Drop one dimension and last episode's conjecture was a theorem ── the template for the proof, and the three things missing in four dimensions
Following this series' discipline #6, Episode 9's sentence was labelled "a candidate, not a solution." So how could that candidate become a solution? Is there a route to a proof, or no idea at all?
That is a big difference. When Episode 8 sorted the unsolved items into three kinds, putting something in the "essential" box meant we do not even know what is missing.
Looking again, that was not so. The template exists, in three dimensions it has already succeeded, and what is missing in four dimensions can be named: three things. And one of those three was something Episode 8 had already written down as unsolved.
In 1998, Strominger derived the entropy of the three-dimensional black hole (BTZ), \(1/4\) included. He used only three steps.
Here is the decisive point. Not a single microstate is identified.
It never says "inside the black hole there are \(e^{A/4G}\) states of such-and-such a kind." Without knowing what was counted, the correct number comes out.
How is that possible? Because Cardy's formula fixes the asymptotic form of the density of states from symmetry alone.
"The state count comes out of symmetry alone" is hard to believe. Let us do it in a case we can check.
Take the simplest 2d CFT ── a single free boson (\(c=1\)). The number of states at energy level \(n\) in this theory is the partition number \(p(n)\) of the integer \(n\): the number of ways to split \(n\) into a sum of positive integers.
\(p(4)=5\)： \(4\)、\(3{+}1\)、\(2{+}2\)、\(2{+}1{+}1\)、\(1{+}1{+}1{+}1\)
this can be computed exactly by a recurrence
\(p(100)=190{,}569{,}292\). Since every last one can be enumerated, we can check it against Cardy's prediction.
Cardy's formula, meanwhile, looks at nothing inside the theory and uses only \(c=1\) ──
$$S=\log p(n)\;\simeq\;2\pi\sqrt{\frac{c\,n}{6}}\qquad(c=1)$$The counted answer, and the answer from symmetry alone. Do they agree?
With the leading term alone the ratio at \(n=100\) is 0.74 ── off. That is to be expected, since Cardy is an asymptotic formula. But include the logarithmic correction (Hardy–Ramanujan) and ──
n log p(n) Cardy leading ratio Cardy+log corr. ratio 50 12.227 18.138 0.674 12.290 0.99484 100 19.066 25.651 0.743 19.110 0.99766 500 49.187 57.357 0.858 49.207 0.99960 1000 72.258 81.116 0.891 72.272 0.99981 2000 105.168 114.715 0.917 105.178 0.99991
0.99991. The answer obtained by counting one by one and the formula obtained from the single number \(c=1\) agree to that extent.
This is what Strominger did. For BTZ he took \(c=3\ell/2G\) from the asymptotic symmetry and put it into Cardy. Without ever looking inside the black hole.
The mechanism is modular invariance.
Put a 2d CFT on a torus and the partition function becomes a function of the torus's shape \(\tau\). But a torus has the symmetry that exchanging its two cycles gives the same shape ── \(\tau\to-1/\tau\).
And that exchange swaps high and low temperature.
On the low-temperature side the partition function is fixed by the ground state alone ── and its energy is \(-c/24\), i.e. the central charge, one number.
Modular invariance translates that to the high-temperature side. So the high-energy density of states is fixed by \(c\) alone.
"Carry the easiest end (the ground state) to the hardest end (high excitation) using symmetry" ── that is what Cardy is. Which is why you need not know the contents.
And three premises are hidden here that will matter later. To say "fixed by the ground state alone," the spectrum must be discrete, must have a gap, and must be unitary. We come back to this in §06.
This was the biggest find of the episode.
Strominger's BTZ is about AdS. So what about flat spacetime? The asymptotic symmetry of 3d asymptotically flat spacetime is BMS₃ ── as Episode 5 showed, the conformal Carroll algebra.
The steps were the same.
| step | in 3d flat spacetime |
|---|---|
| asymptotic symmetry | BMS₃ (= 2d conformal Carroll) |
| central charge | from the contraction of the AdS₃ algebra, \(c_M=3/G\) (the Barnich–Compère central charge) |
| Cardy-type formula | BMS-Cardy: \(S=2\pi L_0\sqrt{c_M/24M_0}\) (in the \(c_M=3/G\) convention; positive control run in Episode 11) |
| result | reproduces the horizon entropy of 3d flat cosmology |
Barnich (2012), and Bagchi–Detournay–Fareghbal–Simón. The 3d version of the sentence written as a conjecture in Episode 9 is a theorem.
"A Carrollian theory living on a null surface of area \(A\) has \(e^{A/4G}\) states"
── this has already been shown in three dimensions.
The standing of the conjecture changes. Not "no idea at all," but "the raised-dimension version of a statement verified one dimension down."
Decompose the three-dimensional success and what four dimensions needs falls straight out.
| what is needed | in 3d | in 4d |
|---|---|---|
| ① the central charge | \(c_M=3/G\) | unknown |
| ② an analogue of modular invariance | established with BMS₃ characters | unsolved (what corresponds on \(\mathbb{R}_u\times S^2\)) |
| ③ a Cardy-type density-of-states formula | BMS-Cardy exists | unsolved |
And about ①, we notice something important.
Here §01's "no microstate is identified" starts to matter.
Episode 8 diagnosed this route's central gap as "there is no independent definition of the boundary theory." No action, no constructive definition, no bootstrap, no identification.
But the Cardy route runs without knowing what the theory is. All it needs is the symmetry, the central charge and a modular analogue; the contents of the Hilbert space are not required.
Episode 8's "no definition" may not be the blocker for the entropy problem.
We thought the definition had to come first, but it may be the reverse ── the entropy may fall first.
And that is what actually happened. BTZ's entropy came out of Cardy in 1998. It was accepted while nobody knew what "the microstates of BTZ" were. In order of events, entropy fell first.
Let us collect the premise hidden at the end of §03.
Cardy's derivation is not unconditional. To say "the low-temperature side is fixed by the ground state alone," it assumes a discrete spectrum, a gap, and unitarity.
And what Episode 8 confirmed was exactly this.
| what Cardy assumes | in celestial / Carrollian CFT |
|---|---|
| the spectrum is discrete | continuous (principal series \(\Delta\in1+i\mathbb{R}\)) |
| there is a gap | being continuous, the ground state is not isolated |
| positivity from unitarity | does not hold in the usual form |
Exactly the same reasons bootstrap does not work. Cardy was among the tools Episode 8 described as having their premises collapse.
Three dimensions worked because an analogue of modular invariance was actually found for BMS₃. Whether the same thing stands for 4d Carrollian, nobody yet knows.
This series' discipline #2 was "take the positive control first." Following it, the order is this.
| step | what to do |
|---|---|
| 1 | Positive control. First reproduce the 3d result with our own tools. From BMS₃'s \(c_M=3/G\) and BMS-Cardy, get the horizon entropy of flat cosmology. If it does not agree here, the tool is broken |
| 2 | Compute the 4d central charge. The central extension of the 3d conformal Carroll algebra ── the counterpart of Brown–Henneaux's \(3\ell/2G\) and BMS₃'s \(3/G\) |
| 3 | Look for a modular analogue on \(\mathbb{R}_u\times S^2\). What can be said with the spectrum still continuous |
| 4 | Produce the density of states and check it against \(A/4G\) |
Step 1 matters most. The same discipline as Episode 1, where we struck \(E_h\) before heading into the unknown. Validate the machine where the answer is known, then take it where it is not.
Finally, let us make the target level explicit.
| method | what you get | example |
|---|---|---|
| Euclidean path integral | \(A/4G\) comes out, but you do not know what was counted | Gibbons–Hawking (1977) |
| Cardy-type | \(A/4G\) comes out. no microstates identified | BTZ (1998), BMS₃ (2012) |
| explicit enumeration | the states themselves are listed | Strominger–Vafa (1996) |
The realistic target for Episode 9's conjecture is the middle one. Since BTZ is accepted at that level, reaching the same level counts as a proof.
The template for the proof already exists. Asymptotic symmetry → central charge → a Cardy-type formula → \(A/4G\). The procedure Strominger applied to BTZ in 1998, and it identifies not a single microstate.
And Cardy really does work. The state count of a free boson (\(c=1\)) is the partition number \(p(n)\), which can be counted exhaustively. The counted \(\log p(2000)=105.168\) against the formula from \(c=1\) alone (with the log correction) \(105.178\) ── ratio 0.99991.
In three-dimensional flat spacetime it is already finished. BMS₃'s central charge \(c_M=3/G\) and the BMS-Cardy formula reproduce the horizon entropy of 3d flat cosmology (Barnich 2012 and others). The 3d version of Episode 9's conjecture is a theorem.
Three things are missing in four dimensions. ① the central charge ② an analogue of modular invariance ③ a Cardy-type density-of-states formula. And ① is what Episode 8 wrote down as "the celestial central charge is unknown" ── not a small hole but the proof's very first input.
The ordering may change. Since the Cardy route demands no definition of the theory, Episode 8's central gap — "there is no independent definition" — may not obstruct the entropy problem. For BTZ too, the entropy fell first while the microstates stayed unknown.
But the obstacle is in the same place. Cardy assumes a discrete spectrum, a gap and unitarity, and Carrollian CFT breaks all three ── exactly the reasons bootstrap fails. Three dimensions went through because a modular analogue was found for BMS₃; whether one stands in four dimensions is unsolved.
This document is Episode 10 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: the Brown–Henneaux central charge \(c=3\ell/2G\); Cardy's formula and its derivation from modular invariance; Strominger's derivation of the BTZ entropy (1998); that the state count of a free boson is the partition number, and the Hardy–Ramanujan asymptotics; the Barnich–Compère central charge \(c_M=3/G\) of BMS₃ and the BMS-Cardy formula, and the resulting reproduction of the horizon entropy of 3d flat cosmology (Barnich 2012, Bagchi–Detournay–Fareghbal–Simón).
On the other hand the 4d central charge, an analogue of modular invariance, and a Cardy-type density-of-states formula are all unsolved. Episode 9's conjecture is therefore not proved in this episode either. The outlook that "the entropy can fall first even with no definition" is also an inference from the 3d precedent, not a guarantee. The \(p(n)\) in the figure is computed exactly by recurrence in the browser, but only for \(n\) that double precision can handle.
Main series: Episode 1｜Episode 2｜Episode 3｜Episode 4｜Episode 5｜Episode 6｜Episode 7｜Episode 8｜Episode 9 ｜ bonus: ①／②／③ ── to print, use your browser's "Print" → "Save as PDF."
Print / PDF: ⌘+P (Ctrl+P on Windows). In the figure you can watch the enumeration and the formula converge.