aainstitutetext: Instituto de Física, Facultad de Ciencias, Igúa 4225,
Montevideo, Uruguaybbinstitutetext: Chennai Mathematical Institute, SIPCOT IT
Park, Siruseri 603103, India

# BMS Algebra, Double Soft Theorems, and All That

Miguel Campiglia b Alok Laddha<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Lie algebra generated by supertranslation and superrotation vector fields
at null infinity, known as the extended Bondi–van der Burg–Metzner–Sachs
(eBMS) algebra is expected to be a symmetry algebra of the quantum gravity
S-matrix. However, the algebra of commutators of the quantized eBMS charges
has been a thorny issue in the literature. On the one hand, recent
developments in celestial holography point towards a symmetry algebra which is
a closed Lie algebra with no central extension or anomaly; and on the other
hand, work of Distler, Flauger and Horn has shown that when these charges are
quantized at null infinity, the commutator of a supertranslation and a
superrotation charge does not close into a supertranslation but gets deformed
by a 2-cocycle term, which is consistent with the original proposal of Barnich
and Troessaert.

In this paper, we revisit this issue in light of recent developments in the
classical understanding of superrotation charges. We show that, for extended
BMS symmetries, a phase space at null infinity is an extension of hitherto
considered phase spaces which also includes a mode associated to the spin-
memory and its conjugate partner. We also show that for holomorphic vector
fields on the celestial plane, quantization of the eBMS charges in the new
phase space leads to an algebra which closes without a 2-cocycle. The
degenerate vacua are labelled by the soft news and a Schawarzian mode which
corresponds to deformations of the celestial metric by superrotations. The
closed eBMS quantum algebra may also lead to a convergence between two
manifestations of asymptotic symmetries, one via asymptotic quantization at
null infinity and the other through celestial holography.

###### Contents

  1. 1 Introduction
  2. 2 Review of the radiative phase space
  3. 3 Radiative phase space with superrotation modes
  4. 4 Kinematical phase space
    1. 4.1 Hard phase space
    2. 4.2 Poisson brackets in the hard phase space
    3. 4.3 Soft phase space
    4. 4.4 Poisson brackets in the soft phase space
  5. 5 Poisson brackets on $\Gamma$
  6. 6 Quantization of $\Gamma$
    1. 6.1 eBMS quantum charges
    2. 6.2 Ward identities of eBMS charges
    3. 6.3 eBMS quantum algebra
    4. 6.4 Consistency with consecutive double soft graviton theorems
  7. 7 Conclusions and outlook
    1. 7.1 Summary of results
    2. 7.2 Comparison with the eBMS algebra in celestial holography
    3. 7.3 Towards gBMS algebra
  8. A Derivation of Dirac brackets

##  1 Introduction

The Bondi–van der Burg–Metzner–Sachs (BMS) symmetry algebra of General
Relativity bms1 ; bms2  has received renewed attention over the past decade
thanks to seminal results by Strominger et. al. strom1 ; stromST ; stromSR ;
stromzhibo . The symmetries of classical as well as perturbative quantum
gravitational scattering (which may include massive scattering particles)
includes a universal BMS symmetry group which acts on all the boundaries of
spacetime including time-like infinity. At quantum level, the existence of BMS
symmetries imply a highly intricate structure of the algebra of boundary
observables which can be used to formulate flat space holography in terms of
such algebra of observables suvrat-flat , paralleling the conception of (far
more developed) holography in asymptotically AdS spacetimes suvrat-ads .

In fact, it is by now becoming increasingly clear that the BMS symmetry
algebra, which is a semi-direct product of supertranslations and celestial
Lorentz transformations, gets enlarged into two possible infinite dimensional
extensions in which the Lorentz algebra is embedded, either the algebra of
meromorphic vector fields btebms  or the algebra of smooth vector fields
clgbms  on the celestial sphere. These two extensions are known in the
literature as extended BMS (eBMS) and generalised BMS (gBMS) respectively. We
will refer to both classes of celestial vector fields, that are either
meromorphic or smooth, as superrotation vector fields.

At the level of the algebra generated by supertranslation and superrotation
vector fields, both extensions of the BMS algebra form an infinite dimensional
Lie algebra. Representation of eBMS algebra is an active area of
investigation. Coadjoint orbits of (e/g)BMS algebra have been studied in
freidelorbit , barnichruzz , paving the way for its quantization via the orbit
method kirilov . In the seminal work cordova , it was shown that gBMS algebra
is faithfully represented in a unitary conformal field theory in Minkowski
spacetime as the algebra of so called light-ray operators on lightlike
boundaries of finite regions.

However the situation is more intricate in gravity at null infinity. As
remarkably shown by Barnich and Troessaert (BT) btchargealgebra , the eBMS
charge algebra at null infinity does not close and the non-closure term is in
fact not a central extension but linear in the asymptotic gravitational field.
Technically, the extension is a 2-cocycle jackiw . That is, it is a closed
2-form on the space of eBMS generators.111A standard example of 2-cocycle
occurs in anomalous gauge theories. That is, the commutator of two gauge
generators in an anomalous gauge theory is a 2-cocycle. This result was
confirmed in a beautiful paper by Distler, Flauger, and Horn (DFH) dfh  by
evaluating the commutator of the quantum eBMS charges on the Fock space of
asymptotic gravitons. Their result complemented an earlier analysis in aar
which showed how Ward identities associated to e/gBMS generators in the so-
called shifted vacuum were consistent with double soft graviton theorem. It
was also proved in dfh  that the eBMS algebra with the BT extension is
consistent with the so called consecutive double soft graviton theorem for
tree-level amplitudes. More in detail, the authors showed that the right hand
side of the commutator of two consecutive double soft theorems up to sub-sub
leading order equals the matrix element of the 2-cocycle extended eBMS
algebra.

However this results appears to be in tension with the following:

  * •

As a consequence of Jacobi identity, we expect the commutators of two
symmetries of the S-matrix to be a symmetry of the S-matrix. A centrally
extended algebra is consistent with this basic fact, but the presence of a
2-cocycle obstructs the interpretation of eBMS algebra as a symmetry algebra
of quantum gravity. In other words, eBMS symmetry appears to be anomalous even
for tree-level S matrix!

  * •

Recent developments in celestial holography have led to a novel realization of
the eBMS algebra ebmsope ; shamik ; alfredo  from conformally soft graviton
theorems confsoft1 ; confsoft2 ; confsoft3 . This realisation provides a
strong evidence that in the infrared sector, flat-space holography is captured
by a conformal field theory (CFT) with a supertranslation current that
generates an abelian ideal and a holographic stress tensor that generates a
Witt algebra with vanishing central charge.
In ebmsope ; alfredo , the celestial eBMS algebra was derived from the
operator product expansion (OPE) involving supertranslation current and
celestial stress tensor, which are the supertranslation and superrotation soft
charges in the celestial basis. This OPE is in turn obtained from conformal
soft graviton theorems. In particular, the OPE between the supertranslation
current $P(z,\bar{z})$ and the (shadow of) superrotation soft mode ${\cal
T}_{ww}$ has a double pole $\frac{1}{(z-w)^{2}}$ ebmsope ; shamik . From this
perspective, the appearance of the BT cocycle anomaly is rather mysterious
since it corresponds to a forth order pole in the corresponding OPE dfh . In
fact, the presence of such a term hints at a possibility that the
supertranslation current is not a primary (we refer to section 5 of dfh  for a
detailed discussion on this issue).

In this paper we take the first steps in resolving these issues. More in
detail, we show that for a class of superrotation vector fields, the
“improved” superrotation charges derived in compere ; cp  lead to a closed
eBMS quantum algebra with no cocycle anomaly. In particular, we consider
superrotations generated by holomorphic (or anti-holomorphic) vector fields on
the celestial sphere which only have singularity at the North pole. This is
equivalent to de-compactifying the celestial sphere to a plane and consider
superrotations generated by holomorphic (or anti-holomorphic) as opposed to
meromorphic vector fields.

Our starting point is the classical analysis of gBMS charges initiated in
compere ; cp ; wbms . In compere  it was shown how the asymptotic phase space
of gravity can be consistently renormalized in order to obtain canonical gBMS
charges at null infinity. This led to several corrections in the original
charge expressions stromSR ; clpilot , in particular due to their non-trivial
dependence on the Geroch tensor geroch . The charge algebra of compere ,
however, still displayed a 2-cocycle. In cp  it was noticed that the 2-cocycle
could be eliminated by a total derivative term in the superrotation charge,
and the corresponding correction on the gravitational symplectic structure was
derived. These anomaly-free gBMS charges were then re-derived from other
approaches lambdabms ; comperenichols , thus giving confidence on the
proposal. Whereas these results were obtained for generalized BMS charges,
they apply equally well to their extended version, leading to an anomaly-free
eBMS charge algebra.

It is not clear, however, if the above classical results carry over to quantum
theory, and if so how they relate to the DFH quantum analysis. A major
obstacle arises because the extended radiative phase of cp  includes soft
modes associated to superrotation symmetry which are not all independent but
are subject to certain non-linear constraints. This complicates the task of
isolating the physical degrees of freedom and computing the corresponding
Poisson brackets. Without a complete understanding of the physical phase
space, the quantization and the connection with the results of dfh  has
remained out of reach.

The analysis of cp  also relied on an extension of the radiative phase space
of GR which did not explicitly include the supertranslation modes. Hence this
extension remained disparate from the phase space derived by He, Lysov, Mitra
and Strominger (HLMS) stromST  in which a _physical_ phase space that included
supertranslation soft modes was derived.

We start by filling both gaps in the present paper. That is, we derive a phase
space at null infinity which we refer to as $\Gamma$ and show that it contains
supertranslation modes, superrotation modes and their conjugate partners.
However the phase space we obtain does not contain all superrotation modes. A
generic superrotation (either meromorphic or smooth) deforms the metric on the
celestial sphere. Quantizing the celestial metric at ${\cal I}$ is subtle and
we do not attempt to do so in this paper. Instead, we consider the (infinite
dimensional) algebra of holomorphic (or anti-holomorphic) superrotations which
have poles only at infinity in the celestial plane.

We then show that we can quantize $\Gamma$ and obtain the quantum version of
the supertranslation and the superrotation charges defined in compere ; cp .
The quantum algebra generated by these charges closes without a 2-cocycle and
show how eBMS is a symmetry of tree-level S matrix for holomorphic
superrotations. As there is no free lunch, there is a price to pay. Namely,
the superrotation soft charge contains in addition to the linear soft operator
(which is proportional to the spin memory), a quadratic operator that depends
on leading soft news and its conjugate constant shear. This operator has a
trivial action on Fock states defined over the trivial vacuum, but has a non-
trivial action on graviton states defined with respect to supertranslation
displaced vacua.

The paper is organised as follows. In section 2, we briefly review the
radiative phase space derived in stromST  which contained an explicit
parametrization of supertranslation modes. In section 3 we review and extend
the analysis of cp  to obtain a phase space $\Gamma$ that explicitly includes
supertanslation as well as superrotation modes. In section 4 we construct an
auxiliary “kinematical” phase space $\Gamma^{\textrm{kin}}$ in which these
“soft” modes are decoupled from the “hard” fields such that $\Gamma$ can be
understood as a constrained phase space inside $\Gamma^{\textrm{kin}}$. This
description is used in section 5 to obtain Poisson brackets in $\Gamma$ via a
Dirac constraint analysis. In section 6 we quantize the physical phase space
and show that the corresponding quantized charges generate a closed eBMS
algebra. We also show how our results are consistent with earlier results in
the literature including those of dfh . We conclude in section 7 with a
summary of results and open questions and offer some preliminary comments on
the relationship between quantized eBMS algebra derived in this paper and the
eBMS algebra in celestial holography.

As we shall see, our focus on holomorphic rather than meromorphic
superrotations greatly simplifies the analysis. The extension of our work to
complete eBMS as well as gBMS requires further investigation. In both of these
cases, the complete radiative phase space should include modes associated to
deformation of the metric on the celestial sphere. In the conclusion section,
we briefly comment on some ideas for extending our results to the case where
superrotations are generated by meromorphic (or smooth) rather than
holomorphic, vector fields on the celestial sphere.

_Conventions:_
We work in units such that $32\pi G=1$. Given a function $f$ on a phase space
with symplectic structure $\Omega$, we define its Hamiltonian vector field
(HVF) $X_{f}$ by the condition $\delta f=\Omega(\delta,X_{f})$. The PB between
two functions $f$ and $g$ is then defined by
$\\{f,g\\}:=\Omega(X_{g},X_{f})=X_{g}(f)=-X_{f}(g)$.

##  2 Review of the radiative phase space

In this section, we briefly review the classical definition of gravitational
radiative phase space in General Relativity as81 . We then review one of the
proposed extensions stromST  in which the soft news and its conjugate partner
are included as part of the phase space degrees of freedom. Our review is
rather brief and the interested reader is encouraged to consult the cited
papers for more details.

The radiative phase space of General Relativity at future null infinity ${\cal
I}^{+}$ was derived by Ashtekar and Streubel (AS) in as81 . Although the
radiative phase space is gauge invariant and can be defined in a coordinate-
independent manner, it can be most easily motivated by analysing the space of
solutions to Einstein equations in Bondi gauge close to future null infinity
${\cal I}^{+}$,

$\displaystyle
ds^{2}\,=\,-\,du^{2}\,-\,2du\,dr\,+\,r^{2}\,(q_{ab}\,+\,\frac{1}{r}\,\sigma_{ab}\,)dx^{a}dx^{b}\,+\,O(\frac{1}{r^{2}})$
(1)

where $(a,b)$ label coordinates on the celestial sphere (we shall later choose
them to be stereographic coordinates), $q_{ab}$ is the unit 2-sphere metric
and $\sigma_{ab}=\sigma_{ab}(u,\hat{x})$ is the shear field associated to the
true degrees of freedom of the radiative gravitational field. We denote by
$(u,\hat{x})$ points on future null infinity, with $u$ the retarded time and
$\hat{x}$ the direction on the celestial sphere.

Ashtekar and Streubel showed that the phase space of General Relativity
without matter sources is parametrised by the shear field
$\sigma_{ab}(u,\hat{x})$ with the symplectic structure

$\Omega_{0}(\delta,\,\delta^{\prime})\,=\,\int_{{\cal
I}^{+}}\,du\,d^{2}\hat{x}\,[\,\delta\,\sigma_{ab}\,\delta^{\prime}\,N^{ab}\,-\,\delta\,\leftrightarrow\,\delta^{\prime}\,]$
(2)

where $N_{ab}\,=\,\partial_{u}\,\sigma_{ab}$ is the news tensor at ${\cal
I}^{+}$. The finiteness of $\Omega_{0}$ is guaranteed by the fall-off
conditions on the shear as we approach the two boundaries of ${\cal I}^{+}$.
AS require minimal fall-offs for this to happen, namely

$N_{ab}(u,\hat{x})\,\overset{u\rightarrow\,\pm\infty}{=}\,O(1/|u|^{1+\epsilon})\quad\text{(AS
fall-offs)}$ (3)

with $\epsilon>0$. However, in order to be able to define superrotation
charges in later sections, we shall require the stronger fall-offs

$N_{ab}(u,\hat{x})\,\overset{u\rightarrow\,\pm\infty}{=}\,O(1/|u|^{2+\epsilon})\quad\text{($\Gamma_{0}$
fall-offs)}$ (4)

with $\epsilon>0$. We will refer to this phase space as $\Gamma_{0}$.222The
discussion in the remainder of this section is however valid for the weaker
fall-offs (3). It is interesting to note that generic radiative fall-offs are
of the form $N_{ab}=O(1/|u|^{2})$ (see e.g. ashoke ; biswajit ). This is
allowed by (3) but forbidden in (4). Fall-offs of the type (4) are found when
computing gravitational radiation at tree-level and correspond to a
$O(\omega^{0})$ subleading soft emission. The $N_{ab}=O(1/|u|^{2})$ behavior
is associated to the so-called tail to the memory effect and corresponds to a
$O(\ln\omega)$ subleading soft emission ashoke ; biswajit . We leave the
generalisation of our work and a completely satisfactory construction of the
radiative phase space that includes tail to the memory modes for future work.
For some progress in this direction we refer the reader to cl1903 ; sayali .

BMS symmetries, that is supertranslations as well as Lorentz transformations
are represented on $\Gamma_{0}$ by their actions on the shear and news
tensors. However, only the latter admits a realization in term Poisson
brackets, see e.g. stromST ; clgbms . In order to fully capture the action of
supertranslations via Poisson brackets, one needs an enhancement of the
radiative phase space. Precisely such an extension was proposed by He, Lysov,
Mitra and Strominger (HLMS) in stromST , with a phase space of the form

$\Gamma_{\textrm{HLMS}}\,\subset\,\Gamma_{0}\,\times\Gamma_{\textrm{s}},$ (5)

where $\Gamma_{\textrm{s}}$ is the “soft sector” parametrized by the soft
(zero frequency component of) the news and its conjugate which is simply the
$u$ independent mode of $\sigma_{ab}$. More in detail, the soft sector is a
$2\,\times\,\infty$ dimensional phase space parametrized by,

$\begin{array}[]{lll}\overset{\scriptscriptstyle
0}{N}\vphantom{N}(\hat{x})\,:\overset{\scriptscriptstyle
0}{N}\vphantom{N}_{ab}(\hat{x})\,=\,D_{a}D_{b}\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}(\hat{x})\\\
C(\hat{x})\,:\,C_{ab}\,=\,D_{a}D_{b}\,C(\hat{x}).\end{array}$ (6)

However as we have not changed the theory, the additional degrees of freedom
are not independent but are related to the fields in $\Gamma_{0}$ via,

$\displaystyle\lim_{u\to-\infty}C_{ab}(u,\hat{x})$ $\displaystyle=$
$\displaystyle D_{a}D_{b}C(\hat{x}),$ (7) $\displaystyle\int
du\,N_{ab}(u,\hat{x})$ $\displaystyle=$ $\displaystyle
D_{a}D_{b}\,\overset{\scriptscriptstyle 0}{N}\vphantom{N}(\hat{x}).$ (8)

Following strom1 , we refer to these conditions as the Christodolou-
Klainermann (CK) constraints. These are the constraints that define the phase
space $\Gamma_{\textrm{HLMS}}$ in (5). Physically they show that the leading
soft news (the displacement memory) is associated to a single function on the
celestial sphere.333They can also be understood in terms of vanishing of the
so-called dual supertranslation charges, see e.g. subsub2 ; pope . As stromST
showed, the CK constraints expressed in this form are second class constraints
and one has to solve them by employing the standard technique of Dirac
brackets. The resulting brackets are such that the soft degrees of freedom
Poisson commute with the finite energy news $N_{ab}(u,\hat{x})$ but they have
non-trivial Poisson brackets with the shear field $\sigma_{ab}(u,\hat{x})$.

##  3 Radiative phase space with superrotation modes

In this section we review the phase space defined in cp , which is an
extension of $\Gamma_{0}$ obtained by adding modes associated to
superrotations. The extension in cp  was defined for $\textrm{Diff}(S^{2})$
superrotations and so the phase space included smooth deformations of the
sphere metric. In this paper however, we restrict attention to holomorphic
vector fields. They leave the 2d metric invariant but act non-trivially on the
space of Geroch tensors comperelong . For simplicity, we will work in a
conformal frame where the celestial 2d metric is the flat one.

We thus start by considering Bondi coordinates $(r,u,z,\bar{z})$ such that the
$r\to\infty$ expansion of the spacetime metric takes the form

$ds^{2}=-2dudr+2r^{2}dzd\bar{z}+r((\sigma_{zz}(u,z,\bar{z})+uT_{zz}(z,\bar{z}))dzdz+c.c.)+\cdots.$
(9)

The metric written above is based on the following conventions and
assumptions:

  * •

We choose the metric on null infinity to be the 2d flat metric,
$q_{z\bar{z}}=1,q_{zz}=q_{\bar{z}\bar{z}}=0$. Whereas our analysis readily
generalizes to non-flat metrics, the underlying plane topology assumption will
play a key simplifying role.444Strictly speaking, our analysis is only be
valid for the case where the celestial surface is a plane rather than a
sphere. We will however continue to use the term ”celestial sphere” to refer
to this surface. See barnichruzz ; celestialtorus  for recent discussions on
the role of the celestial surface topology.

  * •

$\sigma_{zz}$ is the shear and $T_{zz}$ is the so called Geroch tensor. The
news tensor is defined as 555From this perspective, the role of the Geroch
tensor is to substract the non-decaying part of the “naive news”
$N^{\prime}_{ab}\,=\,\partial_{u}\sigma_{ab}+T_{ab}$, thus ensuring finiteness
of various quantities such as the total radiated energy flux. Geroch’s
original derivation of the “corrected” news tensor (10) was motivated by the
need to work with quantities that are covariant under conformal rescalings
geroch .

$N_{zz}=\partial_{u}\sigma_{zz}$ (10)

with

$N_{zz}\stackrel{{\scriptstyle u\to\pm\infty}}{{=}}O(1/|u|^{2+\epsilon})$ (11)

as in (4).

We now analyse the Geroch tensor $T_{ab}$. For simplicity, we focus on the
chiral sector corresponding to $T_{zz}$ but it can be readily generalised to
include the $T_{\bar{z}\bar{z}}$ components as well.

The $T_{zz}\neq 0$ sectors can be generated from the $T_{zz}=0$ one by
considering spacetime diffeomorphisms generated by finite superrotations
$z\to\phi(z)$. Following comperelong  (see also compere ; cp ; sraction ) one
finds the resulting $T_{zz}$ is given by minus the Schwarzian derivative of
$\phi$ wrt to $z$,

$T_{zz}=\frac{\phi^{\prime\prime\prime}(z)}{\phi^{\prime}(z)}-\frac{3}{2}\left(\frac{\phi^{\prime\prime}(z)}{\phi^{\prime}(z)}\right)^{2}.$
(12)

An important property of $T_{zz}$ which will be central to this paper is that
it is holomorphic in the entire complex plane (or in general meromorphic with
poles of order 2)

$\partial_{\bar{z}}T_{zz}=0.$ (13)

The presence of a non-trivial $T$ has non-trivial implications for the
constraints at ${\cal I}^{+}$. In particular, the CK condition (i.e. the
vanishing of the magnetic part of
$\sigma^{\pm}_{zz}=\lim_{u\to\pm\infty}\sigma_{zz}$) is modified in a sector
with $T_{zz}\neq 0$ cp . This modification can be understood as follows.

In the presence of $T$, the sphere derivative $\partial_{z}$ is twisted to a
superrotation-covariant derivative $D_{z}$ (also referred to as Weyl-covariant
derivative barnichruzz ) such that the CK constraints are implemented by the
conditions cp ,

$\sigma^{\pm}_{zz}=-2D^{2}_{z}\sigma^{\pm}=-2\partial_{z}^{2}\sigma^{\pm}+T_{zz}\sigma^{\pm},$
(14)

for some scalars $\sigma^{\pm}$. See subsection 4.3 for further details on the
superrotation-covariant derivative.

We will call by $\Gamma$ the space of all possible fields
$(\sigma_{ab},T_{ab})$, with $ab=zz$ and $\bar{z}\bar{z}$, satisfying
conditions (11), (13), (14) (and their complex conjugated versions),

$\Gamma=\\{(\sigma_{ab},T_{ab}):\eqref{fallnews},\eqref{holT},\eqref{ckcond}\\}.$
(15)

There are two functions on $\Gamma$ that will play a key role in our analysis.
These are the leading and sub-leading soft modes of the news tensor:

$\displaystyle\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}(z,\bar{z})$ $\displaystyle:=$
$\displaystyle\int_{-\infty}^{\infty}N_{zz}(u,z,\bar{z})du,$ (16)
$\displaystyle\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}(z,\bar{z})$ $\displaystyle:=$
$\displaystyle\int_{-\infty}^{\infty}uN_{zz}(u,z,\bar{z})du.$ (17)

Let us now discuss the extended BMS symmetries on $\Gamma$. The spacetime
action of super-translations and superrotations translates into

$\begin{array}[]{llllll}\delta_{f}\sigma_{zz}&=&f\partial_{u}\sigma_{zz}-2D^{2}_{z}f,&\delta_{V}\sigma_{zz}&=&(\mathcal{L}_{V}-\alpha+\alpha
u\partial_{u})\sigma_{zz},\\\
\delta_{f}T_{zz}&=&0,&\delta_{V}T_{zz}&=&\mathcal{L}_{V}T_{zz}-2\partial_{z}^{2}\alpha.\end{array}$
(18)

where $\alpha=1/2(\partial_{z}V^{z}+c.c)$. We remind the reader that $V^{a}$
is an entire vector field with polynomial coefficients. We also note that
contrary to earlier representations btaspects ; stromSR ; clpilot ,
$\sigma_{zz}$ transforms homogeneously under superrotations as the
inhomogeneous term is absorbed in $T_{zz}$. The supertranslation and
superrotation charges are then given by compere ; cp

$\displaystyle P_{f}$ $\displaystyle=$ $\displaystyle\int d^{2}z\int
duN^{zz}\delta_{f}\sigma_{zz}+c.c.$ (19) $\displaystyle J_{V}$
$\displaystyle=$ $\displaystyle\int d^{2}z\left(\int
duN^{zz}\delta_{V}\sigma_{zz}+\tilde{\Pi}^{zz}\delta_{V}T_{zz}\right)+c.c.,$
(20)

where

$\tilde{\Pi}_{zz}:=2\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}+\frac{1}{2}(\sigma^{+}\sigma^{+}_{zz}-\sigma^{-}\sigma^{-}_{zz}).$
(21)

The quadratic term in (21) was introduced in cp  to ensure the consistency
condition

$\delta_{f}J_{V}+\delta_{V}P_{f}=0$ (22)

is satisfied.666This condition only requires the existence of Poisson
brackets, but is otherwise insensitive to the presence or absence of a
2-cocyle in the charge algebra. In particular, the BT charges satisfy (22).
Finally, we can think of $\Gamma$ as a constrained phase space by treating
$\tilde{\Pi}_{ab}$ as the momentum conjugate to $T_{ab}$, subjected to the
constraint equation (21).

The symplectic form on $\Gamma$ implementing the supertranslation and
superrotation charges is given by

$\Omega=\int d^{2}z\left(\int du\delta
N^{zz}\wedge\delta\sigma_{zz}+\delta\tilde{\Pi}^{zz}\wedge\delta
T_{zz}\right)+c.c.$ (23)

which satisfies

$\delta P_{f}=\Omega(\delta,\delta_{f}),\quad\delta
J_{V}=\Omega(\delta,\delta_{V}).$ (24)

We conclude by noting that, in the present case where $\delta T_{zz}$ is
holomorphic and $\delta T_{\bar{z}\bar{z}}$ anti-holomorphic, we can add to
$\tilde{\Pi}_{zz}$ a total $\partial_{z}$ derivative without affecting the
expressions above. We will use this freedom to trade $\tilde{\Pi}_{zz}$ for

$\displaystyle\Pi_{zz}$ $\displaystyle:=$
$\displaystyle\tilde{\Pi}_{zz}-\partial_{z}(\partial_{z}\sigma^{+}\sigma^{-}-\partial_{z}\sigma^{-}\sigma^{+})$
(25) $\displaystyle=$ $\displaystyle 2\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}+\frac{1}{2}(\sigma^{+}+\sigma^{-})\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}.$

We end this section with three remarks.

  * •

$\Pi_{zz}$ defined in Eq. (25) is a sum of leading and sub-leading soft news
terms, such that the coefficient of the leading soft news in the sum is in
fact the constant shear mode $C$. $\Pi_{zz}$ (and $\Pi_{\bar{z}\bar{z}}$) thus
have a cleaner physical interpretation as compared to $\tilde{\Pi}_{zz}$. As
we will show this change from $\tilde{\Pi}_{zz}$ to $\Pi_{zz}$ will play an
important role in simplifying the analysis below, when we determine the
reduced phase space.

  * •

We emphasise that this is one possible parametrisation of $\Pi_{zz}$
indicating one among many possible extensions of $\Gamma_{0}$. This is
consistent with the well known fact that consistency of Poisson brackets and
requirement of faithful representation of asymptotic symmetries does not
uniquely fix the radiative phase space stromST .

  * •

It is important to note that the homogenous change in $T_{zz}$ induced by
infinitesimal holomorphic superrotations,
$\delta_{V}\,T_{zz}\,+\,\partial_{z}^{3}\,V^{z}$, is holomorphic if and only
if the background $T_{zz}$ is holomorphic. Although generically $T_{ab}$ is
meromorphic, in this paper we restrict ourselves to holomorphic $T_{zz}$ (or
anti-holomorphic $T_{\bar{z}\bar{z}}$ ) tensors.

  * •

To summarise, we started with the Ashtekar-Streubel radiative phase space
$\Gamma_{0}$ and extended it by adding modes associated to superrotation
symmetries, namely the pair $(\Pi_{ab},T_{ab})$, where $\Pi_{ab}$ is
_constrained to be_ a sum of leading and subleading soft news (25).

  * •

We can thus think of $\Gamma$ as a subspace in a _kinematical_ phase space

$\Gamma\subset\\{\,\sigma_{ab}(u,\hat{x}),\Pi_{ab}(\hat{x}),\,T_{ab}(\hat{x})\\}.$
(26)

In the same spirit of the HLMS construction, we will use this description in
order to obtain Poisson brackets via the Dirac procedure.

##  4 Kinematical phase space

The phase space $\Gamma$ introduced in the previous section still suffers from
the drawback in the AS phase space $\Gamma_{0}$ regarding the Poisson bracket
realization of supertranslations. In addition, the mode conjugate to $T_{ab}$
has not been determined so far as $\Pi_{ab}$ is associated to the news tensor
via a constraint. Following HLMS, our strategy to solve both issues will be to
isolate the supertranslation and superrotation modes from the phase space
$\Gamma$. This will lead us to a kinematical phase space
$\Gamma^{\textrm{kin}}$ with well defined Poisson brackets that are consistent
with eBMS transformations. The objective of the present section is to
introduce this kinematical phase space and brackets. In the next section we
will follow the Dirac procedure to obtain eBMS-compatible Poisson brackets on
$\Gamma\subset\Gamma^{\textrm{kin}}$.

Our first step will be to isolate the zero mode of the shear
$\sigma_{ab}(u,\hat{x})$.777Our conventions differ from HLMS in two respects:
(i) the zero mode is given by the average of the shear at $u=\pm\infty$ and
(ii) a multiplicative $(-2)$ factor in the definitions of $C$ and
$\overset{\scriptscriptstyle 0}{N}\vphantom{N}$ (30). The second choice
implies $\delta_{f}C=f$ under supertranslations. We thus start by rewriting
the shear as

$\sigma_{zz}(u,z,\bar{z})=\mathring{\sigma}_{zz}(u,z,\bar{z})+C_{zz}(z,\bar{z})$
(27)

with $\mathring{\sigma}_{zz}$ obeying

$\mathring{\sigma}^{+}_{zz}+\mathring{\sigma}^{-}_{zz}=0\quad,\quad\mathring{\sigma}^{\pm}_{zz}:=\lim_{u\to\pm\infty}\mathring{\sigma}_{zz}(u,z,\bar{z}).$
(28)

Note that the news (10) can be entirely written in terms of
$\mathring{\sigma}_{zz}$,

$N_{zz}=\partial_{u}\mathring{\sigma}_{zz}.$ (29)

A simple way to distinguish $\mathring{\sigma}_{ab}$ from $\sigma_{ab}$ is
that $\mathring{\sigma}_{ab}$ contains soft news but no constant shear. The
asymptotic condition (14) can then be written as

$\displaystyle C_{zz}=-2D^{2}_{z}\,C\,,\quad\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}=-2D^{2}_{z}\overset{\scriptscriptstyle
0}{N}\vphantom{N}$ (30)

for scalars $C$ and $\overset{\scriptscriptstyle 0}{N}\vphantom{N}$ such that

$C=\frac{1}{2}(\sigma^{+}+\sigma^{-}),\quad\overset{\scriptscriptstyle
0}{N}\vphantom{N}={\sigma}^{+}-{\sigma}^{-}.$ (31)

In the parametrization (27), the symplectic form (23) now takes the form

$\Omega=\int d^{2}z\left(\int du\delta
N^{zz}\wedge\delta\mathring{\sigma}_{zz}+\delta\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}^{zz}\wedge\delta
C_{zz}+\delta\Pi^{zz}\wedge\delta T_{zz}\right)+c.c.,$ (32)

with $\Pi_{zz}=2\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}+C\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}$.

The eBMS action on the shear sector becomes

$\begin{array}[]{llllll}\delta_{f}\mathring{\sigma}_{zz}&=&f\partial_{u}\mathring{\sigma}_{zz},&\delta_{V}\mathring{\sigma}_{zz}&=&(\mathcal{L}_{V}-\alpha+\alpha
u\partial_{u})\,\mathring{\sigma}_{zz},\\\
\delta_{f}C&=&f,&\delta_{V}C&=&(\mathcal{L}_{V}-\alpha)\,C.\end{array}$ (33)

We note that the inhomogeneous part of supertranslations is absorbed in $C$ so
that $\mathring{\sigma}_{zz}$ transforms homogeneously under both
supertranslations and superrotations.

We will now use the splitting (27) to define auxiliary phase spaces
$\Gamma^{\text{hard}}$ and $\Gamma^{\text{soft}}$, each with its own
symplectic structure and eBMS action. The kinematical phase space will be
defined as
$\Gamma^{\textrm{kin}}=\Gamma^{\text{hard}}\times\Gamma^{\text{soft}}$. The
space $\Gamma$ will then arise as a constrained subspace on this kinematical
space.

###  4.1 Hard phase space

We define $\Gamma^{\text{hard}}$ as the space parametrized by
$\mathring{\sigma}_{ab}$

$\Gamma^{\text{hard}}:=\\{\mathring{\sigma}_{ab}:N_{ab}(u,z,\bar{z})\equiv\partial_{u}\mathring{\sigma}_{zz}(u,z,\bar{z})\,\overset{u\to\pm\infty}{=}\,O(1/|u|^{2+\epsilon})\,\\}$
(34)

with symplectic structure

$\Omega^{\text{hard}}=\int d^{2}zdu\delta
N^{zz}\wedge\delta\mathring{\sigma}_{zz}+c.c.$ (35)

At this stage we do not impose neither the zero mode condition
$\mathring{\sigma}^{+}_{zz}+\mathring{\sigma}^{-}_{zz}=0$ nor the CK
condition. Both conditions will be imposed in the next section as part of the
constraints that define $\Gamma$. Thus, $\Gamma^{\text{hard}}$ appears to be
identical to the AS phase space $\Gamma_{0}$. It however differs in how
supertranslation are represented. Following (33), we define the eBMS action on
$\Gamma^{\text{hard}}$ by

$\delta_{f}\mathring{\sigma}_{zz}=f\partial_{u}\mathring{\sigma}_{zz},\quad\delta_{V}\mathring{\sigma}_{zz}=(\mathcal{L}_{V}-\alpha+\alpha
u\partial_{u})\mathring{\sigma}_{zz},$ (36)

so that supertranslations (and superrotations) act homogenously on
$\Gamma^{\text{hard}}$. It is easy to verify that this action is symplectic
and generated by the charges

$\displaystyle P^{\text{hard}}_{f}$ $\displaystyle=$ $\displaystyle\int
dud^{2}zN^{zz}\delta_{f}\mathring{\sigma}_{zz}+c.c.$ (37) $\displaystyle
J^{\text{hard}}_{V}$ $\displaystyle=$ $\displaystyle\int
dud^{2}zN^{zz}\delta_{V}\mathring{\sigma}_{zz}+c.c.$ (38)

###  4.2 Poisson brackets in the hard phase space

To obtain Poisson brackets (PBs) involving the news, we start by considering
its Hamiltonian vector field (HVF),

$X_{N_{zz}(z,u)}=\frac{1}{2}\frac{\delta}{\delta\mathring{\sigma}_{\bar{z}\bar{z}}(u,z)}.$
(39)

To simplify expressions, here and in the following we will use $z$ rather than
$(z,\bar{z})$ to denote points on the celestial sphere. It is well known that
the shear does not admit a HVF stromST ; clgbms . We can however define its PB
with the news using (39) leading to

$\\{N_{zz}(z,u),\mathring{\sigma}_{\bar{w}\bar{w}}(u^{\prime},w)\\}=-\frac{1}{2}\delta(u-u^{\prime})\delta^{(2)}(z,w),$
(40)

and $\\{N_{zz}(z,u),\mathring{\sigma}_{ww}(u^{\prime},w)\\}=0$. On the other
hand, the non-zero PBs between the news tensor and itself is given by

$\\{N_{zz}(z,u),N_{\bar{w}\bar{w}}(u^{\prime},w)\\}=\frac{1}{2}\delta^{\prime}(u-u^{\prime})\delta^{(2)}(z,w).$
(41)

Other quantities of interest are the leading (16) and subleading (17) soft
modes $\overset{\scriptscriptstyle 0}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}$
and $\overset{\scriptscriptstyle 1}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}$.
Their HVFs are given by888Readers not interested in the rigours of symplectic
geometry of radiative phase space are encouraged to skip these subtleties and
focus solely on the Poisson brackets between different modes.

$X_{\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}}=\int
du\frac{\delta}{\delta\mathring{\sigma}_{\bar{z}\bar{z}}(u,z)},\quad
X_{\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}}=\frac{1}{2}\int
du\,u\frac{\delta}{\delta\mathring{\sigma}_{\bar{z}\bar{z}}(u,z)}$ (42)

Strictly speaking, the second HVF is ill defined on $\Gamma^{\text{hard}}$ as
it does not preserve the fall-offs in (34). We will use it cautiously to
define various PBs, often requiring regularization prescriptions. We shall
then verify that these choices lead to a self-consistent set of Poisson
brackets on the physical phase space.

We now discuss the PBs with the soft modes. We only focus on brackets
involving tensor components of different helicities, since those with the same
helicity are trivially vanishing.

The leading soft mode clearly Poisson commutes with the news and with itself,

$\\{N_{zz}(u,z),\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{\bar{w}\bar{w}}\\}=0,\quad\\{\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz},\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{\bar{w}\bar{w}}\\}=0.$ (43)

The bracket between the news and the subleading mode is found to be

$\\{N_{zz}(u,z),\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{\bar{w}\bar{w}}\\}=\frac{1}{2}\delta^{(2)}(z,w).$
(44)

For the brackets involving a leading and a subleading mode, we take the
prescription $\\{\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz},\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{\bar{w}\bar{w}}\\}:=-X_{\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}}(\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{\bar{w}\bar{w}})=0$.999Notice that
$X_{\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}}(\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{\bar{w}\bar{w}})$ is formally
proportional to $\delta^{(2)}(z,w)\int^{\infty}_{-\infty}du$. Our prescription
may be thought of as a regularization of this divergence. For the brackets
involving the subleading mode with itself, we notice that
$X_{\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz}}(\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{\bar{w}\bar{w}})$ is formally
proportional to $\delta^{(2)}(z,w)\int^{\infty}_{-\infty}udu$. This can
naturally be interpreted as vanishing. Summarizing, our prescription for the
subleading mode is such that it Poisson commutes with the leading mode and
with itself,

$\\{\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz},\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{\bar{w}\bar{w}}\\}=\\{\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{zz},\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{\bar{w}\bar{w}}\\}=0.$ (45)

We conclude the section by bringing attention to the fact that the HVF
operation may not be continuous with respect to expressions involving
$u=\pm\infty$ limits. For example, one has

$\int_{-\infty}^{\infty}du\,X_{N_{zz}(z,u)}=\frac{1}{2}X_{\int_{-\infty}^{\infty}du\,N_{zz}(z,u)}.$
(46)

However, as we will see in subsequent sections, this technicality has no
effect on the final symplectic structure defined on $\Gamma$.

###  4.3 Soft phase space

We now define what we call the soft phase space which is co-ordinatized by
soft modes and their conjugate fields,

$\Gamma^{\text{soft}}:=\\{(C,\overset{\scriptscriptstyle
0}{N}\vphantom{N},\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{ab},T_{ab})\\},$ (47)

where $C$ and $\overset{\scriptscriptstyle 0}{N}\vphantom{N}$ capture,
respectively, the zero mode of the shear and the leading soft news.
$\overset{\scriptscriptstyle 1}{N}\vphantom{N}_{ab}$ represents the sub-
leading soft mode of the news. We note that the vanishing of super-translation
magnetic charge implies that the leading soft news is parametrized by a single
scalar mode on the celestial sphere. However as the superrotation magnetic and
electric charge coincide subsub2 , $\overset{1}{N}_{ab}$ has two independent
modes.

At this stage we treat these fields as independent from
$\Gamma^{\textrm{hard}}$. Their relation with the fields in the hard sector
will be imposed later in terms of the constraints

$\displaystyle\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{ab}$ $\displaystyle=$
$\displaystyle-2D_{a}D_{b}\overset{\scriptscriptstyle 0}{N}\vphantom{N},$ (48)
$\displaystyle\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{ab}$ $\displaystyle=$
$\displaystyle\overset{\scriptscriptstyle 1}{N}\vphantom{N}_{ab}.$ (49)

Similarly, we do not yet require $T_{zz}$ to be holomorphic (and
$T_{\bar{z}\bar{z}}$ antiholomorphic); this condition will also be part of the
constraints to be imposed later.

The action of eBMS on $\Gamma^{\text{soft}}$ is given by

$\begin{array}[]{llllll}\delta_{f}C&=&f,&\delta_{V}C&=&(\mathcal{L}_{V}-\alpha)C\\\
\delta_{f}\overset{\scriptscriptstyle
0}{N}\vphantom{N}&=&0,&\delta_{V}\overset{\scriptscriptstyle
0}{N}\vphantom{N}&=&(\mathcal{L}_{V}-\alpha)\overset{\scriptscriptstyle
0}{N}\vphantom{N}\\\ \delta_{f}\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}&=&2fD^{2}_{z}\overset{\scriptscriptstyle
0}{N}\vphantom{N},&\delta_{V}\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}&=&(\mathcal{L}_{V}-2\alpha)\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\\\
\delta_{f}T_{zz}&=&0,&\delta_{V}T_{zz}&=&\mathcal{L}_{V}T_{zz}-2\partial_{z}^{2}\alpha.\end{array}$
(50)

The expressions for $C$ and $T_{ab}$ were already described in (33) and (18),
and those for $\overset{\scriptscriptstyle 0}{N}\vphantom{N}$ and
$\overset{\scriptscriptstyle 1}{N}\vphantom{N}_{ab}$ can be obtained from the
action on the news cp .

From the splitting of $\Omega$ in (32) and the analgous splitting for the
charges, we define

$\Omega^{\text{soft}}=\int d^{2}z(\delta\overset{\scriptscriptstyle
0}{N}\vphantom{N}^{zz}\wedge\delta C_{zz}+\delta\Pi^{zz}\wedge\delta
T_{zz})+c.c.$ (51) $\displaystyle P^{\text{soft}}_{f}$ $\displaystyle=$
$\displaystyle\int d^{2}z\overset{\scriptscriptstyle
0}{N}\vphantom{N}^{zz}\delta_{f}C_{zz}+c.c.$ (52) $\displaystyle
J^{\text{soft}}_{V}$ $\displaystyle=$ $\displaystyle\int
d^{2}z(\overset{\scriptscriptstyle
0}{N}\vphantom{N}^{zz}\delta_{V}C_{zz}+\Pi^{zz}\delta_{V}T_{zz})+c.c.$ (53)

where

$C_{zz}=-2D^{2}_{z}C,\quad\overset{\scriptscriptstyle
0}{N}\vphantom{N}_{zz}=-2D^{2}_{z}\overset{\scriptscriptstyle
0}{N}\vphantom{N},$ (54) $\Pi_{zz}=2\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}+C\overset{\scriptscriptstyle 0}{N}\vphantom{N}_{zz}.$
(55)

We shall see below that (52), (53) generate the eBMS action (50) on
$\Gamma^{\text{soft}}$.

In order to facilitate computations, in particular to verify that the above
symplectic form and charges lead to a closed eBMS algebra, it will be useful
to isolate $C,\overset{\scriptscriptstyle 0}{N}\vphantom{N}$ from
$\overset{\scriptscriptstyle 1}{N}\vphantom{N}_{ab},T_{cd}$. This will be
achieved through the use of certain identities involving the superrotation-
covariant derivative and the Geroch tensor that we now review.

The superrotation-covariant derivative $D_{a}$ is constructed with the help of
a “potential” $\psi$ for the Geroch tensor (also referred to as Liouville
field compere ) satisfying101010In the parametrization of (12) one has
$\psi(z,\bar{z})=-1/2\ln(\phi^{\prime}(z)\bar{\phi}^{\prime}(\bar{z}))$. There
is change in notation with respect to cp : There, $D_{a}$ denotes the sphere
covariant derivative and $\bar{D}_{a}$ the superrotation-covariant derivative.
The formulas of cp  can be translated to the ones here by the substitutions
$D_{a}\to\partial_{a}$, $\bar{D}_{a}\to D_{a}$ and $\mathcal{R}\to 0$, where
$\mathcal{R}$ is the scalar curvature of the 2d metric.

$T_{zz}=2((\partial_{z}\psi)^{2}+\partial^{2}_{z}\psi).$ (56)

$D_{a}$ acts as the ordinary derivative $\partial_{a}$ plus two contributions:
(i) A multiplicative term of the form $k\partial_{a}\psi$ with $k$ dictated by
the $\alpha$ factor in the action of $\delta_{V}$ (for $C$ and
$\overset{\scriptscriptstyle 0}{N}\vphantom{N}$, $k=-1$) and (ii) Christoffel-
like symbols
$\overset{\scriptscriptstyle\psi}{\Gamma}\vphantom{\Gamma}_{ab}^{c}$ that take
into account the tensorial structure of the field being acted upon. In the
present setting the only non-zero symbols are
$\overset{\scriptscriptstyle\psi}{\Gamma}\vphantom{\Gamma}_{zz}^{z}=-2\partial_{z}\psi$
and
$\overset{\scriptscriptstyle\psi}{\Gamma}\vphantom{\Gamma}_{\bar{z}\bar{z}}^{\bar{z}}=-2\partial_{\bar{z}}\psi$.
For example,

$\displaystyle D_{z}C$ $\displaystyle=$
$\displaystyle\partial_{z}C-\partial_{z}\psi C$ (57) $\displaystyle
D^{2}_{z}C$ $\displaystyle=$
$\displaystyle\partial_{z}D_{z}C-\partial_{z}D_{z}C-\overset{\scriptscriptstyle\psi}{\Gamma}\vphantom{\Gamma}_{zz}^{z}D_{z}C$
(58) $\displaystyle=$ $\displaystyle\partial^{2}_{z}C-\frac{1}{2}T_{zz}C.$
(59)

With these rules one can compute superrotation-covariant derivatives of
arbitrary order. Of particular interest for our purposes is the 4-th order
differential operator

$\displaystyle\mathcal{D}C$ $\displaystyle:=$ $\displaystyle
4(D^{2}_{z}D^{2}_{\bar{z}}+D^{2}_{\bar{z}}D^{2}_{z})C$ (60) $\displaystyle=$
$\displaystyle(8\partial^{2}_{z}\partial^{2}_{\bar{z}}-4(T_{\bar{z}\bar{z}}\partial_{z}^{2}+T_{zz}\partial_{\bar{z}}^{2})+2T_{zz}T_{\bar{z}\bar{z}})C.$
(61)

Note that these operators have the same form when acting on
$\overset{\scriptscriptstyle 0}{N}\vphantom{N}$. From these expressions, one
can verify the identities,

$\displaystyle\delta(\mathcal{D}C)$ $\displaystyle=$
$\displaystyle\mathcal{D}\delta C-4(D^{2}_{z}C\delta
T_{\bar{z}\bar{z}}+c.c.),$ (62) $\displaystyle\delta(D_{z}^{2}C)$
$\displaystyle=$ $\displaystyle D^{2}_{z}\delta C-\frac{1}{2}C\delta T_{zz},$
(63)

and identical identities involving $\overset{\scriptscriptstyle
0}{N}\vphantom{N}$ instead of $C$.

Using the above definitions and identities, one can rewrite the symplectic
form (51) in the following alternative forms

$\displaystyle\Omega^{\text{soft}}$ $\displaystyle=$ $\displaystyle\int
d^{2}z\Big{(}\delta(\mathcal{D}\overset{\scriptscriptstyle
0}{N}\vphantom{N})\wedge\delta C+2(\delta(\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}-2CD^{2}_{z}\overset{\scriptscriptstyle
0}{N}\vphantom{N})\wedge\delta T_{\bar{z}\bar{z}})+c.c.)\Big{)}$ (64)
$\displaystyle=$ $\displaystyle\int
d^{2}z\Big{(}\delta\overset{\scriptscriptstyle
0}{N}\vphantom{N}\wedge\delta(\mathcal{D}C)+2(\delta\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\wedge\delta T_{\bar{z}\bar{z}}+c.c.).\Big{)}$ (65)

Similarly, the soft charges can be rewritten in various ways. For instance

$\displaystyle P^{\text{soft}}_{f}$ $\displaystyle=$ $\displaystyle\int
d^{2}zf\mathcal{D}\overset{\scriptscriptstyle 0}{N}\vphantom{N},$ (66)
$\displaystyle J^{\text{soft}}_{V}$ $\displaystyle=$ $\displaystyle\int
d^{2}z\Big{(}-\mathcal{D}C\delta_{V}\overset{\scriptscriptstyle
0}{N}\vphantom{N}+2(\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\delta_{V}T_{\bar{z}\bar{z}}+c.c.)\Big{)}.$ (67)

We end this section with a few remarks.

  * •

$\Omega^{\text{soft}}$ as written in Eq. (65) makes it explicit that (prior to
imposing the constraints), ${\cal D}C$ is conjugate to
$\overset{\scriptscriptstyle 0}{N}\vphantom{N}$ and $T_{zz}$ is conjugate to
$\overset{\scriptscriptstyle 1}{N}\vphantom{N}_{\bar{z}\bar{z}}$. This will
simplify the constraint imposition enormously.

  * •

The first form of $\Omega^{\text{soft}}$ in Eq. (64) is specially suited for
analyzing supertranslations since

$\delta_{f}(\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}-2CD^{2}_{z}\overset{\scriptscriptstyle
0}{N}\vphantom{N})=0.$ (68)

Together with $\delta_{f}T_{zz}=0$, and using the form (66) for
$P^{\text{soft}}_{f}$ it immediately follows that

$\delta P^{\text{soft}}_{f}=\Omega^{\text{soft}}(\delta,\delta_{f}).$ (69)

  * •

Finally, using (65) and (67) it can be shown after a rather lengthy but
straightforward computation that

$\delta J^{\text{soft}}_{V}=\Omega^{\text{soft}}(\delta,\delta_{V}).$ (70)

###  4.4 Poisson brackets in the soft phase space

In this section we analyse the Poisson brackets of “elementary” fields in
$\Gamma^{\textrm{soft}}$. We need the Green’s function $G(z,w)$ of the
differential operator $\mathcal{D}$ defined in (60) for this purpose,

$\mathcal{D}G(z,w)=\delta^{(2)}(z,w).$ (71)

Although we will not need the explicit form of $G(z,w)$, we know its leading
term in the $T_{ab}$ expansion stromST :

$G(z,w)=\frac{1}{16\pi}|z-w|^{2}\ln|z-w|^{2}+O(T).$ (72)

From the symplectic structure $\Omega^{\text{soft}}$, the HVFs of the
elementary variables are found to be

$\displaystyle X_{C(z)}$ $\displaystyle=$ $\displaystyle-\int
d^{2}wG(z,w)\Big{(}\frac{\delta}{\delta\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w)}+2D^{2}_{w}C(w)\frac{\delta}{\delta\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{ww}(w)}+c.c.\Big{)},$ (73) $\displaystyle
X_{\overset{\scriptscriptstyle 0}{N}\vphantom{N}(z)}$ $\displaystyle=$
$\displaystyle\int d^{2}wG(z,w)\frac{\delta}{\delta C(w)},$ (74)
$\displaystyle X_{\overset{\scriptscriptstyle 1}{N}\vphantom{N}_{zz}(z)}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\frac{\delta}{\delta
T_{\bar{z}\bar{z}}(z)}+2D^{2}_{z}C(z)\int d^{2}wG(w,z)\frac{\delta}{\delta
C(w)},$ (75) $\displaystyle X_{T_{zz}(z)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\delta}{\delta\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{\bar{z}\bar{z}}(z)},$ (76)

The HVFs for $X_{\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{\bar{z}\bar{z}}}$ and $X_{T_{\bar{z}\bar{z}}}$ can be
written similarly. We can now compute elementary Poisson brackets on
$\Gamma^{\text{soft}}$,

$\displaystyle\\{C(z),\overset{\scriptscriptstyle 0}{N}\vphantom{N}(w)\\}$
$\displaystyle=$ $\displaystyle G(z,w),$ (77)
$\displaystyle\\{C(z),\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{ww}(w)\\}$ $\displaystyle=$ $\displaystyle
2D^{2}_{w}C(w)G(z,w),$ (78) $\displaystyle\\{\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}(z),T_{\bar{w}\bar{w}}(w)\\}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\delta^{(2)}(z,w),$ (79)

together with their complex conjugated versions. All remaining brackets
vanish. This completes the characterization of the soft phase space.

##  5 Poisson brackets on $\Gamma$

We now derive the physical phase space at ${\cal I}^{+}$ which includes
supertranslations as well as superrotation modes. Our strategy is the same as
the one employed by HLMS stromST , which is to impose certain constraints on
the extended phase space which express the soft radiative modes as “averages”
of finite energy radiative data. However we now have more constraints thanks
to the sub-leading soft sector parametrized by $(\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{ab},\,T_{ab}\,)$. For the benefit of the reader, we
summarize the $4\times(2\times\infty$) constraints once again:

$\displaystyle\begin{array}[]{lll}F^{0}_{ab}(z,\bar{z})\,:=\,\overset{\scriptscriptstyle
0}{\mathcal{N}}\vphantom{\mathcal{N}}_{ab}(z,\bar{z})\,+\,2\,D_{a}D_{b}\overset{\scriptscriptstyle
0}{N}\vphantom{N}(z,\bar{z})\\\
F^{1}_{ab}(z,\bar{z})\,:=\,\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{ab}(z,\bar{z})\,-\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{ab}(z,\bar{z})\\\
F^{2}_{a}(z,\bar{z})\,:=\,\partial^{b}T_{ab}(z,\bar{z})\\\
F^{3}_{ab}(z,\bar{z})\,:=\,\mathring{\sigma}^{+}_{ab}(z,\bar{z})+\mathring{\sigma}^{-}_{ab}(z,\bar{z})\end{array}$
(84)

where we recall that $ab=zz$ or $\bar{z}\bar{z}$. We regard (84) as functions
on the kinematical phase space

$\Gamma^{\textrm{kin}}=\Gamma^{\text{hard}}\times\Gamma^{\text{soft}}\quad,\quad\Omega^{\textrm{kin}}=\Omega^{\text{hard}}+\Omega^{\text{soft}}.$
(85)

By construction, the constraint surface $\\{F^{I}=0,I=0,1,2,3\\}$ defines the
embedding of the phase space $(\Gamma,\Omega)$ of section 3 inside
$\Gamma^{\textrm{kin}}$, namely,

$\Gamma=\Gamma^{\textrm{kin}}|_{\\{F^{I}=0\\}}\quad,\quad\Omega=\Omega^{\textrm{kin}}|_{\\{F^{I}=0\\}}.$
(86)

Since the Poisson brackets on $\Gamma^{\textrm{kin}}$ are known, we can follow
Dirac’s procedure to obtain Poisson brackets on $\Gamma$. We do so in appendix
A. There, we show that the constraints are indeed second class so that we can
apply Dirac’s formula. It is found that from all possible brackets among
elementary variables, only three get modified, namely

$\displaystyle\begin{array}[]{lll}\\{\,N_{\bar{z}\bar{z}}(u),\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{ww}\,\\}_{\star}\,=\,\\-\,O_{\bar{w}}\,\delta^{(2)}(z,w)\\\
\\{\,C(w),\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\,\\}_{\star}\,=\,2(1\,-\,O_{z})D_{z}^{2}C(z)\,G(w,z)\\\
\\{\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz},\,T_{\bar{w}\bar{w}}\,\\}_{\star}\,=\,-\,\frac{1}{2}(1-O_{w})\delta^{(2)}(z,w),\end{array}$
(90)

where $O_{w}\,=\,\partial_{w}\,\partial_{w}^{-2}\,\partial_{w}$ and
$O_{\bar{w}}\,=\,\partial_{w}\,\partial_{\bar{w}}^{-2}\,\partial_{\bar{w}}$.
Together with their complex conjugated versions. All remaining Dirac brackets
coincide with the kinematical ones, as presented in section 4.

Operators such as $(1\,-\,O_{w}\,)$ ensure that the constraint
$\partial_{w}\,T_{\bar{w}\bar{w}}\,=\,0$ is satisfied. $O_{w},\,O_{\bar{w}}$
are symmetric operators and have the same kernel and co-kernel.
$O_{w}\,=\,\partial_{w}\partial_{w}^{-2}\,\partial_{w}$ is the operator whose
kernel and co-kernel is the space of anti-holomorphic functions on the complex
plane, and such that it reduces to the identity on the space of smooth
functions (modulo the anti-holomorphic kernel).

In particular, this implies that if $T_{\bar{w}\bar{w}}$ is smeared with a
function which is entire anti-holomorphic, than its bracket with
$\overset{\scriptscriptstyle 1}{N}\vphantom{N}_{zz}$ vanishes.

For completeness we present the remaining non-zero Dirac brackets

$\displaystyle\begin{array}[]{lll}\\{N_{zz}(z,u),N_{\bar{w}\bar{w}}(u^{\prime},w)\\}_{\star}=\frac{1}{2}\delta^{\prime}(u-u^{\prime})\delta^{(2)}(z,w),\\\
\\{C(z),\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w)\\}_{\star}=G(z,w),\end{array}$ (93)

whose expression coincide with the kinematical ones.

Eqs. (90), (93) provide the non-vanishing Poisson brackets on $\Gamma$. They
can be thought of as an extension of the HLMS brackets that incorporates the
subleading soft modes $T_{ab}$ and $\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{cd}$. We observe a crucial structural difference with the
HLMS space: The hard modes $N_{ab}(u,z,\bar{z})$ do not Poisson commute with
$\overset{\scriptscriptstyle 1}{N}\vphantom{N}_{ab}$, whereas in
$\Gamma_{\textrm{HLMS}}$ these hard degrees of freedom have vanishing Poisson
brackets with the soft modes $C(z,\bar{z})$ and $\overset{\scriptscriptstyle
0}{N}\vphantom{N}(z,\bar{z})$. As we will see below, this has rather non-
trivial consequences in the quantum theory. The existence of the sub-leading
soft modes imply that the soft Hilbert space and finite energy graviton Fock
space do not factorise.

##  6 Quantization of $\Gamma$

In order to quantize the space $\Gamma$, we start by choosing a family of
elementary phase space functions to be promoted to operators. We consider the
Fourier transform of the news tensor

$N_{zz}(\omega):=\int_{-\infty}^{\infty}due^{i\omega u}N_{zz}(u,z,\bar{z}),$
(94)

together with the soft sector variables. That is, we seek a Hilbert space
realization of operators
$\widehat{N}_{zz}(\omega),\widehat{C}(z),\widehat{N}^{0}(z),\widehat{T}_{zz},\widehat{N}^{1}_{zz}$
with commutators dictated by the Dirac brackets (90), (93):

$[\widehat{N}_{zz}(\omega),\widehat{N}_{\bar{w}\bar{w}}(\omega^{\prime})]=\hbar\pi\,\omega\delta(\omega+\omega^{\prime})\delta^{(2)}(z,w),$
(95)
$[\widehat{N}_{zz}(\omega),\widehat{N}^{1}_{\bar{w}\bar{w}}]=i\hbar\pi\delta(\omega)\,O_{w}\,\delta^{(2)}(z,w)$
(96) $[\widehat{C}(z),\widehat{N}^{0}(w)]=i\hbar\widehat{G}(z,w)$ (97)
$[\widehat{C}(z),\widehat{N}^{1}_{ww}]=2i\hbar(1\,-\,O_{w}\,)\widehat{D}^{2}_{w}\widehat{C}(w)\widehat{G}(z,w),$
(98)
$[\widehat{N}^{1}_{zz},\widehat{T}_{\bar{w}\bar{w}}]=-\frac{i\hbar}{2}(1-O_{w}\,)\delta^{(2)}(z,w),$
(99)

with the corresponding hermitian conjugated versions and with all remaining
commutators being zero. Both the Green’s function and the superrotation-
covariant derivative become operators due to their dependence on the Geroch
tensor. The RHS of (98), however, does not present operator order ambiguities
since $\widehat{C}$ and $\widehat{T}_{zz}$ commute.

We start by constructing a Hilbert space based on the following set of
maximally commuting operators:

$\\{\widehat{N}_{zz}(\omega),\widehat{N}_{\bar{z}\bar{z}}(\omega),\omega>0\\},\quad\widehat{N}^{0}(z),\quad\widehat{T}_{zz},\widehat{T}_{\bar{z}\bar{z}}$
(100)

This choice is motivated by the fact that News tensors and the displacement
memory are physical observables in the theory. We then consider a Hilbert
space of the form

$\mathcal{H}=\mathcal{H}^{\text{hard}}\otimes\mathcal{H}^{\text{soft}},$ (101)

with $\mathcal{H}^{\text{hard}}$ the graviton Fock space associated to the
positive frequency news operators and $\mathcal{H}^{\text{soft}}$ the space of
wavefunctionals $\Psi(N,T)$ where $N\equiv\overset{\scriptscriptstyle
0}{N}\vphantom{N}(z)$ and $T\equiv(T_{zz},T_{\bar{z}\bar{z}})$ label the
eigenvalues of the corresponding soft operators,

$\displaystyle\widehat{N}^{0}(z)\Psi(N,T)$ $\displaystyle=$ $\displaystyle
N(z)\Psi(N,T)$ (102) $\displaystyle\widehat{T}_{zz}\Psi(N,T)$ $\displaystyle=$
$\displaystyle T_{zz}\Psi(N,T)$ (103)
$\displaystyle\widehat{T}_{\bar{z}\bar{z}}\Psi(N,T)$ $\displaystyle=$
$\displaystyle T_{\bar{z}\bar{z}}\Psi(N,T).$ (104)

Thus, a basis for $\mathcal{H}$ is given by states of the form

$|(p_{1},h_{1}),\ldots,(p_{n},h_{n});N,T\rangle,$ (105)

where the $(p_{i},h_{i}),i=1,\ldots,n$ label momenta and helicity of the Fock-
space gravitons and $N,T$ are eigenvalues of the operatos
$\widehat{N}^{0}(z)$, $\widehat{T}_{zz}$ and $\widehat{T}_{\bar{z}\bar{z}}$.

We will occasionally find it convenient to work with states that are smeared
over the soft variables, for which we shall use the notation

$|\\{p_{i},h_{i}\\}\rangle_{\Psi(N,T)}:=\int
dNdT\Psi(N,T)|\\{p_{i},h_{i}\\};N,T\rangle.$ (106)

These states can be interpreted as graviton Fock states “dressed” by soft
wavefunctionals $\Psi(N,T)$, similar to the fiber-bundle structure advocated
in aalectures .

We now describe the remaining operators. The zero-mode shear $\widehat{C}(z)$
is naturally defined as an operator that is trivial on
$\mathcal{H}^{\text{hard}}$ and that on $\mathcal{H}^{\text{soft}}$ acts
according to

$\widehat{C}(z)\Psi(N,T)=i\hbar\int d^{2}wG_{T}(z,w)\frac{\delta}{\delta
N(z)}\Psi(N,T),$ (107)

thus ensuring the commutator (97). We use the notation $G_{T}$ for the Green’s
function in order to make explicit its dependence on the Geroch tensor label
$T$. The subleading news operator is more subtle, as it is sensitive to both
soft and hard sectors. The commutators (96) and (99) suggest the definition

$\widehat{N}^{1}_{zz}|\\{(p_{i},h_{i})\\}\rangle_{\Psi(N,T)}=\big{(}-iO_{z}\,\lim_{\omega\to
0}\partial_{\omega}\widehat{N}_{zz}(\omega)-\frac{i\hbar}{2}(1-O_{z})\,\frac{\delta}{\delta
T_{\bar{z}\bar{z}}}\big{)}|\\{(p_{i},h_{i})\\}\rangle_{\Psi(N,T)}.$ (108)

By construction, this expression satisfies (96) and (99). One can further
check that (108) and (107) satisfy (98). It is also easy to see that the above
operators reproduce the vanishing commutator relations dictated by the Dirac
brackets.

###  6.1 eBMS quantum charges

Promoting the classical expressions to operators we have

$\displaystyle\hat{P}_{f}$ $\displaystyle=$
$\displaystyle\hat{P}^{\text{hard}}_{f}+\hat{P}^{\text{soft}}_{f}$ (109)
$\displaystyle\hat{J}_{V}$ $\displaystyle=$
$\displaystyle\hat{J}^{\text{hard}}_{V}+\hat{J}^{\text{soft}}_{V}$ (110)

We define the hard charges as the standard Fock-space hard charges used in the
literature (see e.g. stromST ; stromSR ; dfh ). These act non-trivially on the
hard Hilbert space factor according to:

$\displaystyle\hat{P}^{\text{hard}}_{f}|\\{(p_{i},h_{i})\\};N,T\rangle$
$\displaystyle=$
$\displaystyle\sum_{i}E_{i}f(z_{i},\bar{z}_{i})|\\{p_{i},h_{i}\\};N,T\rangle$
(111) $\displaystyle\hat{J}^{\text{hard}}_{V}|\\{(p_{i},h_{i})\\};N,T\rangle$
$\displaystyle=$
$\displaystyle-i\hbar\sum_{i}\delta^{i}_{V}|\\{p_{i},h_{i}\\};N,T\rangle$
(112)

where $(E_{i},z_{i},\bar{z}_{i})$ parametrize the momenta of the $i$-th
graviton and
$\delta^{i}_{V}=V^{z_{i}}\partial_{z_{i}}+V^{\bar{z}_{i}}\partial_{\bar{z}_{i}}\pm(\partial_{z_{i}}V^{z_{i}}-\partial_{\bar{z}_{i}}V^{\bar{z}_{i}})-1/2(\partial_{z_{i}}V^{z_{i}}-\partial_{\bar{z}_{i}}V^{\bar{z}_{i}})E_{i}\partial_{E_{i}}$.

For the soft charges we consider

$\displaystyle\hat{P}^{\text{soft}}_{f}$ $\displaystyle=$ $\displaystyle\int
d^{2}zf(z)\widehat{\mathcal{D}}\widehat{N}^{0}(z),$ (113)
$\displaystyle\hat{J}^{\text{soft}}_{V}$ $\displaystyle=$ $\displaystyle\int
d^{2}z\Big{(}-\delta_{V}\widehat{N}^{0}(z)\widehat{\mathcal{D}}\widehat{C}(z)+2(\delta_{V}\widehat{T}_{zz}\widehat{N}^{1}_{\bar{z}\bar{z}}+h.c.)\Big{)}.$
(114)

$\hat{P}^{\text{soft}}_{f}$ is purely multiplicative and so it does not
present operator order ambiguities (the differential operator $\mathcal{D}$
becomes a multiplicative Hilbert-space operator $\widehat{\mathcal{D}}$ due to
its dependence on the Geroch tensor). For $\hat{J}^{\text{soft}}_{V}$ we
choose an operator ordering such that multiplicative operators are on the left
and derivative operators on the right. This ordering choice will be justified
a posteriori, as we will show that the eBMS algebra closes for
$\hat{J}_{V}^{\textrm{soft}}$ defined as in (113).

###  6.2 Ward identities of eBMS charges

Before proving that the quantized eBMS algebra closes, we first analyse the
Ward identity associated to $\hat{P}_{f},\,\hat{J}_{V}$ for tree-level S
matrix. The equivalence of Ward identities for supertranslations and
superrotations with leading and sub-leading soft graviton theorem has been
extensively analysed in the literature, where the quantized soft charges are
defined via soft limits of finite energy insertions, with the seminal
references being stromST ; stromSR . However, in the last few years, the
quantization of $\hat{P}_{f}$ in the extended phase space
$\Gamma_{\textrm{HLMS}}$ stromIR ; akhoury1 ; akhoury2 ; acl ; pate  has led
to a novel implication of the supertranslation Ward identity. That the vacuum
transition from incoming to outgoing scattering state in a generic
gravitational scattering is non-trivial and is constrained by the infinity of
conservation laws. These constraints are of course consistent with Weinberg
soft theorem, but can be interpreted as a statement in a “dual basis” in which
the scattering states are eigen-states of soft news operator rather than being
displaced by it (via a soft graviton insertion).

We review the basic idea of stromIR ; akhoury1 ; akhoury2  below and then show
that the quantization of $\hat{J}_{V}$ on $\Gamma$ leads to an analogous
implication for the superrotation Ward identity.

For the purpose of this section, we adopt the following notation for basis
states. We will denote the generic outgoing scattering state as
$|\,\textrm{out},\,N,T\,\rangle$. Here out denotes an outgoing multi-graviton
state where gravitons have arbitrary helicity and _hard_ momenta,111111Hard
momenta graviton states simply corresponds to states created by
$a_{\omega,\hat{p}}^{\dagger}$ where $\omega\,>\,0$. as in (105). As discussed
earlier, these states are eigenstates of hard graviton momentum operators as
well as leading soft news and Geroch tensor. We will also denote a state with
trivial soft modes, namely $|\,\textrm{in},\,N=0,\,T=0\,\rangle$ as
$|\,\textrm{in}\,\rangle$. In particular, we will consider all the in-coming
states to have trivial soft modes.

The Ward identity associated to $\hat{P}_{f}$ can finally be written as,

$\displaystyle\langle\,\textrm{out},\,N,\,T\,|\,[\hat{P}_{f}^{\textrm{soft}},\,S\,]\,|\,\textrm{in}\,\rangle\,=\,-\,\langle\,\textrm{out},\,N,\,T\,|\,[\hat{P}_{f}^{\textrm{hard}},\,S\,]\,|\,\textrm{in}\,\rangle\,,$
(115)

where the soft charge is defined in Eq. (113). The LHS of the Ward identity in
Eq. (115) can thus be evaluated as,

$\displaystyle\langle\,\textrm{out},\,N,\,T\,|\,[\hat{P}_{f}^{\textrm{soft}},\,S\,]\,|\,\textrm{in}\,\rangle\,=\,\int\,d^{2}z\,{\cal
D}f(z)\,N(z)\,\langle\,\textrm{out},\,N,\,T\,|\,S\,|\,\textrm{in}\,\rangle\,$
(116)

Thus, the supertranslation Ward identity can be written as,

$\displaystyle\int\,d^{2}z\,{\cal
D}f(z)\,N(z)\,\langle\textrm{out},\,N,\,T\,|\,S\,|\,\textrm{in}\,\rangle\,=\,-\,\langle\textrm{out},\,N,\,T\,|\,[\hat{P}_{f}^{\textrm{hard}},\,S\,]\,|\,\textrm{in}\,\rangle\,.$
(117)

As was shown in stromIR ; akhoury1 ; akhoury2 , this identity thus constrains
$N$ in terms of the eigenvalue of the hard supertranslation charge which is
effectively the Weinberg soft factor.

The Ward identity associated to $\hat{J}_{V}$ presents two challenges. Since
the contribution to the soft charge that is linear in
$\overset{\scriptscriptstyle 1}{N}\vphantom{N}_{ab}$ exponentiates to a
“translation operator” in the space of Geroch tensors, the relationship of
this Ward identity with Cachazo-Strominger soft theorem is not immediately
clear.121212This subtlety parallels the subtlety in relating the two versions
of supertranslation Ward identities, namely the one which constrains the soft
vacuum transitions (reviewed above) with the one leading to Weinberg’s soft
theorem. That is, in the standard Fock-space formulation of the Ward identity,
one defines the superrotation soft charge in terms of
$\lim_{\omega\,\rightarrow\,0}\partial_{\omega}\,\omega\langle\,\textrm{out}\,|\,a_{\omega,\hat{p}}\,S\,|\,\textrm{in}\,\rangle$.
But in the new quantization of $\hat{J}_{V}^{\textrm{soft}}$, the term linear
in $\widehat{N}^{1}_{ab}$ is defined as a sum of the zero frequency operator
and $\frac{\delta}{\delta T_{zz}}$ which may be interpreted as
$\langle\,\textrm{out}\,|\,\lim_{\omega\,\rightarrow\,0}\partial_{\omega}\,\omega\,a_{\omega,\hat{p}}\,S\,|\,\textrm{in}\,\rangle$.
Depending on the poles of $V$, only one of the two terms contribute to the
soft charge. Furthermore, $\hat{J}_{V}^{\textrm{soft}}$ contains quadratic
operators involving the leading soft modes and it is not a priori clear
whether the corresponding Ward identity is affected by the inclusion of these
new terms.

In fact, the superrotation soft charge appears to mix the hard and soft
Hilbert space factors due to the subleading soft news operator (108). However,
when one studies the action of $\hat{J}^{\text{soft}}_{V}$ (53) on a state
(106), one finds only one of the two terms in $\hat{N}^{1}_{zz}$ is non-
trivial.

  * •

If $\delta_{V}T$ is holomorphic, then one finds the hard-sector part of
$\widehat{N}^{1}_{zz}$ drops out due to the holomorphicity of
$\delta_{V}T_{zz}$. The action is then found to be given by

$\displaystyle\begin{array}[]{lll}\hat{J}^{\text{soft}}_{V}|\\{(p_{i},h_{i})\\}\rangle_{\Psi(N,T)}=\\\
\hskip 36.135pt-i\hbar\int d^{2}z\Big{(}\delta_{V}N(z)\frac{\delta}{\delta
N(z)}+(\delta_{V}T_{zz}\frac{\delta}{\delta
T_{zz}}+c.c.)\Big{)}|\\{(p_{i},h_{i})\\}\rangle_{\Psi(N,T)},\end{array}$ (120)

where we recall that $\delta_{V}N=(\mathcal{L}_{V}-\alpha)N$ and
$\delta_{V}T_{zz}=\mathcal{L}_{V}T_{zz}-2\partial_{z}^{2}\alpha$.

  * •

Even though $\hat{J}_{V}$ is strictly speaking only defined for holomorphic
$\delta T_{ab}$, we now consider an extension of its domain of validity to
meromorphic vector fields.131313The reason we expect our charges to give the
correct Ward identity for all $V^{a}$ is simply that we do not expect the
subtlety of the change in the sphere metric to affect the single Ward identity
associated to infinitesimal superrotations. The added subtlety would show up
when we consider ‘higher order” structures such as commutators of two charges.
It can be immediately be verfied that if $\delta_{V}T$ has poles at finite
$z$, then the second term in $\hat{N}^{1}_{zz}$ drops out and the soft charge
can be written as,

$\displaystyle\begin{array}[]{lll}\hat{J}^{\text{soft}}_{V}|\\{(p_{i},h_{i})\\}\rangle_{\Psi(N,T)}=\\\
\hskip 14.45377pt-i\hbar\int d^{2}z\Big{(}\delta_{V}N(z)\frac{\delta}{\delta
N(z)}-\,O_{z}\,\delta_{V}T_{zz}\lim_{\omega\to
0}\partial_{\omega}\widehat{N}_{zz}(\omega)\,+c.c.)\Big{)}|\\{(p_{i},h_{i})\\}\rangle_{\Psi(N,T)}\end{array}$
(123)

In the second case, the Ward identity of $\hat{J}_{V}$ implies the Cachazo-
Strominger soft theorem. Hence in the rest of the section, we analyse the
first case in which the soft superrotation charge has a non-trivial action on
$T_{ab}$. We now evaluate the Ward identity between out-states of the form
$|\,\textrm{out},\,N,\,T\,=\,0\,\rangle$ and in-states
$|\,\textrm{in}\,\rangle$ as before. The computation presented here can be
easily generalised to generic scattering states. The action of the soft
superrotation charge on the in and the out states is given by

$\displaystyle\begin{array}[]{lll}\hat{J}^{\text{soft}}_{V}|\textrm{in}\,\rangle=-i\hbar\int
d^{2}z(\delta_{V}T_{zz}\frac{\delta}{\delta
T_{zz}}+c.c.)|\,\textrm{in},\,N=\,0\,T=0\,\rangle\\\
\hat{J}^{\text{soft}}_{V}|\textrm{out},\,N,\,T=0\,\rangle=-i\hbar\int
d^{2}z(\delta_{V}T_{zz}\frac{\delta}{\delta
T_{zz}}+c.c.)|\,\textrm{out},\,N,\,T=0\,\rangle\\\ \hskip
180.67499pt-i\,\hbar\,\int\,\delta_{V}N(z)\frac{\delta}{\delta
N(z)}\,|\,\textrm{out},\,N,\,T\,=\,0\rangle\end{array}$ (127)

As $\delta_{V}\,N(z)\,=\,(L_{V}\,-\,\alpha\,)\,N(z)$, the action of
$\int\,\delta_{V}N(z)\,\frac{\delta}{\delta N(z)}$ on the
$|\,\textrm{in}\,\rangle$ state vanishes.

In fact we note that the state obtained by action of $\,\frac{\delta}{\delta
N(z)}$ on the out state will not contribute to the Ward identity since $N$ is
already constrained by the leading soft theorem. More in detail as,

$\displaystyle\begin{array}[]{lll}\langle\,\textrm{out},\,N+\lambda\,\delta_{V}N,T=0\,|\,S\,|\,\textrm{in},\lambda\,\delta_{V}\,\rangle\,=\,0\,\forall\,\lambda\\\
\implies\,\lim_{\lambda\,\rightarrow\,0}\,\frac{d}{d\lambda}\,\langle\,\textrm{out},\,N+\lambda\,N,T=0\,|\,S\,|\,\textrm{in}\,\rangle\,=\,0\end{array}$
(130)

By combining Eqs. (127), (111) and (130) we get,

$\displaystyle\int\,d^{2}z\,\delta_{V}T_{zz}\,\langle\,\textrm{out}\,N,T=0|\,[\,\frac{\delta}{\delta
T_{zz}},\,S\,]\,|\,\textrm{in},N=0,T=0\,\rangle$ $\displaystyle=$ (131)
$\displaystyle-\,\langle\,\textrm{out},\,N,\,T\,=\,0\,|\,[\,\hat{J}_{V}^{\textrm{hard}},\,S\,]\,|\,\textrm{in}\,\rangle$
(132)

The LHS of (131) can be understood as

$\displaystyle\lim_{\lambda\,\rightarrow\,0}\,\frac{d}{d\lambda}\,[\,\langle\,\textrm{out}\,N,\lambda\,\delta_{V}T|\,S\,|\,\textrm{in}\,\rangle\,-\,\langle\,\textrm{out}\,N,T=0|\,S\,|\,\textrm{in},\,N=0,\lambda\,\delta_{V}T\rangle\,\,]$
(133)

The Ward identity thus implies that the infinitesimal change $\delta_{V}T$ in
the Geroch tensor is equivalent to the action of sub-leading soft graviton
operator on hard scattering states.

As we saw above, Cachazo-Strominger soft graviton theorem is realised as a
Ward identity of the superrotation charge in the case $\delta_{V}T$ is
meromorphic. But if $\delta_{V}T$ is holomorphic (except pole at infinity)
then the relationship between the superrotation Ward identity and the Cachazo-
Strominger soft theorem deserves further scrutiny.

We end this section with some comments.

  * •

The operator action of $\widehat{N}^{1}_{zz}$ is a sum of two operators which
act respectively on the hard and soft Hilbert space factors (108). One is the
“familiar” sub-leading soft graviton insertion of the type
$\lim_{\omega\,\rightarrow\,0}\partial_{\omega}\,\widehat{N}_{zz}(\omega,\hat{x})$
and the other one is the soft operator which is linear in
$\frac{\delta}{\delta T_{zz}}$. We see that for the case rigorously analysed
in this paper, $\widehat{N}^{1}_{zz}$ acts as the latter but our preliminary
analysis suggests that for a generic superrotation, it is precisely one of
these two operators that contribute to the soft superrotation charge. This may
offer a new perspective on the relationship between sub-leading soft graviton
theorem and Ward identities, akin to the relationship between dressed vacua,
Weinberg soft theorem and supertranslation symmetries. We leave a more
thorough analysis of this aspect and its impact on the dressed states for
future work.

  * •

The second class constraint $F^{1}_{ab}\,=\,\overset{\scriptscriptstyle
1}{\mathcal{N}}\vphantom{\mathcal{N}}_{ab}\,-\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{ab}\,=\,0$ was strongly imposed classically by using Dirac
brackets. On the other hand, had we first quantized the extended phase space
and imposed $F^{1}_{ab}$ at quantum level, it would imply (at least formally)
that we identify $\widehat{N}^{1}_{ab}$ with $\widehat{{\cal
N}}^{1}_{ab}\,=-i\lim_{\omega\to 0}\partial_{\omega}\widehat{N}_{zz}(\omega)$.
This could bring the Ward identity in the more “traditional form” of a sub-
leading soft graviton theorem for all superrotations.

  * •

We chose the scattering states to be eigenstates of $\widehat{N}^{0}$ and
$\widehat{T}_{ab}$ since $\widehat{N}^{1}_{ab}$ does not commute with the news
tensor $\widehat{N}_{ab}(\omega,\hat{x})$. However this commutator has support
only at $\omega=0$ and hence it may be possible to work with eigenstates of
$\widehat{N}^{0}$ and $\widehat{N}^{1}_{ab}$ instead. In such case, the
superrotation Ward identity would take a form analogous to (117), equating a
vacuum transition parametrized by the difference of $N^{1}_{ab}(\hat{x})$
between in and out states with an amplitude with no vacuum transition but with
hard states “super-rotated” by the angular momentum operators.

###  6.3 eBMS quantum algebra

We now evaluate the commutator of the quantum eBMS charges. We start by
noticing that the hard part of the operators satisfy the extended BMS algebra
(see e.g. dfh ):

$\displaystyle[\hat{P}_{f}^{\text{hard}},\hat{P}_{f^{\prime}}^{\text{hard}}]$
$\displaystyle=0$ (134)
$\displaystyle[\hat{P}_{f}^{\text{hard}},\hat{J}_{V}^{\text{hard}}]$
$\displaystyle=-i\hbar\hat{P}^{\text{hard}}_{V(f)}$ (135)
$\displaystyle[\hat{J}_{V}^{\text{hard}},\hat{J}_{V^{\prime}}^{\text{hard}}]$
$\displaystyle=i\hbar\hat{J}^{\text{hard}}_{[V,V^{\prime}]}.$ (136)

where $V(f)=(\mathcal{L}_{V}-\alpha)f$.

We now discuss the commutators involving the soft part the operators. The only
missing operator action to complete this computation is that of
$\hat{P}_{f}^{\textrm{soft}}$. The soft supermomentum has a simple
multiplicative action that is only sensitive to the soft Hilbert space,

$\hat{P}^{\text{soft}}_{f}|\\{(p_{i},h_{i})\\}\rangle_{\Psi(N,T)}=P^{\text{soft}}_{f}(N,T)|\\{(p_{i},h_{i})\\}\rangle_{\Psi(N,T)}$
(137)

where

$P^{\text{soft}}_{f}(N,T)=\int d^{2}xf\mathcal{D}_{T}N$ (138)

is the value of the classical soft supermomentum associated with leading news
$N$ and Geroch tensor $T$. We use the notation $\mathcal{D}_{T}$ to make
explicit the $T$-dependence of the differential operator $\mathcal{D}$ (60).

Eqs. (137), (120) imply that the hard and soft charges commute with each
other.141414Notice this is technically different from the individual terms
that contribute to the commutator in dfh . This is related to the fact that
our definition of soft superrotation charge includes a contribution from the
zero mode of the shear that is usually considered as part of the hard charge.
To complete the computation of the charge algebra we need to evaluate the
commutator of the soft charges. Since $\hat{P}^{\text{soft}}_{f}$ acts
multiplicatively it follows that

$[\hat{P}_{f}^{\text{soft}},\hat{P}_{f^{\prime}}^{\text{soft}}]=0.$ (139)

Given the expression (120) for the action of $\hat{J}^{\text{soft}}$ it easy
to show that

$[\hat{J}_{V}^{\text{soft}},\hat{J}_{V^{\prime}}^{\text{soft}}]=i\hbar\hat{J}^{\text{soft}}_{[V,V^{\prime}]}$
(140)

and

$[\hat{P}^{\text{soft}}_{f},\hat{J}^{\text{soft}}_{V}]\Psi[N,T]=i\hbar\delta_{V}(P^{\text{soft}}_{f}(N,T))\Psi[N,T],$
(141)

where $\delta_{V}$ acts on the $N$ and $T$ labels of $P_{f}(N,T)$. Thus, to
prove the algebra closure we need to show that

$\delta_{V}P_{f}(N,T)=-P_{V(f)}(N,T).$ (142)

This identity follows in fact due to the superrotation-covariance of the
classical charge cp :

$\delta_{V}P_{f}(N,T)=\delta_{V}\int d^{2}xf\mathcal{D}_{T}N=\int
d^{2}xf\delta_{V}(\mathcal{D}_{T}N)=-\int d^{2}xV(f)\mathcal{D}_{T}N,$ (143)

where

$\delta_{V}(\mathcal{D}_{T}N)=(\mathcal{L}_{V}+3\alpha)\mathcal{D}_{T}N.$
(144)

Notice that the $T$-dependence of the supermomentum, appearing through the use
of the superrotation-covariant differential operator $\mathcal{D}_{T}$, is
what ensures the transformation property (144) that underlies (142). It is
interesting to compare with the situation where one works in the $T=0$ sector
and does not include $\delta_{V}T$ terms. Denoting such action by
$\delta^{\prime}_{V}$, one finds

$\displaystyle\delta^{\prime}_{V}P_{f}(N,T=0)$ $\displaystyle:=$
$\displaystyle\int d^{2}xf\mathcal{D}_{0}\delta_{V}N$ (145) $\displaystyle=$
$\displaystyle\int d^{2}xf\delta_{V}(\mathcal{D}_{T}N)|_{T=0}-\int
d^{2}xf\delta_{V}(\mathcal{D}_{T})N|_{T=0}$ (146) $\displaystyle=$
$\displaystyle-P_{V(f)}(N,T=0)-\mathcal{K}_{(f,V)}(N,T=0)$ (147)

where we used (143) and defined

$\mathcal{K}_{(f,V)}(N,T):=\int d^{2}xf\delta_{V}(\mathcal{D}_{T})N.$ (148)

From (62) and (18) one finds the non-closure term is given by

$\mathcal{K}_{(f,V)}(N,T)=-4\int
d^{2}xf(\mathcal{L}_{V}T_{\bar{z}\bar{z}}-\partial_{\bar{z}}^{3}V^{\bar{z}})D^{2}_{z}N+c.c..$
(149)

By setting $T=0$, one obtains a 2-cocycle term of the type found in
btchargealgebra .151515The relation of (149) with the BT cocycle is
$\mathcal{K}_{(f,V)}=-4K_{(f,V)}$. The different multiplicative factor is due
to the extra dependence on the Geroch tensor when using
$\overset{\scriptscriptstyle
0}{N}\vphantom{N}_{ab}=-2D_{a}D_{b}\overset{\scriptscriptstyle
0}{N}\vphantom{N}$. The analogue of (149) for the BT case is $K_{(f,V)}=-2\int
d^{2}xf\delta_{V}(D^{a}D^{b})\overset{\scriptscriptstyle
0}{N}\vphantom{N}_{ab}|_{T=0}$.

###  6.4 Consistency with consecutive double soft graviton theorems

In dfh , Distler, Flauger and Horn gave an explicit proof of the non-closure
of eBMS algebra in quantum theory. Their main result can be summarized as
follows. Given the consecutive double soft graviton theorem in tree-level
scattering amplitudes at sub-leading (and sub-sub leading) order, one can
compute the commutator of two such soft limits. The resulting factorisation
theorem is equivalent to the algebra generated by the (quantized)
supertranslation and superrotation charges.

More in detail, the authors in dfh  considered the following commutators
obtained from double soft theorems.

$\displaystyle\begin{array}[]{lll}[\lim_{\omega_{1}\,\rightarrow\,0}\,\omega_{1},\,\lim_{\omega_{2}\,\rightarrow\,0}\,\partial_{\omega_{2}}\,\omega_{2}\,]\,\langle\textrm{out},k_{1},k_{2}\,|\,S\,|\,\textrm{in}\,\rangle\,=\,S(k_{1},k_{2})\,\langle\textrm{out}\,|\,S\,|\,\textrm{in}\,\rangle\\\
\vspace*{0.1in}[\lim_{\omega_{1}\,\rightarrow\,0}\,\partial_{\omega_{1}}\,\omega_{1},\,\lim_{\omega_{2}\,\rightarrow\,0}\,\partial_{\omega_{2}}\,\omega_{2}\,]\,\langle\textrm{out},k_{1},k_{2}\,|\,S\,|\,\textrm{in}\,\rangle\,=\,S^{\prime}(k_{1},k_{2})\,\langle\textrm{out}\,|\,S\,|\,\textrm{in}\,\rangle\end{array}$
(152)

Here $k_{i}\,=\,(\omega_{i},\hat{k}_{i})$ and we have supressed the helicity
information of all the gravitons for the sake of brevity. The two commutators
are associated to consecutive double soft theorems at sub-leading and sub-
subleading orders respectively.

The authors then analysed $S(k_{1},k_{2}),\,S^{\prime}(k_{1},k_{2})$ in
complete detail for all possible helicity configurations of the soft gravitons
and showed that it consists of terms which have
(1) “soft” poles consistent with single soft theorem or
(2) colinear poles arising when $k_{1},\,k_{2}$ become colinear.

The RHS of (152) was shown to equal the following matrix elements

$\displaystyle\begin{array}[]{lll}S(k_{1},k_{2})\,\langle\textrm{out}\,|\,S\,|\,\textrm{in}\,\rangle\,=\,\langle\textrm{out}\,|\,[\,[\hat{P}_{f},\,\hat{Q}_{V}],\,S\,]\,|\,\textrm{in}\,\rangle\\\
\vspace*{0.1in}S^{\prime}(k_{1},k_{2})\,\langle\textrm{out}\,|\,S\,|\,\textrm{in}\,\rangle\,=\,\langle\textrm{out}\,|\,[\,[\hat{Q}_{V_{1}},\,\hat{Q}_{V_{2}}],\,S\,]\,|\,\textrm{in}\,\rangle\end{array}$
(155)

for specific choices of parameters $f,V,V_{1}$ and $V_{2}$ that are dictated
by the helicity and direction of the soft gravitons. In the above equations
out/in are outgoing or incoming graviton states in the “trivial sector”
defined by $N\,=\,T_{ab}\,=\,0$. We have denoted the superrotation charge as
$\hat{Q}_{V}$ instead of $\hat{J}_{V}$ in Eq. (155). This is because in the
$\overset{\scriptscriptstyle 0}{N}\vphantom{N}\,=\,T\,=\,0$ sector,
$\hat{Q}_{V}^{\textrm{soft}}$ differs from $\hat{J}_{V}^{\textrm{soft}}$ by

$\displaystyle\hat{Q}_{V}^{\textrm{soft}}\,=\,\hat{J}_{V}^{\textrm{soft}}\,+\,i\hbar\int
d^{2}z\,\delta_{V}N(z)\frac{\delta}{\delta N(z)}$ (156)

Eq. (155) is already a rather surprising result. As the LHS is simply the soft
factor multiplied by the scattering amplitude and the RHS is the Ward
identities associated to commutators of the symmetry generators. This is
structurally different than the identification between a single soft theorem
and Ward identities.161616We have written the equality in (155) slightly
differently than in dfh . This is simply for the sake of pedagogy and does not
affect the final result which is our primary concern. The interested reader is
encouraged to consult the original reference for more details. In fact in dfh
, the striking nature of the above result was made more manifest as it was
shown that this identity implies non-closure of eBMS algebra as,

$\displaystyle\begin{array}[]{lll}\langle\textrm{out}\,|\,[\,[\hat{P}_{f},\,\hat{Q}_{V}],\,S\,]\,|\,\textrm{in}\,\rangle\,=\,-i\hbar\langle\,\textrm{in}\,|\,[\,\hat{P}_{V(f)}\,+\,\hat{K}_{(f,V)},\,S\,]\,|\,\textrm{out}\,\rangle\\\
\langle\textrm{out}\,|\,[\,[\hat{Q}_{V_{1}},\,\hat{Q}_{V_{2}}],\,S\,]\,|\,\textrm{in}\,\rangle\,=\,i\hbar\langle\,\textrm{in}\,|\,[\,\hat{Q}_{[V_{1},V_{2}]}\,,\,S\,]\,|\,\textrm{out}\,\rangle\\\
\end{array}$ (159)

The analysis by Distler, Flauger and Horn was for the entire eBMS algebra
including holomorphic superrotations. Hence we can compare the result stated
above with our analysis.

Using the modified superrotation charge derived in this paper the above
identities can be rewritten as,

$\displaystyle\begin{array}[]{lll}[\,\hat{P}_{f},\,\hat{J}_{V}\,]\,=\,-i\hbar\hat{P}_{V(f)}\\\
\vspace*{0.1in}[\,\hat{J}_{V_{1}},\,\hat{J}_{V_{2}}\,]\,=\,i\hbar\hat{J}_{[V_{1},V_{2}]}\end{array}$
(162)

We thus observe that first equation in (155) implies that

$\displaystyle
S(k_{1},k_{2})\,\langle\textrm{out}\,|\,S\,|\,\textrm{in}\,\rangle\,=\,\langle\textrm{out}\,|\,[\,[\hat{P}_{f},\,\hat{J}_{V}]\,+\,\hat{K}(f,V),\,S\,]\,|\,\textrm{in}\,\rangle\,$
(163)

With the new superrotation charge which generates closed eBMS algebra (albeit
only for entire vector fields), _the right hand side_ of (the commutator) of
sub-leading double soft theorem can be realised as matrix element of
commutator of S matrix with
$\,[\,\hat{P}_{f},\,\hat{J}_{V}\,]\,+\,\hat{K}(f,V)$.

##  7 Conclusions and outlook

Our main goal in this paper has been to analyse the quantum charge algebra
associated to extended BMS symmetries in light of recent progress in the
understanding of superrotation charges in classical gravity. In this section,
we briefly summarize our results and highlight some of the open issues which
have emerged out of our analysis and require further investigation.

###  7.1 Summary of results

Throughout this paper, we considered the eBMS algebra generated by
supertranslations and holomorphic (anti-holomorphic) superrotations. Although
we expect our analysis to generalise to the complete eBMS and gBMS algebra
(for which the superrotations are generated by meromorphic or smooth vector
fields respectively) we did not attempt to do so in this paper and offer a few
comments in 7.3 below regarding the possible complications we are likely to
encounter in attempting such an extension.

The main ideas in this paper can be summarized as follows: Based on the
analysis in cp , we considered an extension of the radiative phase space of
shear modes which contains the leading and sub-leading soft news and their
conjugate partners. This extended phase space is not physical in the sense
that the new variables are constrained in terms of the shear and news
fields.171717These constraints assumed an invariant complex structure on the
celestial sphere which is only preserved by meromorphic vector fields.
We showed that these constraints are second class and can be solved explicitly
to obtain a physical radiative phase space $\Gamma$ on which both the
supertranslation as well as superrotation charges have a well defined action.
We then quantized the phase space and showed that the quantized charges form a
closed algebra which is simply a faithful (as opposed to projective)
representation of the eBMS Lie algebra at ${\cal I}$.
A natural off-shoot of our analysis is an explicit parametrization of the
space of all vacua which generates (via finite energy excitations) a
representation space for the eBMS algebra. These vacua are labelled by the
soft news $\overset{\scriptscriptstyle 0}{N}\vphantom{N}$ and the Geroch
tensor $T_{ab}$ which is a Schwarzian. This result should thus be viewed as a
synthesis of the structure of vacua analysed in stromST  and sraction .

###  7.2 Comparison with the eBMS algebra in celestial holography

We now offer some brief and rather speculative comments on the implication of
our results for the eBMS algebra in celestial holography which has been
derived and analysed in a number of recent papers (see ebmsope ; confsoft1 ;
confsoft2 ; confsoft3 ; shamik ; alfredo )181818We thank Dileep Jatkar for
extensive discussions on eBMS algebra in celestial CFT.. Our comments are
rather preliminary and only serve to highlight a possible way in which the
quantized eBMS algebra at null infinity in this paper can be reconciled with
the eBMS algebra in celestial CFT.

To be more specific, we consider only positive helicity soft insertions. The
holomorphic supertranslation current and the boundary stress tensor are
defined as,

$\displaystyle\begin{array}[]{lll}P(z)\,=\,P(f)|_{f\,=\,\frac{z-w}{\bar{z}-\overline{w}}}\\\
{\cal
T}_{\bar{z}\bar{z}}\,=\,Q_{Y}^{\textrm{soft}}|_{Y^{\bar{z}}\,=\,\frac{1}{\bar{z}-\overline{w}},Y^{z}\,=\,0}\end{array}$
(166)

We have denoted the stress tensor as ${\cal T}_{ab}$ to distinguish it from
$T_{ab}$ which in this paper denotes the conjugate partner to the sub-leading
soft news.
In the celestial basis, $P(z)$ is a conformal primary with $\triangle=1$ and
${\cal T}_{\bar{z}\bar{z}}$ is the shadow transform of a $\triangle=0$ primary
operator.
In the above equation, $w$ is being integrated over. We also emphasize that
the soft charge $Q_{Y}^{\textrm{soft}}$ used to define the celestial stress
tensor is the “old” soft superrotation charge given in Eq. (156). However in
order to make contact with the results derived in this paper, we need to
consider the following tensor on the celestial sphere

$\displaystyle
t_{\bar{z}\bar{z}}\,=\,J_{Y}^{\textrm{soft}}|_{Y^{\bar{z}}\,=\,\frac{1}{\bar{z}-\overline{w}},Y^{z}\,=\,0}$
(167)

Once again, we have assumed that our definition of $J_{Y}$ is valid for a
meromorphic vector field, as we believe that this has no bearing on the main
point raised in this section. With this tensor, we now know that the
commutator between $[P(z),\,t_{\bar{z}\bar{z}}\,]$ closes in a super-
translation.191919This result can be re-written in the so-called celestial
basis which is obtained by Mellin transforming $P_{z},\,t_{zz}$ to conformal
operators.
But as,

$\displaystyle t_{\bar{z}\bar{z}}\,=\,{\cal T}_{\bar{z}\bar{z}}\,+\,{\cal
O}_{\bar{z}\bar{z}}$ (168)

The algebra generated by $P(z),{\cal T}_{\bar{z}\bar{z}}$ does not close and
has a 2 co-cycle extension consistent with dfh .

We expect that in the celestial basis, the OPE of $t_{ab}$ with conformal
primaries (with $\triangle\,=\,1+i\lambda,\,\lambda\,\neq\,0$) is the same as
that of ${\cal T}_{ab}$. This is because this OPE follows from sub-leading
soft graviton theorem which as a Ward identity is insensitive to the
distinction between the old and new superrotation charges. In other words, we
expect that the following OPE has no singular terms.

$\displaystyle{\cal O}_{ab}\,O_{\triangle,\pm
2}(z,\bar{z})\,=\,\textrm{regular}$ (169)

where $O_{\triangle,\pm 2}$ is the hard graviton in the celestial basis with
conformal dimension $\triangle\,=\,1+i\lambda$. Thus both ${\cal
T}_{ab},\,t_{ab}$ appear to be possible candidates for a stress tensor on the
celestial sphere.

However as the algebra of $t_{ab}$ with $P(z)$ is isomorphic to the eBMS
algebra in ebmsope , we also expect that the OPE of $t_{ab}$ with the
supertranslation current is precisely as expected of an OPE between a CFT
stress tensor and a Kac-Moody current. That is, with $t_{ab}$ as the celestial
stress tensor, $P(z)$ is a level-1 descendent primary field. But we note that
in the celestial basis ${\cal O}_{ab}$ is the Mellin transform of,

$\displaystyle\int\,-\,{\cal D}C\,\delta_{Y}\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}$ (170)

where $Y^{z}\,=\,0,\,Y^{\bar{z}}\,=\,\frac{1}{\bar{z}-\overline{w}}$. Hence
although $O_{ab}$ has a trivial OPE with hard graviton modes, it must have
non-trivial OPE with respect to $P(z)/P(\bar{z})\,=\,O_{\triangle=1,\pm 2}$.
That $P(z)$ is not a descendent primary with respect to ${\cal T}_{ab}$ was
observed in dfh . We refer the reader to section five in dfh  for
details.202020In barnichalgroid  Barnich has put forward a different
perspective on the non-centrally extended eBMS algebra. Namely to view it as a
“centrally extended algebroid” instead of an algebra with a 2-cocycle anomaly.
However the implications of the Lie-algebroid ideas in quantum theory are not
clear to the authors. More in detail, the presence of the 2-cocycle appears to
be in tension with the fact that quantized supertranslation and superrotation
charges are symmetries of the S-matrix. It is not clear to us, how
reinterpreting eBMS algebra as a Lie algebroid alleviates this tension.

Thus we expect that in the celestial CFT, our results may be equivalent to
choosing $t_{ab}$ as the stress tensor which is obtained by deforming ${\cal
T}_{ab}$ by ${\cal O}_{ab}$.

###  7.3 Towards gBMS algebra

Our analysis so far has been restricted to the extended BMS algebra in which
superrotations are generated by (local) conformal Killing vector fields on the
celestial sphere. In this section, we offer a few remarks about the
possibility of generalising the analysis to quantize the gBMS algebra.

  * •

In a seminal paper pdp , the soft charge associated to smooth superrotations
$\textrm{Diff}(S^{2})$ was shown to be a shadow transform of the celestial
stress tensor (this stress tensor is the soft charge for eBMS superrotations
in the conformal primary basis). However a crucial ingredient in this analysis
was to consider generic configurations of $T_{ab}$ which are meromorphic on
the celestial sphere. Our constraint analysis assumed that $T_{ab}$ has poles
only at infinity and hence to use the ideas of pdp , we need to generalise our
analysis to ensure that all meromorphic Geroch tensors are included in the
radiative phase space.

  * •

To allow for smooth superrotations, we need to reincorporate the celestial
metric $q_{ab}$ as part of the kinematical phase space, as originally
presented in cp . In such case, the analogue of the constraints (84) are more
involved. In particular, the constraint $F^{2}_{a}$ becomes

$\displaystyle\textrm{D}^{b}\,T_{ab}\,=\,0$ (171)

where $\textrm{D}_{a}$ is the covariant derivative of the celestial metric,
$\textrm{D}_{a}q_{bc}=0$. Thus, the 2d metric and Geroch tensor are no longer
decoupled but are related through the non-linear constraint (171).

  * •

Once the sphere metric is included in the radiative phase space, it also
contains a mode $p^{ab}$ conjugate to $q_{ab}$ that is constrained to be
certain combination of soft news modes (we refer the reader to cp  for
details). This is likely to make the constraint analysis far more intricate.

An investigation of these constraints and the resulting Dirac brackets will be
pursued elsewhere.

## Acknowledgements

We are grateful to Rafael Flauger for drawing our attention to the issue of
cocycle extension in eBMS algebra and for crucial discussions which led us to
this project. We would also like to thank Anupam A H, Sujay Ashok, Sayali
Bhatkar, Chandramouli Chowdhury, Laurent Freidel, Guzman Hernandez-Chifflet,
Arpan Kundu, Ruchira Mishra, Prahar Mitra, Silvia Nagy, Olga Papadoulaki,
Aneesh P.B., Javier Peraza, Siddharth Prabhu, Suvrat Raju, Biswajit Sahoo,
Ashoke Sen and Pushkal Srivastava for valuable discussions on issues related
to asymptotic symmetries in Quantum Gravity over the years. We are especially
indebted to Dileep Jatkar for patiently explaining to us many facets of the
BMS algebra in celestial CFT.
AL would like to thank the organizers of “Celestial Holography” at PCTS
(Princeton Center for Theoretical Science) where part of this work was
presented. He would also like to thank Laura Donnay, Sabrina Pasterski and
Andrea Puhm for discussions. MC acknowledges support from PEDECIBA and ANII
grant FCE-1-2019-1-155865.

##  Appendix A Derivation of Dirac brackets

In this section, we present a derivation of the “physical” radiative phase
space Poisson brackets presented in section 5. The first step (see e.g. HT )
is to study the matrix with entries given by the Poisson brackets between
constraints. In our case we have $4\times 2\,\times\infty$ constraints given
in Eq. (84). We write the corresponding matrix as

$\begin{pmatrix}\\{F^{I}(z),F^{J}(w)\\}&\\{F^{I}(z),F^{\bar{J}}(w)\\}\\\
\\{F^{\bar{I}}(z),F^{J}(w)\\}&\\{F^{\bar{I}}(z),F^{\bar{J}}(w)\\}\end{pmatrix}$
(172)

where $I,J,\overline{I},\overline{J}=0,1,2,3$ are constraint indices that
distinguish holomorphic and anti-holomorphic components of the constraint
functions. For instance, $F^{0}(z)\equiv F^{0}_{zz}$ and
$F^{\overline{0}}(z)\equiv F^{0}_{\bar{z}\bar{z}}$. We can evaluate the
various entries of this matrix by using the kinematical PBs presented in
section 4. It is immediate to see that brackets of the same helicity type
vanish,

$\\{F^{I}(z),F^{J}(w)\\}=\\{F^{\bar{I}}(z),F^{\bar{J}}(w)\\}=0.$ (173)

Evaluating the PBs between constraints with opposite helicity, we find the
only non-vanishing ones are212121The function $F^{3}_{zz}$ suffers from the
type of ambiguities discussed in section 4.2 when evaluating PBs. We take the
prescription
$\\{F^{0}_{zz},F^{3}_{\bar{w}\bar{w}}\\}=-X_{F^{0}_{zz}}(F^{3}_{\bar{w}\bar{w}})=-2\delta^{(2)}(z,w)$
and
$\\{F^{1}_{zz},F^{3}_{\bar{w}\bar{w}}\\}=\\{F^{3}_{zz},F^{3}_{\bar{w}\bar{w}}\\}=0$.

$\\{F^{0}_{zz},F^{1}_{\bar{w}\bar{w}}\\}=\frac{1}{2}\overset{\scriptscriptstyle
0}{N}\vphantom{N}(z)\delta^{(2)}(z,w),\quad\\{F^{0}_{zz},F^{3}_{\bar{w}\bar{w}}\\}=-2\delta^{(2)}(z,w),\quad\\{F^{1}_{zz},F^{2}_{\bar{w}}\\}=-\frac{1}{2}\partial_{z}\delta^{(2)}(z,w),$
(174)

together with their complex conjugates. In the notation of (172), we write
them as the 4 by 4 matrix

$\\{F^{I}(z),F^{\bar{J}}(w)\\}=\mathcal{F}^{I\bar{J}}(z)\delta^{(2)}(z,w)$
(175)

where

$\mathcal{F}^{I\bar{J}}(z)=-\frac{1}{2}\begin{pmatrix}0&-\overset{\scriptscriptstyle
0}{N}\vphantom{N}(z)&0&4\\\ \overset{\scriptscriptstyle
0}{N}\vphantom{N}(z)&0&\partial_{z}&0\\\ 0&\partial_{\bar{z}}&0&0\\\
-4&0&0&0\\\ \end{pmatrix}.$ (176)

Similarly
$\\{F^{\overline{I}}(z),F^{J}(w)\\}=\mathcal{F}^{\overline{I}J}\delta^{(2)}(z,w)$
with $\mathcal{F}^{\overline{I}J}$ the complex conjugate of (176).

Provided we can invert the differential operator $\partial_{z}$, the matrices
(176) and (172), are invertible, implying the constraints are second-class. We
shall now proceed under this assumption and discuss later the subtleties that
arise in situations where $\partial_{z}$ cannot be inverted.

The inverse of (172) can be written as

$\begin{pmatrix}\\{F^{I}(z),F^{J}(w)\\}&\\{F^{I}(z),F^{\bar{J}}(w)\\}\\\
\\{F^{\bar{I}}(z),F^{J}(w)\\}&\\{F^{\bar{I}}(z),F^{\bar{J}}(w)\\}\end{pmatrix}^{-1}=\begin{pmatrix}0&\mathcal{K}_{I\overline{J}}(z)\\\
\mathcal{K}_{\overline{I}J}(z)&0\end{pmatrix}\delta^{(2)}(z,w),$ (177)

where $\mathcal{K}_{\overline{I}J}$ is the operator-inverse of (176)

$\mathcal{F}^{I\overline{J}}\mathcal{K}_{\overline{J}K}=\delta^{I}_{K},$ (178)

given by

$\mathcal{K}_{\overline{I}J}(z)=-\frac{1}{2}\begin{pmatrix}0&0&0&-1\\\
0&0&4\partial^{-1}_{\bar{z}}&0\\\
0&4\partial^{-1}_{z}&0&\partial^{-1}_{z}\overset{\scriptscriptstyle
0}{N}\vphantom{N}(z)\\\ 1&0&\overset{\scriptscriptstyle
0}{N}\vphantom{N}(z)\partial^{-1}_{\bar{z}}&0\\\ \end{pmatrix},$ (179)

where $\partial^{-1}_{z}\overset{\scriptscriptstyle 0}{N}\vphantom{N}(z)$ and
$\overset{\scriptscriptstyle 0}{N}\vphantom{N}(z)\partial^{-1}_{\bar{z}}$
should be understood as compositions of integral and multiplicative operators.
Similarly $\mathcal{K}_{I\overline{J}}$ is given by the complex conjugate of
(179).

The Dirac bracket between two functions $f$ and $g$ is then given by

$\\{f,g\\}_{\star}=\\{f,g\\}+\\{f,g\\}_{\text{extra}}$ (180)

where

$\\{f,g\\}_{\text{extra}}=\int
d^{2}w\\{f,F^{I}(w)\\}\mathcal{K}_{I\overline{J}}(w)\\{g,F^{\overline{J}}(w)\\}+(I\leftrightarrow\overline{I},J\leftrightarrow\overline{J})$
(181)

We would like to compute the Dirac brackets between the elementary variables

$N_{ab}(u,\hat{x}),\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}(\hat{x}),\,C(\hat{x}),\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{ab}(\hat{x}),\,T_{ab}(\hat{x})\,.$ (182)

We start by displaying the PBs of elementary variables with the constraints.
Given the kinematical PBs of section 4, one finds

$\\{C(z),F^{0}_{ww}\\}=2D^{2}_{w}G(z,w),\quad\\{C(z),F^{1}_{ww}\\}=-2D^{2}_{w}C(w)G(z,w),$
(183) $\\{\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz},F^{0}_{\bar{w}\bar{w}}\\}=\frac{1}{2}\overset{\scriptscriptstyle
0}{N}\vphantom{N}(z)\delta^{(2)}(z,w),\quad\\{\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz},F^{2}_{\bar{w}}\\}=-\frac{1}{2}\partial_{w}\delta^{(2)}(z,w),$
(184)
$\\{T_{zz},F^{1}_{\bar{w}\bar{w}}\\}=-\frac{1}{2}\delta^{(2)}(z,w),\quad\\{N_{zz}(u,z),F^{1}_{\bar{w}\bar{w}}\\}=\frac{1}{2}\delta^{(2)}(z,w),$
(185)

(together with their complex conjugates) with all remaining brackets being
zero. We can now evaluate the extra contribution (181) and hence the Dirac
bracket (180) for all pairs of elementary variables (182). We start by
noticing that since $\overset{\scriptscriptstyle 0}{N}\vphantom{N}$ Poisson
commutes with all constraints, its Dirac bracket with other functions will
coincide with the kinematical one. In particular, the only non-zero Dirac
bracket involving $\overset{\scriptscriptstyle 0}{N}\vphantom{N}$ is

$\\{C(z),\overset{\scriptscriptstyle 0}{N}\vphantom{N}(w)\\}_{\star}=G(z,w).$
(186)

Another pair of brackets which is left unchanged is that of the news tensor
with itself. The reason being that the news tensor has non-trivial brackets
only with $F^{1}$, but the corresponding components of
$\mathcal{K}_{I\overline{J}}$ vanish. Thus,

$\\{N_{zz}(z,u),N_{\bar{w}\bar{w}}(u^{\prime},w)\\}_{\star}=\frac{1}{2}\delta^{\prime}(u-u^{\prime})\delta^{(2)}(z,w).$
(187)

For the remaining variables, one finds the non-trivial brackets involve the
composition of derivative and inverse-derivative operators. As we explain
below, we will deal with such terms through the use of projection operators
defined on the space of functions,

$\displaystyle\begin{array}[]{lll}O_{w}\,=\,\partial_{w}\,\partial_{w}^{-2}\,\partial_{w}\\\
O_{\bar{w}}\,=\,\partial_{\bar{w}}\,\partial_{\bar{w}}^{-2}\,\partial_{\bar{w}}\end{array}$
(190)

The remaining non-trivial brackets are then written as

$\displaystyle\\{C(z),\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{ww}\\}_{\star}$ $\displaystyle=$ $\displaystyle
2(1-O_{w})D^{2}_{w}C(w)G(z,w),$ (191)
$\displaystyle\\{\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz},T_{\bar{w}\bar{w}}\\}_{\star}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(1\,-\,O_{w}\,)\delta^{(2)}(z,w),$ (192)
$\displaystyle\\{N_{zz}(z,u),\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{\bar{w}\bar{w}}\\}_{\star}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,O_{w}\,\delta^{(2)}(z,w)$ (193)

The reason we get $O_{w}$ or its orthogonal complement is the following.
Up to this stage, we had assumed $\partial_{w}$ was invertible and hence
$O_{w}\,=\,\partial_{w}\partial^{-2}_{w}\partial_{w}=1$. We however note that
if we work with smeared variables, the nature of the smearing function would
alter this conclusion. In particular, for an anti-holomorphic smearing, the
$\partial_{w}$ derivatives would vanish. To allow for this possibility, we
work with the expressions above interpreting
$\partial_{w}\partial^{-2}_{w}\partial_{w}$ as a symmetric operator that
annihilates anti-holomorphic functions and such that is the identity on the
space of smooth functions modulo anti-holomorphic kernel. We note that our
choice of $O_{w}\,=\,\partial_{w}\partial_{w}^{-2}\partial_{w}$ instead of an
operator such as $\partial_{w}^{-1}\,\partial_{w}$ ensures that it is
symmetric with respect to its action on smearing of either functions. The
latter however has orthogonal kernel and co-kernel. Its kernel is the space of
anti-holomorphic functions of ${\bf C}$ and it has a trivial cokernel.

Since the computation of Dirac brackets is subtle, We conclude the appendix by
verifying the Dirac brackets satisfy the Jacobi identity. It is easy to see
that the only non-trivial condition comes from

$\displaystyle\\{\,\\{\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz},\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w)\,\\},\,C(x)\,\\}_{\star}\,+\,\\{\,\\{\,C(x),\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\,\\},\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w)\,\\}_{\star}\,+\,\\{\,\\{\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w),\,C(x)\,\\},\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\,\\}_{\star}\,\,=\,0$ (194)

Since $\\{\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz},\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w)\,\\}_{\star}$ vanishes, the Jacobi identity will be
satisfied if,

$\displaystyle\\{\,\\{\,C(x),\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\,\\}_{\star},\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w)\,\\}_{\star}\,=\,-\,\\{\,\\{\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w),\,C(x)\,\\}_{\star},\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\,\\}_{\star}.$ (195)

Computing each side of (195) we find

$\displaystyle\\{\,\\{\,C(x),\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\,\\}_{\star},\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w)\,\\}_{\star}\,$ $\displaystyle=$ $\displaystyle
2(1-O_{z})\\{D^{2}_{z}C(z)G(x,z),\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w)\,\\}_{\star}\,$ (196) $\displaystyle=$ $\displaystyle
2(1-O_{z})D^{2}_{z}G(z,w)G(x,z)$ (197)
$\displaystyle-\,\\{\,\\{\,\overset{\scriptscriptstyle
0}{N}\vphantom{N}(w),\,C(x)\,\\}_{\star},\,\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\,\\}_{\star}$ $\displaystyle=$
$\displaystyle\\{G(x,w),\overset{\scriptscriptstyle
1}{N}\vphantom{N}_{zz}\,\\}_{\star}$ (198) $\displaystyle=$
$\displaystyle\frac{1}{2}(1-O_{z})\frac{\delta G(x,w)}{\delta
T_{\bar{z}\bar{z}}}.$ (199)

The last derivative can be evaluated from the condition
$\delta(\mathcal{D}G)=\delta\mathcal{D}G+\mathcal{D}\delta G=0$, together with
the equation (62) for $\delta\mathcal{D}$. One finds

$\frac{\delta G(x,w)}{\delta T_{\bar{z}\bar{z}}}=4D^{2}_{z}G(z,w)G(x,z),$
(200)

from which (195) immediately follows.

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