Celestial CFT was throwing away too much information ── a geometry with zero speed of light gets it back
In Bonus ③ we actually computed and confirmed that celestial CFT's four-point function becomes a distribution supported on \(\delta(z-\bar z)\). The cause was "2→2 scattering is planar" ── elementary kinematics.
There I wrote that "a breakage whose mode of breaking is understood can be handled." This episode is that handling.
And the fix was not a repair to some detail. The dimension of the boundary itself was wrong. The correct boundary is three-dimensional, and it carries "a geometry with zero speed of light" ── a world, named after Lewis Carroll, in which nothing can move.
First, let us reread Bonus ③'s diagnosis one level deeper.
The standard form of holography is AdS/CFT. There, a \(d+1\)-dimensional bulk ↔ a \(d\)-dimensional boundary. The boundary is at codimension 1 in the bulk. But ──
| bulk | boundary | codim | |
|---|---|---|---|
| AdS/CFT | AdS\(_{d+1}\) | CFT\(_d\) | 1 |
| celestial | 4d flat spacetime | 2d celestial sphere \(S^2\) | 2 |
Celestial alone has codimension 2. That is anomalous, and it is no accident. Recall what the Mellin transform did.
$$\tilde{\mathcal{M}}(\Delta,z,\bar z)=\int_0^\infty d\omega\;\omega^{\Delta-1}\,\mathcal{M}(\omega,z,\bar z)$$It integrates the energy \(\omega\) away. One dimension's worth of the four-dimensional information was crushed by the transform. That is why the boundary came out two-dimensional.
The distributional behaviour is not "because it is a weird CFT."
The container is one dimension short, so the leftover information overflows as a delta function.
Then the prescription is obvious ── do not delete \(\omega\).
Instead of the Mellin transform, move \(\omega\) to its Fourier conjugate, the retarded time \(u\). No information is discarded.
Then the place the operators live becomes this.
$$\mathcal{O}(u,z,\bar z)\qquad\text{──}\qquad \mathscr{I}^+ \cong \mathbb{R}_u\times S^2$$Null infinity \(\mathscr{I}^+\) is a three-dimensional manifold: the celestial sphere \(S^2\) times the \(u\) direction. The sphere was only a "slice" of it.
| celestial | Carrollian | |
|---|---|---|
| boundary | celestial sphere \(S^2\) | all of \(\mathscr{I}\) |
| dimension | 2 | 3 |
| codimension | 2 (anomalous) | 1 (standard) |
| \(\omega\) is | integrated away | kept as \(u\) |
The codimension is back to 1. What remains is "what theory sits on a three-dimensional boundary?" This is the strange and interesting part.
Let us try putting a metric on \(\mathscr{I}^+\). Something awkward happens.
The \(u\) direction is null. It is the direction light runs, so its length is zero. Hence the metric on \(\mathscr{I}\) is ──
$$ds^2\Big|_{\mathscr{I}}=\underbrace{0\cdot du^2}_{\text{zero length in the time direction}}+\;d\Omega^2$$Degenerate. There is no inverse. Neither ordinary Riemannian nor Lorentzian geometry applies.
So what is it? It has a name.
The Poincaré group has two limits. Both are group contractions (Inönü–Wigner contractions).
| limit | group obtained | form of the boost | what is absolute |
|---|---|---|---|
| \(c\to\infty\) | Galilei group | \(x'=x-vt,\quad t'=t\) | time |
| \(c\to 0\) | Carroll group | \(t'=t-b\!\cdot\!x,\quad x'=x\) | space |
Neatly dual. In Galilei, time is absolute and space mixes. In Carroll it is the reverse ── space is absolute, and it is time that mixes.
Lévy-Leblond named it (1965), taking it from the Red Queen in Lewis Carroll's Through the Looking-Glass ── "you have to run as fast as you can just to stay in the same place."
Why that quotation? See the figure.
At \(c\to\infty\) the light cone opens out flat and surfaces of simultaneity become absolute (Galilei). Conversely at \(c\to0\) ──
The light cone collapses onto the time axis. The set of reachable places shrinks to the single point you are at.
However much time passes, you can go nowhere.
That is the meaning of the Red Queen. And an important physical consequence follows ── Carrollian dynamics is "ultra-local". Since no signal propagates, each point of space evolves in time independently.
Now back to Episode 4. What was a supertranslation?
$$u\;\longrightarrow\;u-f(\theta,\phi)$$Shift \(u\) independently in each direction on the celestial sphere.
Read it again. This is "each point of space evolves in time independently." An ultra-local time translation ── precisely the form Carroll geometry demands.
And in fact an exact isomorphism holds (Duval–Gibbons–Horvathy and others).
What surprised us in Episode 4 — "the symmetry of flat spacetime is infinite-dimensional" — turns out to be the conformal group at zero speed of light. It is infinite-dimensional because of ultra-locality ── each point being independent, there is one generator per point.
And back to the distributions. In an ultra-local theory, a delta function in the correlator of separated points is not pathology. Since no signal propagates, of course the points are not connected.
The delta function that looked like "something weird" in the celestial picture becomes, in Carrollian language, a basic property of the theory. The same formula stopped being an anomaly the moment it was put in the right container.
Let us head off a misunderstanding. The celestial computation done in Bonus ③ is not wasted.
What "Bridging Carrollian and Celestial Holography" (2022–23) showed is that the two are related by an integral transform. Not different theories but a change of basis for the scattering amplitude.
what has been confirmed
Map Carrollian operators to celestial ones and the Carrollian Ward identities reproduce those of the 2d celestial CFT. Furthermore, the Carrollian-side Ward identities are precisely the BMS flux-balance laws.
So ── it is not that one is right and the other wrong. It is only that the Carrollian side has not thrown information away. Celestial is the "projection" obtained by integrating out \(u\).
| celestial | Carrollian | |
|---|---|---|
| ease of use | the toolkit of 2d CFT applies | 3d; the tools are still developing |
| codimension | 2 | 1 |
| delta functions | look anomalous | natural, as ultra-locality |
| identity of the symmetry | \(SL(2,\mathbb{C})\) is visible | BMS, directly |
The story so far has been all "not enough," so here is one piece of strong positive evidence.
Episode 4 dealt with the soft theorem, but in fact only its leading term. There is an infinite tower.
$$\underbrace{q^{-1}}_{\text{supertranslations}},\quad\underbrace{q^{0}}_{\text{superrotations}},\quad\underbrace{q^{1}}_{?},\quad\underbrace{q^{2}}_{?},\quad\cdots$$What Strominger (2021) showed is that the entire tower of positive-helicity soft gravitons organizes into a single chiral 2d Kac–Moody symmetry based on the wedge subalgebra of \(w_{1+\infty}\). And the towers of soft photons and gluons transform irreducibly under it.
Why does this matter? \(w_{1+\infty}\) is not an algebra built for this purpose.
| where \(w_{1+\infty}\) had already appeared | |
|---|---|
| area-preserving diffeomorphisms of a 2d surface | pure geometry |
| the symmetry of self-dual gravity | a solvable sector of gravity |
| twistor theory | Penrose's programme |
| 2d solvable systems | integrability |
From an entirely different entrance — the soft limit of scattering amplitudes — out came a known infinite-dimensional algebra with a name and a representation theory. Bonus ③ wrote that "there is not enough content"; this is evidence in the opposite direction ── a sign that there is content.
The codimension is fixed. The meaning of the delta functions is understood. Strong algebraic evidence has appeared. So is it finished?
No. The central gap remains exactly as it was.
An independent definition of the boundary theory.
What makes AdS/CFT strong is that the boundary side, \(\mathcal{N}=4\) supersymmetric Yang–Mills, is a theory defined with no reference to gravity, so both sides can be computed separately and cross-checked.
Carrollian CFT has no such "independent definition" yet. No action, no lattice, no other construction.
That the object of the search has changed from "a weird 2d CFT" to "a 3d Carrollian CFT" is progress. The place we are looking has become the right one. But it has not been found.
In addition, let us list the technical holes that remain.
| what remains | status |
|---|---|
| massive particles | massless works cleanly, but with mass they live on the hyperboloid \(H^3\) in momentum space and need separate treatment. partial |
| loops and IR divergences | loop amplitudes are IR divergent, so a prescription for dressed states is needed. active |
| the central charge | there is a stress tensor, yet what the central charge is (or whether it is zero) is unknown |
| the flat limit of AdS | the conservative route of deriving it as \(\ell\to\infty\). the limit is singular and the CFT has not been identified |
| de Sitter | still the main prize. \(\Lambda>0\) is unsolved |
The diagnosis was wrong. Bonus ③'s distributional behaviour was not "because it is a weird CFT" but because the container is one dimension short. The Mellin transform was integrating the energy away, making the codimension 2 ── anomalous against AdS/CFT's 1.
Keep \(\omega\) as \(u\) and the boundary becomes three-dimensional. \(\mathscr{I}\cong\mathbb{R}_u\times S^2\). The codimension returns to 1. The celestial sphere was only a slice of it.
That three-dimensional boundary carries a geometry with zero speed of light. The \(u\) direction being null, the metric degenerates and becomes Carroll geometry. At \(c\to0\) the light cone collapses and you can go nowhere (the Red Queen). Hence the dynamics is ultra-local.
And BMS's identity came out. \(\text{BMS}_4\cong\) the 3d conformal Carroll group. That a supertranslation is "shift time independently in each direction" is ultra-locality itself. The infinite-dimensionality that surprised us in Episode 4 meant "one generator per point." And the delta function became a basic property rather than an anomaly.
Strong positive evidence has appeared too. The infinite tower of soft theorems organizes into \(w_{1+\infty}\) ── an algebra that had already shown up in area-preserving diffeomorphisms, self-dual gravity and twistors came out of an entirely different entrance. But only at tree level, in the self-dual sector.
Even so, the centre is still empty. There is no independent definition of the boundary theory. What advanced is that the place we are looking became the right one, not that anything was found.
This document is Episode 5 of the "Lattice We Build" series, a reading piece for high-school and university students who love physics. Where the sister series "That Clicks" explains known physics, this series shows the work itself.
Established material: that the metric on \(\mathscr{I}\) is degenerate; that the Galilei and Carroll groups are the \(c\to\infty\) / \(c\to0\) contractions of the Poincaré group; the naming of the Carroll group (Lévy-Leblond 1965); the isomorphism between the BMS group and the 3d conformal Carroll group; that Carrollian and celestial are related by an integral transform (Donnay–Fiorucci–Herfray–Ruzziconi 2022, and the Bridging papers 2022–23); that the Carrollian Ward identities reproduce the BMS flux-balance laws; and that the tower of soft gravitons forms the wedge subalgebra of \(w_{1+\infty}\) (Strominger 2021).
Carrollian holography is an ongoing research programme, and an independent construction of the boundary theory does not yet exist. The \(w_{1+\infty}\) results are established mainly at tree level in the self-dual sector; the extension to full gravity and to loops is unsolved. The figure is a schematic of how the light cone opens, and the contraction limits hold exactly only at the two ends of the slider.
Main series: Episode 1｜Episode 2｜Episode 3｜Episode 4 ｜ bonus: ① Could physics be written more simply?／② How far does "ct = constant" get you?／③ Not a CFT yet ── to print, use your browser's "Print" → "Save as PDF."
Print / PDF: ⌘+P (Ctrl+P on Windows). Swing the slider in the figure all the way left to see Carroll's country.