We end up this manuscript by providing future potentially interesting directions for this work.

(i) The formalism of sourced quantum field theory introduced in Sec. IV was applied in this work to flat space holography and its associated asymptotic dynamics of the bulk spacetime. We believe that this setup describing a sourced system could be applied to a much broader scope than the one presented here. Indeed, any gravitational system with leaky boundary conditions could in principle be holographically described using this formalism. For instance, it would be worth applying this framework in the case of hypersurfaces at finite distance (see e.g. [196,199,212-217]), such as black hole horizons, to deduce some insights on holography for finite spacetime regions.

(ii) As mentioned in the introduction, most of the results obtained in Carrollian holography are deduced from AdS/CFT by taking a flat limit in the bulk, leading to an ultrarelativistic limit at the boundary. However, as highlighted in [61-65,218], one has to start from leaky boundary conditions in 4d AdS if one wants to obtain radiative spacetimes in the limit. This contrasts with the standard Dirichlet boundary conditions that are usually considered in AdS/CFT. Therefore, the first step would be to obtain a holographic description of AdS spacetimes with leaky boundary conditions by coupling the dual CFT with some external sources and using the formalism of Sec. IV. Then, taking a flat limit in the bulk will imply an ultrarelativistic limit at the boundary. The hope is that one might get an explicit realization of the sourced Carrollian CFT using this procedure. It would also be interesting to revisit in this context the flat limit procedure of scattering amplitudes in AdS in the spirit of the works done in [219-222] and relate it to our setup.

(iii) The newly uncovered $w_{1+\infty}$ symmetries in the CCFT [98] have not yet been given a clear interpretation in the Carrollian CFT. As suggested by the recent analysis of [103], the information on these symmetries might be encoded in the subleading orders of the bulk metric. In our analysis, we have only considered Carrollian stress tensor or source operators. However, nothing would prevent us to consider Carrollian fields that are holographically identified with subleading orders in the expansion of the bulk metric. It might be instructive to revisit these $w_{1+\infty}$ symmetries in those terms and provide an interpretation at null infinity.

(iv) Finally, it would be a great progress if one could provide an explicit example of Carrollian CFT living at null infinity that would holographically capture some features of gravity in the bulk (for the celestial approach, see e.g. [223-226] for recent top-down

models). A good starting point is the BMS geometric action constructed in [32] which, together with [227], furnishes an effective description of the dual Carrollian CFT for nonradiative spacetimes. The source sector of the Carrollian CFT is not yet known but we believe that the analysis provided in the present paper imposes strong constraints on it. For instance, the Carrollian weights of the source operators are completely fixed via (5.17). Moreover, the 2-point correlation function for source operators is identified with the bulk 2-point amplitude as in (5.23). In particular, this tells us that the propagator of source operators is u-dependent, which suggests that the source sector of the Carrollian CFT is a timelike electric-type of Carrollian theory [41,46,190,228,229].

## ACKNOWLEDGMENTS

We would like to thank Arjun Bagchi, Glenn Barnich, Nicolas Boulanger, Andrea Campoleoni, Geoffrey Compère, Florian Ecker, Erfan Esmaeili, Laurent Freidel, Gaston Giribet, Daniel Grumiller, Carlo Heissenberg, Kévin Nguyen, Sabrina Pasterski, Marios Petropoulos, Stefan Prohazka, Andrea Puhm, and Jakob Salzer, for useful conversations. We are also grateful to all the participants of the Carroll Workshops in Vienna and Mons for inspiring and stimulating discussions. L. D. is partially supported by INFN Iniziativa Specifica ST&FI. A. F. and R. R. are supported by Austrian Science Fund (FWF), Project No. P 32581-N. L. D., A. F., and R. R. also acknowledge support from the FWF START Project No. Y 1447-N.

## APPENDIX A: BONDI COORDINATES FOR MINKOWSKI SPACETIME

In the main text, we make an extensive use of the parametrization of Minkowski space by Bondi coordinates. We found convenient to work with a flat representative of the conformal boundary metric, which amounts to perform a boundary Weyl rescaling with respect to the usual choice of round boundary representative. The aim of this appendix is to install our conventions and notations regarding this choice of coordinates.

### 1. Bondi coordinates with round boundary representative

In retarded Bondi coordinates {$u_0, r_0, z_0, \bar{z}_0$} ($u_0 \in \mathbb{R}$, $r_0 \in \mathbb{R}^+$, $z_0 \in \mathbb{C}$), the Minkowski line elements reads as

$$ds^2 = -du_0^2 - 2du_0dr_0 + \frac{4r_0^2}{(1 + z_0 \bar{z}_0)^2} dz_0 d\bar{z}_0. \quad (A1)$$

Cuts of constant $u_0$ of future null infinity $\mathcal{I}^+ = \{r_0 \to +\infty\}$ are spheres on which the line element is the unit
round-sphere metric in stereographic coordinates $z_\circ = e^{iq_\circ} \cot \theta_\circ$. The change of coordinates from the Cartesian chart $X^\mu = \{t, \bar{x}\}$ is given by

$$X^\mu = u_\circ \delta_0^\mu + r_\circ \frac{\sqrt{2}}{1+z_\circ \bar{z}_\circ} q^\mu(z_\circ, \bar{z}_\circ), \quad (A2)$$

where

$$q^\mu(z, \bar{z}) = \frac{1}{\sqrt{2}}(1+z\bar{z}, z+\bar{z}, -i(z-\bar{z}), 1-z\bar{z}) \quad (A3)$$

is the standard parametrization of a null direction pointing toward the direction $(z_\circ, \bar{z}_\circ)$ on the celestial sphere. In a similar way, one defines advanced Bondi coordinates $\{v_\circ, r'_\circ, z'_\circ, \bar{z}'_\circ\}$ ($v_\circ \in \mathbb{R}$, $r'_\circ \in \mathbb{R}^+$, $z'_\circ \in \mathbb{C}$) in which the Minkowski line element is

$$ds^2 = -dv_\circ^2 + 2dv_\circ dr'_\circ + \frac{4r_\circ^2}{(1+z'_\circ \bar{z}'_\circ)^2} dz'_\circ d\bar{z}'_\circ. \quad (A4)$$

In terms of the Cartesian coordinates, we have

$$X^\mu = v_\circ \delta_0^\mu - r'_\circ \frac{\sqrt{2}}{1+z'_\circ \bar{z}'_\circ} q^\mu(z'_\circ, \bar{z}'_\circ). \quad (A5)$$

For any finite value of $r_\circ, r'_\circ$, both Bondi coordinate systems are related as

$$v_\circ = u_\circ + 2r_\circ, \quad r'_\circ = r_\circ, \quad z'_\circ = -\frac{1}{\bar{z}_\circ}, \quad (A6)$$

where in particular the holomorphic coordinates are related by the antipodal map $(\theta, \varphi) \mapsto (\pi - \theta, \varphi + \pi)$, or $z_\circ \mapsto -\frac{1}{z_\circ}$. This particularly convenient choice [151] allows one to label a light ray crossing the spacetime with the same value of the angular coordinates in both coordinate systems. In other words, a light ray originating from a point $z'_\circ = a$ on the celestial sphere at past null infinity will pierce again the celestial sphere at future null infinity at the antipodal point identified by $z_\circ = a$.

## 2. Bondi coordinates with flat boundary representative

We now introduce the coordinate system $\{u, r, z, \bar{z}\}$ by trading the unit round-sphere metric on the boundary for the flat complex plane metric. The diffeomorphism that implements the boundary Weyl rescaling has been worked out e.g. in [230] and reads as

$$r = \frac{\sqrt{2}}{1+z_\circ \bar{z}_\circ} r_\circ + \frac{u_\circ}{\sqrt{2}}, \quad u = \frac{1+z_\circ \bar{z}_\circ}{\sqrt{2}} u_\circ - \frac{z_\circ \bar{z}_\circ u_\circ^2}{2r}, \\ z = z_\circ - \frac{z_\circ u_\circ}{\sqrt{2}r}, \quad (A7)$$

for retarded coordinates. Here $u, r \in \mathbb{R}$ and $z \in \mathbb{C}$. The relation with Cartesian coordinates is now given by

$$X^\mu = u \partial_z \partial_{\bar{z}} q^\mu(z, \bar{z}) + r q^\mu(z, \bar{z}) \quad (A8)$$

and the Minkowski line element reads as

$$ds^2 = -2dudr + 2r^2dzd\bar{z}. \quad (A9)$$

These coordinates are such that $\mathcal{I}^+$ and $\mathcal{I}^-$ are respectively obtained by taking the limits $r \to +\infty$ and $r \to -\infty$. Cuts of $\mathcal{I}^+$ are now complex planes endowed with a flat metric. Lines obtained by keeping $(u, z, \bar{z})$ fixed form a null geodesic congruence extending from past to future null infinity. The matching between past and future null infinities is therefore here immediate.

Similarly, we can also flatten the boundary representative in advanced Bondi coordinates to reach the coordinates $\{v, r', z', \bar{z}'\}$. The diffeomorphism implementing the boundary Weyl rescaling is given explicitly by

$$r' = \frac{\sqrt{2}}{1+z_\circ \bar{z}_\circ} r'_\circ - \frac{v_\circ}{\sqrt{2}}, \quad v = \frac{1+z_\circ \bar{z}_\circ}{\sqrt{2}} v_\circ + \frac{z_\circ \bar{z}_\circ}{2r'_\circ} v_\circ^2, \\ z' = z'_\circ + \frac{z_\circ v_\circ}{\sqrt{2} r'_\circ}. \quad (A10)$$

In these coordinates, related to the Cartesian coordinates via

$$X^\mu = v \partial_{z'} \partial_{\bar{z}'} q^\mu(z', \bar{z}') - r' q^\mu(z', \bar{z}'), \quad (A11)$$

the Minkowski line element is

$$ds^2 = 2dvd\bar{r} + 2r'^2dz'd\bar{z}'. \quad (A12)$$

Here $\mathcal{I}^+$ is reached in the limit $r' \to -\infty$ while $\mathcal{I}^-$ is obtained by taking $r' \to +\infty$. Inverting (A7) recalling that $r_\circ \in \mathbb{R}^+$ and taking into account the diffeomorphism (A6), one can show that (A10) yields

$$v = u, \quad r' = -r, \quad z' = z. \quad (A13)$$

The Bondi coordinates with flat boundary representative are asymptotically related to those with round boundary representative as follows. In the limit where $r_\circ$ goes to infinity in the retarded Bondi coordinates, we deduce from (A7) that

$$u = u_\circ \frac{1+z_\circ \bar{z}_\circ}{\sqrt{2}} + \mathcal{O}(r_\circ^{-1}), \quad r = r_\circ \frac{\sqrt{2}}{1+z_\circ \bar{z}_\circ} + \mathcal{O}(r_\circ^0), \\ z = z_\circ + \mathcal{O}(r_\circ^{-1}), \quad (A14)$$

while, in the limit where $r'_\circ \equiv r_\circ$ goes to infinity in the advanced Bondi coordinates, (A10) together with the matching (A13) imply
$$v = v_{\circ} \frac{1 + z'_{\circ} \bar{z}'_{\circ}}{\sqrt{2}} + \mathcal{O}(r_{\circ}^{-1}), \qquad r = -r_{\circ} \frac{\sqrt{2}}{1 + z'_{\circ} \bar{z}'_{\circ}} + \mathcal{O}(r_{\circ}^{0}), \\ z = z'_{\circ} + \mathcal{O}(r_{\circ}^{-1}). \qquad (A15)$$

In this sense, the coordinate system {$u, r, z, \bar{z}$} with flat boundary representative interpolates between the advanced and retarded coordinate systems with boundary round representatives at large distances. In particular points situated at $z_\circ$ on the future celestial sphere are identified with points at $z'_\circ = -\frac{1}{\bar{z}_\circ}$ on the past celestial sphere through a null ray defined with constant $(u, z, \bar{z})$ such that $z \equiv z_\circ$ at future null infinity. Similar considerations apply for the advanced

coordinate system up to performing the simple change of coordinates (A13).

## APPENDIX B: ISOMORPHISM BETWEEN GLOBAL CONFORMAL CARROLLIAN AND POINCARÉ ALGEBRAS

In Cartesian coordinates $X^\mu = \{t, x^i\}$, the Poincaré generators on Minkowski spacetime are

(i) Translations: $P_0 = \partial_t, P_i = \partial_{x^i}, i = 1, 2, 3$.

(ii) Rotations: $R_i = x^j \partial_{x^k} - x^k \partial_{x^j}$, $(i, j, k) = (1, 2, 3)$, $(2, 1, 3)$, $(3, 1, 2)$.

(iii) Special Lorentz transformations: $B_i = x^i \partial_t + t \partial_{x^i}$, $i = 1, 2, 3$.

They satisfy the well-known iso(3, 1) algebra

$$[P_i, P_j] = 0 = [P_0, P_i], \qquad [R_i, P_j] = -\epsilon_{ijk} P_k, \qquad [R_i, P_0] = 0, [B_i, P_j] = -\delta_{ij} P_0, \\ [B_i, P_0] = -P_i, \qquad [R_i, R_j] = -\epsilon_{ijk} R_k, \qquad [R_i, B_j] = -\epsilon_{ijk} B_k, \qquad [B_i, B_j] = \epsilon_{ijk} R_k. \qquad (B1)$$

Performing the change of coordinates to retarded Bondi gauge with flat conformal frame at the boundary, one finds that the Poincaré generators can be expressed as (4.31) on $\mathcal{I}^+$ with particular functions $\mathcal{T}, \mathcal{Y}^z$ and $\mathcal{Y}^\bar{z}$ given in Table I.

From the intrinsic point of view, these generators do not necessarily seem natural but can be related to the standard generators of a conformal Carrollian symmetry algebra (see Sec. IV D 1) thanks to the following isomorphism:

$$\mathcal{C}\mathcal{C}\mathcal{A}\mathcal{R}\mathcal{R}_3 \simeq \mathcal{C}\mathcal{O}\mathcal{N}\mathcal{F}_2 \oplus \mathbb{R}^4 \simeq \mathcal{I}\mathcal{S}\mathcal{O}(3, 1) \oplus \mathbb{R}^4 \equiv \mathcal{I}\mathcal{S}\mathcal{O}(3, 1). \qquad (B2)$$

The dictionary is as follows:

$$P_0 = \frac{1}{\sqrt{2}}(\bar{P}_0 + \bar{P}_3), \qquad P_1 = -\frac{i}{2}(\bar{R}_1 + i\bar{R}_2 + i\bar{B}_1 + \bar{B}_2), \qquad P_2 = \frac{i}{2}(\bar{R}_1 - i\bar{R}_2 - i\bar{B}_1 + \bar{B}_2), \\ J = i\bar{R}_3, D = \bar{B}_3, \qquad B_1 = -\frac{1}{\sqrt{2}}(\bar{P}_1 - i\bar{P}_2), \qquad B_2 = -\frac{1}{\sqrt{2}}(\bar{P}_1 + i\bar{P}_2), \\ K_0 = -\sqrt{2}(\bar{P}_0 - \bar{P}_3), \qquad K_1 = -i(\bar{R}_1 + i\bar{R}_2 - i\bar{B}_1 - \bar{B}_2), \qquad K_2 = i(\bar{R}_1 - i\bar{R}_2 + i\bar{B}_1 - \bar{B}_2). \qquad (B3)$$

The bar over the Poincaré generators means again their restriction to future null infinity. Applying the redefinitions (B3) on (B1) gives the algebra (4.32) and (4.33).

<table><caption>TABLE I. Poincaré generators in retarded Bondi gauge.</caption><thead><tr><th>Generator</th><th>T(z, z̄)</th><th>Y<sup>z</sup>(z)</th><th>Y<sup>z̄</sup>(z̄)</th></tr></thead><tbody><tr><td>P<sub>0</sub></td><td><sup>1</sup>&frasl;<sub>&radic;2</sub>(1 + zz̄)</td><td>0</td><td>0</td></tr><tr><td>P<sub>1</sub></td><td>-<sup>1</sup>&frasl;<sub>&radic;2</sub>(z + z̄)</td><td>0</td><td>0</td></tr><tr><td>P<sub>2</sub></td><td><sup>1</sup>&frasl;<sub>&radic;2</sub>i(z - z̄)</td><td>0</td><td>0</td></tr><tr><td>P<sub>3</sub></td><td><sup>1</sup>&frasl;<sub>&radic;2</sub>(1 - zz̄)</td><td>0</td><td>0</td></tr><tr><td>R<sub>1</sub></td><td>0</td><td><sup>1</sup>&frasl;<sub>2</sub>i(1 - z<sup>2</sup>)</td><td>-<sup>1</sup>&frasl;<sub>2</sub>i(1 - z̄<sup>2</sup>)</td></tr><tr><td>R<sub>2</sub></td><td>0</td><td><sup>1</sup>&frasl;<sub>2</sub>(1 + z<sup>2</sup>)</td><td><sup>1</sup>&frasl;<sub>2</sub>(1 + z̄<sup>2</sup>)</td></tr><tr><td>R<sub>3</sub></td><td>0</td><td>iz</td><td>-iz̄</td></tr><tr><td>B<sub>1</sub></td><td>0</td><td><sup>1</sup>&frasl;<sub>2</sub>(1 - z<sup>2</sup>)</td><td><sup>1</sup>&frasl;<sub>2</sub>(1 - z̄<sup>2</sup>)</td></tr><tr><td>B<sub>2</sub></td><td>0</td><td><sup>1</sup>&frasl;<sub>2</sub>i(1 + z<sup>2</sup>)</td><td>-<sup>1</sup>&frasl;<sub>2</sub>i(1 + z̄<sup>2</sup>)</td></tr><tr><td>B<sub>3</sub></td><td>0</td><td>z</td><td>z̄</td></tr></tbody></table>
APPENDIX C: CONSTRAINTS ON THE CARROLLIAN STRESS TENSOR

In this appendix, we detail the proof of the identities (4.38) obeyed by the Carrollian stress tensor $C^a_b$ at the classical level. All of them stem from the flux-balance law (4.37).

For Carrollian translations, $\xi^a = \delta^a_b$, hence $\partial_a C^a_b = F_b$ is immediate from (4.37). Invariance under Carrollian rotation imposes

$$0 = \partial_a (-C^a_z z + C^a_{\bar{z}} \bar{z}) + z F_z - \bar{z} F_{\bar{z}} = -z(\partial_a C^a_z - F_z) + \bar{z}(\partial_a C^a_{\bar{z}} - F_{\bar{z}}) - C^z_{\bar{z}} + C^z_{\bar{z}}, \quad (C1)$$

which gives $C^z_{\bar{z}} = C^{\bar{z}}_{\bar{z}}$ using the invariance by translation, i.e. the first condition in (4.38). For the Carrollian boosts,

$$0 = \partial_a (C^a_u x^A) - x^A F_u = (\partial_a C^a_u - F_u)x^A + C^A_u \Rightarrow C^A_u = 0. \quad (C2)$$

The invariance under Carrollian dilatations finally gives

$$0 = \partial_a (C^a_b x^b) - F_b x^b = (\partial_a C^a_b - F_b)x^b + C^a_u \Rightarrow C^a_u = 0, \quad (C3)$$

which concludes the demonstration of (4.38).

It remains to show that the Carrollian special conformal transformations $K_0 = -2z\bar{z}\partial_u$, $K_1 = 2u\bar{z}\partial_u + 2\bar{z}^2\partial_{\bar{z}}$ and $K_2 = 2uz\partial_u + 2z^2\partial_z$ do not impose further constraints. Let us prove this statement for $K_2$ (the proof for $K_0$ and $K_1$ is similar). The flux-balance law is particularized as

$$\begin{aligned} 0 &= \partial_a (C^a_u uz + C^a_z z^2) - F_u uz - F_z z^2 \\ &= (\partial_a C^a_u - F_u) uz + (\partial_a C^a_z - F_z) z^2 + C^u_u z + C^z_u u + 2C^z_z z \\ &= (C^u_u + C^z_z + C^{\bar{z}}_{\bar{z}}) z = 0. \end{aligned} \quad (C4)$$

The second equality holds by virtue of the invariance by translation and boosts, the third one holds thanks to the invariance by rotation while the last one uses the invariance by dilatation.

Let us finally prove that the global conformal Carrollian symmetries are enough to completely constrain $C^a_b$, i.e. (4.37) is automatically satisfied by the pure supertranslation and superrotation currents provided (4.38) holds. Considering first a generic supertranslation $\xi^u = T(z, \bar{z})$, $\xi^A = 0$, we have

$$0 = \partial_a (C^a_u T) - F_u T = (\partial_a C^a_u - F_u) T + C^A_u \partial_A T = 0 \quad (C5)$$

using successively the invariance by Carrollian translations and boosts. For superrotations $\xi^u = \frac{u}{2}\partial_A \mathcal{Y}^A$, $\xi^z = \mathcal{Y}^z(z)$ and $\xi^{\bar{z}} = \mathcal{Y}^{\bar{z}}(\bar{z})$, we have

$$\begin{aligned} & \partial_a \left( C^a_u \frac{u}{2} \partial_A \mathcal{Y}^A + C^a_z \mathcal{Y}^A \right) - F_u \frac{u}{2} \partial_A \mathcal{Y}^A - F_A \mathcal{Y}^A \\ &= \frac{u}{2} \partial_A \mathcal{Y}^A (\partial_a C^a_u - F_u) + \mathcal{Y}^A (\partial_a C^a_z - F_z) + \frac{1}{2} C^u_u \partial_A \mathcal{Y}^A + \frac{u}{2} C^B_u \partial_B \partial_A \mathcal{Y}^A + C^B_A \partial_B \mathcal{Y}^A \\ &= \frac{1}{2} C^u_u (\partial_z \mathcal{Y}^z + \partial_{\bar{z}} \mathcal{Y}^{\bar{z}}) + C^z_{\bar{z}} \partial_z \mathcal{Y}^z + C^{\bar{z}}_{\bar{z}} \partial_{\bar{z}} \mathcal{Y}^{\bar{z}} \\ &= \frac{1}{2} C^u_u (\partial_z \mathcal{Y}^z + \partial_{\bar{z}} \mathcal{Y}^{\bar{z}}) + \frac{1}{2} (C^z_{\bar{z}} + C^{\bar{z}}_{\bar{z}}) \partial_z \mathcal{Y}^z + \frac{1}{2} (C^z_{\bar{z}} + C^{\bar{z}}_{\bar{z}}) \partial_{\bar{z}} \mathcal{Y}^{\bar{z}} \\ &= \frac{1}{2} (C^u_u + C^z_z + C^{\bar{z}}_{\bar{z}}) (\partial_z \mathcal{Y}^z + \partial_{\bar{z}} \mathcal{Y}^{\bar{z}}) = 0. \end{aligned} \quad (C6)$$

The third equality uses the invariance by translations and boosts. It also uses the fact that superrotations are holomorphic. The fourth equality uses the invariance by rotation while the last one invokes the invariance by dilatation. This concludes the proof of (4.38).
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