Bridging Carrollian and Celestial Holography

Laura Donnay♣,♠111<EMAIL_ADDRESS>Adrien Fiorucci♥222
<EMAIL_ADDRESS>Yannick Herfray♠333 yannick.herfray@univ-
tours.fr, Romain Ruzziconi♥444<EMAIL_ADDRESS>
${}^{\text{\char 168}}$International School for Advanced Studies (SISSA),
Via Bonomea 265, 34136 Trieste, Italy

${}^{\text{{\color[rgb]{0.75390625,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75390625,0,0}\char
169}}}$Istituto Nazionale di Fisica Nucleare (INFN) — Sezione di Trieste,
Via Valerio 2, 34127 Trieste, Italy

${}^{\text{{\color[rgb]{0.75390625,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75390625,0,0}\char
170}}}$Institute for Theoretical Physics, Technische Universität Wien,
Wiedner Hauptstrasse 8, A-1040 Vienna, Austria

${}^{\text{\char 171}}$Institut Denis Poisson UMR 7013, Université de Tours,
Parc de Grandmont, 37200 Tours, France

###### Abstract

Gravity in $4d$ asymptotically flat spacetime constitutes the archetypal
example of a gravitational system with leaky boundary conditions. Pursuing our
analysis of [1], we argue that the holographic description of such a system
requires the coupling of the dual theory living at null infinity to some
external sources encoding the radiation reaching the conformal boundary and
responsible for the non-conservation of the charges. In particular, we show
that the sourced Ward identities of a conformal Carrollian field theory living
at null infinity reproduce the BMS flux-balance laws. We also derive the
general form of low-point correlation functions for conformal Carrollian field
theories and exhibit a new branch of solutions, which is argued to be the
relevant one for holographic purposes. We then relate our Carrollian approach
to the celestial holography proposal by mapping the Carrollian Ward identities
to those constraining celestial operators through a suitable integral
transform.

###### Contents

  1. 1 Introduction
  2. 2 Massless scattering in flat spacetime
    1. 2.1 Bulk operators
    2. 2.2 Boundary operators
    3. 2.3 From null infinity to the celestial sphere
    4. 2.4 $\mathcal{S}$-matrix in the three scattering bases
  3. 3 Asymptotic symmetries in flat spacetime
    1. 3.1 Scalar electrodynamics
    2. 3.2 Gravity
  4. 4 Sourced quantum field theory
    1. 4.1 Generalized Noether symmetries
    2. 4.2 Sourced Ward identities
    3. 4.3 Sourced $U(1)$ Ward identities
    4. 4.4 Sourced conformal Carrollian Ward identities
  5. 5 Holographic conformal Carrollian field theory
    1. 5.1 Time-ordered conformal Carrollian manifolds
    2. 5.2 Carrollian holographic correspondence
    3. 5.3 Holographic Ward identities for a massless scattering
    4. 5.4 Conformal Carrollian invariant correlation functions
  6. 6 Relation with celestial holography
    1. 6.1 Ward identities of $2d$ celestial CFT currents
    2. 6.2 From Carrollian to celestial holography
  7. 7 Discussion
  8. Appendices
    1. A Bondi coordinates for Minkowski spacetime
    2. B Isomorphism between global conformal Carrollian and Poincaré algebras
    3. C Constraints on the Carrollian stress tensor

##  1 Introduction

The program of flat space holography consists in building a holographic
duality between gravity in asymptotically flat spacetime and a lower-
dimensional field theory. The motivations are twofold. Firstly, from a purely
theoretical perspective, this program is enshrined in a broader context that
aims at understanding how general is the holographic principle. Does it extend
beyond the framework of the celebrated AdS/CFT correspondence [2, 3, 4]?
Secondly, asymptotically flat spacetimes provide realistic models to describe
a huge range of physical processes occurring in our universe, all the way up
to astrophysical scales which are below the cosmological scale.

In spite of repeated early efforts from various points of views [5, 6, 7, 8,
9, 10], it became clear very soon that flat space holography would not merely
immediately follow from AdS/CFT via a simple limit procedure where the AdS
radius is sent to infinity. But fortunately, the last decades have also taught
us that the symmetries of flat spacetime turn out to be way richer and more
subtle than originally thought [11, 12, 13, 14, 15, 16]. Building on the
constraints imposed by the BMS symmetries, two bottom-up roads towards flat
space holography have emerged over the years: Carrollian and celestial
holography.

The Carrollian approach to flat space holography proposes that the role of the
dual theory is played by a conformal Carrollian field theory (or Carrollian
CFT for short) that lives on the codimension-one boundary of spacetime (null
infinity $\mathscr{I}$). Alternatively, one could refer to this theory as a
BMS-invariant field theory, as it has been known for quite some time that the
BMS group is isomorphic to the conformal Carroll group [17]; but it is fair to
say that the “Carrollian” name, due to Lévy-Leblond [18], has struck
physicists’ imagination.

This first approach has proven to be very successful in the context of three-
dimensional ($3d$) gravity: among other results, one can find $(i)$ a matching
between the entropy of asymptotically flat cosmological solutions and entropy
computed with a Cardy-like formula for a Carrollian CFT [19, 20]; $(ii)$ a
computation of the entanglement entropy in the Carrollian CFT from the bulk
geometry using some extension of the Ryu-Takayanagi prescription [21, 22, 23];
$(iii)$ the form of the correlation functions in the dual theory [24, 25, 26,
27, 28]; $(iv)$ an effective action for the dual Carrollian CFT [29, 30, 31,
32]. This construction of effective action has been repeated in $4d$ gravity
and shown to describe the dynamics of non-radiative spacetime [33]. A
complementary approach was adopted earlier in [34] where an action encoding
the radiative modes at $\mathscr{I}$ was explicitly written. Let us also
mention that Carrollian holography is well understood in the fluid/gravity
correspondence for both $3d$ and $4d$ spacetimes using a suitable flat limit
procedure [35, 36, 37, 38, 39]. The dual fluid is a Carrollian fluid [40, 41,
42, 43, 44] at $\mathscr{I}$ whose properties are deduced by taking the ultra-
relativistic limit of a relativistic fluid.

The main difficulties inherent to the Carrollian holographic approach are
related to two key differences compared to what one is used to encounter in
AdS/CFT. $(i)$ The conformal boundary $\mathscr{I}$ is a null hypersurface,
implying that the dual theory involves some Carrollian or “ultralocal” physics
[45, 46, 47, 48, 49, 50] which intrinsically defines the unconventional
features of Carrollian CFTs [51, 52, 53, 42, 54, 55, 1, 56], about which very
little is known so far. This contrasts with the AdS/CFT correspondence where
the boundary is timelike and the dual theory is an honest CFT whose
proprieties have been studied for decades. $(ii)$ The gravitational charges at
null infinity are generically non-conserved due to the presence of matter or
gravitational radiation [57, 11, 58, 12, 59, 60]. This constitutes the plain
vanilla example of a gravitational system with leaky boundary conditions.
Describing the non-conservation of the charges from the point of view of the
dual theory requires the development of new techniques such as the coupling
with external sources [1]. On the contrary, Dirichlet-type of boundary
conditions are usually imposed in AdS, preventing leaks through the conformal
boundary, which thus acts as a reflective cavity. Let us however mention that
leaky boundary conditions can also be imposed in AdS by relaxing the usual
boundary conditions and turning on boundary sources [61, 62, 63, 64, 65]. This
relaxation is necessary in order to recover a radiative phase space and BMS
symmetries from a large radius limit of AdS.

In the celestial holography paradigm, the proposed dual theory to gravity in
flat space is a _celestial CFT_ (or CCFT for short) living on a codimension-
two boundary, the celestial sphere. It is anchored in the fact that
$\mathcal{S}$-matrix elements in the bulk, once rewritten in boost eigenstate
basis, enjoy conformal invariance in a manifest way [66, 67, 68, 69, 70, 14,
71]. The advantages of this approach is that one can use some of the very
powerful CFT techniques to study the celestial dual, such as operator product
expansions (OPEs) [72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83], conformal
block decomposition [84, 85, 86, 87, 88], state-operator correspondence [89],
null states and conformal multiplets [90, 91, 92, 93], crossing symmetry [94]
or shadow formalism [70, 95, 96, 97]. This CFT language has recently allowed
to uncover new $w_{1+\infty}$ symmetries in CCFT [98] (see also [99, 100,
101]). This remarkable finding suggests that gravity might exhibit much more
symmetries than one could have expected. Patterns of these symmetries have
been found in the subleading orders of the gravitational solution space in
[102, 103]. However, there is always a price to pay for the emergence of CFT-
like features in flat space, and this is manifested in some of the exotic
features that CCFTs exhibit. It is indeed not clear for the moment exactly to
which extent they differ from standard (e.g. unitary and compact) CFTs or what
could be an axiomatic definition of CCFTs. Moreover, the fact that the dual
theory is codimension-two with respect to the bulk makes the link with the
AdS/CFT correspondence more nebulous, since it would require more involved
steps than simply getting celestial correlators from a flat limit of those in
AdS (see however [66, 69, 104, 105, 106, 107, 108] for connections to AdS and
[109] to wedge holography). Finally, because the celestial encoding favors
conformal transformations over time translations (which we recall are not lost
but rather reshuffled into shifts in the conformal dimension of celestial
operators), the dynamics of the gravitational theory such as the Bondi mass
loss formula is not easily interpreted in the celestial CFT.

Though these two approaches to flat space holography seem in apparent tension,
it has been argued in [1] (see also [56]) that they are in fact complementary
to each other, as depicted in Fig. 1. Explicit links between them can be
established; in particular, Carrollian source operators $\sigma_{(k,\bar{k})}$
living at null infinity can be mapped to CCFT operators
$\mathcal{O}_{\Delta,J}$ living on the celestial sphere after using an
appropriate integral transform. This allows to relate the correlation
functions between the sourced Carrollian CFT and those of the CCFT. In
particular, the Ward identities of the sourced Carrollian CFT are then found
to be equivalent to those of the CCFT encoding the bulk soft theorem. In this
paper, we will pursue our previous analysis of [1] and provide more details
about the interplay between the Carrollian and the celestial approaches to
flat space holography.

Evolution along $\mathscr{I}$ Scattering data on $\mathscr{I}$ 3d Carrollian
CFT 2d Celestial CFT Y1.0e0]ptCarrollian holographyY1.0e0]ptCelestial
holography Two roads towards flat holography Complementary points of view vs
Carrollian operatorsConformal operators $\sigma_{(k,\bar{k})}(x^{a})$
$\mathcal{O}_{\Delta,J}(x^{A})$Y1.0e0]ptY1.0e0]pt$i^{0}$Y1.0e0]pt$i^{+}$Y1.0e0]pt$i^{-}$Y1.0e0]pt$i^{0}$Y1.0e0]pt$\mathscr{I}^{+}$Y1.0e0]pt$\mathscr{I}^{-}$Y1.0e0]pt$\frac{dQ}{du}\neq
0$Y1.0e0]pt$\mathscr{I}^{-}$Y1.0e0]pt$Q(u_{1})$Y1.0e0]pt$Q(u_{2})$Y1.0e0]ptY1.0e0]pt$i^{0}$Y1.0e0]pt$i^{+}$Y1.0e0]pt$i^{-}$Y1.0e0]pt$i^{0}$Y1.0e0]pt$\mathscr{I}^{+}$Y1.0e0]pt$\mathscr{I}^{-}$Y1.0e0]pt$\mathscr{I}^{+}$Y1.0e0]pt$\mathscr{I}^{-}$Y1.0e0]pt$\langle
out|$Y1.0e0]pt$\mathcal{S}$Y1.0e0]pt$|in\rangle$ Figure 1: Holographic nature
of null infinity: two equivalent visions. $(i)$ On the left picture,
$\mathscr{I}^{+}$ is seen as a boundary along which there is a Carrollian time
evolution. This naturally leads to the Carrollian holography proposal where
the putative dual theory is a $3d$ Carrollian CFT. This picture is well
adapted to describe the dynamics of the system through the flux-balance laws
encoding the non-conservation of the charges $Q$. $(ii)$ On the right picture,
$\mathscr{I}^{+}$ is seen as a portion of a Cauchy hypersuface at late time.
This naturally leads to the celestial holography proposal where the putative
dual theory is a $2d$ CCFT. This second approach is particularly natural when
considering a scattering process in flat spacetime.

##### Summary of the paper

The aims of this paper are threefold. The first objective is to review the key
ingredients needed to bridge Carrollian and celestial holography. These both
build up on the fact that the $\mathcal{S}$-matrix is in a sense holographic
by nature, namely scattering amplitudes in the bulk can be identified with
correlation functions at the boundary. Depending on the choice of basis for
the fields, position space or Mellin space, one ends up with the Carrollian or
the celestial approach. We will also review the asymptotic symmetry analysis
of both electrodynamics and gravity. These will constitute the two concrete
examples that we shall discuss in our set-up.

After this review part, the second objective is to improve the Carrollian
holography proposal. As mentioned above, one of the main obstructions to this
approach (obstruction $(ii)$) is the non-conservation of the asymptotic
charges $Q$ at null infinity due to the presence of radiation, as illustrated
in the left part of Fig. 1. We argue that, from the point of view of the dual
theory, these dissipative properties can be encoded by coupling the theory to
some external sources. We discuss the general framework needed to treat
symmetries in that context. This allows us to write flux-balance laws and Ward
identities for a sourced field theory. Applying these considerations to a
sourced Carrollian CFT, we show that the sourced Ward identities are able to
describe the asymptotic dynamics of $4d$ asymptotically flat spacetimes given
an appropriate holographic map between the bulk and boundary quantities. In
particular, the external sources are argued to encode holographically the
radiation in the bulk, which in gravity is contained in the asymptotic shear
and the Bondi news tensor.

While the insertion of external sources locally spoils the symmetries of the
theory, leading to non-conservation of the currents and dissipation, one can
still extract some useful constraints implied by the symmetries on the
correlators by using some holographic inputs. A crucial ingredient to
implement this constraint is to consider that the Carrollian CFT is not living
on $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$ separately, but on the whole
conformal boundary $\hat{\mathscr{I}}=\mathscr{I}^{+}\sqcup\mathscr{I}^{-}$
obtained by gluing antipodally $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$ along
$\mathscr{I}^{+}_{-}$ and $\mathscr{I}^{-}_{+}$, which provides a geometric
implementation of the antipodal matching from the Carrollian point of view,
see Fig. 2.

$i^{0}$$i^{0}$$i^{+}$$i^{-}$$\mathscr{I}^{+}$$\mathscr{I}^{-}$$\mathscr{I}^{+}\
$$\mathscr{I}^{-}\
$$\mathscr{I}^{-}$$\mathscr{I}^{+}$$\mathscr{I}^{+}_{+}$$\mathscr{I}^{-}_{-}$$\mathscr{I}^{+}_{-}$$\mathscr{I}^{-}_{+}$
Antipodal gluing Y1.0e0]ptCarrollian CFT Figure 2: Geometric implementation of
the antipodal matching.

After providing details of this construction, we show that the sourced Ward
identities of the Carrollian CFT integrated over $\hat{\mathscr{I}}$ determine
the general form of low-point ($1$-, $2$\- and $3$-point) correlation
functions of this theory. In particular, for the $2$-point function, we
exhibit a new branch of solutions that does not seem to have appeared in
previous literature. We then argue that this new branch is precisely the one
that is relevant for holographic Carrollian CFT. We relate it to the usual
bulk propagator via Fourier transform.

The third objective is to provide further details on the relation between the
Carrollian and the celestial approaches to flat space holography. We show that
the Carrollian source operators are mapped on celestial operators through an
integral transform, coined as the $\mathcal{B}$-transform, which is the
combination of a Mellin and a Fourier transform. This allows to relate the
correlation functions in the two theories. After providing the properties of
the $\mathcal{B}$-transform, we demonstrate the equivalence between the
sourced Ward identities in the Carrollian CFT that holographically encode the
asymptotic bulk dynamics and the Ward identities in the CCFT that encode the
leading and subleading soft theorems. For each of these two points, we
consider both electrodynamics and gravity in the bulk.

##### Organization of the paper

The remaining of the paper is organized as follows. In Section 2, we review
the scattering of massless particles in flat spacetime, which serves to
introduce important notations and conventions for the remaining of the
article. In Section 3, we review the asymptotic symmetry analysis of
electrodynamics and gravity in $4d$ asymptotically flat spacetime. In Section
4 we develop a framework to treat the symmetries and their associated flux-
balance laws / Ward identities in presence of external sources. We then
specify this formalism to sourced theories exhibiting $U(1)$ or conformal
Carrollian symmetries. In Section 5 we provide more insights about the nature
of the proposed $3d$ holographic sourced conformal Carrollian field theory. We
derive the low-point functions for this theory using the sourced Ward
identities. We also propose a holographic correspondence between bulk metric
and boundary Carrollian stress tensor, which allows to show that the sourced
Ward identities of the Carrollian CFT encode the BMS flux-balance laws. In
Section 6, we relate the sourced conformal Carrollian field theory with the
celestial CFT. We provide the integral transform that maps Carrollian source
operators to celestial operators and show that the BMS Ward identities in both
theories are equivalent. The low-point functions are also related with each
other. Finally, in Section 7, we conclude our analysis with some comments and
future directions. This paper is also complemented with some appendices:
Appendix A describes our coordinate conventions; Appendix B reviews the
isomorphism between global conformal Carroll algebra in three dimensions and
the Poincaré algebra in four dimensions; finally, the last Appendix C provides
details on the derivation of the classical constraints on the Carrollian
stress tensor.

##  2 Massless scattering in flat spacetime

This section is a review of salient features of scattering of massless fields
in flat spacetime. We consider a scattering of massless bosonic spin-$s$
fields through the $\mathcal{S}$-matrix approach via perturbation theory in
Minkowski spacetime. We describe the asymptotic free states in terms of plane
wave and conformal primary wave function bases. The boundary operators are
obtained by taking the large radius expansion of the bulk fields as they
approach null infinity. Reciprocally, the bulk fields can be reconstructed
from the boundary operators using the Kirchhoff-d’Adhémar formula that defines
a boundary-to-bulk propagator. Finally, we discuss various integral transforms
that relate position space, momentum space and Mellin space. This allows us to
formulate the scattering problem in position and Mellin spaces, which will be
suitable for the subsequent discussion on flat space holography. To write this
section, we abundantly used references [68, 70, 14, 110, 111, 112, 113] where
complementary material can be found.

###  2.1 Bulk operators

####  2.1.1 Massless fields in Minkowski spacetime and plane wave expansion

Let us consider a free massless bosonic spin-$s$ Fronsdal field
$\phi^{(s)}_{I}$ in Minkowski spacetime [114, 115, 116] ($s=0,1,2,\dots$).
$I=(\mu_{1}\mu_{2}\dots\mu_{s})$ is a symmetrized multi-index notation,
$\mu_{i}\in\\{0,\dots,3\\}$, and spacetime is covered by standard Cartesian
coordinates $X^{\mu}=\\{t,\vec{x}\\}$. The massless spin-$s$ field
$\phi^{(s)}_{I}$ can be conveniently put in the De Donder traceless gauge

$\partial^{\nu}\phi^{(s)}_{\nu\mu_{2}\dots\mu_{s}}(X)=0,\qquad\eta^{\mu\nu}\phi^{(s)}_{\mu\nu\mu_{3}\dots\mu_{s}}(X)=0,$
(2.1)

in which the equations of motion reduce to

$\partial^{\mu}\partial_{\mu}\phi^{(s)}_{I}(X)=0$ (2.2)

and the residual gauge transformations are

$\displaystyle\delta_{\lambda}\phi^{(s)}_{\mu_{1}\dots\mu_{s}}(X)=\partial_{(\mu_{1}}\lambda_{\mu_{2}\dots\mu_{s})}(X)$
(2.3) $\displaystyle\text{such that
}\partial^{\nu}\lambda_{\nu\mu_{3}\dots\mu_{s}}(X)=0\ \text{and}\
\eta^{\mu\nu}\lambda_{\mu\nu\mu_{4}\dots\mu_{s}}(X)=0.$

The field can be quantized in the Heisenberg representation and expanded in
Fourier modes. Each mode is labelled by an on-shell null $4$-momentum vector
(see e.g. [14])

$p^{\mu}(\omega,w,\bar{w})=\omega\,q^{\mu}(w,\bar{w}),\quad
q^{\mu}(w,\bar{w})=\frac{1}{\sqrt{2}}\Big{(}1+w\bar{w},w+\bar{w},-i(w-\bar{w}),1-w\bar{w}\Big{)}$
(2.4)

parametrized by the light-cone energy $\omega>0$ and coordinates $(w,\bar{w})$
on the complex plane. Let $\varepsilon^{\pm}_{\mu}(\vec{q})$ be the
polarization co-vectors,

$\begin{split}\varepsilon^{+}_{\mu}(\vec{q})&=\partial_{w}q_{\mu}=\frac{1}{\sqrt{2}}\big{(}-\bar{w},1,-i,-\bar{w}\big{)},\\\
\varepsilon^{-}_{\mu}(\vec{q})&=[\varepsilon^{+}_{\mu}(\vec{q})]^{*}=\partial_{\bar{w}}q_{\mu}=\frac{1}{\sqrt{2}}\big{(}-w,1,i,-w\big{)}.\end{split}$
(2.5)

We can complete these into a null co-tetrad
$\mathcal{N}=\\{q_{\mu},n_{\mu},\varepsilon^{+}_{\mu},\varepsilon^{-}_{\mu}\\}$
satisfying

$q^{\mu}n_{\mu}=-1,\quad\varepsilon_{+}^{\mu}\varepsilon^{-}_{\mu}=1,\quad
q^{\mu}\varepsilon^{\pm}_{\mu}=0=n^{\mu}\varepsilon^{\pm}_{\mu},$ (2.6)

by setting $n_{\mu}\equiv\partial_{w}\partial_{\bar{w}}q_{\mu}$. The spin-$s$
field in De Donder gauge (2.1) can be expanded in Fourier modes as

$\phi^{(s)}_{I}(X)=K_{(s)}\sum_{\alpha=\pm}\int\frac{d^{3}p}{(2\pi)^{3}\,2p^{0}}\Big{[}\varepsilon_{I}^{*\alpha}(\vec{q})\,a^{(s)}_{\alpha}(\vec{p})\,e^{ip^{\mu}X_{\mu}}+\varepsilon_{I}^{\alpha}(\vec{q})\,a^{(s)}_{\alpha}(\vec{p})^{\dagger}\,e^{-ip^{\mu}X_{\mu}}\Big{]}$
(2.7)

after choosing a Lorentz-invariant measure in momentum space and defining the
polarization tensors

$\varepsilon^{\pm}_{\mu_{1}\dots\mu_{s}}(\vec{q})=\varepsilon^{\pm}_{\mu_{1}}(\vec{q})\varepsilon^{\pm}_{\mu_{2}}(\vec{q})\dots\varepsilon^{\pm}_{\mu_{s}}(\vec{q}),$
(2.8)

which are fully symmetric and transverse tensors, i.e.
$q^{\mu_{i}}\varepsilon^{\pm}_{\mu_{1}\dots\mu_{i}\dots\mu_{s}}(\vec{q})$ for
any $i=1,\dots,s$. The overall constant $K_{(s)}\in\mathbb{R}^{+}_{0}$ may
depend on the coupling constant for the relevant spin (e.g. the elementary
charge $e$ for the spin-$1$ field or the Newton-Cavendish constant $G$ for the
spin-$2$ field). One usually takes

$K_{(1)}=e\sqrt{\hbar},\quad K_{(2)}=\sqrt{32\pi G\hbar}.$ (2.9)

Ladder operators obey the usual commutation relations

$\left[a^{(s)}_{\alpha}(\vec{p}),a^{(s)}_{\alpha^{\prime}}(\vec{p}^{\prime})^{\dagger}\right]=(2\pi)^{3}\,2p^{0}\,\delta^{(3)}(\vec{p}-\vec{p}^{\prime})\,\delta_{\alpha,\alpha^{\prime}}$
(2.10)

as induced by the canonical commutation relations of the field (2.7).

Importantly, a residual gauge transformation (2.3) driven by
$\lambda_{\mu_{2}...\mu_{s}}(X)$ admitting a well defined Fourier transform
acts on the field (2.7) by addition of a term of the form

$\partial_{(\mu_{1}}\lambda_{\mu_{2}\dots\mu_{s})}(X)=\int\frac{d^{3}p}{(2\pi)^{3}\,2p^{0}}\Big{[}p_{(\mu_{1}}\hat{\lambda}_{\mu_{2}\dots\mu_{s})}(\vec{p})\,e^{ip^{\mu}X_{\mu}}+p_{(\mu_{1}}\hat{\lambda}^{*}_{\mu_{2}\dots\mu_{s})}(\vec{p})\,e^{-ip^{\mu}X_{\mu}}\Big{]}.$
(2.11)

This impacts the polarization tensors as

$\delta_{\lambda}\varepsilon^{\alpha}_{\mu_{1}\dots\mu_{s}}(\vec{q})=q_{(\mu_{1}}\tilde{\lambda}^{\alpha}_{\mu_{2}\dots\mu_{s})}(\vec{p})$
(2.12)

writing
$\hat{\lambda}_{\mu_{2}\dots\mu_{s}}(\vec{p})\equiv\frac{K_{(s)}}{\omega}\tilde{\lambda}^{\alpha}_{\mu_{2}\dots\mu_{s}}(\vec{p})a_{\alpha}^{(s)}(\vec{p})$
and $\eta^{\mu\nu}\tilde{\lambda}^{\alpha}_{\mu\nu\mu_{4}\dots\mu_{s}}=0$, but
the ladder operators $a^{(s)}_{\alpha}$ are left invariant by these gauge
transformations. However, as we will review later, asymptotic gauge symmetries
(whose generating parameters do not have finite energy) do act non trivially
on the operators.

Transforming the integration measure for the parametrization (2.4), we can
write

$\phi^{(s)}_{I}(X)=\frac{K_{(s)}}{16\pi^{3}}\sum_{\alpha=\pm}\int\omega\,d\omega\,d^{2}w\Big{[}a^{(s)}_{\alpha}(\omega,w,\bar{w})\varphi_{I}^{*\alpha}(\omega,w,\bar{w}|X)+a^{(s)}_{\alpha}(\omega,w,\bar{w})^{\dagger}\varphi_{I}^{\alpha}(\omega,w,\bar{w}|X)\Big{]}$
(2.13)

where $d^{2}w=i\,dwd\bar{w}$ denotes the integration measure on the complex
plane with local holomorphic coordinates $(w,\bar{w})$ and

$\varphi_{I}^{*\alpha}(\omega,w,\bar{w}|X)\equiv\varepsilon_{I}^{*\alpha}(w,\bar{w})\,e^{i\omega
q^{\mu}X_{\mu}}$ (2.14)

are the basis vectors of plane waves. In this parametrization, the canonical
commutation relations (2.10) become

$\left[a^{(s)}_{\alpha}(\omega,w,\bar{w}),a^{(s)}_{\alpha^{\prime}}(\omega^{\prime},w^{\prime},\bar{w}^{\prime})^{\dagger}\right]=16\pi^{3}\,\omega^{-1}\delta(\omega-\omega^{\prime})\,\delta^{(2)}(w-w^{\prime})\,\delta_{\alpha,\alpha^{\prime}}.$
(2.15)

At the quantum level, $a^{(s)}_{+}(\omega,w,\bar{w})^{\dagger}$ (resp.
$a^{(s)}_{-}(\omega,w,\bar{w})^{\dagger}$) creates a massless particle of spin
$s$ and helicity $J=+s$ (resp. $J=-s$), with energy $\omega$ and a null
momentum pointing towards the direction $q^{\mu}(w,\bar{w})$.

Poincaré transformations $X^{\prime\mu}={\Lambda^{\mu}}_{\nu}X^{\nu}+t^{\mu}$
act on the gauge field as

$\phi^{(s)}_{\mu_{1}\mu_{2}\dots\mu_{s}}(X)\mapsto\phi^{\prime(s)}_{\mu_{1}\mu_{2}\dots\mu_{s}}(X^{\prime})={\Lambda_{\mu_{1}}}^{\nu_{1}}{\Lambda_{\mu_{2}}}^{\nu_{2}}\dots\Lambda_{\mu_{s}}^{\nu_{s}}\phi^{(s)}_{\nu_{1}\nu_{2}\dots\nu_{s}}(X).$
(2.16)

Lorentz transformations induce a $SL(2,\mathbb{C})$ Möbius transformation

$w\mapsto w^{\prime}(w)=\frac{aw+b}{cw+d}$ (2.17)

with $ad-bc=1$ on the complex coordinates determining the direction of the
null momentum $q^{\mu}$ with the embedding (2.4) of the Riemann sphere into
the light cone. The expression of the matrix ${\Lambda^{\mu}}_{\nu}$ in terms
of the Möbius parameters $(a,b,c,d)$ can be found e.g. in [117]. Since
$p^{\mu}$ is a Lorentz vector, one has

$\omega^{\prime}=\left|\frac{\partial w^{\prime}}{\partial
w}\right|^{-1}\omega,\quad
q^{\mu}(w^{\prime},\bar{w}^{\prime})=\left|\frac{\partial w^{\prime}}{\partial
w}\right|{\Lambda^{\mu}}_{\nu}q^{\nu}(w,\bar{w}).$ (2.18)

The second equation states the fact that a Lorentz transformation on the light
cone induces the corresponding Möbius transformation on the Riemann sphere via
the embedding (2.4). Owing to (2.5) and (2.18), one can show that
$\varepsilon_{\mu}^{\pm}(w,\bar{w})$ do not transform homogeneously under the
action of (2.17) but the supplementary terms are part of the residual gauge
freedom (2.12), i.e.

$\varepsilon^{\prime\pm}_{\mu}(w^{\prime},\bar{w}^{\prime})=\left(\frac{\partial
w^{\prime}}{\partial
w}\right)^{\mp\frac{1}{2}}\left(\frac{\partial\bar{w}^{\prime}}{\partial\bar{w}}\right)^{\pm\frac{1}{2}}{\Lambda_{\mu}}^{\nu}\varepsilon_{\nu}^{\pm}(w,\bar{w})+\mathscr{A}(w,\bar{w}){\Lambda_{\mu}}^{\nu}q_{\nu}(w,\bar{w}),$
(2.19)

where $\mathscr{A}$ is a fixed function of $(w,\bar{w})$. For the expansion
(2.13)–(2.14) recalling that the integration measure and the plane wave are
Lorentz-invariant, we deduce from (2.19) that the transformation of ladder
operators under the Poincaré group is

$a_{\pm}^{\prime(s)}(\omega^{\prime},w^{\prime},\bar{w}^{\prime})=\left(\frac{\partial
w^{\prime}}{\partial
w}\right)^{-\frac{J}{2}}\left(\frac{\partial\bar{w}^{\prime}}{\partial\bar{w}}\right)^{\frac{J}{2}}\,e^{-i\omega
q^{\mu}(w,\bar{w}){\Lambda^{\nu}}_{\mu}t_{\nu}}\,a_{\pm}^{(s)}(\omega,w,\bar{w}).$
(2.20)

In particular, one recovers that the ladder operators are eigenvectors of
translations, which is expected since they are assumed to create/annihilate
energy eigenstates. The infinitesimal version of (2.20) can be obtained by
setting $X^{\prime\mu}=X^{\mu}-\epsilon\xi^{\mu}$, with
$\xi^{\mu}={\varpi^{\mu}}_{\nu}X^{\nu}+\tau^{\mu}$
($\varpi_{\mu\nu}=\varpi_{[\mu\nu]}$) and
$w^{\prime}(w)=w-\epsilon\mathcal{Y}^{w}(w)$, and retaining only the linear
terms in $\epsilon$. One concludes that

$\begin{split}&\delta_{\xi(\mathcal{T},\mathcal{Y})}a_{\pm}^{(s)}(\omega,w,\bar{w})\\\
&=\left[-i\omega\mathcal{T}+\mathcal{Y}^{w}\partial_{w}+\mathcal{Y}^{\bar{w}}\partial_{\bar{w}}+\frac{J}{2}\partial_{w}\mathcal{Y}^{w}-\frac{J}{2}\partial_{\bar{w}}\mathcal{Y}^{\bar{w}}-\frac{\omega}{2}(\partial_{w}\mathcal{Y}^{w}+\partial_{\bar{w}}\mathcal{Y}^{\bar{w}})\partial_{\omega}\right]a_{\pm}^{(s)}(\omega,w,\bar{w})\end{split}$
(2.21)

where $\xi(\mathcal{T},\mathcal{Y})$ is now parametrized by the function
$\mathcal{T}(w,\bar{w})=-q^{\mu}(w,\bar{w})\tau_{\mu}$ and the vector
$\mathcal{Y}=\mathcal{Y}^{w}(w)\partial_{w}+\mathcal{Y}^{\bar{w}}(\bar{w})\partial_{\bar{w}}$
on the Riemann sphere, while acting in momentum space covered by coordinates
$(\omega,w,\bar{w})$.

####  2.1.2 Mellin transform and conformal primary basis

In (2.13), we expanded the massless spin-$s$ field in the plane wave basis
(2.14) whose elements are energy eigenstates. Another convenient choice is the
conformal primary wave function basis [66, 69, 68, 70] whose elements are
boost eigenstates. This basis trades the energy parameter $\omega$ for the
eigenvalue $\Delta$ of the Lorentz boost along the direction fixed by the null
momentum $p^{\mu}$.

The map between the two bases is given by the Mellin transform defined as

$F(\Delta)=\mathcal{M}[f(\omega),\Delta]\equiv\int_{0}^{+\infty}d\omega\,\omega^{\Delta-1}\,f(\omega)$
(2.22)

for $f:\mathbb{R}^{+}\to\mathbb{C}$ where $\Delta=c+i\nu$ is generically
complex (see e.g. [118] for the precise assumptions that make the integral
(2.22) well defined). According to the Mellin inversion theorem, if
$F(\Delta)$ is analytic in the complex strip defined by $c\in\ ]a,b[\
\subset\mathbb{R}$ and the integral

$f(\omega)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}d\nu\,\omega^{-(c+i\nu)}F(c+i\nu)$
(2.23)

converges for any $c\in]a,b[$, then it defines the inverse Mellin transform
$f(\omega)\equiv\mathcal{M}^{-1}[F(\Delta),\omega]$ acting on $F(\Delta)$.

The Mellin transform of the plane waves (2.14), for $\omega>0$, yields (see
e.g. [66, 68, 70])

$\begin{split}V_{I}^{*\alpha}(\Delta,w,\bar{w}|X)&=\lim_{\epsilon\to
0^{+}}\varepsilon^{*\alpha}_{I}(w,\bar{w})\int_{0}^{+\infty}d\omega\,\omega^{\Delta-1}\,e^{i\omega
q^{\mu}X_{\mu}-\epsilon\omega}\\\ &=\lim_{\epsilon\to
0^{+}}\varepsilon^{*\alpha}_{I}(w,\bar{w})\frac{i^{\Delta}\Gamma[\Delta]}{(q^{\mu}X_{\mu}+i\epsilon)^{\Delta}}\end{split}$
(2.24)

where $\epsilon>0$ is a regulator that can be arbitrarily close to zero. The
computation of the integral (2.24) is made particularly easy if one uses
Ramanujan’s master theorem, stating that if $f(\omega)$ is a complex-valued
function on the positive real axis admitting a power-series expansion of the
form $f(\omega)=\sum_{k=0}^{+\infty}\frac{\varphi(k)}{k!}(-\omega)^{k}$ in
some neighborhood of the origin, then its Mellin transform is simply given by
$\mathcal{M}[f(\omega),\Delta]=\Gamma[\Delta]\varphi(-\Delta)$ in a certain
range of validity $\Delta\in\mathcal{R}\subset\mathbb{C}$. For the plane wave,
the regulation of the integral (2.24) by a decaying exponential function
$\sim\,e^{-\epsilon\omega}$ allows it to converge in the whole half complex
plane $\mathcal{R}=\\{c+i\nu\,|\,c>0,\nu\in\mathbb{R}\\}$. Using the identity

$\int_{0}^{+\infty}d\omega\,\omega^{i\nu-1}=2\pi\,\delta(\nu),$ (2.25)

one can show [70] that the statement that plane waves form a delta-function
normalizable basis for the Klein-Gordon inner product translates, after Mellin
transform, into requiring that $\Delta$ lays on the principal continuous
series of the irreducible unitary representations of the Lorentz group, i.e.
$\Delta=1+i\nu$.

Let us first consider the expansion of the bulk field (2.13), where we recall
that the polarization is taken as products of (2.5). In terms of the Mellin
representatives (2.24), such an expansion reads [110, 119]

$\phi^{(s)}_{I}(X)=\frac{K_{(s)}}{32\pi^{4}}\sum_{\alpha=\pm}\int
d\nu\,d^{2}w\left[a^{(s)}_{2-\Delta,\alpha}(w,\bar{w})V_{I}^{*\alpha}(\Delta,w,\bar{w}|X)+a^{(s)}_{2-\Delta,\alpha}(w,\bar{w})^{\dagger}V^{\alpha}_{I}(\Delta,w,\bar{w}|X)\right]$
(2.26)

where we defined the ladder operators in the Mellin basis as in [93]

$\displaystyle a_{\Delta,\alpha}^{(s)}(w,\bar{w})$
$\displaystyle\equiv\mathcal{M}\big{[}a^{(s)}_{\alpha}(\omega,w,\bar{w}),\Delta\big{]}=\int_{0}^{+\infty}d\omega\,\omega^{\Delta-1}\,a^{(s)}_{\alpha}(\omega,w,\bar{w}),$
(2.27) $\displaystyle a_{\Delta,\alpha}^{(s)}(w,\bar{w})^{\dagger}$
$\displaystyle\equiv\mathcal{M}\big{[}a^{(s)}_{\alpha}(\omega,w,\bar{w})^{\dagger},\Delta\big{]}=\int_{0}^{+\infty}d\omega\,\omega^{\Delta-1}\,a^{(s)}_{\alpha}(\omega,w,\bar{w})^{\dagger}.$

These last relations can be inverted as

$\begin{split}a^{(s)}_{\alpha}(\omega,w,\bar{w})&=\frac{1}{2\pi}\int_{-\infty}^{+\infty}d\nu\,\omega^{-\Delta}\,a^{(s)}_{\Delta,\alpha}(w,\bar{w}),\\\
a^{(s)}_{\alpha}(\omega,w,\bar{w})^{\dagger}&=\frac{1}{2\pi}\int_{-\infty}^{+\infty}d\nu\,\omega^{-\Delta}\,a^{(s)}_{\Delta,\alpha}(w,\bar{w})^{\dagger}.\end{split}$
(2.28)

Crucially, applying the Mellin transforms (2.27) to (2.20), it has been
observed that these operators transform as

$a^{\prime(s)}_{\Delta,\alpha}(w^{\prime},\bar{w}^{\prime})=\left(\frac{\partial
w^{\prime}}{\partial
w}\right)^{-\frac{\Delta+J}{2}}\left(\frac{\partial\bar{w}^{\prime}}{\partial\bar{w}}\right)^{-\frac{\Delta-J}{2}}a^{(s)}_{\Delta,\alpha}(w,\bar{w}).$
(2.29)

under the action of the Lorentz group, which is precisely the definition of a
conformal primary field of conformal dimension $\Delta$ and $2d$ spin
$J=\alpha s$ (see e.g. [120]). The action of the Poincaré translations in the
Mellin basis is given by [121]

$a_{\Delta,\alpha}^{\prime(s)}(w^{\prime},\bar{w}^{\prime})=e^{-iq^{\mu}(w,\bar{w})t_{\mu}\hat{\partial}_{\Delta}}a^{(s)}_{\Delta,\alpha}(w,\bar{w})=\sum_{n=0}^{+\infty}\frac{\left(-iq^{\mu}(w,\bar{w})t_{\mu}\right)^{n}}{n!}a^{(s)}_{\Delta+n,\alpha}(w,\bar{w})$
(2.30)

defining the discrete derivative operator
$\hat{\partial}_{\Delta}F(\Delta)\equiv F(\Delta+1)$. The infinitesimal action
of Poincaré transformations in the Mellin basis reads as

$\begin{split}&\delta_{\xi(\mathcal{T},\mathcal{Y})}a^{(s)}_{\Delta,\alpha}(w,\bar{w})=\left(-i\mathcal{T}\hat{\partial}_{\Delta}+\mathcal{Y}^{w}\partial_{w}+\mathcal{Y}^{\bar{w}}\partial_{\bar{w}}+\frac{\Delta+J}{2}\partial_{w}\mathcal{Y}^{w}+\frac{\Delta-J}{2}\partial_{\bar{w}}\mathcal{Y}^{\bar{w}}\right)a^{(s)}_{\Delta,\alpha}(w,\bar{w}).\end{split}$
(2.31)

##### Remark

The functions $V_{I}^{\alpha}(\Delta,w,\bar{w}|X)$ defined in (2.24) are not
strictly speaking boost eigenstates since they do not transform covariantly
under the action of the Lorentz group: this is due to the transformation law
of the polarization vector (2.19). Instead, one can define the conformal
primary wave functions [70] by rescaling the functions
$V_{I}^{\alpha}(\Delta,w,\bar{w}|X)$ and performing a gauge transformation

$A_{I}^{\alpha}(\Delta,w,\bar{w}|X)=c^{(s)}(\Delta)V_{I}^{\alpha}(\Delta,w,\bar{w}|X)+\partial_{(\mu_{1}}f^{(s)\alpha}_{\mu_{2}\dots\mu_{s})}(\Delta,w,\bar{w}|X).$
(2.32)

The precise expressions of scaling factor $c^{(s)}(\Delta)$ and gauge
parameter $f^{(s)\alpha}_{\mu_{2}\dots\mu_{s}}(\Delta,w,\bar{w}|X)$ will
depend on the spin $s$ (explicit expressions for $s=1,2$ can be found e.g. in
[70, 119]). It is convenient to use the compact form [112]

$A_{I}^{\pm}(\Delta,w,\bar{w}|X)=m^{\pm}_{\mu_{1}}(w,\bar{w}|X)\dots
m^{\pm}_{\mu_{s}}(w,\bar{w}|X)\frac{1}{(q_{\nu}X^{\nu})^{\Delta}},$ (2.33)

for any (integer) spin $s$, where $m_{\mu}^{\pm}(w,\bar{w}|X)$ represent the
modified polarization co-vectors

$m^{\pm}_{\mu}(w,\bar{w}|X)\equiv\varepsilon^{\pm}_{\mu}(w,\bar{w})-\frac{\varepsilon^{\pm}_{\nu}X^{\nu}}{q_{\rho}X^{\rho}}q_{\mu}(w,\bar{w}).$
(2.34)

One can show that the set of functions (2.33) are actual boost eigenstates,
because now (2.34) transforms homogeneously under $SL(2,\mathbb{C})$
transformations (2.17), i.e.

$m^{\prime\pm}_{\mu}(w^{\prime},\bar{w}^{\prime}|X^{\prime})=\left(\frac{\partial
w^{\prime}}{\partial
w}\right)^{\mp\frac{1}{2}}\left(\frac{\partial\bar{w}^{\prime}}{\partial\bar{w}}\right)^{\pm\frac{1}{2}}{\Lambda_{\mu}}^{\nu}m^{\pm}_{\nu}(w,\bar{w}|X).$
(2.35)

While the basis elements $V_{I}^{\alpha}(\Delta,w,\bar{w}|X)$ only satisfy the
De Donder gauge fixing conditions (2.1) because
$q^{\mu}\varepsilon^{\pm}_{\mu}=0$, the new basis functions
$A_{I}^{\alpha}(\Delta,w,\bar{w}|X)$ also satisfy the radial (or Fock-
Schwinger) gauge condition
$X^{\mu}A^{\alpha}_{\mu\mu_{2}\dots\mu_{s}}(\Delta,w,\bar{w}|X)=0$ because
$X^{\mu}m_{\mu}^{\pm}=0$ [70]. The price to pay for trading
$\varepsilon^{\pm}_{\mu}$ for $m_{\mu}^{\pm}$ is thus the breaking of manifest
translation invariance, because the supplementary gauge fixing is sensitive to
the choice of an origin. Notice that the fixation of the radial gauge is
compatible with the De Donder gauge, as argued in [122]. The so-called
celestial operators are defined as [119]

$\mathscr{O}^{(s)}_{\Delta,\alpha}(w,\bar{w})\equiv\left\langle
A_{I}^{\alpha}(\Delta,w,\bar{w}|X),\phi_{I}^{(s)}(X)\right\rangle$ (2.36)

where $\langle\cdot,\cdot\rangle$ denotes the relevant spin-$s$ inner-product.
For scalar fields ($s=0$), it will be the usual Klein-Gordon inner product;
more involved expressions are needed for $s=1,2$ which can be constructed from
symplectic structures available in the literature (see [119] and references
therein). Notice that the operators
$\mathscr{O}^{(s)}_{\Delta,\alpha}(w,\bar{w})$ and
$a^{(s)}_{\Delta,\alpha}(w,\bar{w})$ are proportional to each other [119, 93]
(see also [123] for more details). Hence we will persist in using consistently
the Mellin ladder operators $a^{(s)}_{\Delta,\alpha}(w,\bar{w})$ in the rest
of the paper.

###  2.2 Boundary operators

####  2.2.1 From bulk to boundary: the large-r expansion

In this section, we review how to construct boundary operators by performing a
large-$r$ expansion around $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$ [14].

We first analyze the late-time behavior of free massless fields around
$\mathscr{I}^{+}$: these will ultimately approximate outgoing free fields of
the $\mathcal{S}$-matrix. We start from the Fourier expansion

$\phi^{(s)}_{I}(X)=\frac{K_{(s)}}{16\pi^{3}}\int\omega\,d\omega\,d^{2}w\,\Big{[}\varepsilon_{I}^{*\alpha}(w,\bar{w})\,a^{(s)}_{\alpha}(\omega,w,\bar{w})\,e^{i\omega
q^{\mu}X_{\mu}}+\varepsilon_{I}^{\alpha}(w,\bar{w})\,a^{(s)}_{\alpha}(\omega,w,\bar{w})^{\dagger}\,e^{-i\omega
q^{\mu}X_{\mu}}\Big{]}$ (2.37)

and use retarded Bondi coordinates $\\{u,r,z,\bar{z}\\}$ with flat boundary
representatives, as defined in Appendix A, to perform an asymptotic expansion
in $r$ near $\mathscr{I}^{+}=\\{r\to+\infty\\}$. Rewriting $X^{\mu}$ as (A.8)
in terms of retarded Bondi coordinates one has $q^{\mu}X_{\mu}=-u-r|z-w|^{2}$.
We now introduce polar coordinates in the complex plane as
$z-w=\rho\,e^{i\vartheta}$. The integration measure becomes $d^{2}w\to
2\,\rho\,d\rho\,d\vartheta$ and altogether (2.37) can be rewritten as

$\phi^{(s)}_{I}(X)=\frac{K_{(s)}}{8\pi^{3}}\int_{0}^{+\infty}d\omega\,\omega\int_{0}^{+\infty}d\rho\,\rho\int_{0}^{2\pi}d\vartheta\left[\varepsilon^{*\alpha}_{I}(w,\bar{w})a^{(s)}_{\alpha}(\omega,w,\bar{w})e^{-i\omega
u-i\omega r\rho^{2}}+\text{h.c.}\right],$ (2.38)

where “h.c.” stands for hermitian conjugation. The expansion of (2.38) in the
limit $r\to+\infty$ (as approaching $\mathscr{I}^{+}$) is given by the
stationary phase approximation of the $\rho$-integral around the point
$\rho=0$ corresponding to the situation where $q$ and $x$ are collinear.
Evaluating the integral in $\rho$ for large-$r$ gives the property [113]

$\rho\,e^{-i\omega
r\rho^{2}}=-\frac{i}{2r\omega}\delta(\rho)+\mathcal{O}(r^{-2}).$ (2.39)

In Bondi retarded coordinates, the expressions of the elements of the co-
tetrad $\mathcal{N}$ are

$\begin{split}q_{\mu}dX^{\mu}&=-du-|z-w|^{2}dr-r(\bar{z}-\bar{w})dz-r(z-w)d\bar{z},\quad
n_{\mu}dX^{\mu}=-dr,\\\
\varepsilon_{\mu}^{+}dX^{\mu}&=(\bar{z}-\bar{w})\,dr+r\,d\bar{z},\quad\varepsilon_{\mu}^{-}dX^{\mu}=(z-w)\,dr+r\,dz,\end{split}$
(2.40)

and, in the collinear limit $w=z$, we have
$\varepsilon_{\mu}^{+}dX^{\mu}=r\,d\bar{z}$ and
$\varepsilon_{\mu}^{-}dX^{\mu}=r\,dz$. Using (2.39), the leading components of
$\phi^{(s)}_{I}(X)$ near $\mathscr{I}^{+}$ are thus

$\phi^{(s)}_{z\dots
z}(X)=-\frac{iK_{(s)}}{8\pi^{2}}r^{s-1}\int_{0}^{+\infty}d\omega\left[a_{+}^{(s)}(\omega,z,\bar{z})e^{-i\omega
u}-a_{-}^{(s)}(\omega,z,\bar{z})^{\dagger}e^{+i\omega
u}\right]+\mathcal{O}(r^{s-2}),$ (2.41)

and its complex conjugated component $\phi^{(s)}_{\bar{z}\dots\bar{z}}(X)$ as
well as components of the form $\phi^{(s)}_{rz\dots z}(X)$,
$\phi^{(s)}_{rrz\dots z}(X)$ etc., all generically of order
$\mathcal{O}(r^{s-1})$ as well. Indeed, the subleading $\mathcal{O}(r^{-2})$
terms in (2.39) contribute in the radial component of in (2.40), recalling
that $dr=-r^{2}d(r^{-1})$, producing a contribution at leading order. One can
extract the _boundary value_ of the field by

$\begin{split}&\bar{\phi}^{(s)}_{z\dots
z}(u,z,\bar{z})\,dz\otimes\cdots\otimes
dz+\bar{\phi}^{(s)}_{\bar{z}\dots\bar{z}}(u,z,\bar{z})\,d\bar{z}\otimes\cdots\otimes
d\bar{z}\\\
&\equiv\lim_{r\to+\infty}\iota^{*}\left(r^{1-s}\,\phi^{(s)}_{\mu_{1}\dots\mu_{s}}dX^{\mu_{1}}\otimes\dots\otimes
dX^{\mu_{s}}\right),\end{split}$ (2.42)

where $\iota^{*}$ denotes the pull-back on the constant $r$ hypersurfaces,
hence deleting the leading radial components. Importantly, the resulting
fields are independent of the residual gauge ambiguity (2.11)–(2.12) as can be
seen from the fact that $q_{\mu}dX^{\mu}=\mathcal{O}(1)$ in the collinear
limit while $\varepsilon_{\mu}^{\pm}dX^{\mu}=\mathcal{O}(r)$ (for a discussion
on this phenomenon, see [124]).

To summarize, the asymptotic behavior of the field $\phi^{(s)}_{I}(X)$ is
encoded by the boundary value (2.42): its Fourier expansion can be directly
read off from (2.41) as [125, 14]

$\bar{\phi}^{(s)}_{z\dots
z}(u,z,\bar{z})=-\frac{iK_{(s)}}{8\pi^{2}}\int_{0}^{+\infty}d\omega\left[a_{+}^{(s)}(\omega,z,\bar{z})e^{-i\omega
u}-a_{-}^{(s)}(\omega,z,\bar{z})^{\dagger}e^{+i\omega u}\right],$ (2.43)

while the Fourier modes for
$\bar{\phi}^{(s)}_{\bar{z}\dots\bar{z}}=(\bar{\phi}^{(s)}_{z\dots
z})^{\dagger}$ are obtained from (2.43) by exchanging $a^{(s)}_{\pm}\to
a^{(s)}_{\mp}$.

##### Remark

As stated below equation (2.42), boundary values (2.43) of the fields are left
invariant under residual gauge transformations (2.3) whose parameters
$\lambda_{\mu_{2}\dots\mu_{s}}(X)$ admit a Fourier decomposition (2.11).
Notice that there exist gauge transformations which do not admit such Fourier
decomposition and do modify the boundary value of the fields. For instance,
picking $s=1$, the action of a gauge transformation
$\delta_{\lambda}\phi^{(1)}_{\mu}=\partial_{\mu}\lambda$ of parameter
$\lambda$ satisfying $\partial^{\mu}\partial_{\mu}\lambda=0$, which is solved
asymptotically by
$\lambda(u,r,z,\bar{z})=\lambda^{(0)}(z,\bar{z})+u\,\partial_{z}\partial_{\bar{z}}\lambda^{(0)}\,r^{-1}\ln
r+\mathcal{O}(r^{-1})$, would shift the boundary gauge field by

$\delta_{\lambda}\bar{\phi}^{(1)}_{z}(u,z,\bar{z})=\partial_{z}\lambda^{(0)}(z,\bar{z}).$
(2.44)

Another point of view is to re-interpret this shift as the following change in
the zero mode of the ladder operators

$\delta_{\lambda}^{S}a^{(s)}_{+}(\omega,z,\bar{z})=-\delta_{\lambda}^{S}a^{(s)}_{-}(\omega,z,\bar{z})^{\dagger}\equiv\frac{8\pi^{2}i}{K_{(1)}}\,\partial_{z}\lambda^{(0)}(z,\bar{z})\,\delta(\omega).$
(2.45)

Hence the operator $\delta_{\lambda}^{S}$ (where $S$ stands for “soft”) only
affects the “Goldstone mode” (i.e. the zero-energy mode transforming in a pure
inhomogeneous way) but leaves the bulk field (2.37) unchanged since
$\omega\,\delta(\omega)\simeq 0$ in the sense of distributions. Notice that
this mode has to be distinguished from the leading soft photon mode [126]

$\mathcal{N}^{(0)}_{z}(z,\bar{z})\equiv\int_{-\infty}^{+\infty}du\,\partial_{u}\bar{\phi}^{(1)}_{z}(u,z,\bar{z})=\bar{\phi}^{(1)}_{z}(u,z,\bar{z})\big{|}_{u\to+\infty}-\bar{\phi}^{(1)}_{z}(u,z,\bar{z})\big{|}_{u\to-\infty}$
(2.46)

whose expression in terms of ladder operators is obtained using (2.43) and
$\int_{-\infty}^{+\infty}du\,e^{\pm i\omega u}=2\pi\delta(\omega)$ and gives

$\mathcal{N}^{(0)}_{z}(z,\bar{z})=-\frac{K_{(1)}}{8\pi}\lim_{\omega\to
0^{+}}\left[\omega\,a_{+}(\omega,z,\bar{z})+\omega\,a_{-}(\omega,z,\bar{z})^{\dagger}\right].$
(2.47)

The non-triviality of the result corresponds to the presence of a pole in the
ladder operators in Fourier space. The discussion is easily extended to any
spin $s\geq 1$.

The commutation relation (2.15) implies (see e.g. [127, 125, 126])

$\begin{split}\big{[}\bar{\phi}^{(s)}_{z\dots
z}(u_{1},z_{1},\bar{z}_{1}),\bar{\phi}^{(s)}_{\bar{z}\dots\bar{z}}(u_{2},z_{2},\bar{z}_{2})\big{]}&=-\frac{i}{4}K_{(s)}^{2}\,\text{sign}(u_{1}-u_{2})\,\delta^{(2)}(z_{1}-z_{2}),\\\
\big{[}\partial_{u_{1}}\bar{\phi}^{(s)}_{z\dots
z}(u_{1},z_{1},\bar{z}_{1}),\bar{\phi}^{(s)}_{\bar{z}\dots\bar{z}}(u_{2},z_{2},\bar{z}_{2})\big{]}&=-\frac{i}{2}K_{(s)}^{2}\,\delta(u_{1}-u_{2})\,\delta^{(2)}(z_{1}-z_{2}).\end{split}$
(2.48)

In our notations, $\text{sign}(x)$ is the sign distribution related to the
Heaviside and Dirac distributions $\Theta(x)$, $\delta(x)$ as

$\text{sign}(x)=2\,\Theta(x)-1,\quad\text{sign}^{\prime}(x)=2\,\delta(x),$
(2.49)

for the particular choice $\Theta(0)=\frac{1}{2}$. Owing to (2.18), the
Fourier transform of (2.20) gives

$\bar{\phi}^{\prime(s)}_{z\dots
z}(u^{\prime},z^{\prime},\bar{z}^{\prime})=\left(\frac{\partial
z^{\prime}}{\partial
z}\right)^{-\frac{1+J}{2}}\left(\frac{\partial\bar{z}^{\prime}}{\partial\bar{z}}\right)^{-\frac{1-J}{2}}\bar{\phi}^{(s)}_{z\dots
z}(u,z,\bar{z})$ (2.50)

in the collinear limit $w=z$, where $u^{\prime}=\left|\frac{\partial
z^{\prime}}{\partial
z}\right|\big{(}u-q^{\mu}(z,\bar{z}){\Lambda_{\mu}}^{\nu}t_{\nu}\big{)}$ and
$J=s$. The complex conjugated component
$\bar{\phi}^{(s)}_{\bar{z}\dots\bar{z}}$ transforms in the same way but with
the flipped helicity $J=-s$. This translates infinitesimally into

$\begin{split}&\delta_{\xi(\mathcal{T},\mathcal{Y})}\bar{\phi}_{z...z}^{(s)}(u,z,\bar{z})=\left(\mathcal{T}+\frac{u}{2}\left(\partial_{z}\mathcal{Y}^{z}+\partial_{\bar{z}}\mathcal{Y}^{\bar{z}}\right)\right)\partial_{u}\bar{\phi}_{z...z}^{(s)}(u,z,\bar{z})\\\
&\quad+\left(\mathcal{Y}^{z}\partial_{z}+\mathcal{Y}^{\bar{z}}\partial_{\bar{z}}+\frac{1+J}{2}\partial_{z}\mathcal{Y}^{z}+\frac{1-J}{2}\partial_{\bar{z}}\mathcal{Y}^{\bar{z}}\right)\bar{\phi}_{z...z}^{(s)}(u,z,\bar{z}).\end{split}$
(2.51)

The same analysis can be performed for operators in the vicinity of past null
infinity $\mathscr{I}^{-}$. The first step consists in trading the retarded
Bondi coordinates $\\{u,r,z,\bar{z}\\}$ for the advanced Bondi coordinates
$\\{v,r^{\prime},z^{\prime},\bar{z}^{\prime}\\}$, also with flat boundary
representative (see Appendix A) in order to perform the large-$r^{\prime}$
expansion around $\mathscr{I}^{-}=\\{r^{\prime}=+\infty\\}$. According to the
change of coordinates (A.13), this amounts effectively to make the permutation
$\\{u\mapsto v,r\mapsto-r^{\prime},z\mapsto z^{\prime}\\}$ in all expressions.
This induces a sign flip in the polarization co-vectors
$\varepsilon_{\mu}^{\pm}$ in (2.40). As now
$q^{\mu}X_{\mu}=-v+r^{\prime}|z^{\prime}-w|^{2}$, another global minus sign is
induced by the evaluation of the $(w,\bar{w})$ integral in (2.13) in the
stationary phase approximation around the collinear configuration
$z^{\prime}=w$, see (2.39). Therefore, the large-$r^{\prime}$ limit of the
field goes in a parallel way as before and provides the following expansion of
operators near $\mathscr{I}^{-}$:

$\phi^{(s)}_{z\dots z}(X)=(-1)^{s}\frac{iK_{(s)}}{8\pi^{2}}r^{\prime
s-1}\int_{0}^{+\infty}d\omega\left[a_{+}^{(s)}(\omega,z^{\prime},\bar{z}^{\prime})e^{-i\omega
v}-a_{-}^{(s)}(\omega,z^{\prime},\bar{z}^{\prime})^{\dagger}e^{+i\omega
v}\right]+\mathcal{O}(r^{\prime s-2}).$ (2.52)

Because the considered coordinate systems $\\{u,r,z,\bar{z}\\}$ and
$\\{v,r^{\prime},z^{\prime},\bar{z}^{\prime}\\}$ both interpolate between
$\mathscr{I}^{-}$ and $\mathscr{I}^{+}$, the most natural choice to extract
the boundary value at $\mathscr{I}^{-}$ is

$\begin{split}&\bar{\phi}^{(s)}_{z\dots
z}(v,z,\bar{z})\,dz\otimes\cdots\otimes
dz+\bar{\phi}^{(s)}_{\bar{z}\dots\bar{z}}(v,z,\bar{z})\,d\bar{z}\otimes\cdots\otimes
d\bar{z}\\\
&\equiv\lim_{r\to-\infty}\iota^{*}\left(r^{1-s}\,\phi^{(s)}_{\mu_{1}\dots\mu_{s}}dX^{\mu_{1}}\otimes\dots\otimes
dX^{\mu_{s}}\right)\end{split}$ (2.53)

as before, where the primes on the $z$’s have been dropped because of (A.13).
This implies

$\bar{\phi}^{(s)}_{z\dots
z}(v,z,\bar{z})=-\frac{iK_{(s)}}{8\pi^{2}}\int_{0}^{+\infty}d\omega\Big{[}a^{(s)}_{+}(\omega,z,\bar{z})e^{-i\omega
v}-a^{(s)}_{-}(\omega,z,\bar{z})^{\dagger}e^{i\omega v}\Big{]},$ (2.54)

and in particular $\bar{\phi}^{(s)}_{z\dots
z}(u,z,\bar{z})\equiv\bar{\phi}^{(s)}_{z\dots z}(v,z,\bar{z})$, which is also
consistent with the change of coordinates (A.13). This finally shows that the
boundary value at $\mathscr{I}^{-}$ also transforms as (2.50)–(2.51) up to the
mere replacement $u\mapsto v$.

####  2.2.2 From boundary to bulk: the Kirchhoff-d’Adhémar formula

Up to this stage, we have reviewed how a bulk free field induces a boundary
field in the asymptotic region. We now make the link with the
Kirchhoff-d’Adhémar formula [128, 129] that describes how to reconstruct the
bulk free field from its boundary value via a boundary-to-bulk propagator.

Inverting the Fourier transform (2.43), we get

$\begin{split}a_{+}^{(s)}(\omega,z,\bar{z})&=\frac{4\pi
i}{K_{(s)}}\int_{-\infty}^{+\infty}du\,e^{i\omega u}\,\bar{\phi}^{(s)}_{z\dots
z}(u,z,\bar{z}),\\\ a_{-}^{(s)}(\omega,z,\bar{z})^{\dagger}&=-\frac{4\pi
i}{K_{(s)}}\int_{-\infty}^{+\infty}du\,e^{-i\omega
u}\,\bar{\phi}^{(s)}_{z\dots z}(u,z,\bar{z}),\end{split}$ (2.55)

for $\omega>0$. Rewriting the expression for the field (2.13) as

$\begin{split}\phi_{I}^{(s)}(X)&=\frac{K_{(s)}}{16\pi^{3}}\int\omega\,d\omega\,d^{2}w\,\varepsilon_{I}^{*+}(w,\bar{w})\left[a^{(s)}_{+}(\omega,w,\bar{w})e^{i\omega
q^{\mu}X_{\mu}}+a^{(s)}_{-}(\omega,w,\bar{w})^{\dagger}e^{-i\omega
q^{\mu}X_{\mu}}\right]+\text{h.c.}\end{split}$ (2.56)

and inserting (2.55) gives

$\displaystyle\phi_{I}^{(s)}(X)$
$\displaystyle=\frac{i}{4\pi^{2}}\int\omega\,d\omega\,d^{2}w\,\varepsilon^{*+}_{I}(w,\bar{w})\int_{-\infty}^{+\infty}d\tilde{u}\left[e^{i\omega(q^{\mu}X_{\mu}+\tilde{u})}-e^{-i\omega(q^{\mu}X_{\mu}+\tilde{u})}\right]\bar{\phi}_{z\dots
z}^{(s)}(\tilde{u},w,\bar{w})+\text{h.c.}$
$\displaystyle=\frac{i}{4\pi^{2}}\int
d^{2}w\,\varepsilon^{*+}_{I}(w,\bar{w})\int_{-\infty}^{+\infty}d\tilde{u}\int_{-\infty}^{+\infty}d\omega\,\omega\,e^{i\omega(q^{\mu}X_{\mu}+\tilde{u})}\,\bar{\phi}_{z\dots
z}^{(s)}(\tilde{u},w,\bar{w})+\text{h.c.}$ (2.57)

One concludes, making use of the identity
$\int_{-\infty}^{+\infty}dx\,x\,e^{ipx}=-2\pi i\,\partial_{p}\delta(p),$ that

$\displaystyle\phi^{(s)}_{I}(X)$
$\displaystyle=\frac{1}{2\pi}\int\,d^{2}w\,d\tilde{u}\;\varepsilon^{*+}_{I}(w,\bar{w})\Big{[}\partial_{\tilde{u}}\delta\left(\tilde{u}+q^{\mu}X_{\mu}\right)\bar{\phi}^{(s)}_{z\dots
z}(\tilde{u},w,\bar{w})\Big{]}+\text{h.c.}$ (2.58)

In this sense, the boundary-to-bulk propagator is
$\mathcal{P}(\tilde{u},w,\bar{w}|X)=\partial_{\tilde{u}}\delta(q^{\mu}X_{\mu}+\tilde{u})$.
Integrating out the $\delta$-distribution, we obtain the Kirchhoff-d’Adhémar
formula [129, 128]

$\phi^{(s)}_{I}(X)=-\frac{1}{2\pi}\int\,d^{2}w\;\varepsilon^{*+}_{I}(w,\bar{w})\;\partial_{\tilde{u}}\bar{\phi}_{z\dots
z}^{(s)}(\tilde{u}=-q^{\mu}X_{\mu},w,\bar{w})+\text{h.c.}$ (2.59)

which allows to reconstruct the bulk field $\phi^{(s)}_{I}(X)$ from its
boundary value at $\mathscr{I}^{+}$. A similar relation holds for the boundary
value at $\mathscr{I}^{-}$.

##### Remark

Consistently with the remark of the previous section, we recover from (2.59)
that a shift of the form (2.44) does not alter the bulk field. Notice that, by
contrast, changing the value of the pole in the Fourier transform
$a_{\pm}^{(s)}(\omega,z,\bar{z})\mapsto
a_{\pm}^{(s)}(\omega,z,\bar{z})+\Delta_{\eta}a_{\pm}^{(s)}(\omega,z,\bar{z})$
with

$\Delta_{\eta}a_{+}^{(s)}(\omega,z,\bar{z})=-\Delta_{\eta}a^{(s)}_{-}(\omega,z,\bar{z})^{\dagger}\equiv-\frac{4\pi}{K_{(s)}}\eta(z,\bar{z})\frac{1}{\omega},$
(2.60)

which corresponds to shifting the boundary field by
$\Delta_{\eta}\bar{\phi}^{(s)}_{z\dots
z}(u,z,\bar{z})=\frac{1}{2}\eta(z,\bar{z})\text{sign}(u)$, does affect the
bulk field as

$\Delta_{\eta}\phi^{(s)}_{I}(X)=-\frac{1}{2\pi}\int\,d^{2}w\;\varepsilon^{*+}_{I}(w,\bar{w})\eta(w,\bar{w})\delta(q^{\mu}X_{\mu})+\text{h.c.}$
(2.61)

###  2.3 From null infinity to the celestial sphere

####  2.3.1 From position space to Mellin basis

Combining the results of the previous sections, we can construct a dictionary
between boundary operators $\bar{\phi}^{(s)}_{z\dots z}(u,z,\bar{z})$ (resp.
$\bar{\phi}^{(s)}_{z\dots z}(v,z,\bar{z})$) evolving in retarded (resp.
advanced) time and the ladder operators in the Mellin basis
$a^{(s)}_{\Delta,\alpha}(z,\bar{z})$ and
$a^{(s)}_{\Delta,\alpha}(z,\bar{z})^{\dagger}$ living on the celestial sphere.
Injecting (2.55) into (2.27), and using (2.24) to compute the integral on
$\omega$, yields

$\begin{split}a^{(s)}_{\Delta,+}(z,\bar{z})&=\frac{4\pi
i}{K_{(s)}}\,\lim_{\epsilon\to
0^{+}}\int_{0}^{+\infty}d\omega\,\omega^{\Delta-1}\int_{-\infty}^{+\infty}du\,e^{i\omega
u-\omega\epsilon}\,\bar{\phi}^{(s)}_{z\dots z}(u,z,\bar{z})\\\
&=\frac{4\pi}{K_{(s)}}\,i^{\Delta+1}\Gamma[\Delta]\,\lim_{\epsilon\to
0^{+}}\int_{-\infty}^{+\infty}du\,(u+i\epsilon)^{-\Delta}\,\bar{\phi}^{(s)}_{z\dots
z}(u,z,\bar{z}),\end{split}$ (2.62)

where $\Delta=c+i\nu$, $c>0$. The integral transform (2.62) coincides with the
one discussed in the extrapolate-style dictionary presented in [93]. It
essentially trades the time dependence $u$ of the operators in position space
for the conformal dimension $\Delta$ of the operators in Mellin space. This
motivates introducing

$\boxed{\mathcal{B}_{\pm}\big{[}f(u),\Delta\big{]}\equiv\kappa_{\Delta}^{\pm}\lim_{\epsilon\to
0^{+}}\int_{-\infty}^{+\infty}du\,(u\pm i\epsilon)^{-\Delta}\,f(u)}$ (2.63)

for any smooth functions $f$ defined on $\mathscr{I}^{+}$, with
$\kappa_{\Delta}^{\pm}=4\pi\,(\pm i)^{\Delta+1}\Gamma[\Delta]$. The integral
(2.63), here referred to as the $\mathcal{B}$-transform, is merely the
composition of a Fourier transform (which maps $u\mapsto\omega$) and a Mellin
transform (which maps $\omega\mapsto\Delta$), see (2.62). One should stress
that the $\mathcal{B}_{+}$-transform (resp. $\mathcal{B}_{-}$) alone is not
invertible since it projects out the positive (resp. negative) frequency modes
of $f$, i.e. it annihilates the second (resp. first) term in the decomposition

$f(u)=\int_{0}^{+\infty}d\omega\big{[}f_{+}(\omega)\,e^{-i\omega
u}+f_{-}(\omega)\,e^{i\omega u}\big{]}.$ (2.64)

This is easily seen from the first line of (2.62) where the integral in $u$
effectively gets rid of all positive frequency modes of
$\bar{\phi}^{(s)}_{z\dots z}(u,z,\bar{z})$. However $\mathcal{B}_{+}$ (resp.
$\mathcal{B}_{-}$) is invertible when restricted to functions with negative
(resp. positive) frequency only. This comes from the fact that composing a
Fourier and a Mellin transform is only possible if the integral over
frequencies is taken over $\mathbb{R}^{+}$, which amounts to use “unilateral”
(hence improper) Fourier transforms that are also not invertible. The physical
meaning of this choice of integration domain is again that the asymptotic
excitations of $\phi^{(s)}_{I}$ have strictly positive energy, $\omega>0$.

Keeping this in mind, we have

$a^{(s)}_{\Delta,+}(z,\bar{z})=\frac{1}{K_{(s)}}\,\mathcal{B}_{+}\Big{[}\bar{\phi}_{z\dots
z}^{(s)}(u,z,\bar{z}),\Delta\Big{]},\quad
a^{(s)}_{\Delta,-}(z,\bar{z})^{\dagger}=\frac{1}{K_{(s)}}\,\mathcal{B}_{-}\Big{[}\bar{\phi}_{z\dots
z}^{(s)}(u,z,\bar{z}),\Delta\Big{]},$ (2.65)

and

$a^{(s)}_{\Delta,-}(z,\bar{z})=\frac{1}{K_{(s)}}\,\mathcal{B}_{+}\Big{[}\bar{\phi}_{\bar{z}\dots\bar{z}}^{(s)}(u,z,\bar{z}),\Delta\Big{]},\quad
a^{(s)}_{\Delta,+}(z,\bar{z})^{\dagger}=\frac{1}{K_{(s)}}\,\mathcal{B}_{-}\Big{[}\bar{\phi}_{\bar{z}\dots\bar{z}}^{(s)}(u,z,\bar{z}),\Delta\Big{]}.$
(2.66)

One can write similar expressions at $\mathscr{I}^{-}$ for
$\bar{\phi}_{z...z}^{(s)}(v,z,\bar{z})$ expanded as (2.54) by making the
replacement $u\mapsto v$.

####  2.3.2 From Mellin basis to position space

We now want to discuss how to reconstruct the boundary field
$\bar{\phi}_{z...z}^{(s)}(u,z,\bar{z})$ from the celestial operators
$a^{(s)}_{\Delta,\pm}(z,\bar{z})$. To achieve this, let us first define

$\boxed{\mathcal{B}^{-1}_{\pm}\big{[}F(\Delta),u\big{]}\equiv-\frac{1}{16\pi^{3}}\lim_{\epsilon\to
0^{+}}\int_{-\infty}^{+\infty}d\nu\,\frac{(\pm
i)^{\Delta}\Gamma[1-\Delta]}{(u\mp i\epsilon)^{1-\Delta}}\,F(\Delta).}$ (2.67)

These operators are effectively obtained by the successive action of two
integral transforms

$\mathcal{B}^{-1}_{\pm}[F(\Delta),u]=\frac{\mp
i}{16\pi^{3}}\int_{0}^{+\infty}d\omega\,e^{\mp i\omega
u}\int_{-\infty}^{+\infty}d\nu\,\omega^{-\Delta}F(\Delta).$ (2.68)

The second integral in (2.68) is the inverse Mellin transform (2.23), while
the first integral means that functions in the image of $\mathcal{B}_{+}^{-1}$
(resp. $\mathcal{B}_{+}^{-1}$) have negative (resp. positive) frequencies
only. Now, one always has

$\displaystyle\mathcal{B}_{\pm}\circ\mathcal{B}^{-1}_{\pm}\big{[}F(\Delta)\big{]}=F(\Delta).$
(2.69)

This relation can checked explicitly by providing a realization of the
$\delta$-distribution on conformal weights as

$\lim_{\epsilon\to
0^{+}}\int_{-\infty}^{+\infty}du\,\frac{i^{\Delta}\Gamma[\Delta]}{(u+i\epsilon)^{\Delta}}\frac{(-i)^{1-\Delta^{\prime}}\Gamma[1-\Delta^{\prime}]}{(u-i\epsilon)^{1-\Delta^{\prime}}}=4\pi^{2}\delta(\nu-\nu^{\prime})$
(2.70)

where $\Delta=c+i\nu$ and $\Delta^{\prime}=c+i\nu^{\prime}$. This identity is
the analog of (2.25) but adapted for the $\mathcal{B}$-transform instead of
the Mellin transform: it can be derived by first rewriting each of the
fractions in the integrand of (2.70) as Mellin transforms of plane waves, then
performing the integral in $u$ to obtain a $\delta$-function and concluding
using (2.25).

However, as we already emphasized, the transforms $\mathcal{B}_{\pm}$ are not
invertible. Rather, if we decompose $f(u)$ in positive/negative frequency
modes as in (2.64), then

$\mathcal{B}_{\pm}^{-1}\circ\mathcal{B}_{\pm}\big{[}f(u)\big{]}=\int_{0}^{+\infty}d\omega\,e^{\mp
i\omega u}f_{\pm}(\omega),$ (2.71)

i.e. $\mathcal{B}_{\pm}^{-1}\circ\mathcal{B}_{\pm}$ are the projectors on
negative/positive frequency modes. This follows from

$\lim_{\epsilon\to
0^{+}}\int_{-\infty}^{+\infty}d\nu\,\frac{i^{\Delta}\Gamma[\Delta]}{(u+i\epsilon)^{\Delta}}\frac{(-i)^{1-\Delta}\Gamma[1-\Delta]}{(u^{\prime}-i\epsilon)^{1-\Delta}}=2\pi\int_{0}^{+\infty}d\omega\,e^{i\omega(u-u^{\prime})},$
(2.72)

which is not $\delta(u-u^{\prime})$ because of the restricted domain of
integration. This is responsible for the fact that
$\mathcal{B}_{\pm}^{-1}\circ\mathcal{B}_{\pm}$ is a projection rather than the
identity map. Equation (2.72) can be derived by rewriting the second fraction
in the integrand as the Mellin transform of a plane wave and recognizing the
resulting integral over $\nu$ as an inverse Mellin transform acting on the
first fraction.

From all these remarks, we conclude that the relations (2.65) are inverted as
follows:

$\bar{\phi}_{z\dots
z}^{(s)}(u,z,\bar{z})=K_{(s)}\,\mathcal{B}^{-1}_{+}\Big{[}a^{(s)}_{\Delta,+}(z,\bar{z}),u\Big{]}+K_{(s)}\,\mathcal{B}^{-1}_{-}\Big{[}a^{(s)}_{\Delta,-}(z,\bar{z})^{\dagger},u\Big{]}.$
(2.73)

with a similar expression for $\bar{\phi}_{z\dots z}^{(s)}(v,z,\bar{z})$
obtained by making the replacement $u\mapsto v$.

###  2.4 $\mathcal{S}$-matrix in the three scattering bases

We now turn to the scattering of interacting massless spin-$s$ fields in flat
spacetime. This discussion will be repeated in the three scattering bases
[113]: momentum space, Mellin space and position space. Even though the
$\mathcal{S}$-matrix is known to be trivial for higher spins $s>2$ [130, 131,
132, 133, 134] (see e.g. [135] for a recent review), it will be valuable to
keep the spin arbitrary throughout to highlight some general structure. In
fact, some patterns of an infrared triangle [14] have been shown to persist
for higher spin theories [136, 137, 138, 139], which makes the analysis for
$s>2$ interesting by itself. Moreover, there exist non-trivial (interacting)
chiral higher-spin theories [140, 141, 142, 143, 144, 145, 146] (see [147,
148] for the relevance of these theories in the context of celestial OPEs).
The presence of interactions in those theories does not necessarily mean that
the $\mathcal{S}$-matrix is non-trivial: typically, the usual theorems apply
and, as a result of the higher-spin symmetries, scattering amplitudes vanish.
However there are examples of chiral higher-spin theories with a non-trivial
$\mathcal{S}$-matrix [149, 150].

####  2.4.1 Asymptotically free fields for null scattering processes

As usual in the $\mathcal{S}$-matrix picture, the interacting theory is taken
to be asymptotically free. More explicitly, in retarded coordinates (A.8), we
suppose that the bulk operator $\phi_{I}^{(s)}$ of the full interacting theory
can be expanded around $\mathscr{I}^{+}$ as

$\phi_{I}^{(s)}(X)=\phi_{I}^{(s)}(X)^{\text{out}}+\mathcal{O}(r^{s-2}),$
(2.74)

and in advanced coordinates (A.11) around $\mathscr{I}^{-}$ as

$\phi_{I}^{(s)}(X)=\phi_{I}^{(s)}(X)^{\text{in}}+\mathcal{O}(r^{s-2}).$ (2.75)

Here $\phi_{I}^{(s)}(X)^{\text{out/in}}$ are the free fields discussed in the
previous sections and $\mathcal{O}(r^{s-2})$ stands for the contributions of
the interactions in the bulk that are not seen near the boundary.

Under these assumptions, the boundary values (2.43) and (2.54) of the
interacting field at $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$ are finite and
coincide with the boundary values of $\phi_{I}^{(s)}(X)^{\text{out/in}}$,

$\displaystyle\bar{\phi}^{(s)}_{z\dots z}(u,z,\bar{z})^{\text{out}}$
$\displaystyle=-\frac{iK_{(s)}}{8\pi^{2}}\int_{0}^{+\infty}d\omega\Big{[}a^{(s)\,\text{out}}_{+}(\omega,z,\bar{z})e^{-i\omega
u}-a_{-}^{(s)\,\text{out}}(\omega,z,\bar{z})^{\dagger}e^{i\omega u}\Big{]},$
(2.76) $\displaystyle\bar{\phi}^{(s)}_{z\dots z}(v,z,\bar{z})^{\text{in}}$
$\displaystyle=-\frac{iK_{(s)}}{8\pi^{2}}\int_{0}^{+\infty}d\omega\Big{[}a^{(s)\,\text{in}}_{+}(\omega,z,\bar{z})e^{-i\omega
v}-a_{-}^{(s)\,\text{in}}(\omega,z,\bar{z})^{\dagger}e^{i\omega v}\Big{]}.$
(2.77)

We therefore asymptotically end up with two free theories related by a non-
trivial interaction process. In the free theory, our conventions (2.42)–(2.53)
mean that incoming and outgoing boundary values are equal, up to the mere
replacement $u\mapsto v$. In presence of interactions, the antipodal matching
conditions of [151, 126] read

$\bar{\phi}^{(s)}_{z\dots
z}(u,z,\bar{z})^{\text{out}}\big{|}_{\mathscr{I}^{+}_{-}}=\bar{\phi}^{(s)}_{z\dots
z}(v,z,\bar{z})^{\text{in}}\big{|}_{\mathscr{I}^{-}_{+}},$ (2.78)

where $\mathscr{I}^{+}_{-}=\\{X\in\mathscr{I}^{+}\,|\,u\to-\infty\\}$ and
$\mathscr{I}^{-}_{+}=\\{X\in\mathscr{I}^{-}\,|\,v\to+\infty\\}$. This is
because the coordinates $z=z^{\prime}$ define antipodal directions on the
celestial sphere between $\mathscr{I}^{-}$ and $\mathscr{I}^{+}$.

In Section 2.3, three descriptions of these fields have been presented: the
direct interpretation in position space and their decomposition in terms of
ladder operators both in Fourier and Mellin spaces. In the following, we
relate the asymptotic states in these three equivalent descriptions by
integral transforms.

####  2.4.2 Scattering amplitudes in momentum space

In momentum space, we can construct an incoming or outgoing state,
respectively denoted as $\ket{\omega,z,\bar{z},s,\alpha}$ and
$\bra{\omega,z,\bar{z},s,\alpha}$, representing a particle of light-cone
energy $\omega$, spin $s$ and helicity $J=\alpha s=\pm s$ coming from or
heading to the point $(z,\bar{z})$ on the celestial sphere, by acting on the
respective vacua with the ladder operators
$a^{(s)\,\text{in/out}}_{\alpha}(\omega,z,\bar{z})$, i.e.

$\displaystyle\bra{\omega,z,\bar{z},s,\alpha}$
$\displaystyle=-\frac{iK_{(s)}}{4\pi}\bra{0}a_{\alpha}^{(s)\,\text{out}}(\omega,z,\bar{z}),$
(2.79) $\displaystyle\ket{\omega,z,\bar{z},s,\alpha}$
$\displaystyle=\frac{iK_{(s)}}{4\pi}a_{\alpha}^{(s)\,\text{in}}(\omega,z,\bar{z})^{\dagger}|0\rangle.$

The perhaps unusual choice of normalization of energy eigenstates in (2.79) is
inspired by the relations (2.55) and will be motivated in Section 2.4.4. The
scattering amplitudes in momentum space involving $N$ massless particles – $n$
of which are outgoing – are given by the $\mathcal{S}$-matrix elements
$\mathcal{A}_{N}(p_{1};\dots;p_{N})=\braket{\text{out}}{\text{in}}_{\text{mom}}$
with

$\bra{\text{out}}=\bra{\omega_{1},z_{1},\bar{z}_{1},s_{1},\alpha_{1}}\otimes\dots\otimes\bra{\omega_{n},z_{n},\bar{z}_{n},s_{n},\alpha_{n}}$
(2.80)

and

$\ket{\text{in}}=\ket{\omega_{n+1},z_{n+1},\bar{z}_{n+1},s_{n+1},\alpha_{n+1}}\otimes\dots\otimes|\omega_{N},z_{N},\bar{z}_{N},s_{N},\alpha_{N}\rangle.$
(2.81)

####  2.4.3 Scattering amplitudes in Mellin space

Alternatively, as it was pointed out in [68, 70] (see [111] for a review), one
can express the $\mathcal{S}$-matrix elements in Mellin space using the change
of representation discussed in Section 2.1.2. The Mellin transform (2.27)
turns the in/out asymptotic ladder operators in momentum space to asymptotic
operators in the Mellin basis. Correspondingly, we obtain the asymptotic
states on the celestial sphere from the usual momentum eigenstates by

$\bra{\Delta,z,\bar{z},s,\alpha}=-\frac{iK_{(s)}}{4\pi}\bra{0}a_{\Delta,\alpha}^{(s)\,\text{out}}(z,\bar{z})=\int_{0}^{+\infty}d\omega\,\omega^{\Delta-1}\bra{\omega,z,\bar{z},s,\alpha}$
(2.82)

for outgoing particles, and

$\ket{\Delta,z,\bar{z},s,\alpha}=\frac{iK_{(s)}}{4\pi}a_{\Delta,\alpha}^{(s)\,\text{in}}(z,\bar{z})^{\dagger}\ket{0}=\int_{0}^{+\infty}d\omega\,\omega^{\Delta-1}\ket{\omega,z,\bar{z},s,\alpha}$
(2.83)

for incoming particles. Because of (2.29), states defined as (2.82)–(2.83) are
boost eigenstates. The so-called “celestial amplitudes”
$\mathcal{M}_{N}=\braket{\text{out}}{\text{in}}_{\text{boost}}$ involving $N$
inserted particles on the celestial sphere are now obtained as (dropping the
spin indices)

$\begin{split}&\mathcal{M}_{N}(\Delta_{1},z_{1},\bar{z}_{1};\dots;\Delta_{N},z_{N},\bar{z}_{N})=\int_{0}^{+\infty}d\omega_{1}\,\omega_{1}^{\Delta_{1}-1}\dots\int_{0}^{+\infty}d\omega_{N}\,\omega_{N}^{\Delta_{N}-1}\mathcal{A}_{N}(p_{1};\dots;p_{N}).\end{split}$
(2.84)

####  2.4.4 Scattering amplitudes in position space

Finally, we can define the asymptotic quantum states directly in position
space. Outgoing states at $\mathscr{I}^{+}$ are naturally defined as

$\displaystyle\bra{u,z,\bar{z},s,+}$
$\displaystyle=\bra{0}\bar{\phi}^{(s)}_{z\dots
z}(u,z,\bar{z})=\frac{1}{2\pi}\int_{0}^{+\infty}d\omega\,e^{-i\omega
u}\langle\omega,z,\bar{z},s,+|,$ (2.85) $\displaystyle\bra{u,z,\bar{z},s,-}$
$\displaystyle=\bra{0}\bar{\phi}^{(s)}_{\bar{z}\dots\bar{z}}(u,z,\bar{z})=\frac{1}{2\pi}\int_{0}^{+\infty}d\omega\,e^{-i\omega
u}\langle\omega,z,\bar{z},s,-|,$

because of (2.76)–(2.77). The boundary field $\bar{\phi}^{(s)}_{z\dots
z}(u,z,\bar{z})$ creates outgoing spin-$s$ particles with positive helicity
and destroys outgoing spin-$s$ particles with negative helicity, while
$\bar{\phi}^{(s)}_{\bar{z}\dots\bar{z}}(u,z,\bar{z})$ $=$
$\bar{\phi}^{(s)}_{z\dots z}(u,z,\bar{z})^{\dagger}$ acts in the opposite way.
The normalization of (2.79) allows to represent the creation and annihilation
operators in position space (2.85) without extra normalization factors.
Playing the same game at $\mathscr{I}^{-}$, we construct the incoming states
in position space as

$\displaystyle\ket{v,z,\bar{z},s,+}$ $\displaystyle=\bar{\phi}^{(s)}_{z\dots
z}(v,z,\bar{z})^{\dagger}\ket{0}=\frac{1}{2\pi}\int_{0}^{+\infty}d\omega\,e^{i\omega
v}|\omega,z,\bar{z},s,+\rangle,$ (2.86) $\displaystyle\ket{v,z,\bar{z},s,-}$
$\displaystyle=\bar{\phi}^{(s)}_{\bar{z}\dots\bar{z}}(v,z,\bar{z})^{\dagger}\ket{0}=\frac{1}{2\pi}\int_{0}^{+\infty}d\omega\,e^{i\omega
v}|\omega,z,\bar{z},s,-\rangle.$

We then introduce “position space amplitudes”
$\mathcal{C}_{N}=\braket{\text{out}}{\text{in}}_{\text{pos}}$, which are
obtained from the usual momentum representation of the $\mathcal{S}$-matrix as

$\boxed{\begin{aligned}
&\mathcal{C}_{N}(u_{1},z_{1},\bar{z}_{1};\dots;u_{n},z_{n},\bar{z}_{n};v_{n+1},z_{n+1},\bar{z}_{n+1};\dots;v_{N},z_{N},\bar{z}_{N})\\\
&=\frac{1}{(2\pi)^{N}}\prod_{k=1}^{n}\int_{0}^{+\infty}d\omega_{k}\,e^{-i\omega_{k}u_{k}}\prod_{\ell=n+1}^{N}\int_{0}^{+\infty}d\omega_{\ell}\,e^{i\omega_{\ell}v_{\ell}}\mathcal{A}_{N}(p_{1};\dots;p_{N}).\end{aligned}}$
(2.87)

These amplitudes can be translated in the Mellin representation thanks to the
$\mathcal{B}$-transform (2.63) by noticing that

$\begin{split}\bra{\Delta,z,\bar{z},s,\alpha}&=\frac{1}{4\pi
i}\mathcal{B}_{+}\big{[}\bra{u,z,\bar{z},s,\alpha},\Delta\big{]},\quad\ket{\Delta,z,\bar{z},s,\alpha}=-\frac{1}{4\pi
i}\mathcal{B}_{-}\big{[}\ket{v,z,\bar{z},s,\alpha},\Delta\big{]},\end{split}$
(2.88)

owing to the definitions (2.82)–(2.83) and (2.65). Hence

$\begin{split}&\mathcal{M}_{N}(\Delta_{1},z_{1},\bar{z}_{1};\dots;\Delta_{N},z_{N},\bar{z}_{N})\\\
&=\frac{(-1)^{N-n}}{(4\pi
i)^{N}}\mathcal{B}_{-}^{(N-n)}\Big{[}\mathcal{B}_{+}^{(n)}\big{[}\mathcal{C}_{N}(u_{1},z_{1},\bar{z}_{1};\dots;v_{N},z_{N},\bar{z}_{N}),\\{\Delta_{1},\dots,\Delta_{n}\\}\big{]},\\{\Delta_{n+1},\dots,\Delta_{N}\\}\Big{]},\end{split}$
(2.89)

where $\mathcal{B}^{(k)}_{+}$ represents $k$ successive applications of the
$\mathcal{B}$-transform.

####  2.4.5 Low-point amplitudes

Let us make more concrete the action of the various integral transforms
introduced so far by considering propagation of one particle, ignoring quantum
loop effects. In momentum basis, the tree-level amplitude $\mathcal{A}_{2}$ is

$\mathcal{A}_{2}(\omega_{1},z_{1},\bar{z}_{1};\omega_{2},z_{2},\bar{z}_{2})=K_{(s)}^{2}\,\pi\,\frac{\delta(\omega_{1}-\omega_{2})}{\omega_{1}}\,\delta^{(2)}(z_{1}-z_{2})\,\delta_{\alpha_{1},\alpha_{2}},$
(2.90)

for $\omega_{i}>0$. This simply describes the travel of a particle of helicity
$J=\alpha s$ inserted at $\mathscr{I}^{-}$ crossing Minkowski spacetime
towards $\mathscr{I}^{+}$.

Recalling the definition (2.82)–(2.83) and using the transformation (2.84),
the two-point amplitude in Mellin space can be deduced from (2.90) and is
given by [71]

$\begin{split}&\mathcal{M}_{2}(\Delta_{1},z_{1},\bar{z}_{1};\Delta_{2},z_{2},\bar{z}_{2})=\int_{0}^{+\infty}d\omega_{1}\,\omega_{1}^{\Delta_{1}-1}\int_{0}^{+\infty}d\omega_{2}\,\omega_{2}^{\Delta_{2}-1}\,\mathcal{A}_{2}(\omega_{1},z_{1},\bar{z}_{1};\omega_{2},z_{2},\bar{z}_{2})\\\
&=K^{2}_{(s)}\,\pi\int_{0}^{+\infty}d\omega\,\omega^{\Delta_{1}+\Delta_{2}-3}\,\delta^{(2)}(z_{1}-z_{2})\,\delta_{\alpha_{1},\alpha_{2}}=2\pi^{2}\,K_{(s)}^{2}\,\delta(\nu_{1}+\nu_{2})\,\delta^{(2)}(z_{1}-z_{2})\,\delta_{\alpha_{1},\alpha_{2}},\end{split}$
(2.91)

where $\Delta_{i}=1+i\nu_{i}$ is assumed for the integral over $\omega$ to be
converging, as prescribed by (2.25).

The position space amplitude $\mathcal{C}_{2}$ can also be computed from
(2.90) as

$\begin{split}\mathcal{C}_{2}(u_{1},z_{1},\bar{z}_{1};u_{2},z_{2},\bar{z}_{2})&=\frac{1}{4\pi^{2}}\int_{0}^{+\infty}d\omega_{1}\int_{0}^{+\infty}d\omega_{2}\,e^{-i\omega_{1}u_{1}}e^{i\omega_{2}u_{2}}\mathcal{A}_{2}(\omega_{1},z_{1},\bar{z}_{1};\omega_{2},z_{2},\bar{z}_{2})\\\
&=\frac{K^{2}_{(s)}}{4\pi}\underbrace{\int_{0}^{+\infty}\frac{d\omega}{\omega}\,e^{-i\omega(u_{1}-u_{2})}}_{\equiv\,\mathcal{I}_{0}(u_{1}-u_{2})}\,\delta^{(2)}(z_{1}-z_{2})\,\delta_{\alpha_{1},\alpha_{2}},\end{split}$
(2.92)

owing to (2.87) and using (A.13) to trade $v$ for $u$. The integral
$\mathcal{I}_{0}(u_{1}-u_{2})$ is divergent but can nevertheless be regulated
noticing that $\displaystyle\mathcal{I}_{0}(u_{1}-u_{2})=\lim_{\beta\to
0^{+}}\mathcal{I}_{\beta}(u_{1}-u_{2})$ with the definition

$\begin{split}\mathcal{I}_{\beta}(x)&=\lim_{\epsilon\to
0^{+}}\int_{0}^{+\infty}d\omega\,\omega^{\beta-1}\,e^{-i\omega
x-\omega\epsilon}=\lim_{\epsilon\to
0^{+}}\frac{\Gamma[\beta](-i)^{\beta}}{(x-i\epsilon)^{\beta}}.\end{split}$
(2.93)

In the limit $\beta\to 0^{+}$, we obtain

$\mathcal{I}_{\beta}(x)=\frac{1}{\beta}-\left[\gamma+\ln|x|+\frac{i\pi}{2}\text{sign}(x)\right]+\mathcal{O}(\beta),$
(2.94)

where $\gamma$ is the Euler-Mascheroni constant. So (2.92) evaluates to

$\boxed{\mathcal{C}_{2}(u_{1};u_{2})=\frac{K^{2}_{(s)}}{4\pi}\left[\frac{1}{\beta}-\left(\gamma+\ln|u_{1}-u_{2}|+\frac{i\pi}{2}\text{sign}(u_{1}-u_{2})\right)\right]\delta^{(2)}(z_{1}-z_{2})\,\delta_{\alpha_{1},\alpha_{2}}+\mathcal{O}(\beta).}$
(2.95)

We show in Section 5.4 that (2.95) is a Poincaré invariant object as a
consequence of (2.50): importantly, the pole in $1/\beta$ shall be kept
because it is essential to ensure boost invariance. Let us stress that,
fundamentally, this divergence can be interpreted as an infrared pole. Indeed,
if we choose to regularize the integral $\mathcal{I}_{0}(x)$ as

$\mathcal{I}_{0}(x)=\lim_{\epsilon\to
0^{+}}\int_{\epsilon}^{+\infty}\frac{d\omega}{\omega}\,e^{-i\omega
x}\equiv\lim_{\epsilon\to 0^{+}}E_{1}(ix\epsilon)$ (2.96)

where $E_{1}(z)$ represents the principal branch
$\text{Arg}(z)\in\,\,]-\pi,\pi[$ of the complex exponential integral, for any
$z\notin\mathbb{R}_{-}$, we have the following series [152]

$E_{1}(z)=-\gamma-\ln
z-\sum_{k=1}^{+\infty}\frac{1}{k!}\frac{(-z)^{k}}{k}\quad\Rightarrow\quad\mathcal{I}_{0}(x)=\ln\frac{1}{\epsilon}-\gamma-\ln|x|-\text{sign}(x)\,\frac{i\pi}{2}+\mathcal{O}(\epsilon)$
(2.97)

in the limit $\epsilon\to 0^{+}$. Boost invariance requires that
$\ln(\epsilon^{-1})$ behaves around zero as the pole of the Euler Gamma
function, hence $\beta^{-1}\simeq\ln(\epsilon^{-1})$ and both treatments of
the divergence agree on the final result.

Let us now observe thanks to (2.85)–(2.86) that

$\begin{split}\mathcal{C}_{2}(u_{1},z_{1},\bar{z}_{1};u_{2},z_{2},\bar{z}_{2})\equiv\langle
0|\bar{\phi}^{(s)}_{z\dots z}(u_{1},z_{1},\bar{z}_{1})\bar{\phi}^{(s)}_{z\dots
z}(u_{2},z_{2},\bar{z}_{2})^{\dagger}|0\rangle\end{split}$ (2.98)

picking $\alpha=+$ for definiteness (similar considerations apply, mutatis
mutandis, for $\alpha=-$). In Section 5, we will interpret this object as a
(holographic) boundary two-point function: it results from the double limit of
the bulk two-point function where one point is sent to $\mathscr{I}^{+}$ and
the other one to $\mathscr{I}^{-}$. Although (2.98) has an infrared
divergence, we directly observe from (2.94) that the difference
$\mathcal{I}_{\beta}(u_{1}-u_{2})-\mathcal{I}_{\beta}(u_{2}-u_{1})$ is finite
in the limit $\beta\to 0^{+}$ and reproduces the commutation relation (2.48),
i.e.

$\begin{split}&\langle 0|\big{[}\bar{\phi}^{(s)}_{z\dots
z}(u_{1},z_{1},\bar{z}_{1}),\bar{\phi}^{(s)}_{z\dots
z}(u_{2},z_{2},\bar{z}_{2})^{\dagger}\big{]}|0\rangle\\\ &=\lim_{\beta\to
0^{+}}\frac{K^{2}_{(s)}}{4\pi}\left[\mathcal{I}_{\beta}(u_{1}-u_{2})-\mathcal{I}_{\beta}(u_{2}-u_{1})\right]\,\delta^{(2)}(z_{1}-z_{2})\\\
&=-\frac{iK_{(s)}^{2}}{4}\,\text{sign}(u_{1}-u_{2})\,\delta^{(2)}(z_{1}-z_{2}).\end{split}$
(2.99)

The behavior (2.95) of the boundary two-point function (2.98) has also been
established independently in [153].

Finally, let us close the loop by showing that applying twice the
$\mathcal{B}$-transform as prescribed in (2.89) gives us back the celestial
amplitude. Keeping (2.88) in mind, we first compute

$\begin{split}&\mathcal{B}_{+}\big{[}\mathcal{I}_{0}(u_{1}-u_{2}),\Delta_{1}\big{]}=4\pi\,i^{\Delta_{1}+1}\Gamma[\Delta_{1}]\lim_{\epsilon\to
0^{+}}\int_{-\infty}^{+\infty}\frac{du_{1}}{(u_{1}+i\epsilon)^{\Delta_{1}}}\mathcal{I}_{0}(u_{1}-u_{2})\\\
&=4\pi i\lim_{\epsilon\to
0^{+}}\int_{-\infty}^{+\infty}du_{1}\int_{0}^{+\infty}\omega^{\Delta_{1}-1}\,e^{i\omega
u_{1}-\omega\epsilon}\int_{0}^{+\infty}\frac{d\omega^{\prime}}{\omega^{\prime}}e^{-i\omega^{\prime}(u_{1}-u_{2})}\\\
&=8\pi^{2}i\lim_{\epsilon\to
0^{+}}\int_{0}^{+\infty}d\omega\,\omega^{\Delta_{1}-2}\,e^{i\omega
u_{2}-\omega\epsilon}=8\pi^{2}i\lim_{\epsilon\to
0^{+}}\frac{i^{\Delta_{1}-1}\Gamma[\Delta_{1}-1]}{(u_{2}+i\epsilon)^{\Delta_{1}-1}}.\end{split}$
(2.100)

Computing the second $\mathcal{B}$-transform leads to

$\begin{split}&\mathcal{B}_{-}\Big{[}\mathcal{B}_{+}\big{[}\mathcal{I}_{0}(u_{1}-u_{2}),\Delta_{1}\big{]},\Delta_{2}\Big{]}\\\
&=32\pi^{3}\lim_{\epsilon\to
0^{+}}\int_{-\infty}^{+\infty}du_{2}\,\frac{i^{\Delta_{1}-1}\Gamma[\Delta_{1}-1]}{(u_{2}+i\epsilon)^{\Delta_{1}-1}}\frac{(-i)^{\Delta_{2}}\Gamma[\Delta_{2}]}{(u_{2}-i\epsilon)^{\Delta_{2}}}=128\pi^{5}\delta(\nu_{1}+\nu_{2}),\end{split}$
(2.101)

as a corollary of (2.70) assuming $\Delta_{i}=1+i\nu_{i}$. Therefore, using
the dictionary (2.89),

$\begin{split}\mathcal{M}_{2}(\Delta_{1},z_{1},\bar{z}_{1};\Delta_{2},z_{2},\bar{z}_{2})&=\frac{1}{(4\pi)^{2}}\mathcal{B}_{-}\Big{[}\mathcal{B}_{+}\big{[}\mathcal{C}_{2}(u_{1},z_{1},\bar{z}_{1};u_{2},z_{2},\bar{z}_{2}),\Delta_{1}\big{]},\Delta_{2}\Big{]}\\\
&=2\pi^{2}\,K^{2}_{(s)}\,\delta(\nu_{1}+\nu_{2})\,\delta^{(2)}(z_{1}-z_{2})\,\delta_{\alpha_{1},\alpha_{2}},\end{split}$
(2.102)

we recover (2.91) as it should.

A similar treatment could be applied to three-point amplitudes but kinematic
constraints imply that the latter have to vanish in Lorentzian signature
$(-,+,+,+)$. This issue can be circumvented by working in split signature
$(-,+,-,+)$ in which $z$ and $\bar{z}$ are no longer related by complex
conjugation (see e.g. [154]) and the description of amplitudes in position
space can be adapted to this framework. These considerations, as well as the
computation of the higher-point amplitudes ($N\geq 4$) will be discussed
elsewhere.

####  2.4.6 The three bases for scattering amplitudes in flat spacetime

From the above discussion, it follows that one can work in three different
spaces to discuss the boundary fields:

  1. $(i)$

The position space involving the boundary field
$\bar{\phi}^{(s)}(u/v,z,\bar{z})$,

  2. $(ii)$

The Fourier space providing the usual description by ladder operators
$a_{\pm}^{(s)\,\text{out/in}}(\omega,z,\bar{z})$,

  3. $(iii)$

The Mellin space with the celestial ladder operators
$a^{(s)\,\text{out/in}}_{\Delta,\pm}(z,\bar{z})$,

see Figure 3 for a summary. This observation was already pointed out in [113]
where it was argued that the three spaces, or “bases”, are useful, depending
on the question of interest. For instance, to discuss infrared issues, Fourier
space will be suited to write the soft theorems, Mellin space will be more
convenient to discuss symmetries through some Ward identities, and position
space will be appropriate to highlight the memory effects.

Position space $\bar{\phi}(u/v,z,\bar{z})$ Mellin space
$a^{\text{out/in}}_{\Delta}(z,\bar{z})$ Momentum space
$a^{\text{out/in}}(\omega,z,\bar{z})$ $\mathcal{B}$-transform (2.65)–(2.73)
Fourier transform(2.43)–(2.55)Mellin transform(2.27)–(2.28) Figure 3:
Interplay between the three bases of scattering in flat spacetime.

We observe that the $\mathcal{S}$-matrix is holographic by nature, in the
sense that its elements can be re-interpreted as correlations between boundary
operators. Interestingly, as we will argue in Section 5, the boundary
operators in position space can be interpreted as operators sourcing a dual
Carrollian CFT. The $\mathcal{S}$-matrix elements are then just seen as
correlation functions for these operators. This points towards the Carrollian
approach of flat space holography. Similarly, the boundary operators in Mellin
space can be interpreted as operators in the CCFT, pointing towards the
celestial holography proposal. Therefore, one can already deduce that the
$\mathcal{B}$-transform discussed in Section 2.3 relates the Carrollian CFT
and the CCFT. It trades the null time dependence of the Carrollian operators
for the conformal dimension of the CCFT operators. We will come back to this
observation in Section 6 when discussing the link between Carrollian and
celestial holographies.

##  3 Asymptotic symmetries in flat spacetime

This section reviews the asymptotic symmetry analysis of electrodynamics and
gravity. This allows us to fix the notations and conventions that will be
useful in order to discuss the holographic Carrollian correspondence in
Section 5. The Bondi-Metzner-Sachs (BMS) asymptotic symmetries of gravity will
play the role of global spacetime symmetries in the dual Carrollian CFT, while
the $U(1)$ asymptotic symmetries of electrodynamics will provide an example of
global internal symmetry.

###  3.1 Scalar electrodynamics

We start by reviewing the asymptotic symmetry structure of electrodynamics on
a flat background (see [126, 155] for the original references and [14] for a
review). We denote the electromagnetic potential as $A_{\mu}(X)$ and the
matter current as $\mathpzc{J}_{\mu}(X)$. For definiteness, we focus on
complex scalar matter field $\phi(X)$ carrying a charge $Qe$, where
$Q\in\mathbb{R}$ and $e$ is the elementary charge. The electromagnetic current
is thus

$\mathpzc{J}_{\mu}=iQe\big{(}\phi^{*}\partial_{\mu}\phi-\phi\partial_{\mu}\phi^{*}\big{)}.$
(3.1)

In the retarded coordinates $\\{u,r,z,\bar{z}\\}$, radiative boundary
conditions for the potential $A_{\mu}$ near future null infinity are

$\begin{split}A_{z}(u,r,z,\bar{z})&=A_{z}^{(0)}(u,z,\bar{z})+\mathcal{O}(r^{-1}),\\\
A_{r}(u,r,z,\bar{z})&=\mathcal{O}(r^{-2}),\quad
A_{u}(u,r,z,\bar{z})=\mathcal{O}(r^{-1}),\end{split}$ (3.2)

consistently with Section 2. Comparing with (2.43), the role of the boundary
field is played here by $A_{z}^{(0)}(u,z,\bar{z})$. These boundary conditions
are preserved by gauge transformations with
$\lambda(u,r,z,\bar{z})=\lambda^{(0)}(z,\bar{z})+\mathcal{O}(r^{-1})$ such
that $\delta_{\lambda}A^{(0)}_{z}=\partial_{z}\lambda^{(0)}$. On the matter
side, the massless scalar field decays as

$\phi(u,r,z,\bar{z})=\frac{\phi^{(0)}(u,z,\bar{z})}{r}+\mathcal{O}(r^{-2})$
(3.3)

at future null infinity. The boundary field $\phi^{(0)}(u,z,\bar{z})$ is left
unconstrained by the massless Klein-Gordon equation and transforms as

$\delta_{\lambda}\phi^{(0)}(u,z,\bar{z})=-i\lambda^{(0)}(z,\bar{z})Qe\phi^{(0)}(u,z,\bar{z})$
(3.4)

under the gauge symmetry.

Expanding the field strength
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and current
$\mathpzc{J}_{\mu}$ in $1/r$ near $\mathscr{I}^{+}$ taking (3.2) into account,
we have

$\begin{split}F_{uz}(u,r,z,\bar{z})&=F_{uz}^{(0)}(u,z,\bar{z})+\mathcal{O}(r^{-1}),\\\
F_{ru}(u,r,z,\bar{z})&=F_{ru}^{(2)}(u,z,\bar{z})r^{-2}+\mathcal{O}(r^{-3}),\\\
F_{z\bar{z}}(u,r,z,\bar{z})&=F_{z\bar{z}}^{(0)}(u,z,\bar{z})+\mathcal{O}(r^{-1}),\\\
\mathpzc{J}_{u}(u,r,z,\bar{z})&=\mathpzc{J}_{u}^{(2)}(u,z,\bar{z})r^{-2}+\mathcal{O}(r^{-3}).\end{split}$
(3.5)

In addition, one requires that the magnetic field is vanishing at
$\mathscr{I}^{+}_{\pm}$ (see [156] for a consideration of magnetic
contributions at $\mathscr{I}^{+}_{\pm}$), i.e.

$\begin{split}F_{z\bar{z}}^{(0)}\Big{|}_{\mathscr{I}^{+}_{\pm}}=(\partial_{z}A^{(0)}_{\bar{z}}-\partial_{\bar{z}}A^{(0)}_{z})\Big{|}_{\mathscr{I}^{+}_{\pm}}=0.\end{split}$
(3.6)

The leading equation of motion is

$\partial_{u}F_{ru}^{(2)}+\partial_{\bar{z}}F_{uz}^{(0)}+\partial_{z}F_{u\bar{z}}^{(0)}+e^{2}\mathpzc{J}^{(2)}_{u}=0.$
(3.7)

The surface charges associated with the residual gauge symmetry driven by
$\lambda^{(0)}$ can be derived as [126, 155, 14]

$Q_{\lambda}[A]=-\frac{1}{e^{2}}\int_{\Sigma}d^{2}z\,\lambda^{(0)}(z,\bar{z})\,F^{(2)}_{ru}(u,z,\bar{z}),$
(3.8)

where $\Sigma$ is a $\\{u=\text{const.}\\}$ cut of $\mathscr{I}^{+}$. As
retarded time evolves along $\mathscr{I}^{+}$, these charges are not conserved
because of the presence of electromagnetic radiation, encoded in the leading
of piece of the Maxwell field $A^{(0)}_{z}$, and the null matter current
$\mathpzc{J}^{(2)}_{u}$. Indeed, using (3.7), one has the following flux-
balance law:

$\frac{dQ_{\lambda}}{du}=\frac{1}{e^{2}}\int_{\Sigma}d^{2}z\,\lambda^{(0)}\left(\partial_{\bar{z}}F_{uz}^{(0)}+\partial_{z}F_{u\bar{z}}^{(0)}+e^{2}\,\mathpzc{J}^{(2)}_{u}\right)\equiv\int_{\Sigma}d^{2}z\,F_{\lambda}.$
(3.9)

In order to distinguish between hard (finite energy) and soft (zero energy)
radiative excitations of the gauge field, one can split it as [126]

$A^{(0)}_{z}(u,z,\bar{z})=\tilde{A}_{z}^{(0)}(u,z,\bar{z})+\partial_{z}\chi(z,\bar{z})$
(3.10)

where $\chi(z,\bar{z})$ is related to the early and late-time values of the
boundary field as

$\partial_{z}\chi(z,\bar{z})=\frac{1}{2}\left(A_{z}^{(0)}\Big{|}_{\mathscr{I}^{+}_{+}}+A_{z}^{(0)}\Big{|}_{\mathscr{I}^{+}_{-}}\right).$
(3.11)

Because $\chi(z,\bar{z})$ transforms by an inhomogeneous shift
($\delta_{\lambda}\chi=\lambda^{(0)}$), it is interpreted as the Goldstone
mode of asymptotic symmetry breaking while $\tilde{A}_{z}^{(0)}(u,z,\bar{z})$
is taken to be gauge-invariant (i.e. $\delta_{\lambda}\tilde{A}_{z}^{(0)}=0$).
The latter encodes the hard modes of the boundary value $A^{(0)}_{z}$ of the
gauge field $A_{\mu}$, which are captured by the Fourier expansion (2.43) for
$s=1$. In particular, it transforms as (2.51) with $J=1$ under Poincaré
symmetries. Defining
$\mathcal{N}^{(0)}_{z}\equiv\int_{-\infty}^{+\infty}du\,F_{uz}^{(0)}$ and
recalling the boundary condition (3.6), we have

$\begin{split}\partial_{z}\mathcal{N}^{(0)}_{\bar{z}}-\partial_{\bar{z}}\mathcal{N}^{(0)}_{z}=\int_{-\infty}^{+\infty}du\,\partial_{u}F_{z\bar{z}}^{(0)}=F_{z\bar{z}}^{(0)}\Big{|}_{\mathscr{I}^{+}_{+}}-F_{z\bar{z}}^{(0)}\Big{|}_{\mathscr{I}^{+}_{-}}=0,\end{split}$
(3.12)

where we used the Bianchi identity $\partial_{[\mu}F_{\nu\rho]}=0$ expressed
for the leading components of the Faraday tensor. This equation is solved if
there exists a scalar field $N(z,\bar{z})$ such that

$\partial_{z}N(z,\bar{z})\equiv\frac{1}{e^{2}}\int_{-\infty}^{+\infty}du\,F^{(0)}_{uz}(u,z,\bar{z})=\frac{1}{e^{2}}\left(A_{z}^{(0)}\Big{|}_{\mathscr{I}^{+}_{+}}-A_{z}^{(0)}\Big{|}_{\mathscr{I}^{+}_{-}}\right).$
(3.13)

Notice that the so-defined $N(z,\bar{z})$, referred to as the memory mode, is
gauge-invariant. Hence the radiative variables are organized in hard and soft
fields as

$\Gamma^{H}=\\{\tilde{A}^{(0)}_{z},\partial_{u}\tilde{A}^{(0)}_{z},\tilde{A}^{(0)}_{\bar{z}},\partial_{u}\tilde{A}^{(0)}_{\bar{z}}\\},\quad\Gamma^{S}=\\{\partial_{z}\chi,\partial_{z}N,\partial_{\bar{z}}\chi,\partial_{\bar{z}}N\\}.$
(3.14)

The evaluation of the symplectic structure at the boundary $\mathscr{I}^{+}$
has been detailed e.g. in [126, 14] and gives rise to the following Poisson
brackets:

$\displaystyle\left\\{\partial_{u_{1}}\tilde{A}_{z_{1}}^{(0)}(u_{1},z_{1},\bar{z}_{1}),\tilde{A}_{\bar{z}_{2}}^{(0)}(u_{2},z_{2},\bar{z}_{2})\right\\}=-\frac{e^{2}}{2}\,\delta(u_{1}-u_{2})\,\delta^{(2)}(z_{1}-z_{2}),$
(3.15)
$\displaystyle\left\\{\partial_{z_{1}}N(z_{1},\bar{z}_{1}),\partial_{\bar{z}_{2}}\chi(z_{2},\bar{z}_{2})\right\\}=-\frac{1}{2}\,\delta^{(2)}(z_{1}-z_{2})$
(3.16)

for the hard and soft symplectic pairs
$(\tilde{A}^{(0)}_{z},\partial_{u}\tilde{A}^{(0)}_{\bar{z}})$ and
$(\partial_{z}\chi,\partial_{\bar{z}}N)$. The quantum version of the bracket
(3.15) is nothing but (2.48) for the choice of normalization (2.9). Because
the theory is linear, the hard flux only encompasses matter contributions
while the flux terms depending on the gauge field are soft, i.e.

$\displaystyle\mathcal{F}^{H}_{\lambda}$
$\displaystyle=\int_{\mathscr{I}^{+}}du\,d^{2}z\,\lambda^{(0)}\,\mathpzc{J}_{u}^{(2)},$
(3.17) $\displaystyle\mathcal{F}^{S}_{\lambda}$
$\displaystyle=\frac{1}{e^{2}}\int_{\mathscr{I}^{+}}du\,d^{2}z\,\lambda^{(0)}\left(\partial_{\bar{z}}F_{uz}^{(0)}+\partial_{z}F_{u\bar{z}}^{(0)}\right)=2\int_{\Sigma}d^{2}z\,\lambda^{(0)}\,\partial_{z}\partial_{\bar{z}}N,$
(3.18)

where the second equality of (3.18) holds because of (3.12) and (3.13). From
(3.15)–(3.16) together with the canonical commutation relation
$\\{\phi^{(0)}(u,z,\bar{z}),\partial_{u^{\prime}}\phi^{(0)}(u^{\prime},z^{\prime},\bar{z}^{\prime})\\}=\delta(u-u^{\prime})\,\delta^{(2)}(z-z^{\prime})$
for the matter field, one deduces the very important properties

$\begin{split}&\\{\mathcal{F}_{\lambda}^{H},\Gamma^{S}\\}=0=\\{\mathcal{F}_{\lambda}^{S},\Gamma^{H}\\},\\\
&\\{\mathcal{F}_{\lambda}^{H},\tilde{A}_{z}^{(0)}\\}=0,\quad\\{\mathcal{F}_{\lambda}^{S},\partial_{z}\chi\\}=\partial_{z}\lambda^{(0)},\\\
&\\{\mathcal{F}_{\lambda}^{H},\phi^{(0)}\\}=\delta_{\lambda}\phi^{(0)},\quad\\{\mathcal{F}_{\lambda}^{S},\phi^{(0)}\\}=0.\end{split}$
(3.19)

Because $N$ and $\mathpzc{J}_{\mu}$ are gauge-invariant, we also have that
hard and soft fluxes represent separately the $U(1)$ algebra, i.e.
$\delta_{\lambda_{1}}\mathcal{F}_{\lambda_{2}}^{H/S}=\\{\mathcal{F}_{\lambda_{1}}^{H/S},\mathcal{F}_{\lambda_{2}}^{H/S}\\}=0$.

A similar analysis can be performed at $\mathscr{I}^{-}$, choosing the
advanced Bondi coordinates $\\{v,r,z,\bar{z}\\}$ as defined in Appendix A. The
equation of motion (3.7) becomes

$-\partial_{v}F_{rv}^{(2)}+\partial_{\bar{z}}F_{vz}^{(0)}+\partial_{z}F_{v\bar{z}}^{(0)}+e^{2}\mathpzc{J}_{v}^{(2)}=0.$
(3.20)

The induced surface charges are

$Q_{\lambda}[A]=-\frac{1}{e^{2}}\int_{\Sigma}d^{2}z\,\lambda^{(0)}(z,\bar{z})\,F_{rv}^{(2)}(v,z,\bar{z}),$
(3.21)

with no sign change with respect to (3.8). The fluxes get a sign flip because
of the corresponding sign flip in the equation of motion (3.20), i.e.

$\frac{dQ_{\lambda}}{dv}=-\frac{1}{e^{2}}\int_{\Sigma}d^{2}z\,\lambda^{(0)}\left(\partial_{\bar{z}}F_{vz}^{(0)}+\partial_{z}F_{v\bar{z}}^{(0)}+e^{2}\,\mathpzc{J}^{(2)}_{v}\right)=\int_{\Sigma}d^{2}z\,F_{\lambda}.$
(3.22)

Using an homologous split between hard and soft modes in the boundary value
$A^{(0)}_{z}(v,z,\bar{z})$ of the gauge field, it can be shown that the
property (3.19) also holds at past null infinity without any relative change
of sign. Notice finally that the convention (2.78) implies
$A^{(0)}_{z}(u\to-\infty,z,\bar{z})=A^{(0)}_{z}(v\to+\infty,z,\bar{z})$, i.e.
the antipodal matching condition around spatial infinity as stated in [157].

###  3.2 Gravity

We now switch to gravity in $4d$ asymptotically flat spacetimes. We follow the
same notations and conventions than [158, 1, 159]. In retarded Bondi
coordinates $\\{u,r,x^{A}\\}$, $x^{A}=(z,\bar{z})$, the solution space of
four-dimensional asymptotically flat metrics reads as

$\begin{split}ds^{2}=\,&\left(\frac{2M}{r}+\mathcal{O}(r^{-2})\right)du^{2}-2\left(1+\mathcal{O}(r^{-2})\right)dudr\\\
&+\left(r^{2}\mathring{q}_{AB}+r\,C_{AB}+\mathcal{O}(r^{0})\right)dx^{A}dx^{B}\\\
&+\left(\frac{1}{2}\partial^{B}C_{AB}+\frac{2}{3r}(N_{A}+\frac{1}{4}C_{AB}\partial_{C}C^{BC})+\mathcal{O}(r^{-2})\right)dudx^{A},\end{split}$
(3.23)

where $\mathring{q}_{AB}dx^{A}dx^{B}=2dzd\bar{z}$ is the flat metric on the
punctured complex plane $\mathbb{C}^{*}$. The topology of future null infinity
is taken to be $\mathscr{I}^{+}\simeq\mathbb{R}\times\mathbb{C}^{*}$ in order
to allow meromorphic superrotations among the set of asymptotic symmetries
(see e.g. [160, 161, 158] for details on the precise set-up). Indices $A$, $B$
are lowered and raised by $\mathring{q}_{AB}$ and its inverse. The asymptotic
shear $C_{AB}$ is a symmetric trace-free tensor ($\mathring{q}^{AB}C_{AB}=0$).
The Bondi mass aspect $M(u,z,\bar{z})$ and the angular momentum aspect
$N_{A}(u,z,\bar{z})$ satisfy the time evolution/constraint equations

$\begin{split}\partial_{u}M&=-\frac{1}{8}N_{AB}N^{AB}+\frac{1}{4}\partial_{A}\partial_{B}N^{AB}-4\pi
G\,T_{uu}^{m(2)},\\\
\partial_{u}N_{A}&=\partial_{A}M+\frac{1}{16}\partial_{A}(N_{BC}C^{BC})-\frac{1}{4}N^{BC}\partial_{A}C_{BC}-8\pi
G\,T_{uA}^{m(2)}\\\
&\quad-\frac{1}{4}\partial_{B}(C^{BC}N_{AC}-N^{BC}C_{AC})-\frac{1}{4}\partial_{B}\partial^{B}\partial^{C}C_{AC}+\frac{1}{4}\partial_{B}\partial_{A}\partial_{C}C^{BC},\end{split}$
(3.24)

with $N_{AB}=\partial_{u}C_{AB}$ the Bondi news tensor encoding the outgoing
radiation and $T_{\mu\nu}^{m}$ the null matter stress tensor whose expansion
near null infinity is taken to be

$\begin{split}T_{uu}^{m}(u,r,z,\bar{z})&=T_{uu}^{m(2)}(u,z,\bar{z})\frac{1}{r^{2}}+\mathcal{O}(r^{-3}),\\\
T_{uA}^{m}(u,r,z,\bar{z})&=T_{uA}^{m(2)}(u,z,\bar{z})\frac{1}{r^{2}}+\mathcal{O}(r^{-3}).\end{split}$
(3.25)

The diffeomorphisms preserving the solution space (3.23) are generated by
vectors fields
$\xi=\xi^{u}\partial_{u}+\xi^{z}\partial_{z}+\xi^{\bar{z}}\partial_{\bar{z}}+\xi^{r}\partial_{r}$
whose leading order components read

$\begin{split}&\xi^{u}=\mathcal{T}+\frac{u}{2}(\partial_{z}\mathcal{Y}^{z}+\partial_{\bar{z}}\mathcal{Y}^{\bar{z}}),\\\
&\xi^{z}=\mathcal{Y}^{z}+\mathcal{O}(r^{-1}),\quad\xi^{\bar{z}}=\mathcal{Y}^{\bar{z}}+\mathcal{O}(r^{-1}),\\\
&\xi^{r}=-\frac{r}{2}(\partial_{z}\mathcal{Y}^{z}+\partial_{\bar{z}}\mathcal{Y}^{\bar{z}})+\mathcal{O}(r^{0}),\end{split}$
(3.26)

where $\mathcal{T}=\mathcal{T}(z,\bar{z})$ is the supertranslation parameter
and $\mathcal{Y}^{z}=\mathcal{Y}^{z}(z)$,
$\mathcal{Y}^{\bar{z}}=\mathcal{Y}^{\bar{z}}(\bar{z})$ are the superrotation
parameters satisfying the conformal Killing equation

$\partial_{\bar{z}}\mathcal{Y}^{z}=0,\quad\partial_{z}\mathcal{Y}^{\bar{z}}=0.$
(3.27)

Using the modified Lie bracket
$[\xi_{1},\xi_{2}]_{\star}\equiv[\xi_{1},\xi_{2}]-\delta_{\xi_{1}}\xi_{2}+\delta_{\xi_{2}}\xi_{1}$,
where the last two terms take into account the field-dependence of (3.26) at
subleading orders in $r$ [13], these asymptotic Killing vectors satisfy the
commutation relations

$\big{[}\xi(\mathcal{T}_{1},\mathcal{Y}^{z}_{1},\mathcal{Y}^{\bar{z}}_{1}),\xi(\mathcal{T}_{2},\mathcal{Y}^{z}_{2},\mathcal{Y}^{\bar{z}}_{2})\big{]}_{\star}=\xi(\mathcal{T}_{12},\mathcal{Y}^{z}_{12},\mathcal{Y}^{\bar{z}}_{12}),$
(3.28)

with

$\begin{split}\mathcal{T}_{12}&=\mathcal{Y}^{z}_{1}\partial_{z}\mathcal{T}_{2}-\frac{1}{2}\partial_{z}\mathcal{Y}^{z}_{1}\mathcal{T}_{2}+\text{c.c.}-(1\leftrightarrow
2)\,,\\\
\mathcal{Y}^{z}_{12}&=\mathcal{Y}^{z}_{1}\partial_{z}\mathcal{Y}^{z}_{2}-(1\leftrightarrow
2)\,,\quad\mathcal{Y}^{\bar{z}}_{12}=\mathcal{Y}^{\bar{z}}_{1}\partial_{\bar{z}}\mathcal{Y}^{\bar{z}}_{2}-(1\leftrightarrow
2).\end{split}$ (3.29)

This corresponds to the (extended) BMS algebra [162, 13, 163]. The six
globally well-defined solutions of (3.27) and the four linearly-independent
solutions of $(\partial_{A}\partial_{B}\mathcal{T})^{TF}=0$ generate the
Poincaré subgroup. The latter condition, where $TF$ means the tracefree part
with respect to the boundary metric $\mathring{q}_{AB}$, is generally solved
as $\mathcal{T}(z,\bar{z})=b_{\mu}q^{\mu}(z,\bar{z})$ where $b^{\mu}$ are the
constant parameters of an infinitesimal bulk translation in Cartesian
coordinates.

The infinitesimal transformation of the solution space is induced by the Lie
derivative of the bulk metric along BMS generators (3.26). The most crucial
transformation for the purpose of this paper is the variation of the
asymptotic shear, which reads as

$\begin{split}&\delta_{\xi}C_{zz}=\delta^{H}_{\xi}C_{zz}+\delta^{S}_{\xi}C_{zz},\\\
&\delta^{H}_{\xi}C_{zz}=\left[\left(\mathcal{T}+\frac{u}{2}(\partial_{z}\mathcal{Y}^{z}+\partial_{\bar{z}}\mathcal{Y}^{\bar{z}})\right)\partial_{u}+\mathcal{Y}^{z}\partial_{z}+\mathcal{Y}^{\bar{z}}\partial_{\bar{z}}+\frac{3}{2}\partial_{z}\mathcal{Y}^{z}-\frac{1}{2}\partial_{\bar{z}}\mathcal{Y}^{\bar{z}}\right]C_{zz},\\\
&\delta^{S}_{\xi}C_{zz}=-2\partial_{z}^{2}\mathcal{T}-u\,\partial_{z}^{3}\mathcal{Y}^{z},\end{split}$
(3.30)

together with the complex conjugate relations for $C_{\bar{z}\bar{z}}$. The
hard and soft parts of the transformation are denoted by
$\delta_{\xi}^{H}C_{AB}$ and $\delta_{\xi}^{S}C_{AB}$, respectively, and are
built up from the respective homogeneous and inhomogeneous terms appearing in
the induced variation.

Following the discussion of [164, 165, 62, 65, 158, 166, 102, 103, 159], the
BMS charges are taken to be

$Q_{\xi}[g]=\frac{1}{16\pi
G}\int_{\Sigma}d^{2}z\,\big{(}4\mathcal{T}\bar{M}+2\mathcal{Y}^{A}\bar{N}_{A}\big{)},$
(3.31)

where the redefined mass $\bar{M}$ and angular momentum aspect $\bar{N}_{A}$
are given by

$\begin{split}\bar{M}=M&+\frac{1}{8}N_{AB}C^{AB},\\\
\bar{N}_{A}=N_{A}&-u\,\partial_{A}\bar{M}+\frac{1}{4}C_{AB}\partial_{C}C^{BC}+\frac{3}{32}\partial_{A}(C_{BC}C^{BC})\\\
&+\frac{u}{4}\partial^{B}\big{[}(\partial_{B}\partial_{C}-\frac{1}{2}N_{BC}){C_{A}}^{C}\big{]}-\frac{u}{4}\partial^{B}\big{[}(\partial_{A}\partial_{C}-\frac{1}{2}N_{AC}){C_{B}}^{C}\big{]}.\end{split}$
(3.32)

The latter can be rewritten in a more elegant way in terms of Newman-Penrose
coefficients [167, 168] as the compact expressions
$\bar{M}=-\frac{1}{2}(\Psi_{2}^{0}+\bar{\Psi}_{2}^{0})$,
$\bar{N}_{z}=-\Psi_{1}^{0}+u\,\partial_{z}\Psi_{2}^{0}$ and
$\bar{N}_{\bar{z}}=\bar{N}_{z}^{*}$; see [159]. Using the time
evolution/constraint equations (3.24), these charges satisfy the flux-balance
laws

$\frac{dQ_{\xi}}{du}=\int_{\Sigma}d^{2}z\,F_{\xi(\mathcal{T},\mathcal{Y})},$
(3.33)

where the local fluxes (respectively associated with supertranslations and
superrotations) read as

$\displaystyle F_{\xi(\mathcal{T},0)}$ $\displaystyle=\frac{1}{16\pi
G}\,\mathcal{T}\left[\partial_{z}^{2}N_{\bar{z}\bar{z}}+\frac{1}{2}C_{\bar{z}\bar{z}}\partial_{u}N_{zz}+\text{c.c.}\right]-\mathcal{T}\,T_{uu}^{m(2)},$
(3.34) $\displaystyle F_{\xi(0,\mathcal{Y})}$ $\displaystyle=\frac{1}{16\pi
G}\,\mathcal{Y}^{z}\left[-u\partial^{3}_{z}N_{\bar{z}\bar{z}}+C_{zz}\partial_{z}N_{\bar{z}\bar{z}}-\frac{u}{2}\partial_{z}C_{zz}\partial_{u}N_{\bar{z}\bar{z}}-\frac{u}{2}C_{zz}\partial_{z}\partial_{u}N_{\bar{z}\bar{z}}\right]$
$\displaystyle\quad-\mathcal{Y}^{z}T^{m(2)}_{uz}+\frac{u}{2}\mathcal{Y}^{z}\partial_{z}T_{uu}^{m(2)}+\text{c.c.}$
(3.35)

As in electrodynamics, a careful split between hard and soft boundary degrees
of freedom can be performed, taking into account the factorization of the
radiative phase space between hard and soft sectors [169, 159, 165]. At
quantum level, this split allows for the BMS Ward identities to correctly
reproduce the leading and subleading soft theorems [125, 170] at all orders of
perturbation [159, 171]. For completeness, we provide here a mere summary of
the construction and refer to [159] for details.

For the analysis of the radiative phase spaces including superrotations, the
following early and late time behavior of the radiative fields are imposed
[169, 165, 62, 158, 65]:

$C_{AB}\Big{|}_{\mathscr{I}^{+}_{\pm}}=(u+C_{\pm})N^{vac}_{AB}-2(\partial_{A}\partial_{B}C_{\pm})^{TF}+o(u^{-1}),\quad
N_{AB}=N_{AB}^{vac}+o(u^{-2}),$ (3.36)

where $C_{\pm}(z,\bar{z})$ correspond to the values of the supertranslation
field for $u\to\pm\infty$ and whose difference encodes the displacement memory
effect, and $N_{AB}^{vac}(z,\bar{z})$ is the vacuum news tensor [172, 164],
identified with the tracefree part of the Geroch tensor [173, 165, 174]. The
latter can be expressed as

$N_{zz}^{vac}=\frac{1}{2}(\partial_{z}\varphi)^{2}-\partial^{2}_{z}\varphi,\quad
N_{\bar{z}\bar{z}}^{vac}=\frac{1}{2}(\partial_{\bar{z}}\bar{\varphi})^{2}-\partial^{2}_{\bar{z}}\bar{\varphi},$
(3.37)

where $\varphi(z)$, $\bar{\varphi}(\bar{z})$ are the holomorphic superboost
fields encoding the velocity kick (or refraction) memory effect [164]. For
conformal primary fields $\psi_{k,\bar{k}}(z,\bar{z})$ of weights
$(k,\bar{k})$ that transform as

$\delta_{\mathcal{Y}}\psi_{(k,\bar{k})}(z,\bar{z})=\left(\mathcal{Y}^{z}\partial_{z}+\mathcal{Y}^{\bar{z}}\partial_{\bar{z}}+k\,\partial_{z}\mathcal{Y}^{z}+\bar{k}\,\partial_{\bar{z}}\mathcal{Y}^{\bar{z}}\right)\psi_{(k,\bar{k})}(z,\bar{z})$
(3.38)

under meromorphic superrotations, it is convenient to introduce the
conformally covariant derivative operators [160, 165]

$\begin{split}&\mathscr{D}_{z}:(k,\bar{k})\to(k+1,\bar{k}):\psi_{(k,\bar{k})}\mapsto\mathscr{D}_{z}\psi_{(k,\bar{k})}=[\partial_{z}-k\,\partial_{z}\varphi]\psi_{(k,\bar{k})}\\\
&\mathscr{D}_{\bar{z}}:(k,\bar{k})\to(k,\bar{k}+1):\psi_{(k,\bar{k})}\mapsto\mathscr{D}_{\bar{z}}\psi_{(k,\bar{k})}=[\partial_{\bar{z}}-\bar{k}\,\partial_{\bar{z}}\bar{\varphi}]\psi_{(k,\bar{k})}\end{split}$
(3.39)

satisfying $[\mathscr{D}_{z},\mathscr{D}_{\bar{z}}]\psi_{(k,\bar{k})}=0$. With
these definition, the split between hard and soft variables works as follows.
Defining

$C_{zz}\equiv uN_{zz}^{vac}+C^{(\circ)}_{zz}+\tilde{C}_{zz},\quad
N_{zz}=N_{zz}^{vac}+\tilde{N}_{zz}$ (3.40)

where
$C_{zz}^{(\circ)}(z,\bar{z})=-2\mathscr{D}_{z}^{2}C^{(\circ)}(z,\bar{z})$ for
$C^{(\circ)}=\frac{1}{2}(C_{+}+C_{-})$, the Goldstone mode of
supertranslation. The falloff conditions (3.36) imply that
$\tilde{N}_{zz}=\partial_{u}\tilde{C}_{zz}=o(u^{-2})$ and that the asymptotic
shear is “purely electric” at early and late times

$(\mathscr{D}_{z}^{2}C_{\bar{z}\bar{z}}-\mathscr{D}_{\bar{z}}^{2}C_{zz})\big{|}_{\mathscr{I}^{+}_{\pm}}=0\quad\Rightarrow\quad(\mathscr{D}_{z}^{2}\tilde{C}_{\bar{z}\bar{z}}-\mathscr{D}_{\bar{z}}^{2}\tilde{C}_{zz})\big{|}_{\mathscr{I}^{+}_{\pm}}=0.$
(3.41)

The implication comes from the fact that $\mathscr{D}_{z}$ and
$\mathscr{D}_{\bar{z}}$ commute, hence
$\mathscr{D}_{z}C^{(\circ)}_{\bar{z}\bar{z}}-\mathscr{D}_{\bar{z}}C^{(\circ)}_{zz}=0$
and by definition
$\mathscr{D}_{z}N^{vac}_{\bar{z}\bar{z}}=0=\mathscr{D}_{\bar{z}}N^{vac}_{zz}$.
The condition (3.41) is the analog of (3.12) for gravity and is solved by

$\tilde{C}_{zz}\big{|}_{\mathscr{I}^{+}_{\pm}}=\mp
2\mathscr{D}^{2}_{z}N^{(\circ)},\quad
N^{(\circ)}(z,\bar{z})=\frac{1}{2}(C_{+}-C_{-}).$ (3.42)

The hard variables of the radiative phase space are collectively denoted as

$\Gamma^{H}=\\{\tilde{C}_{zz},\tilde{N}_{zz},\tilde{C}_{\bar{z}\bar{z}},\tilde{N}_{\bar{z}\bar{z}}\\}.$
(3.43)

On the other hand, soft variables are identified as

$\Gamma^{S}=\\{C_{zz}^{(\circ)},C_{\bar{z}\bar{z}}^{(\circ)},\mathcal{N}^{(0)}_{zz},\mathcal{N}^{(0)}_{\bar{z}\bar{z}},\Pi_{zz},\Pi_{\bar{z}\bar{z}},N^{vac}_{zz},N^{vac}_{\bar{z}\bar{z}}\\},\quad\Pi_{zz}\equiv
2\mathcal{N}_{zz}^{(1)}+C^{(0)}\mathcal{N}_{zz}^{(0)},$ (3.44)

where $\mathcal{N}^{(0)}_{zz}$ and $\mathcal{N}^{(1)}_{zz}$ are respectively
the leading and subleading soft news [125, 170] given by

$\mathcal{N}^{(0)}_{zz}(z,\bar{z})\equiv\int_{-\infty}^{+\infty}du\,\tilde{N}_{zz}(u,z,\bar{z}),\quad\mathcal{N}^{(1)}_{zz}(z,\bar{z})\equiv\int_{-\infty}^{+\infty}du\,u\,\tilde{N}_{zz}(u,z,\bar{z}).$
(3.45)

The boundary condition $\tilde{N}_{AB}=o(u^{-2})$ implies that both integrals
converge and in particular
$\mathcal{N}^{(0)}_{zz}(z,\bar{z})=-4\mathscr{D}^{2}_{z}N^{(\circ)}(z,\bar{z})$
because of (3.42). The transformations of all these objects, which can be
worked out from (3.30), are reproduced in Equation (3.6) of [159]. The fields
in (3.43) are functions of $(u,z,\bar{z})$ and transform homogeneously under
the action of extended BMS symmetries. For instance,

$\delta_{\xi(\mathcal{T},\mathcal{Y})}\tilde{C}_{zz}=\left(\mathcal{T}+\frac{u}{2}(\partial_{z}\mathcal{Y}^{z}+\partial_{\bar{z}}\mathcal{Y}^{\bar{z}}\right)\tilde{N}_{zz}+\left(\mathcal{Y}^{z}\partial_{z}+\mathcal{Y}^{\bar{z}}\partial_{\bar{z}}+\frac{3}{2}\partial_{z}\mathcal{Y}^{z}-\frac{1}{2}\partial_{\bar{z}}\mathcal{Y}^{\bar{z}}\right)\tilde{C}_{zz}.$
(3.46)

The quantized modes of $\tilde{C}_{zz}(u,z,\bar{z})$ are simply captured by
the expansion (2.43) for $s=2$. The fields in (3.44), on the other hand, are
functions of $(z,\bar{z})$ only, defined at the corners of $\mathscr{I}^{+}$
and their transformation laws

$\displaystyle\delta_{\xi(\mathcal{T},\mathcal{Y})}C_{zz}^{(\circ)}$
$\displaystyle=\left(\mathcal{Y}^{z}\partial_{z}+\mathcal{Y}^{\bar{z}}\partial_{\bar{z}}+\frac{3}{2}\partial_{z}\mathcal{Y}^{z}-\frac{1}{2}\partial_{\bar{z}}\mathcal{Y}^{\bar{z}}\right)C_{zz}^{(\circ)}-2\mathscr{D}_{z}^{2}\mathcal{T},$
(3.47) $\displaystyle\delta_{\xi(\mathcal{T},\mathcal{Y})}N^{vac}_{zz}$
$\displaystyle=\left(\mathcal{Y}^{z}\partial_{z}+2\partial_{z}\mathcal{Y}^{z}\right)N_{zz}^{vac}-\partial_{z}^{3}\mathcal{Y}^{z},$
(3.48)

account for the inhomogeneous pieces in (3.30). As shown in [159, 169],
(3.43)–(3.44) constitute Darboux variables parametrizing the radiative phase
space of asymptotically flat gravity at $\mathscr{I}^{+}$. Using them, the
suitable split of the integrated BMS fluxes
$\mathcal{F}_{\xi(\mathcal{T},\mathcal{Y})}=\int_{\mathscr{I}^{+}}du\,d^{2}z\,F_{\xi(\mathcal{T},\mathcal{Y})}$
computed from (3.34)–(3.35) into pure hard and soft parts has been proposed to
be

$\begin{split}&\mathcal{F}^{H}_{\xi(\mathcal{T},0)}=-\frac{1}{16\pi
G}\int_{\mathscr{I}^{+}}du\,d^{2}z\,\mathcal{T}\,\big{[}\tilde{N}_{zz}\tilde{N}_{\bar{z}\bar{z}}\big{]}-\int_{\mathscr{I}^{+}}du\,d^{2}z\,\mathcal{T}\,T^{m(2)}_{uu},\\\
&\mathcal{F}^{S}_{\xi(\mathcal{T},0)}=\frac{1}{8\pi
G}\int_{\Sigma}d^{2}z\,\mathcal{T}\,\big{[}\mathscr{D}_{z}^{2}\mathcal{N}^{(0)}_{\bar{z}\bar{z}}\big{]},\\\
&\mathcal{F}_{\xi(0,\mathcal{Y})}^{H}=\frac{1}{16\pi
G}\int_{\mathscr{I}^{+}}du\,d^{2}z\,\mathcal{Y}^{z}\,\left[\frac{3}{2}\tilde{C}_{zz}\partial_{z}\tilde{N}_{\bar{z}\bar{z}}+\frac{1}{2}\tilde{N}_{zz}\partial_{z}\tilde{C}_{zz}+\frac{u}{2}\partial_{z}\left(\tilde{N}_{zz}\tilde{N}_{\bar{z}\bar{z}}\right)\right]\\\
&\hskip
45.0pt-\int_{\mathscr{I}^{+}}du\,d^{2}z\,\mathcal{Y}^{z}\left[T^{m(2)}_{uz}+\frac{u}{2}\,\partial_{z}\mathcal{Y}^{z}\,T^{m(2)}_{uu}\right]+\text{c.c.},\\\
&\mathcal{F}_{\xi(0,\mathcal{Y})}^{S}=\frac{1}{16\pi
G}\int_{\Sigma}d^{2}z\,\mathcal{Y}^{z}\,\left[-\mathscr{D}^{3}_{z}\mathcal{N}^{(1)}_{\bar{z}\bar{z}}+\frac{3}{2}C_{zz}^{(\circ)}\mathscr{D}\mathcal{N}_{\bar{z}\bar{z}}^{(0)}+\frac{1}{2}\mathcal{N}_{\bar{z}\bar{z}}^{(0)}\mathscr{D}_{z}C_{zz}^{(\circ)}\right]+\text{c.c.}\end{split}$
(3.49)

where (3.41) was used to simplify the expressions of soft fluxes. Unlike the
case of electromagnetism, the nonlinearity of Einstein theory implies that
hard fluxes do include pure gravitational terms, notably the first term in
$F^{H}_{\xi(1,0)}$, responsible for the Bondi mass loss. Inverting the
symplectic structure yields the following Poisson brackets

$\displaystyle\big{\\{}\tilde{N}_{z_{1}z_{1}}(u_{1},z_{1},\bar{z}_{1}),\tilde{C}_{\bar{z}_{2}\bar{z}_{2}}(u_{2},z_{2},\bar{z}_{2})\big{\\}}=-16\pi
G\,\delta(u_{1}-u_{2})\,\delta^{(2)}(z_{1}-z_{2}),$ (3.50)
$\displaystyle\begin{aligned}
&\big{\\{}\mathcal{N}^{(0)}_{z_{1}z_{1}}(z_{1},\bar{z}_{1}),C^{(0)}_{\bar{z}_{2}\bar{z}_{2}}(z_{2},\bar{z}_{2})\big{\\}}=-16\pi
G\,\delta^{(2)}(z_{1}-z_{2}),\\\
&\big{\\{}\Pi_{z_{1}z_{1}}(z_{1},\bar{z}_{1}),N^{vac}_{\bar{z}_{2}\bar{z}_{2}}(z_{2},\bar{z}_{2})\big{\\}}=-16\pi
G\,\delta^{(2)}(z_{1}-z_{2}),\end{aligned}$ (3.51)

from which one can derive the very important property that the hard and soft
parts of the fluxes act independently on (3.43) and (3.44) as

$\begin{split}&\big{\\{}\mathcal{F}^{H}_{\xi(\mathcal{T},\mathcal{Y})},\Gamma^{H}\big{\\}}=\delta_{\xi(\mathcal{T},\mathcal{Y})}\Gamma^{H},\quad\big{\\{}\mathcal{F}^{H}_{\xi(\mathcal{T},\mathcal{Y})},\Gamma^{S}\big{\\}}=0,\\\
&\big{\\{}\mathcal{F}^{S}_{\xi(\mathcal{T},\mathcal{Y})},\Gamma^{S}\big{\\}}=\delta_{\xi(\mathcal{T},\mathcal{Y})}\Gamma^{S},\quad\big{\\{}\mathcal{F}^{S}_{\xi(\mathcal{T},\mathcal{Y})},\Gamma^{H}\big{\\}}=0,\end{split}$
(3.52)

and form separately two representations of the extended BMS algebra
(3.28)–(3.29) [158, 159].

As usual, a similar analysis can be performed at past null infinity
$\mathscr{I}^{-}$ by trading the retarded coordinates for the advanced ones.
At the vicinity of $\mathscr{I}^{-}$, the solution space is expanded as

$\begin{split}ds^{2}=\,&\left(\frac{2M^{(-)}}{r^{\prime}}+\mathcal{O}(r^{\prime-2})\right)dv^{2}+2\left(1+\mathcal{O}(r^{\prime-2})\right)dvdr^{\prime}\\\
&+\left(r^{\prime
2}\mathring{q}_{AB}+r^{\prime}\,C_{AB}^{(-)}+\mathcal{O}(r^{\prime
0})\right)dx^{A}dx^{B}\\\
&-\left(-\frac{1}{2}\partial^{B}C_{AB}^{(-)}-\frac{2}{3r^{\prime}}(N_{A}^{(-)}+\frac{1}{4}C^{(-)}_{AB}\partial_{C}C^{BC}_{(-)})+\mathcal{O}(r^{\prime-2})\right)dvdx^{A},\end{split}$
(3.53)

still denoting $x^{A}=(z^{\prime},\bar{z}^{\prime})=(z,\bar{z})$. The Bondi
time evolution/constraint equations for the Bondi mass aspect
$M^{(-)}(v,z,\bar{z})$ and the angular momentum aspect
$N_{A}^{(-)}(v,z,\bar{z})$ at $\mathscr{I}^{-}$ read as

$\begin{split}\partial_{v}M^{(-)}&=\frac{1}{8}N_{AB}^{(-)}N^{AB}_{(-)}+\frac{1}{4}\partial_{A}\partial_{B}N^{AB}_{(-)}+4\pi
G\,T^{m(2,-)}_{vv}\,,\\\
\partial_{v}N_{A}^{(-)}&=-D_{A}M^{(-)}+\frac{1}{16}\partial_{A}(N_{BC}^{(-)}C^{BC}_{(-)})-\frac{1}{4}N^{BC}_{(-)}D_{A}C_{BC}^{(-)}-8\pi
G\,T^{m(2,-)}_{vA}\\\
&\quad-\frac{1}{4}\partial_{B}(C^{BC}_{(-)}N_{AC}^{(-)}-N^{BC}_{(-)}C_{AC}^{(-)})+\frac{1}{4}\partial_{B}\partial^{B}\partial^{C}C_{AC}^{(-)}-\frac{1}{4}\partial_{B}\partial_{A}\partial_{C}C^{BC}_{(-)},\end{split}$
(3.54)

with the (past) Bondi news tensor $N^{(-)}_{AB}=\partial_{v}C^{(-)}_{AB}$
encoding the incoming radiation. Under the BMS symmetries acting at
$\mathscr{I}^{-}$

$\xi\Big{|}_{\mathscr{I}^{-}}=\left(\mathcal{T}(z,\bar{z})+\frac{v}{2}(\partial_{z}\mathcal{Y}^{z}+\partial_{\bar{z}}\mathcal{Y}^{\bar{z}})\right)\partial_{v}+\mathcal{Y}^{z}(z)\partial_{z}+\mathcal{Y}^{\bar{z}}(\bar{z})\partial_{\bar{z}},$
(3.55)

the variation of the shear is still given by (3.30) with $u\mapsto v$ and a
sign flip in $\delta_{\xi}^{S}C_{zz}$. The covariant phase space analysis and
a similar split between hard and soft boundary degrees of freedom yield again
(3.52). Around spatial infinity, the requirement (2.78) now implies
$C_{zz}(u\to-\infty,z,\bar{z})=-C_{zz}^{(-)}(v\to+\infty,z,\bar{z})$, where
the minus sign is due to $r^{\prime}=-r$ and the convention (2.53). This
agrees with the antipodal matching in the sense of [151, 170].

##  4 Sourced quantum field theory

As discussed in Section 3 the charges associated with the asymptotic
symmetries are not conserved due to the radiation leaking through
$\mathscr{I}$. We argued in [1] (see also Section 5) that the non-conservation
of the bulk charges can be holographically interpreted as a coupling of the
dual theory with some external sources. In this section, we discuss a general
framework that allows to deal with symmetries in presence of external sources.
The details on the newly introduced notions and the resulting construction
will be presented in the upcoming paper [175] (see also [176, 177]). We argue
that, at the classical level, the inclusion of external sources in field
theories allows to derive a generalized version of Noether’s theorem giving an
interpretation of flux-balance laws for the Noether currents from symmetry
principle. We then obtain the quantum analogue of this result by writing the
sourced Ward identities using a path integral formulation. Finally, we
exemplify these sourced Ward identities for a sourced theory exhibiting $U(1)$
symmetries and BMS symmetries.

###  4.1 Generalized Noether symmetries

In this section, we briefly discuss a generalization of Noether’s theorem that
allows to write some flux-balance laws associated with (generalized)
symmetries in presence of external sources. We consider a theory living on a
$n$-dimensional manifold $\mathscr{M}$ with coordinates $x^{a}$ and boundary
$\partial\mathscr{M}$. The dynamical fields are denoted by $\Psi^{i}(x)$ and
the external sources by $\sigma^{m}(x)$. The latter are defined as local
functions with no associated equations of motion, contrarily to $\Psi^{i}(x)$.
The action reads as

$S[\Psi|\sigma]=\int_{\mathscr{M}}d^{n}x\,L[\Psi|\sigma].$ (4.1)

To derive flux-balance laws for the Noether charges from first principles, it
will be useful to allow variations of the sources on the phase space. The
variation of the action reads as

$\delta S=\int_{\mathscr{M}}d^{n}x\,\frac{\delta
S}{\delta\Psi^{i}}\delta\Psi^{i}+\int_{\mathscr{M}}d^{n}x\,\frac{\delta
S}{\delta\sigma^{m}}\delta\sigma^{m},$ (4.2)

discarding the boundary terms if the fields and sources are sufficiently
decaying while approaching $\partial\mathscr{M}$. For a fixed set of sources
$\sigma^{m}(x)$, the action is stationary for arbitrary variations
$\delta\Psi^{i}$ if and only if the equations of motion $\frac{\delta
S}{\delta\Psi^{i}}=0$ are obeyed by the dynamical fields $\Psi^{i}(x)$.

If the theory admits some Noether symmetries $\delta_{K}\Psi^{i}=K^{i}[\Psi]$
in absence of external sources, turning on the sources will usually break
these symmetries. However, the Noetherian symmetries
$\delta_{K}\Psi^{i}=K^{i}[\Psi]$ of the theory without source can be promoted
to generalized symmetries of the sourced theory in the sense of [176] (notice
that the present notion of generalized symmetries shall not be confused with
the concept of higher-form symmetries discussed in quantum field theory and
that goes under the same name, see e.g. [178]). The detailed analysis of the
implications of the existence of such symmetries requires some care in the
definition of a generalized symmetry and is beyond the scope of this article.
This analysis will be presented elsewhere [175]. For the purpose of the
present work we will only rely on the following features, which we will take
as a naive definition for a generalized symmetry: the joint action on both the
fields and sources

$\delta_{K}\Psi^{i}=K^{i}[\Psi|\sigma],\quad\delta_{K}\sigma^{m}=K^{m}[\sigma]$
(4.3)

is then a symmetry in the sense that it is required to be a symmetry of the
sourced equations of motion

$\frac{\delta
S}{\delta\Psi^{i}}=0\quad\Longrightarrow\quad\delta_{K}\left(\frac{\delta
S}{\delta\Psi^{i}}\right)=0$ (4.4)

but is not required to preserve the action

$\delta_{K}L=\partial_{a}B^{a}_{K}+V_{K}[\sigma],\quad V_{K}[\sigma=0]=0.$
(4.5)

As the second equation of (4.3) expresses that the external sources transform
among themselves, this translates into the fact that the generalized
symmetries break the invariance of the action in (4.5) only by terms depending
on the sources which vanish when $\sigma=0$. Consequently, in the absence of
source ($\sigma=0$), the symmetries are fully restored as rightful variational
or Noether symmetries. Hence one can say that generalized symmetries are
symmetries of the sourced equations of motion extending the Noetherian
symmetries of the source-free action. In this framework, writing the usual
Noether current as $\bm{j}_{K}=j^{a}_{K}\,(d^{n-1}x)_{a}$, the following
relation can be obtained

$\partial_{a}j^{a}_{K}=K^{i}\frac{\delta S}{\delta\Psi^{i}}+F_{K},$ (4.6)

where $\bm{F}_{K}=F_{K}\,(d^{n}x)$ is a flux term whose explicit form is given
by [175]

$F_{K}=K^{m}\frac{\delta L}{\delta\sigma^{m}}-V_{K}.$ (4.7)

On-shell, (4.6) leads to the flux-balance law:

$\boxed{d\bm{j}_{K}=\bm{F}_{K}[\Psi|\sigma].}$ (4.8)

The latter generalizes the first Noether theorem in presence of external
sources: the conservation of the Noether current $\bm{j}_{K}$ associated with
a symmetry of characteristics $K$ is broken by the flux term $\bm{F}_{K}$. If
one turns off the sources, we have $F_{K}[\Psi|\sigma=0]=0$ and one recovers
the standard conservation law $d\bm{j}_{K}=0$.

###  4.2 Sourced Ward identities

Let us now derive Ward identities associated with the generalized symmetries
(4.3). The derivation follows the usual steps by properly taking into account
the presence of external sources (see e.g. [120] for the standard derivation
of Ward identities from the path integral). For a fixed source $\sigma$, the
partition function reads as

$\mathcal{Z}_{\sigma}[J_{i}]=\int\mathcal{D}[\Psi]_{\sigma}\,\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}\big{)},$
(4.9)

where we allow the path integral measure $\mathcal{D}[\Psi]_{\sigma}$ to
depend upon the sources $\sigma^{m}$ and we use the convenient notation
$J_{A}\Psi^{A}\equiv\int_{\mathscr{M}}d^{n}x\,J_{i}(x)\Psi^{i}(x)$. Here
$J_{i}(x)$ are classical sources introduced in the definition of the partition
function. They are not meant to be quantized and will be sent to zero when
evaluating the correlation function. We stress that they should be
distinguished from the sources $\sigma^{m}$ that have a physical meaning and
break the conservation of Noether currents: the sources $\sigma^{m}$ will be
integrated in the final path integral and will generate a flux.

Then, we suggest to consider that the partition function of the full sourced
quantum theory should be computed by integrating (4.9) over the sources as

$\mathcal{Z}[J_{i},J_{m}]=\int\mathcal{D}[\sigma]\,\mathcal{Z}_{\sigma}[J_{i}]\,\exp\frac{i}{\hbar}J_{M}\sigma^{M}.$
(4.10)

We used again the shorthand
$J_{M}\sigma^{M}\equiv\int_{\mathscr{M}}d^{n}x\,J_{m}(x)\sigma^{m}(x)$ and
introduced a second bunch of classical sources $J_{m}$ as part of the
definition of the partition function. The sources $\sigma^{m}$ are therefore
promoted as source operators and will play an important role in the Carrollian
holographic correspondence discussed in Section 5.2. A very similar procedure
was adopted in [179] in the AdS/CFT context where the path integral was
performed over the boundary sources. We will further comment on this in the
discussion closing the paper (Section 7). Notice that, even though the source
operators $\sigma^{m}$ and field operators $\Psi^{i}$ seem on the same footing
in (4.10), they are distinguished by their role played in the generalized
symmetries introduced in Section 4.1. In the absence of source, the symmetries
of the partition functions are restored. We now investigate the implications
of generalized symmetries on the correlation functions when the sources are
turned on. Inserting the off-shell relation (4.6) in the path integral, we get

$\displaystyle\frac{\partial}{\partial
x^{a}}\int\mathcal{D}[\sigma]\int\mathcal{D}[\Psi]_{\sigma}\,j^{a}_{K}(x)\,\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}$
$\displaystyle=\int\mathcal{D}[\sigma]\int\mathcal{D}[\Psi]_{\sigma}\,\left(K^{i}[\Psi(x)|\sigma(x)]\frac{\delta
S}{\delta\Psi^{i}(x)}+F_{K}(x)\right)\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}.$
(4.11)

The first term in the right-hand side can be reworked by noticing that

$\begin{split}&\frac{\hbar}{i}\frac{\delta}{\delta\Psi^{i}(x)}\,\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}\\\
&=\frac{\delta}{\delta\Psi^{i}(x)}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}\big{)}\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}\\\
&=\left(\frac{\delta
S}{\delta\Psi^{i}(x)}+J_{i}(x)\right)\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)},\end{split}$
(4.12)

and assuming that $\delta(0)=0$ and the path integral is invariant under field
translation, i.e.
$\mathcal{D}[\Psi+\delta\Psi]_{\sigma}=\mathcal{D}[\Psi]_{\sigma}$ for any
variation $\delta\Psi^{i}$:

$\begin{split}&\int\mathcal{D}[\sigma]\int\mathcal{D}[\Psi]_{\sigma}\left(K^{i}[\Psi(x)|\sigma(x)]\frac{\delta
S}{\delta\Psi^{i}(x)}\right)\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}\\\
&=-\int\mathcal{D}[\sigma]\int\mathcal{D}[\Psi]_{\sigma}\,J_{i}(x)K^{i}[\Psi(x)|\sigma(x)]\,\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}.\end{split}$
(4.13)

As a conclusion, (4.11) is simply

$\begin{split}&\frac{\partial}{\partial
x^{a}}\int\mathcal{D}[\sigma]\int\mathcal{D}[\Psi]_{\sigma}\,j^{a}_{K}(x)\,\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}\\\
&=\int\mathcal{D}[\sigma]\int\mathcal{D}[\Psi]_{\sigma}\,\Big{[}-J_{i}(x)K^{i}[\Psi(x)|\sigma(x)]+F_{K}(x)\Big{]}\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}.\end{split}$
(4.14)

Acting on the left-hand side with $N$ successive derivatives with respect to
$J_{i}(x)$ and setting $J_{i}(x)=0=J_{m}(x)$ afterwards gives

$\boxed{\begin{aligned} \frac{\partial}{\partial
x^{a}}\big{\langle}j^{a}_{K}(x)\,X_{N}^{\Psi}\big{\rangle}=\frac{\hbar}{i}\sum_{k=1}^{N}\delta^{(n)}(x-x_{k})\,\delta_{K^{i_{k}}}\big{\langle}X_{N}^{\Psi}\big{\rangle}+\big{\langle}F_{K}(x)\,X_{N}^{\Psi}\big{\rangle},\end{aligned}}$
(4.15)

where we have introduced

$X_{N}^{\Psi}\equiv\Psi^{i_{1}}(x_{1})\dots\Psi^{i_{N}}(x_{N}),\quad\delta_{K^{i_{k}}}X_{N}^{\Psi}=\Psi^{i_{1}}(x_{1})\dots
K^{i_{k}}[\Psi(x_{k})]\dots\Psi^{i_{N}}(x_{N})$ (4.16)

to denote a collection of $N$ quantum insertions and their transformation
under the symmetry of characteristics $K$ and recalled that, by definition,

$\big{\langle}X_{N}^{\Psi}\big{\rangle}\equiv\left(\frac{\hbar}{i}\right)^{N}\frac{\delta}{\delta
J_{i_{1}}(x_{1})}\cdots\frac{\delta}{\delta
J_{i_{N}}(x_{N})}\mathcal{Z}[J]\Big{|}_{J=0}.$ (4.17)

Now acting on the left-hand side of (4.14) with $N$ successive derivatives
with respect to the other $J_{m}(x)$ yields

$\boxed{\begin{aligned} \frac{\partial}{\partial
x^{a}}\big{\langle}j^{a}_{K}(x)\,X_{N}^{\sigma}\big{\rangle}=\big{\langle}F_{K}(x)\,X_{N}^{\sigma}\big{\rangle},\end{aligned}}$
(4.18)

where

$X_{N}^{\sigma}\equiv\sigma^{m_{1}}(x_{1})\dots\sigma^{m_{N}}(x_{N})$ (4.19)

denotes a collection of $N$ insertions of quantum sources. The equations
(4.15) and (4.18) are the local version of the infinitesimal Ward identities
in presence of external sources. With no field insertion in the correlators,
one finds

$\partial_{a}\big{\langle}j^{a}_{K}(x)\big{\rangle}=\big{\langle}F_{K}(x)\big{\rangle},$
(4.20)

which reproduces the classical flux-balance equation (4.8). Integrating (4.15)
on the whole manifold $\mathscr{M}$ with boundary $\partial\mathscr{M}$ gives

$\boxed{\sum_{k=1}^{N}\delta_{K^{i_{k}}}\big{\langle}X_{N}^{\Psi}\big{\rangle}=\frac{i}{\hbar}\Big{\langle}\left(\int_{\mathscr{M}}\bm{F}_{K}-\int_{\partial\mathscr{M}}\bm{j}_{K}\right)\,X_{N}^{\Psi}\Big{\rangle}.}$
(4.21)

The standard textbook result is recovered by setting the sources to zero and
assuming that the Noether currents vanish at the boundary, so that the right-
hand side of (4.21) vanishes. Finally integrating (4.18) on $\mathscr{M}$
leads to

$\boxed{\Big{\langle}\left(\int_{\mathscr{M}}\bm{F}_{K}-\int_{\partial\mathscr{M}}\bm{j}_{K}\right)\,X_{N}^{\sigma}\Big{\rangle}=0.}$
(4.22)

##### Remark

At this stage, and since the partition function (4.10) can be rewritten as

$\mathcal{Z}[J_{i},J_{m}]=\int\mathcal{D}[\sigma]\,\int\mathcal{D}[\Psi]_{\sigma}\,\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)},$
(4.23)

one might wonder what is responsible for the asymmetry between the Ward
identities for the fields (4.15) and the sources (4.18). In the following
lines, we wish to highlight that, besides a possible dependence of the measure
$\mathcal{D}[\Psi]_{\sigma}$ in the sources, this asymmetry can ultimately be
traced back to the fact that the generalized symmetries treat fields and
sources differently. To convince oneself of this point, it is instructive to
consider the sub-case where the measures $\mathcal{D}[\sigma]$ and
$\mathcal{D}[\Psi]$ are assumed to commute and use this property to rewrite
the Ward identities in a form that puts sources and fields on an equal
footing. To achieve this, we only need to reintroduce the explicit form of the
flux (4.7), which has not yet been used in the derivation of the local version
of the Ward identities (4.15) and (4.18). Injecting (4.7) into (4.14), we have

$\begin{split}&\frac{\partial}{\partial
x^{a}}\int\mathcal{D}[\sigma]\int\mathcal{D}[\Psi]_{\sigma}\,j^{a}_{K}(x)\,\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}\\\
&=\int\mathcal{D}[\sigma]\int\mathcal{D}[\Psi]_{\sigma}\,\left(-K^{i}[\Psi(x)|\sigma(x)]J_{i}(x)+K^{m}[\sigma(x)]\frac{\delta
S}{\delta\sigma^{m}(x)}-V_{K}[\sigma(x)]\right)\times\cdots\\\
&\qquad\times\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}.\end{split}$
(4.24)

Assuming that one can commute the measures $\mathcal{D}[\sigma]$ and
$\mathcal{D}[\Psi]$ and that the path integral for the sources is invariant
under translation as well, i.e.
$\mathcal{D}[\sigma+\delta\sigma]=\mathcal{D}[\sigma]$, the second term in the
right-hand side can be reworked as

$\begin{split}&\frac{\partial}{\partial
x^{a}}\int\mathcal{D}[\sigma]\int\mathcal{D}[\Psi]_{\sigma}\,j^{a}_{K}(x)\,\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}\\\
&=\int\mathcal{D}[\sigma]\int\mathcal{D}[\Psi]_{\sigma}\,\left(-K^{i}[\Psi(x)|\sigma(x)]J_{i}(x)-K^{m}[\sigma(x)]J_{m}(x)-V_{K}[\sigma(x)]\right)\times\cdots\\\
&\qquad\times\exp\frac{i}{\hbar}\big{(}S[\Psi|\sigma]+J_{A}\Psi^{A}+J_{M}\sigma^{M}\big{)}.\end{split}$
(4.25)

Deriving $N$ times with respect to $J_{i}(x)$ and then setting $J_{m}=0=J_{i}$
gives

$\frac{\partial}{\partial
x^{a}}\big{\langle}j^{a}_{K}(x)\,X_{N}^{\Psi}\big{\rangle}=\frac{\hbar}{i}\sum_{k=1}^{N}\delta^{(n)}(x-x_{k})\,\delta_{K^{i_{k}}}\big{\langle}X_{N}^{\Psi}\big{\rangle}-\big{\langle}V_{K}(x)\,X_{N}^{\Psi}\big{\rangle}.$
(4.26)

Similarly, deriving $N$ times with respect to $J_{m}(x)$ and then setting
$J_{m}=0=J_{i}$ gives

$\frac{\partial}{\partial
x^{a}}\big{\langle}j^{a}_{K}(x)\,X_{N}^{\sigma}\big{\rangle}=\frac{\hbar}{i}\sum_{k=1}^{N}\delta^{(n)}(x-x_{k})\,\delta_{K^{m_{k}}}\big{\langle}X_{N}^{\sigma}\big{\rangle}-\big{\langle}V_{K}(x)\,X^{\sigma}_{N}\big{\rangle}.$
(4.27)

Under the above assumptions, these expressions are equivalent to (4.15) and
(4.18) respectively. They highlight that the asymmetry between fields and
sources in the treatment of generalized symmetries is always present as a
result of the appearance of $V_{K}[\sigma(x)]$ in the right-hand sides. In
particular, these Ward identities are distinctly different from the usual Ward
identities of a Noetherian symmetry.

###  4.3 Sourced $U(1)$ Ward identities

As a warm-up, we apply the above framework to the simple case of a field
theory exhibiting a $U(1)$ symmetry. We assume that the fields $\Psi_{Q}$
inserted in the correlators transform as

$\delta_{\lambda}\Psi_{Q}=-i\lambda Q\Psi_{Q}$ (4.28)

under the $U(1)$ symmetry parametrized by some arbitrary function
$\lambda(x)$. The infinitesimal Ward identity (4.15) in presence of external
sources then simply reads as

$\begin{split}\frac{\partial}{\partial
x^{a}}\Big{\langle}j^{a}_{\lambda}(x)\,\Psi^{i_{1}}_{Q_{i_{1}}}(x_{1})\dots\Psi^{i_{N}}_{Q_{i_{N}}}(x_{N})\Big{\rangle}&-\hbar\sum_{k=1}^{N}\lambda(x_{k})\,Q_{i}\,\delta^{(n)}(x-x_{k})\,\big{\langle}\Psi^{i_{1}}_{Q_{i_{1}}}(x_{1})\dots\Psi^{i_{N}}_{Q_{i_{N}}}(x_{N})\big{\rangle}\\\
&=\Big{\langle}F_{\lambda}(x)\,\Psi^{i_{1}}_{Q_{i_{1}}}(x_{1})\dots\Psi^{i_{N}}_{Q_{i_{N}}}(x_{N})\Big{\rangle}.\end{split}$
(4.29)

While this example is quite trivial, it will be relevant when discussing
electrodynamics from the Carrollian holographic perspective.

###  4.4 Sourced conformal Carrollian Ward identities

We now make the same exercise for the case of $3d$ conformal Carrollian field
theory. After reviewing conformal Carrollian symmetries and introducing
associated primary fields, we write the sourced Ward identities involving the
Carrollian momenta. Notice importantly that we make here a slight abuse of
terminology, as the very meaning of what exactly is a quantum Carrollian field