# Carrollian Perspective on Celestial Holography

Laura Donnay<EMAIL_ADDRESS>Institute for Theoretical Physics, TU
Wien
Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria  Adrien Fiorucci
<EMAIL_ADDRESS>Institute for Theoretical Physics, TU Wien
Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria  Yannick Herfray
<EMAIL_ADDRESS>Service de Physique de l’Univers, Champs et
Gravitation, Université de Mons
20 place du Parc, 7000 Mons, Belgium  Romain Ruzziconi
<EMAIL_ADDRESS>Institute for Theoretical Physics, TU Wien
Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria

###### Abstract

We show that a $3d$ sourced conformal Carrollian field theory has the right
kinematic properties to holographically describe gravity in $4d$
asymptotically flat spacetime. The external sources encode the leaks of
gravitational radiation at null infinity. The Ward identities of this theory
are shown to reproduce those of the $2d$ celestial CFT after relating
Carrollian to celestial operators. This suggests a new set of interplays
between gravity in asymptotically flat spacetime, sourced conformal Carrollian
field theory and celestial CFT.

††preprint: APS/123-QED

##  I Introduction

The holographic principle [1, 2] states that gravity in a given spacetime
region can be encoded on a lower-dimensional boundary of that region.
Extending this paradigm beyond the celebrated Anti-de Sitter (AdS) / Conformal
field theory (CFT) correspondence [3, 4, 5] to the more realistic model of
asymptotically flat spacetimes is part of an intensive ongoing research
effort, referred to as flat space holography (see [6, 7, 8, 9, 10, 11, 12] for
early works). The asymptotic symmetries preserving the boundary structure of
asymptotically flat spacetimes form the Bondi-van der Burg-Metzner-Sachs (BMS)
group [13, 14, 15], which is an infinite-dimensional enhancement of the
Poincaré group with supertranslations.

Two different roads, referred here as “Carrollian” and “Celestial”
holographies, have emerged in order to describe quantum gravity in $4d$
asymptotically flat spacetime and might seem in apparent tension.

In the first picture, the dual theory is proposed to be a BMS field theory
living on the $3d$ null boundary of the spacetime [16, 17, 18, 19, 20, 21, 22,
23] or, equivalently [24, 25], a conformal Carrollian field theory. While this
$4d$ bulk / $3d$ boundary theory point of view follows the familiar pattern of
a codimension-one holographic duality, establishing such a flat space
holographic dictionary presents new challenges that one is not used to
encounter in AdS/CFT. Key differences include, on the one hand, the null
nature of the conformal boundary and, on the other hand, the presence of
radiative flux leaking through this boundary. However, this approach is
suggested by a flat limit process in the bulk, which consists of taking the
cosmological constant to zero. This implies an ultra-relativistic limit on the
boundary theory contracting the conformal symmetries into BMS symmetries, see
[26, 27, 28, 29, 30, 31, 32, 33] for successful applications in $3d$ gravity
and [34, 35, 29, 36, 37, 38, 39, 40, 41] for a fluid/gravity perspective.

In the second proposal, the holographic dual of gravity in $4d$ asymptotically
flat spacetime is a two-dimensional “celestial conformal field theory” (CCFT)
living on the conformal sphere at infinity. The celestial holography program
is rooted on the observation that gravitational $S$-matrix elements written in
a boost eigenstate basis take the form of conformal correlation functions [9,
42, 43]. Quantum field theory soft theorems can be elegantly encoded in CCFT
in terms of celestial currents associated with asymptotic symmetry generators,
see e.g. [44, 45, 46, 47, 48, 49, 50, 51, 52] and [53] for more references.
The obvious advantage of the celestial paradigm is that it provides a
framework where one can readily make use of the plethora of powerful CFT
techniques.

The main goals of this paper are $(i)$ to deepen the first picture by
providing a precise proposal of a holographic description which properly takes
into account leaks of gravitational radiation through the boundary and $(ii)$
to initiate a dialog between these two different approaches to flat space
holography.

Figure 1: Carrollian approach _vs_ Celestial approach to flat space
holography.

The paper is organized as follows. In section II, after a brief review of $4d$
asymptotically flat spacetimes, we argue that the coexistence of two proposals
for flat space holography relies on the two complementary descriptions of the
spacetime boundary. In section III, we propose that the dual field theory in
the first picture is a conformal Carrollian field theory (CCarrFT) coupled
with some external sources encoding the radiation leaking through the
conformal boundary. In section IV, we show that the Ward identities of the
sourced CCarrFT reproduce those of the CCFT after taking the appropriate
integral transform with respect to the advanced/retarded time. This suggests a
rich set of interplays, as depicted in Figure 1. In section V, we summarize
the results and discuss some implications of this work.

##  II Asymptotically flat spacetimes

In this section, we review the analysis of four-dimensional asymptotically
flat spacetimes at null infinity, denoted $\mathscr{I}^{+}$, and its relation
with Carrollian geometry.

###  II.1 Solution space

In (retarded) Bondi coordinates $(u,r,x^{A})$, $x^{A}=(z,\bar{z})$ [13, 14,
15], the solution space of four-dimensional asymptotically flat metrics reads
as [54, 55]

$\displaystyle
ds^{2}=\left(\frac{2M}{r}+\mathcal{O}(r^{-2})\right){d}u^{2}-2\left(1+\mathcal{O}(r^{-2})\right){d}u{d}r$
$\displaystyle+\left(r^{2}\mathring{q}_{AB}+r\,C_{AB}+\mathcal{O}(r^{-1})\right)dx^{A}dx^{B}$
(1)
$\displaystyle+\left(\frac{1}{2}\partial_{B}C^{B}_{A}+\frac{2}{3r}(N_{A}+\frac{1}{4}C_{A}^{B}\partial_{C}C^{C}_{B})+\mathcal{O}(r^{-2})\right)dudx^{A}\,,$

where the asymptotic shear $C_{AB}(u,z,\bar{z})$ is a $2$-dimensional
symmetric trace-free tensor. For simplicity, we chose the transverse boundary
metric to be the flat metric, namely
$\mathring{q}_{AB}dx^{A}dx^{B}=2dzd\bar{z}$. The Bondi mass $M(u,z,\bar{z})$
and angular momentum aspects $N_{A}(u,z,\bar{z})$ in (1) satisfy the time
evolution/constraint equations

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

with $N_{AB}=\partial_{u}C_{AB}$ the Bondi news tensor encoding the
gravitational radiation.

###  II.2 BMS and conformal Carrollian symmetries

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

$\begin{split}&\xi^{u}=\mathcal{T}+u\alpha\,,\quad\alpha=\frac{1}{2}(\partial\mathcal{Y}+\bar{\partial}\bar{\mathcal{Y}})\,,\\\
&\xi^{z}=\mathcal{Y}+\mathcal{O}(r^{-1})\,,\qquad\xi^{\bar{z}}=\bar{\mathcal{Y}}+\mathcal{O}(r^{-1})\,,\\\
&\xi^{r}=-r\alpha+\mathcal{O}(r^{0})\,,\end{split}$ (3)

where $\mathcal{T}=\mathcal{T}(z,\bar{z})$ is the supertranslation parameter
and $\mathcal{Y}=\mathcal{Y}(z)$,
$\bar{\mathcal{Y}}=\bar{\mathcal{Y}}(\bar{z})$ are the superrotation
parameters satisfying the conformal Killing equation. Using a modified Lie
bracket [55], the asymptotic Killing vectors (3) satisfy the (extended) BMS
algebra.

The infinitesimal transformation of the asymptotic shear $C_{AB}$ under BMS
symmetries can be split into hard and soft pieces
$\delta_{\xi}C_{zz}=\delta^{H}_{\xi}C_{zz}+\delta^{S}_{\xi}C_{zz}$ which are
respectively homogeneous and inhomogeneous in $C_{zz}$ [56, 57]. This reads
explicitly as

$\displaystyle\delta^{H}_{\xi}C_{zz}=\left[\left(\mathcal{T}+u\alpha\right)\partial_{u}+\mathcal{Y}\partial+\bar{\mathcal{Y}}\bar{\partial}+\frac{3}{2}\partial\mathcal{Y}-\frac{1}{2}\bar{\partial}\bar{\mathcal{Y}}\right]C_{zz}\,,$
$\displaystyle\delta^{S}_{\xi}C_{zz}=-2\partial^{2}\mathcal{T}-u\,\partial^{3}\mathcal{Y}\,,$
(4)

together with the complex conjugate relations for $C_{\bar{z}\bar{z}}$.

From a geometric perspective, BMS symmetries are the conformal symmetries of a
Carrollian structure on $\mathscr{I}^{+}$ with coordinates
$x^{a}=(u,z,\bar{z})$ [24, 25] (see also [58, 59, 60, 61, 62, 63, 64, 65,
66]). This Carrollian structure is given by a degenerate metric $q_{ab}$ and a
vector field $n^{a}$ in the kernel of $q_{ab}$, i.e. $q_{ab}n^{b}=0$. From
(1), it reads explicitly $q_{ab}dx^{a}dx^{b}=0du^{2}+2dzd\bar{z}$ and
$n^{a}\partial_{a}=\partial_{u}$. The conformal Carrollian symmetries are
generated by vector fields $\bar{\xi}=\bar{\xi}^{a}\partial_{a}$ on
$\mathscr{I}^{+}$ satisfying

$\mathcal{L}_{\bar{\xi}}q_{ab}=2\alpha
q_{ab}\,,\qquad\mathcal{L}_{\bar{\xi}}n^{a}=-\alpha n^{a}\,.$ (5)

The solution $\bar{\xi}$ of (5) is precisely given by the restriction to
$\mathscr{I}^{+}$ of the asymptotic Killing vectors (3), i.e.

$\bar{\xi}=(\mathcal{T}+u\alpha)\partial_{u}+\mathcal{Y}\partial+\bar{\mathcal{Y}}\bar{\partial}\,.$
(6)

The standard Lie bracket on $\mathscr{I}^{+}$ of these vector fields
reproduces the BMS algebra.

###  II.3 BMS charges

At each cut $\mathcal{S}\equiv\\{u=\text{constant}\\}$ of $\mathscr{I}^{+}$,
the charges associated with the BMS symmetries (3) are given by [56, 57, 67,
68, 69, 70, 71, 72, 73]

$\displaystyle
Q_{\xi}=\kappa\int_{\mathcal{S}}dzd\bar{z}\,\left(4\mathcal{T}M+2\mathcal{Y}^{A}\bar{N}_{A}\right),$
$\displaystyle\bar{N}_{A}=N_{A}-u\,\partial_{A}M+\frac{1}{4}C_{A}^{B}\partial_{C}C_{B}^{C}+\frac{3}{32}\partial_{A}(C_{B}^{C}C^{B}_{C})$
$\displaystyle\qquad+\frac{u}{4}\partial^{B}\partial_{B}\partial_{C}C_{A}^{C}-\frac{u}{4}\partial^{B}\partial_{A}\partial_{C}C_{B}^{C}$
(7)

where $\mathcal{Y}^{A}=(\mathcal{Y},\bar{\mathcal{Y}})$ and $\kappa=(16\pi
G)^{-1}$. These charges differ from the prescriptions considered e.g. in [74,
75, 76, 77]; we refer to [78] for a recent discussion on the relation between
the various proposals. Using the appropriate bracket [76] (see also [79, 69,
80, 81, 82, 83, 84]), these charges satisfy an algebra from which flux-balance
laws can be deduced. The charges (7) are conserved in absence of outgoing
radiation, namely when $N_{AB}=0$.

Notice that the analysis of asymptotically flat spacetimes can be performed at
$\mathscr{I}^{-}$ in an analogous manner by working in advanced Bondi
coordinates $(v,r,x^{A})$.

###  II.4 Holographic nature of null infinity

From the analysis of $4d$ asymptotically flat spacetimes, two complementary
pictures emerge, which correspond to the two possible roads to flat space
holography, as discussed in the introduction (see Figure 1):

  * $\rhd$

In the first picture, $\mathscr{I}^{+}$ (resp. $\mathscr{I}^{-}$) is seen as a
boundary along which there is retarded (resp. advanced) time evolution. This
is suited to describe the dynamics of the system through flux-balance
equations such as the famous Bondi mass loss formula originally derived in
[13, 14, 15]. In other terms, equations (2) are interpreted as evolution
equations, suggesting a $4d$ bulk / $3d$ boundary holographic correspondence,
where the dual field theory lives at null infinity and obeys Carrollian
physics. Since the charges are generically not conserved due to the outgoing
(resp. ingoing) radiation going through null infinity, it is tempting to think
of the dual theory as coupled to some external sources responsible for the
dissipation. We elaborate on this proposal in section III.

  * $\rhd$

In the second picture, $\mathscr{I}^{+}$ (resp. $\mathscr{I}^{-}$) is seen as
a portion of a Cauchy surface in the asymptotic future (resp. past). This
point of view is well adapted to describe the scattering problem in
asymptotically flat spacetime between $\mathscr{I}^{-}$ and $\mathscr{I}^{+}$
and provides information about the state of the system at early and late
times. The equivalence between BMS Ward identities and soft theorems was
established in this picture [56, 57, 67]. Equations (2) are now seen as
constraint equations in the Hamiltonian framework. Scattering amplitudes in
the bulk can be rewritten as correlation functions on the celestial sphere
obeying some Ward identities encoding the information on soft theorems. This
suggests a $4d$ bulk / $2d$ boundary correspondence, with a $2d$ CCFT as
holographic dual. We come back on this proposal in section IV, where we relate
it to the Carrollian framework.

##  III Sourced conformal Carrollian field theory

In this section, we write the Ward identities for a sourced quantum field
theory. We then specify this result for a sourced CCarrFT and argue that it
holographically encodes the asymptotic dynamics of gravity.

###  III.1 Sourced Ward identities

Let us start with a theory of fields $\Phi^{i}$ on a $n$-dimensional manifold
$\mathscr{M}$ with coordinates $x^{a}$ and admitting a well-defined
variational principle. We assume that the theory exhibits some global
symmetries $\delta_{K}\Phi^{i}=K^{i}[\Phi]$ with associated conserved Noether
currents $j^{a}_{K}$. If one couples the theory to external sources $\sigma$
which are non-dynamical fields, the infinitesimal transformations
$\delta_{K}\Phi^{i}=K^{i}[\Phi]$ are generically no longer symmetries due to
the presence of the sources (see e.g. [79, 85, 86]). This translates into the
fact that the Noether currents are no longer conserved, namely

$\partial_{a}j^{a}_{K}(x)=F_{K}(x)\,,$ (8)

where the local flux $F_{K}(x)$, which generically depends on $\Phi$ and
$\sigma$, vanishes when $\sigma=0$. At the quantum level, taking the presence
of external sources into account in the standard derivation of the Ward
identities111See e.g. [87] for the standard derivation of Ward identities. The
consideration of external sources in this derivation will be detailed in an
upcoming work., we obtain

$\displaystyle\partial_{a}\langle
j^{a}_{K}(x)X\rangle+\frac{\hbar}{i}\sum_{i=1}^{N}\delta^{(n)}(x-x_{i})\,\delta_{K^{i}}\braket{X}$
(9) $\displaystyle\qquad=\braket{F_{K}(x)X}\,,$

where $X\equiv\Phi^{i_{1}}(x_{1})\dots\Phi^{i_{N}}(x_{N})$ denotes the
collection of quantized fields and
$\delta_{K^{i}}\braket{X}\equiv\braket{\Phi^{i_{1}}(x_{1})\dots
K^{i}[\Phi(x_{i})]\dots\Phi^{i_{N}}(x_{N})}$. This generalizes the local
version of the infinitesimal Ward identities in presence of external sources.
In particular, with no field insertion in the correlators, it implies

$\partial_{a}\langle j^{a}_{K}(x)\rangle=\braket{F_{K}(x)}\,,$ (10)

which reproduces the classical flux-balance equation (8). Integrating (9) over
the manifold $\mathscr{M}$ with boundary $\partial\mathscr{M}$, we get the
integrated version of the infinitesimal Ward identities with external sources:

$\begin{split}&\sum_{i=1}^{N}\delta_{K^{i}}\braket{X}=\frac{i}{\hbar}\Big{\langle}\left(\int_{\mathscr{M}}\bm{F}_{K}-\int_{\partial\mathscr{M}}\bm{j}_{K}\right)X\Big{\rangle}\,,\end{split}$
(11)

where bold letters denote the forms associated with the objects defined above,
e.g. $\bm{F}_{K}=F_{K}\,(d^{n}x)$ and $\bm{j}_{K}=j_{K}^{a}\,(d^{n-1}x)_{a}$.
The standard result of the invariance of the correlators under symmetry
transformations is recovered by turning off the sources and assuming that the
Noether currents vanish at the boundary.

###  III.2 Application to conformal Carrollian field theory

We now apply these general results to the case of a $3d$ CCarrFT coupled to
some external sources.

BMS symmetries are to $3d$ CCarrFT what Virasoro symmetries are to $2d$ CFT,
while the $4d$ Poincaré group is on the same footing as the Möbius group
$SL(2,\mathbb{C})$. “Global” subgroups inside BMS and Virasoro groups are not
unique and correspond to Poincaré and Möbius subgroups, respectively.
Accordingly, conformal Carrollian primary fields [37, 38, 61] will be taken to
transform infinitesimally as

$\begin{split}&\delta_{\bar{\xi}}\Phi_{(k,\bar{k})}=[(\mathcal{T}+u\alpha)\partial_{u}+\mathcal{Y}\partial+\bar{\mathcal{Y}}\bar{\partial}\\\
&\qquad\qquad\quad+k\,\partial\mathcal{Y}+\bar{k}\,\bar{\partial}\bar{\mathcal{Y}}]\Phi_{(k,\bar{k})}\end{split}$
(12)

under full conformal Carroll symmetries (6) while quasi-primary fields are
only required to transform properly under the global subgroup. Here the
Carrollian weights $(k,\bar{k})$ are some integers or half-integers.

Noether currents associated with the conformal Carrollian symmetries (6) are
taken to be of the form

$j^{a}_{\bar{\xi}}={\mathcal{C}^{a}}_{b}\bar{\xi}^{b}$ (13)

where ${\mathcal{C}^{a}}_{b}$ is the analogue of the stress-energy tensor for
a CCarrFT encoding the Carrollian momenta [33, 88, 37, 38, 89, 90, 91, 92, 93,
94] as follows:

$\displaystyle{\mathcal{C}^{a}}_{b}=\left[\begin{array}[]{cc}\mathcal{M}&\mathcal{N}_{B}\\\
\mathcal{B}^{A}&{\mathcal{A}^{A}}_{B}\end{array}\right].$ (16)

The Noether currents (13) associated with the global subalgebra of the
conformal Carrollian algebra verify the flux-balance law (8) provided
${\mathcal{C}^{a}}_{b}$ satisfies

$\begin{array}[]{rcl}\text{Translations}\,(\partial_{a})&\Rightarrow&\partial_{a}{\mathcal{C}^{a}}_{b}=F_{b}\,,\\\
\text{Rotation}\,(-z\partial+\bar{z}\bar{\partial})&\Rightarrow&{\mathcal{C}^{z}}_{z}={\mathcal{C}^{\bar{z}}}_{\bar{z}}\,,\\\
\text{Boosts}\,(x^{A}\partial_{u})&\Rightarrow&{\mathcal{C}^{A}}_{u}=0\,,\\\
\text{Dilatation}\,(x^{a}\partial_{a})&\Rightarrow&{\mathcal{C}^{a}}_{a}=0\,,\end{array}$
(17)

where we assumed that the flux is linear in the parameters $\bar{\xi}^{a}$ and
can be written as $F_{\bar{\xi}}=F_{a}\bar{\xi}^{a}$ in the right-hand side of
(8), which is sufficient for the purpose of this paper. Notice that the term
${\mathcal{C}^{a}}_{b}\partial_{a}\bar{\xi}^{b}$ does not contribute to the
left-hand side of (8) as a consequence of (6) and (17). The Carrollian special
conformal transformations $K_{0}=-2z\bar{z}\partial_{u}$,
$K_{1}=2u\bar{z}\partial_{u}+2\bar{z}^{2}\bar{\partial}$ and
$K_{2}=2uz\partial_{u}+2z^{2}\partial$ do not impose further constraints.
Furthermore, the above global conformal Carrollian symmetries are enough to
completely constrain ${\mathcal{C}^{a}}_{b}$, i.e. (8) is automatically
satisfied by the supertranslation (and superrotation) currents provided (17)
holds. In terms of the Carrollian momenta, the constraints (17) imply

$\begin{array}[]{rl}\partial_{u}\mathcal{\mathcal{M}}=F_{u}\,,&\quad\mathcal{B}^{z}=0\,,\\\
\partial_{u}\mathcal{N}_{z}-\frac{1}{2}\partial\mathcal{M}+\bar{\partial}{\mathcal{A}^{\bar{z}}}_{z}=F_{z}\,,&\quad
2{\mathcal{A}^{z}}_{z}+\mathcal{M}=0\,,\end{array}$ (18)

together with the complex conjugate relations.

At the quantum level, one can write the infinitesimal Ward identities (9) for
the specific case of a CCarrFT. Assuming that the operators inserted in the
correlators are (quasi-)conformal Carrollian primary fields, we obtain

$\displaystyle\partial_{u}\braket{\mathcal{M}\,X}+\frac{\hbar}{i}\sum_{i}\delta^{(3)}(x-x_{i})\partial_{u_{i}}\braket{X}=\braket{F_{u}\,X}\,,$
$\displaystyle\partial_{u}\braket{\mathcal{N}_{z}\,X}-\frac{1}{2}\partial\braket{\mathcal{M}\,X}+\bar{\partial}\braket{{\mathcal{A}^{\bar{z}}}_{z}\,X}$
$\displaystyle+\frac{\hbar}{i}\sum_{i}\left[\delta^{(3)}(x-x_{i})\partial_{i}\braket{X}-\partial\left(\delta^{(3)}(x-x_{i})\,k_{i}\braket{X}\right)\right]$
$\displaystyle=\braket{F_{z}\,X}\,,\qquad\braket{\mathcal{B}^{z}\,X}=0\,,$
$\displaystyle\braket{({\mathcal{A}^{z}}_{z}+\frac{1}{2}\mathcal{M})X}+\frac{\hbar}{i}\sum_{i}\delta^{(3)}(x-x_{i})\,k_{i}\,\braket{X}=0\,,$
(19)

together with the complex conjugate relations. With no field insertion in the
correlators, the expectation value of the operators reproduce the classical
relations (18). At any stage of this analysis, one can turn off the sources by
setting $F_{a}=0$ to obtain the Ward identities of an honest $3d$ CCarrFT.

###  III.3 Holographic correspondence

We now argue that a quantum CCarrFT coupled with external sources would be a
right candidate to describe holographically gravity in $4d$ asymptotically
flat spacetimes reviewed in section II (few explicit examples of quantum
CCarrFT are known, see e.g. [95, 96, 97, 98, 22]). We propose the following
correspondence between Carrollian momenta and gravitational data at
$\mathscr{I}^{+}$:

$\begin{split}&\langle\mathcal{M}\rangle=4\,\kappa\,M\,,\\\
&\langle\mathcal{N}_{A}\rangle=2\,\kappa\,\Big{(}N_{A}+\frac{1}{4}C_{A}^{B}\partial_{C}C_{B}^{C}+\frac{3}{32}\partial_{A}(C_{B}^{C}C^{B}_{C})\\\
&\qquad\qquad+\frac{u}{4}\partial^{B}\partial_{B}\partial_{C}C_{A}^{C}-\frac{u}{4}\partial^{B}\partial_{A}\partial_{C}C_{B}^{C}\Big{)}\,,\\\
&\langle{\mathcal{A}^{A}}_{B}\rangle+\frac{1}{2}{\delta^{A}}_{B}\langle\mathcal{M}\rangle=0\,.\end{split}$
(20)

The factors are fixed by demanding that the gravitational charges (7)
correspond to the Noether currents (13) of the CCarrFT integrated on a section
$u=\text{constant}$. The correspondence (20) is reminiscent of the AdS/CFT
dictionary where the holographic stress-energy tensor of the CFT is identified
with some subleading order in the expansion of the bulk metric [99, 100]. It
would be interesting to push this analogy further and see if the Carrollian
momenta can be obtained by varying a bulk partition function with respect to
the boundary sources at null infinity (see e.g. [30, 32] for discussions along
these lines in $3d$ gravity).

The Bondi news tensor $N_{AB}$ is a free datum at $\mathscr{I}^{+}$ that
encodes the outgoing gravitational radiation. It is responsible for the non-
conservation of the BMS charges (7) at null infinity. It is therefore
suggestive to holographically identify the external sources $\sigma_{AB}$ at
the boundary with the Bondi news tensor $N_{AB}$ as

$\sigma_{AB}=N_{AB}\,.$ (21)

The external sources are responsible for the dissipation in the CCarrFT
through the fluxes

$\displaystyle F_{u}$
$\displaystyle=-\kappa\Big{[}\sigma^{zz}\sigma_{zz}-2(\bar{\partial}^{2}\sigma_{zz}+\partial^{2}\sigma_{\bar{z}\bar{z}})\Big{]}\,,$
(22) $\displaystyle F_{z}$
$\displaystyle=\frac{\kappa}{2}\Big{[}\partial(\sigma^{zz}\Phi_{zz})+2\Phi_{zz}\partial\sigma^{zz}+u\partial(\bar{\partial}^{2}\sigma_{zz}-\partial^{2}\sigma_{\bar{z}\bar{z}})\Big{]}\,.$

Here $\Phi_{zz}$ denotes the operator associated with a perturbation in the
asymptotic shear: $\langle\Phi_{AB}\rangle=C_{AB}$. Comparing (4) with (12),
one deduces that $\Phi_{zz}$ is a quasi-conformal Carrollian primary fields of
weights $(\frac{3}{2},-\frac{1}{2})$. It constitutes a particular example of a
correlator insertion. Notice that, from the boundary perspective, the momentum
$\Pi_{AB}=\partial_{u}\Phi_{AB}$ conjugated to $\Phi_{AB}$ should be
distinguished from the sources $\sigma_{AB}$. They are identified only through
VEV as $\langle\Pi_{AB}\rangle=\sigma_{AB}$.

Taking the identifications (20) and (22) into account, one can then check
explicitly that the time evolution equations in the sourced Ward identities
(19) reproduce the gravitational retarded time evolution equations (2) when
there is no insertion in the correlators.

Let us emphasize that a similar identification can be performed with the
solution space in advanced Bondi coordinates $(v,r,x^{A})$ at
$\mathscr{I}^{-}$. It is therefore natural to assume that the dual sourced
CCarrFT is living on $\hat{\mathscr{I}}=\mathscr{I}^{-}\sqcup\mathscr{I}^{+}$,
where the two manifolds $\mathscr{I}^{-}$ and $\mathscr{I}^{+}$ are glued
together by identifying antipodally $\mathscr{I}^{+}_{-}$ with
$\mathscr{I}^{-}_{+}$. Geometrically, the gluing 2-sphere on
$\hat{\mathscr{I}}$ is distinguished by the vanishing of the vector field
$n^{a}$ defining the Carrollian structure. Indeed, $\mathscr{I}^{+}_{-}$ and
$\mathscr{I}^{-}_{+}$ are stable under supertranslation. The gluing is
consistent with the antipodal matching conditions proposed in [56, 57, 67] and
confirmed in [101, 102, 103, 104, 105, 106] by an analysis at spacelike
infinity. In particular, the Carrollian data are identified with the solution
space of the retarded (resp. advanced) Bondi gauge at $\mathscr{I}^{+}$ (resp.
$\mathscr{I}^{-}$), with a continuous interpolation in the gluing region
thanks to the antipodal matching. The conformal Carrollian symmetries act on
the whole $\hat{\mathscr{I}}$ and correspond to the diagonal BMS symmetries
identified in [56]. They are generated by (5) on $\mathscr{I}^{+}$ and the
analogue antipodally matched symmetries on $\mathscr{I}^{-}$.

##  IV Relation with celestial holography

In this section, we show that the Ward identities of the sourced CCarrFT
reproduce the BMS Ward identities of the celestial CFT after performing the
right integral transformations. This constitutes a central argument to relate
the two approaches of flat space holography, see Figure 1.

Specifying the integrated version of the sourced Ward identities (11) to the
conformal Carrollian symmetries of the theory living on $\hat{\mathscr{I}}$
suggested in the previous section, we obtain

$\begin{split}&\delta_{\bar{\xi}}\langle
X\rangle=\frac{i}{\hbar}\Big{\langle}\left(\int_{\hat{\mathscr{I}}}{\bm{F}}_{\bar{\xi}}-\int_{\mathscr{I}^{+}_{+}}{\bm{j}}_{\bar{\xi}}+\int_{\mathscr{I}^{-}_{-}}{\bm{j}}_{\bar{\xi}}\right)\,X\Big{\rangle}\,,\end{split}$
(23)

where
$X\equiv\Phi^{out}_{(k_{1},\bar{k}_{1})}(x_{1})\dots\Phi^{out}_{(k_{m},\bar{k}_{m})}(x_{m})\Phi^{in}_{(k_{1},\bar{k}_{1})}(x_{1})\dots$
$\Phi^{in}_{(k_{n},\bar{k}_{n})}(x_{n})$,
$\Phi^{out}_{(k_{i},\bar{k}_{i})}(x_{i})$ and
$\Phi^{in}_{(k_{j},\bar{k}_{j})}(x_{j})$ denoting insertions at
$\mathscr{I}^{+}$ and at $\mathscr{I}^{-}$, respectively. To simplify the
discussion, let us assume that we are describing a scattering of massless
particles, so that the current $\bm{j}_{\bar{\xi}}$ vanishes at
$\mathscr{I}^{+}_{+}$ and $\mathscr{I}^{-}_{-}$. In addition, we require that
the integrated flux on $\mathscr{I}^{+}$ is equal to minus the integrated flux
on $\mathscr{I}^{-}$ at the level of the operators, namely

$\int_{\mathscr{I}^{-}}{\bm{F}}_{\bar{\xi}}=-\int_{\mathscr{I}^{+}}{\bm{F}}_{\bar{\xi}}\,.$
(24)

This constraint on the sources is compatible with the classical bulk
requirement that the integrated ingoing flux is equal to the integrated
outgoing flux for the massless scattering considered. Taking these assumptions
into account, the integrated Ward identities (23) imply

$\delta_{\bar{\xi}}\langle X\rangle=0\,.$ (25)

This is the statement that the correlators are conformal Carroll invariant.
The consequence of this relation have been studied e.g. in [22, 107, 108].

To relate (25) with the CCFT Ward identities, we use similar technical steps
than those advocated in [56, 57, 67] to relate BMS Ward identities and soft
graviton theorems in the bulk. Let us first specify (25) for the
supertranslation symmetries to recover the corresponding Ward identities of
the CCFT. We split the variation into hard and soft parts
$\delta_{\mathcal{T}}\langle X\rangle=\delta^{H}_{\mathcal{T}}\langle
X\rangle+\delta^{S}_{\mathcal{T}}\langle X\rangle$ and rewrite the soft part
of the transformation as the soft charge insertion by using the quantum
commutator

$\begin{split}&\left[\Pi_{zz}(u,z,\bar{z}),\Phi_{\bar{w}\bar{w}}(u^{\prime},w,\bar{w})\right]=\frac{i\hbar}{\kappa}\,\delta(u-u^{\prime})\,\delta^{(2)}(z-{w})\,,\end{split}$
(26)

which ensures the compatibility with the Poisson bracket on the radiative
phase space [109, 110]. Then we specify the relation for
$\mathcal{T}(z,\bar{z})=\frac{1}{z-w}$ and introduce the supertranslation
current $P(z,\bar{z})$ [56] through

$P(z,\bar{z})=\frac{1}{4G}\left(\int du+\int
dv\right)\bar{\partial}\Pi_{zz}\,.$ (27)

Furthermore, we obtain the CCFT operators
$\mathcal{O}^{out}_{\Delta,J}(z,\bar{z})$ and
$\mathcal{O}^{in}_{\Delta,J}(z,\bar{z})$ of conformal dimension $\Delta$ and
spin $J$ from the conformal Carrollian operators
$\Phi^{out}_{(k,\bar{k})}(u,z,\bar{z})$ and
$\Phi^{in}_{(k,\bar{k})}(v,z,\bar{z})$ of (12) through the integral transform

$\begin{split}\mathcal{O}^{out}_{\Delta,J}(z,\bar{z})&=i^{\Delta}\Gamma[\Delta]\int^{+\infty}_{-\infty}du\,u^{-\Delta}\,\Phi^{out}_{(k,\bar{k})}(u,z,\bar{z})\,,\\\
\mathcal{O}^{in}_{\Delta,J}(z,\bar{z})&=i^{\Delta}\Gamma[\Delta]\int^{+\infty}_{-\infty}dv\,v^{-\Delta}\,\Phi^{in}_{(k,\bar{k})}(v,z,\bar{z})\,.\end{split}$
(28)

The above integral is the composition of a Fourier transform (from position to
momentum space) and a Mellin transform which maps energy to boost eigenstates
[42, 43]. It trades the time dependence of the Carrollian operators for the
conformal dimension of the CCFT operators. Importantly, Carrollian weights are
related to the $2d$ spin via

$k=\frac{1}{2}(1+J)\,,\qquad\bar{k}=\frac{1}{2}(1-J)\,,$ (29)

which can be seen to be consistent with the radiative falloffs in the
conformal compactification. Taking into account the aforementioned steps, we
recover the CCFT Ward identity for supertranslations [56, 57, 45] ($N=m+n$)

$\begin{split}&\Big{\langle}P(z,\bar{z})\prod_{i=1}^{N}\mathcal{O}_{\Delta_{i},J_{i}}(z_{i},\bar{z}_{i})\Big{\rangle}\\\
&+\hbar\sum_{q=1}^{N}\frac{1}{z-z_{q}}\Big{\langle}\dots\mathcal{O}_{\Delta_{q}+1,J_{q}}(z_{q},\bar{z}_{q})\dots\Big{\rangle}=0\,,\end{split}$
(30)

which is the celestial encoding of the leading soft graviton theorem.

Now, one can specify (25) for superrotations and follow the same series of
steps. Choosing $\mathcal{Y}(z)=\frac{1}{z-w}$ and defining the $2d$ stress-
tensor as

$T(z)=-\frac{i}{8\pi G}\int\frac{dwd\bar{w}}{z-w}\left(\int du\,u+\int
dv\,v\right)\partial^{3}\Pi_{\bar{w}\bar{w}}\,,$ (31)

we recover the $2d$ CFT Ward identities [111, 112, 46, 113] after performing
(28):

$\displaystyle\Big{\langle}T(z)\prod_{i=1}^{N}\mathcal{O}_{\Delta_{i},J_{i}}(z_{i},\bar{z}_{i})\Big{\rangle}$
(32)
$\displaystyle+{\hbar}\sum_{q=1}^{N}\Big{[}\frac{\partial_{q}}{z-z_{q}}+\frac{h_{q}}{(z-z_{q})^{2}}\Big{]}\,\Big{\langle}\prod_{i=1}^{N}\mathcal{O}_{\Delta_{i},J_{i}}(z_{i},\bar{z}_{i})\Big{\rangle}=0\,,$

where $h_{q}=\tfrac{1}{2}(\Delta_{q}+J_{q})$ (similar results hold for the
anti-holomorphic stress tensor). This result encodes the information on the
subleading soft graviton theorem in the bulk.

##  V Discussion

Here, we have argued that the null nature of $\mathscr{I}$ leads to two
complementary roads to flat space holography (see Figure 1): Carrollian versus
Celestial. We have presented a holographic description in the first picture in
terms of a codimension-one CCarrFT coupled with external sources encoding
gravitational radiation at null infinity. The Ward identities have been shown
to reproduce those of the celestial CFT, by relating celestial primary
operators to conformal Carrollian fields living at $\mathscr{I}$. This
provides a bridge between two different approaches to the ambitious program of
finding a holographic description of quantum gravity for realistic spacetimes.

This work raises new questions for the future. For instance, it would be
interesting $(i)$ to deduce the low-point correlation functions in the CCarrFT
from the Ward identities and match them with those of the CCFT, $(ii)$ to
understand if the sourced CCarrFT can be obtained in the flat limit of the
AdS/CFT correspondence (building on the works [114, 71, 115]), $(iii)$ to
write a concrete proposal for the sourced CCarrFT reproducing the features
described in this work. We let this research program for future
investigations.

###### Acknowledgements.

We would like to thank Glenn Barnich, Geoffrey Compère, Florian Ecker, Gaston
Giribet, Daniel Grumiller and Marios Petropoulos for discussions. LD, AF and
RR are supported by the Austrian Science Fund (FWF) START project Y 1447-N. AF
and RR also acknowledge support from the FWF, project P 32581-N. YH has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No
101002551). The authors also acknowledge support of the Erwin Schrödinger
Institute (ESI) in Vienna where part of this work was conducted during the
thematic program “Geometry for Higher Spin Gravity$\\_$FCC$\\_$2021”.

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