}^{2m}\int\mathrm{d}u\,\partial_{u}N_{ab}\left(N_{cd}N^{cd}\right)^{m}\,\,\delta^{2}(\hat{x}-\hat{y})~{}.$
(142)

The algebra of Poisson brackets amongst $I^{\\{a_{n}\\}}$ is compatible with
the Poisson algebra derived by fixing the gauge, (131).

We can act arbitrary Diff-$S^{2}$ derivatives on any of these Weyl invariant
quantities, and with appropriate factors of $\sqrt{q}$, we obtain more Weyl
invariant tensors. The reduced phase space is thus parameterized by the soft
sector and the integrals $I^{\\{a_{n}\\}}$. These constitute an over-complete
set of invariants, and characterizing the reduced phase space by picking out a
basis from this set of invariants will be a challenging problem.

##  5 Conclusions

The recent studies in asymptotic symmetries beyond the ones generated by BMS
vector fields typically relies on the relaxed boundary conditions in which the
celestial sphere metric is allowed to fluctuate. However, our experience with
asymptotic quantization of radiative phase spaces in which the celestial
metric is dynamical is rather limited am . This is because passing to quantum
theory requires that we find the conjugate pairs in the soft and the hard
sector. The broad goal in this paper has been to provide examples of such
radiative phase spaces, which are in principle amenable to asymptotic
quantization. We have analysed two scenarios in which the sphere metric is
dynamical: gBMS under the restricted setting of linearized gravity and Weyl
BMS where we treat the Weyl scalings as pure gauge.

We obtained the physical radiative phase space for gBMS in the context of
linearized gravity. Even though the generic Dirac bracket involves an abstract
Green’s function, we have identified a certain combination of Geroch tensor
and celestial metric that is conjugate to the sub-leading soft News tensor.
The final phase space does not factorize neatly into the hard and soft
sectors, as the brackets amongst these soft modes and hard modes are non-
vanishing. The results of this analysis parallel those of am , indicating the
robustness of those results.

In the linearized gravity approach to the gBMS phase space, we showed that
keeping Geroch tensor $T_{ab}$ as a constant mode of the News tensor leads to
simplifications in our analysis. A similar approach may help in the non-linear
analysis as well.

We showed that the usual method of obtaining Poisson brackets from the
symplectic form via Dirac’s method of eliminating second class constraints
runs into some issues when applied to the symplectic form for gBMS in mp . The
core obstruction to the program is the fact that the determinant of the Dirac
matrix is an operator that has functional dependence on soft modes. We believe
that it will require new ideas to solve the constraints in this case as an
inversion of an operator-valued Dirac matrix appears to be a rather
intractable problem. Given the importance of obtaining the radiative phase
space to perform (asymptotic) quantization and define a complete set of soft
vacua in quantum gravity, the problem merits a serious investigation.

We then focused on a toy model involving the Weyl BMS group but excluding the
superrotation subgroup. Restricting our attention to the special case with
vanishing Weyl flux, we obtained a symplectic form that is degenerate over the
orbit of the pure Weyl transformations of the Weyl BMS group. We checked that
the _Christodoulou-Klainerman (CK)_ condition is invariant under the action of
the Weyl rescaling and obtained the reduced phase space for the gauged Weyl
model. It is an intriguing observation that the reduced phase space of the
HLMS phase space augmented with Weyl scalings as gauge, is not HLMS anymore,
as it lacks supertranslations.

The gauge invariant observables are precisely those that are fixed under Weyl
rescalings. In the soft sector, this transformation acts via an overall
scaling and thus we have specified a generating set of gauge invariant
quantities by supplementing the soft modes by appropriate factors of sphere
area element $\sqrt{q}$. However, in the hard sector, there is no notion of
local hard News and the supertranslations are not well defined. As a result,
Weyl scalings can not be a gauge redundancy in any physical setting. Also, it
leads to the fact that in the hard sector, only the Weyl invariant densities
constructed out of the $\mathcal{I}$ integrals of hard News survive in the
reduced phase space. We have identified a set of such quantities that form a
closed algebra, and presumably, form an overcomplete set of Weyl invariants.

It would be natural to explore how our work in both settings connects with the
flat space holography program (see Pasterski2016 ; ccft  for a review).

The reduced phase space defined in the gauged Weyl setting may have an
interesting connection with the space of all asymptotically locally flat
geometries which were analyzed in nishant . The phase space description we
provide in this work may prove useful to better understand these locally flat
geometries.

Our reduced phase space analysis for linearized gBMS could prove useful in a
variety of settings. For instance, the double soft graviton theorems are
sensitive to the details of charge algebras distler fh ; anupam kundu . It
would be useful to revisit the double soft theorems in light of conservation
laws emerging out of the gBMS phase space.

###### Acknowledgements.

We are grateful to Alok Laddha for suggesting the problem and thank him for
many fruitful discussions and constant encouragement. We would like to thank
Sujay Ashok for the discussions and for providing feedback on the draft. We
thank Miguel Campiglia, Laurent Freidel, and Daniele Pranzetti for the helpful
correspondence. We also appreciate the long and helpful discussions with PB
Aneesh, Nishant Gupta, and Dileep Jatkar over the past year.

##  Appendix A Dirac Analysis

Given a dynamical system and the corresponding Poisson brackets, if we wish to
impose some constraints $F_{k}=0$ on the dynamical variables, then we follow
Dirac’s analysis. (We refer to d  for a comprehensive review.) The brackets of
the dynamical quantities on the constrained surface are called _Dirac
brackets_. Given the Poisson brackets amongst the dynamical variables, we can
find brackets amongst the constraints as well. If a constraint commutes with
all the other constraints (valued on the constrained surface), then it is
called a first-class constraint. Constraints that are not first class are
termed second class.

Given a set of constraints, consider the maximal set of second-class
constraints. Let us denote them by $\mathcal{F}_{k}$. We denote the
kinematical brackets as $\big{[}\cdot\,,\,\cdot\big{]}$, and the Dirac
brackets as $\big{[}\cdot\,,\,\cdot\big{]}_{*}$. The prescription due to Dirac
for computing these new brackets is :

$\displaystyle\big{[}\varphi,\alpha\big{]}_{*}=\big{[}\varphi,\alpha\big{]}-\sum_{i,k}\big{[}\varphi,\mathcal{F}_{i}\big{]}\mathcal{W}^{ik}\big{[}\mathcal{F}_{k},\alpha\big{]}\
.$ (143)

Here $\varphi$ and $\alpha$ are arbitrary functions on phase space and
$\mathcal{W}^{ik}$ is the inverse of the commutator of constraints, defined as
follows :

$\displaystyle\sum_{i}\big{[}\mathcal{F}_{l}\,,\,\mathcal{F}_{i}\big{]}\mathcal{W}^{ik}=\delta^{k}_{l}$
$\displaystyle=\sum_{i}\mathcal{W}^{ki}\big{[}\mathcal{F}_{i},\mathcal{F}_{l}\big{]}~{}.$
(144) $\displaystyle\text{Note that}\hskip
56.9055pt\big{[}\cdot,\mathcal{F}_{i}\big{]}_{*}$ $\displaystyle=0\ .$ (145)

When the phase space is finite-dimensional, one can check that the Jacobi
identity is satisfied. Hence the final brackets are antisymmetric and satisfy
the Jacobi identity by construction.

The above construction can be generalized to the continuous case. In that
case, the sum over constraints gets additional integrals appropriately as
follows:

$\displaystyle\sum_{K}\int\mathrm{d}^{2}\hat{z}\,\big{[}\mathcal{F}_{I}(\hat{x})\,,\,\mathcal{F}_{K}(\hat{z})\big{]}\,W^{KL}(\hat{z},\hat{y})=\delta^{L}_{I}\,\delta^{2}(\hat{x}-\hat{y})~{}.$
(146)

Recall that the antisymmetry of the Dirac matrix manifests itself as follows:

$\displaystyle
M_{IK}(\hat{x},\hat{z})\equiv\big{[}\mathcal{F}_{I}(\hat{x})\,,\,\mathcal{F}_{K}(\hat{z})\big{]}=-\big{[}\mathcal{F}_{K}(\hat{z})\,,\,\mathcal{F}_{I}(\hat{x})\big{]}=-M_{KI}(\hat{z},\hat{x})~{}.$
(147)

The same holds for the inverse as well.

##  Appendix B Dirac Matrix in the gBMS Case

Let us reiterate the symplectic form for the gBMS case (88):171717Note that
the Geroch tensor is explicitly present, unlike the case of Section 3.1. This
is due to $\Pi^{ab}$ being intricate in the non-linear case and there being no
canonical way to incorporate $\delta\Pi\wedge\delta T$ into the hard sector.

$\displaystyle\Omega$ $\displaystyle=\int_{\mathcal{I}}\sqrt{q}\ \delta
N_{ab}\wedge\delta\overset{o}{\sigma}{}^{ab}+\int_{S^{2}}\sqrt{q}\
\left(\delta\overset{o}{N}\wedge\
\delta\mathfrak{C}+\delta\Pi^{ab}\wedge\delta T_{ab}+\delta p^{ab}\wedge\delta
q_{ab}\right)~{},$ (148)
$\displaystyle\quad\text{with}\qquad\mathfrak{C}=\left(-2\nabla_{a}\nabla_{b}+q_{ab}\Delta+T_{ab}\right)\left(-2\nabla^{a}\nabla^{b}+T^{ab}\right)C~{}.$
(149)

We have traded off $C$ in favor of $\mathfrak{C}$. Given $\mathfrak{C}$, we
can invert the differential operator and obtain $C$ in terms of the
appropriate Green function. The information regarding the boundary modes of
shear is completely encoded in $\mathfrak{C}$, rather than $C$. After finding
the brackets of $\mathfrak{C}$, we can use (149) to obtain the brackets of $C$
as well.

The variables $p^{ab}$ and $\Pi^{ab}$ are both tracefree. This is because
their conjugates also have two degrees of freedom each, as they satisfy
$\text{det }q_{ab}=1$ and $q^{ab}T_{ab}=0$. One can see that by going to the
local complex coordinates. In abstract index notation, since $q_{ab}$ is
dynamical, the tracelessness condition itself is intricate, as can be seen in
the following illustration:

$\displaystyle\delta p^{ab}\wedge\delta q_{ab}$ $\displaystyle=\delta
p^{ab}{}^{\text{TF}}\wedge\delta
q_{ab}+\frac{1}{2}\delta(p\,q^{ab})\wedge\delta q_{ab}$ (150)
$\displaystyle=\delta p^{ab}{}^{\text{TF}}\wedge\delta
q_{ab}+\frac{1}{2}\left(\delta p\wedge q^{ab}\delta q_{ab}+p\,\delta
q^{ab}\wedge\delta q_{ab}\right)$ (151) $\displaystyle=\delta
p^{ab}{}^{\text{TF}}\wedge\delta q_{ab}~{}.$ (152) $\displaystyle\text{Where
we used}\quad\qquad q^{ab}\delta q_{ab}$ $\displaystyle\sim\delta\sqrt{q}=0\ \
,\qquad\delta q^{ab}\wedge\delta q_{ab}\sim\delta(\delta\sqrt{q})=0~{}.$

To obtain the kinematical brackets, we calculate the HVFs.181818Note that in
the following, we treat $\overset{o}{\sigma}_{ab}$ as the fundamental field
and $\overset{o}{\sigma}{}^{ab}$ as a functional of $\overset{o}{\sigma}_{ab}$
and $q_{ab}$.

$\displaystyle X_{N_{ab}}$
$\displaystyle=\frac{1}{2}\,\mathbb{I}_{ab,mn}\,\frac{\delta}{\delta\overset{o}{\sigma}_{mn}}-\,N^{n}{}_{a}\frac{\delta}{\delta
p^{nb}}\ ,$ (153) $\displaystyle X_{\overset{o}{\mathcal{N}}_{ab}}$
$\displaystyle=\mathbb{I}_{ab,mn}\,\int\mathrm{d}u\,\frac{\delta}{\delta\overset{o}{\sigma}_{mn}}\
,$ (154) $\displaystyle X_{\overset{1}{\mathcal{N}}{}^{ab}}$
$\displaystyle=\frac{1}{2}\int\mathrm{d}u\,u\,\frac{\delta}{\delta\overset{o}{\sigma}_{mn}}+q^{am}\overset{1}{\mathcal{N}}{}^{bn}\frac{\delta}{\delta
p^{mn}}\ ,$ (155) $\displaystyle X_{T_{ab}}$
$\displaystyle=-\frac{\delta}{\delta\Pi^{ab}}\ ,\hskip 85.35826pt$ (156)
$\displaystyle X_{\Pi^{ab}}$ $\displaystyle=\frac{\delta}{\delta T_{ab}}\ ,$
(157) $\displaystyle X_{\mathfrak{C}}$
$\displaystyle=-\frac{\delta}{\delta\overset{o}{N}}\ ,\hskip 91.04872pt$ (158)
$\displaystyle X_{\overset{o}{N}}$
$\displaystyle=\frac{\delta}{\delta\mathfrak{C}}\ ,$ (159) $\displaystyle
X_{q_{ab}}$ $\displaystyle=-\frac{\delta}{\delta p^{ab}}\ ,\hskip 85.35826pt$
(160) $\displaystyle X_{p^{ab}}$ $\displaystyle=\frac{\delta}{\delta
q_{ab}}+2\int_{\mathcal{I}}q^{(ma}\overset{o}{\sigma}{}^{bn)}X_{N_{mn}}\ .$
(161)

The relevant kinematic brackets among the phase space variables follow from
the HVFs, and here we state the results :

$\displaystyle\big{[}\overset{o}{\sigma}_{ab}(u,\hat{x})\,,\,N_{cd}(u^{\prime},\hat{y})\big{]}$
$\displaystyle=\frac{1}{2}\delta(u-u^{\prime})\,\mathbb{I}_{ab,cd}\,\delta^{2}(\hat{x}-\hat{y})\
,$ (162)
$\displaystyle\big{[}\overset{o}{\sigma}_{ab}(u,\hat{x})\,,\,\overset{o}{\mathcal{N}}_{cd}(\hat{y})\big{]}$
$\displaystyle=\mathbb{I}_{ab,cd}\ \delta^{2}(\hat{x}-\hat{y})\ ,$ (163)
$\displaystyle\big{[}\overset{o}{\sigma}_{ab}(u,\hat{x})\,,\,\overset{1}{\mathcal{N}}{}^{cd}(\hat{y})\big{]}$
$\displaystyle=\frac{1}{2}u\,\mathbb{I}^{cd}_{ab}\
\delta^{2}(\hat{x}-\hat{y})\ ,$ (164)
$\displaystyle\big{[}N_{ab}(u,\hat{x})\,,\,p^{cd}(\hat{y})\big{]}$
$\displaystyle=N_{a}^{(c}\delta^{d)}_{b}\,\delta^{2}(\hat{x}-\hat{y})\ ,$
(165) $\displaystyle\big{[}T_{cd}(\hat{x})\,,\,\Pi^{ab}(\hat{y})\big{]}$
$\displaystyle=\mathbb{I}^{ab}_{cd}\ \delta^{2}(\hat{x}-\hat{y})\ ,$ (166)
$\displaystyle\big{[}q_{cd}(\hat{x})\,,\,p^{ab}(\hat{y})\big{]}$
$\displaystyle=\delta^{ab}_{cd}\ \delta^{2}(\hat{x}-\hat{y})\ ,$ (167)
$\displaystyle\big{[}p^{cd}(\hat{x})\,,\,\overset{1}{\mathcal{N}}{}^{ab}(\hat{y})\big{]}$
$\displaystyle=\left(\,\overset{1}{\mathcal{N}}{}^{b(d}q^{ac)}-\frac{1}{2}q^{cd}\overset{1}{\mathcal{N}}{}^{ab}\right)\delta^{2}(\hat{x}-\hat{y})\
.$ (168)

The kinematical structure is thus well understood. We now have to impose the
constraints (89) - (93) on this kinematical phase space. We can compute the
kinematical brackets of the constraints yielding the Dirac matrix. At generic
points in the phase space, these constraints are rather unwieldy, so as a
first step we consider the region $\Gamma_{ts}$ defined as the subspace of the
phase space where the modes $\overset{o}{N}$ and
$\overset{1}{\mathcal{N}}_{ab}$ vanish. In this special case, we note that the
brackets of the bi-linear terms in $C$ and $\overset{o}{N}$ vanish, and hence
these terms have no bearing on what follows.191919In addition to all the
brackets that vanish, we have omitted any bracket involving the modes $C$ and
$\overset{o}{N}$. This is because by virtue of (159) the only non-zero bracket
the soft mode $\overset{o}{N}$ has is the $[C,\overset{o}{N}]$ bracket. But
since in the constraints, $C$ and $\overset{o}{N}$ always come in pairs, all
the brackets involving these vanish when $\overset{o}{N}$ is set to zero. Thus
we may ignore the terms from the constraints that are bi-linear in $C$,
$\overset{o}{N}$. We then show that the determinant of the resulting matrix is
zero. This indicates that the determinant of the Dirac matrix is functionally
dependent on the News tensor, unlike the case in am .

Although the precise elements of the Dirac matrix can be computed using these
kinematical brackets, these expressions are cumbersome and hence are omitted
from this document. For our purposes here, we merely need to keep track of
whether the element is non-zero or not, on the points where
$\overset{1}{\mathcal{N}}_{ab}$ and $\overset{o}{N}$ are set to zero. Crucial
to our results will be the vanishing of three kinematical brackets :
$[p^{ab},p^{cd}]=[p^{ab},\Pi^{cd}]=[\overset{1}{\mathcal{N}}{}^{ab},\overset{1}{\mathcal{N}}{}^{cd}]=0$.

We begin with $[\mathcal{F}_{1}^{ab},\mathcal{F}_{1}^{cd}]$. Notice that every
term in $\mathcal{F}_{1}$ (modulo expressions involving the $C$ mode) has an
explicit $\overset{1}{\mathcal{N}}{}^{ab}$ in it. Since the only non-vanishing
bracket involving $\overset{1}{\mathcal{N}}{}^{ab}$, of the form
$\big{[}p^{cd},\overset{1}{\mathcal{N}}{}^{ab}\big{]}$, happens to be
proportional to $\overset{1}{\mathcal{N}}{}^{cd}$, we deduce that
$[\mathcal{F}_{1}^{ab},\mathcal{F}_{1}^{cd}]$ vanishes when the sub-leading
News tensor is set to zero.

The story is similar with $[\mathcal{F}_{1}^{ab},\mathcal{F}_{2}^{cd}]$ :
since $[p^{ab},\Pi^{cd}]$ and
$[\overset{1}{\mathcal{N}}{}^{ab},\overset{1}{\mathcal{N}}{}^{cd}]$ vanish,
the leftover bracket is precisely
$\big{[}p^{cd},\overset{1}{\mathcal{N}}{}^{ab}\big{]}$, which is proportional
to $\overset{1}{\mathcal{N}}{}^{ab}$. Of the remaining brackets, note that
$[(\mathcal{F}_{5})_{a},(\mathcal{F}_{5})_{b}]$ and
$[\mathcal{F}_{2}^{ab},\mathcal{F}_{2}^{cd}]$ vanish (modulo terms with the
$C$ mode). The brackets $[\mathcal{F}_{1}^{ab},(\mathcal{F}_{5})_{c}]$ and
$[\mathcal{F}_{2}^{ab},(\mathcal{F}_{5})_{c}]$ are independent of
$\overset{1}{\mathcal{N}}{}^{cd}$ and $\overset{o}{N}$. Their precise form
will not be required for our arguments. The last brackets are those involving
$\mathcal{F}_{3}$ and $\mathcal{F}_{4}$ : in this case, the condition
$\overset{o}{N}=0$ implies that these constraints decouple from the rest and
the Dirac matrix block diagonalizes. The $\mathcal{F}_{3}$,$\mathcal{F}_{4}$
block is invertible and is functionally independent of the shear tensor, see
(163). The following is a block of the Dirac matrix, formed by constraints
$\mathcal{F}_{1}$, $\mathcal{F}_{2}$, and $\mathcal{F}_{5}$.

$\displaystyle\left(\begin{matrix}\big{[}\mathcal{F}_{1}\,,\,\mathcal{F}_{1}\big{]}&\big{[}\mathcal{F}_{1}\,,\,\mathcal{F}_{2}\big{]}&\big{[}\mathcal{F}_{1}\,,\,\mathcal{F}_{5}\big{]}\\\
\big{[}\mathcal{F}_{2}\,,\,\mathcal{F}_{1}\big{]}&\big{[}\mathcal{F}_{2}\,,\,\mathcal{F}_{2}\big{]}&\big{[}\mathcal{F}_{2}\,,\,\mathcal{F}_{5}\big{]}\\\
\big{[}\mathcal{F}_{2}\,,\,\mathcal{F}_{1}\big{]}&\big{[}\mathcal{F}_{5}\,,\,\mathcal{F}_{2}\big{]}&\big{[}\mathcal{F}_{5}\,,\,\mathcal{F}_{5}\big{]}\\\
\end{matrix}\right)=\left(\begin{matrix}\left(\ldots\overset{1}{\mathcal{N}}\ldots\right)&\left(\ldots\overset{1}{\mathcal{N}}\ldots\right)&\left(\ldots\right)\\\
\left(\ldots\overset{1}{\mathcal{N}}\ldots\right)&0&\left(\ldots\right)\\\
\left(\ldots\right)&\left(\ldots\right)&0\\\ \end{matrix}\right)$ (169)

Each entry in this $3\times 3$ matrix is a $2\times 2$ block. In the matrix
shown in (169), $\ldots\overset{1}{\mathcal{N}}\ldots$ denotes terms that are
linear in subleading soft News and vanish when it is set to zero. Note that
$\overset{o}{N}$ has already been set to zero. If
$\overset{1}{\mathcal{N}}{}^{ab}$ is also zero, the top four rows of the Dirac
matrix form a set of 4 vectors, each of which is $2$ dimensional. Thus, the
determinant of this matrix must vanish. On the other hand, one can explicitly
check that determinants of matrices with non-zero entries in the locations
represented by $\ldots\overset{1}{\mathcal{N}}\ldots$ do not trivially vanish.
Thus we conclude that the determinant of the Dirac matrix is functionally
dependent on the Bondi News tensor, as claimed in Section 3.2.

The aforementioned determinant also happens to be a differential operator
(acting on the Dirac delta function of $S^{2}$). Since the inverse matrix
involves the inverse of the determinant, we would expect it to involve Green’s
function for this operator. The problem of finding this Green’s function,
however, is intractable, as the operator is dependent on the soft modes means
that its coefficients are arbitrary functions on the celestial sphere. We thus
see that, one can not solve the Dirac constraints globally on the phase space
and that they can only be solved locally in phase space (for fixed soft
modes.) However, this analysis is outside the scope of the present paper.
Though, the extended covariant phase space of linearized theory that includes
(in addition to the canonical phase space of linearized gravity) the radiative
phase space of leading and sub-leading soft modes turns out to be tractable,
as described in Section 3.1.

The entries of the inverse of the Dirac matrix, if it exists, are precisely
the functionals of physical brackets among the soft sector modes. For instance
consider the brackets $\big{[}q_{ab}(\hat{x}),q_{cd}(\hat{y})\big{]}_{*}$.
(Kinematically, $q_{ab}$ and $T_{ab}$ only have non-zero brackets with
$p^{ab}$ and $\Pi^{ab}$ respectively.)

$\displaystyle\big{[}q_{ab}(\hat{x}),q_{cd}(\hat{y})\big{]}_{*}$
$\displaystyle=-\int\mathrm{d}^{2}\hat{z}_{1}\mathrm{d}^{2}\hat{z}_{2}\,\big{[}q_{ab}(\hat{x}),\mathcal{F}^{ij}_{1}(\hat{z}_{1})\big{]}\,W^{11}{}_{ij,kl}(\hat{z}_{1},\hat{z}_{2})\,\big{[}\mathcal{F}^{kl}_{1}(\hat{z}_{2}),q_{cd}(\hat{y})\big{]}$
(170)
$\displaystyle=-\int\mathrm{d}^{2}\hat{z}_{1}\mathrm{d}^{2}\hat{z}_{2}\,\big{[}q_{ab}(\hat{x}),p^{ij}(\hat{z}_{1})\big{]}\,\mathcal{W}^{11}{}_{ij,kl}(\hat{z}_{1},\hat{z}_{2})\,\big{[}p^{kl}(\hat{z}_{2}),q_{cd}(\hat{y})\big{]}$
(171) $\displaystyle=\mathcal{W}^{11}{}_{ab,cd}(\hat{x},\hat{y})\ .$ (172)

Similarly, for $T_{ab}$, we have:

$\displaystyle\big{[}T_{ab}(\hat{x}),T_{cd}(\hat{y})\big{]}_{*}=\mathcal{W}^{22}_{ab,cd}(\hat{x},\hat{y})\
.$ (173)

Thus, finding the physical brackets is about as hard a problem as solving the
set of coupled PDEs coming from the constraint analysis.

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