"The central charge is zero" ── but in three dimensions that one was zero as well
Last time we found the template for proving Episode 9's conjecture ── asymptotic symmetry → central charge → Cardy. And what is missing in four dimensions narrowed to three items. This episode goes after the leading one, the central charge.
What came out was, read plainly, a dead end. But decompose the three-dimensional success once more and we notice that the same dead end existed in three dimensions too. And the entropy still came out.
This episode reaches no conclusion. What it can deliver is putting the question into the right form. I think that is still worth writing ── what this series has confirmed again and again is that with the wrong way of looking, nothing comes out.
The literature says two things about the four-dimensional central charge. Neither is welcome.
In constructions of extended BMS₄ (with shadow symmetry), the Virasoro central charge is reported to be \(c=0\).
Put \(c=0\) into Cardy's formula \(S=2\pi\sqrt{cL_0/6}\) and \(S=0\). The entropy vanishes.
And the central extension of BMS₄ is not a constant central charge in the first place. What Barnich treated was a centrally extended BMS₄ Lie algebroid ── a field-dependent structure, not "a single number" like BMS₃'s \(c_M=3/G\).
There is no number to put in, and what there is, is zero. Read plainly, the Cardy route is blocked in four dimensions.
Let us decompose last episode's success one level further.
The BMS algebra is a semidirect product ── superrotations (Virasoro) acting on supertranslations.
$$\text{BMS}=\underbrace{\text{superrotations}}_{L_n}\;\ltimes\;\underbrace{\text{supertranslations}}_{M_n}$$Being a semidirect product, there are two places a central term can enter.
there are two central charges
\(c_L\) enters the bracket of two Virasoros; \(c_M\) enters the bracket that straddles superrotations and supertranslations. The latter is called the Barnich–Compère central charge.
And here are the values for three-dimensional Einstein gravity.
| \(c_L\) (Virasoro side) | \(c_M\) (supertranslation side) | |
|---|---|---|
| BMS₃ (3d flat gravity) | \(0\) | \(3/G\) |
\(c_L=0\). In three dimensions too, the Virasoro-side central charge was zero.
And BMS-Cardy still worked. What made it work was the other side.
The Cardy-type formula for BMS₃ has both central charges in it.
In the literature this formula is sometimes written \(S=2\pi[\,c_L\sqrt{M_0/2c_M}+L_0\sqrt{c_M/2M_0}\,]\), with 2 in the denominator. That is the form in the convention where the central term is written bare as \(c\,m(m^2-1)\) (Bagchi et al., \(c_{LM}=1/4\)).
This article writes the algebra in §02 as \(\frac{c}{12}m^3\) and uses \(c_M=3/G\). In that convention \(c_{LL}=c_L/12,\;c_{LM}=c_M/12\), so on substitution both terms get denominator \(2\to24\).
Mix them and you are off by \(\sqrt{12}=2\sqrt3\approx3.46\). This article did mix them on first publication ── caught by the positive control below and corrected.
In Einstein gravity \(c_L=0\), so the first term drops entirely. Only the second remains ──
$$S=2\pi L_0\sqrt{\frac{c_M}{24\,M_0}}\qquad(c_L=0)$$The entropy is carried entirely by \(c_M\). Move the figure and check.
# check against the known answer for 3d flat space cosmology # metric ds^2 = 8GM du^2 - 2 du dr + 8GJ du dphi + r^2 dphi^2 [Barnich 1208.4371] # horizon r_C = sqrt(2 G J^2 / M) geometry side S = 2 pi r_C / 4G = pi|J| / sqrt(2GM) # field theory side c_L = 0, c_M = 3/G, M_0 = M, L_0 = J G M J | geometry S BMS-Cardy ratio ----------------------------------------------------------------- 10 3 | 2.1074444193 2.1074444193 1.000000000000000 100 17 | 3.7764504974 3.7764504974 1.000000000000000 1e+04 250 | 5.5536036727 5.5536036727 1.000000000000000 1e+06 3e+04 | 66.6432440724 66.6432440724 1.000000000000000 2.5 1.3 | 1.8264518301 1.8264518301 1.000000000000000 max relative deviation = 2.2e-16 → positive control PASSED (with denominator 2 instead of 24, the ratio lines up at 3.4641016151 = 2*sqrt(3) on every row)
This check is not decoration. If we intend to use the same template in four dimensions, we must first see the tool work correctly where the three-dimensional answer is known. And indeed one misplaced coefficient turned up here ── \(\sqrt{12}\) is not a size you notice by eye.
Press "Einstein gravity" and the grey bar (the \(c_L\) contribution) goes to zero, leaving only the blue bar (the \(c_M\) contribution). This is what actually produced the entropy in three dimensions.
Conversely, set \(c_M\) to zero and no matter how far you raise \(c_L\) ── the first term goes as \(1/\sqrt{c_M}\), so it diverges and the formula itself breaks. \(c_M\) is not optional decoration.
Let us reread §01's "bad news" in the light of §02–03.
| \(c_L\) (Virasoro side) | \(c_M\) analogue (supertranslation side) | |
|---|---|---|
| BMS₃ | \(0\) | \(3/G\) ← carried the entropy |
| BMS₄ | \(0\) (as reported) | ? |
"The four-dimensional Virasoro central charge is zero" was also true in three dimensions. What we took for bad news was in fact the same situation as the success case.
If so, what we should be asking changes.
what is the central charge of celestial CFT
does the supertranslation sector of BMS₄ have an analogue of \(c_M\)?
Item ②, "an analogue of modular invariance," also looks different on investigation.
The mechanism of 2d Cardy was the \(SL(2,\mathbb{Z})\) of the torus \(T^2\) (Episode 10, §03). Exchange the two cycles and high and low temperature swap.
But the four-dimensional boundary is \(\mathbb{R}_u\times S^2\). Wrap \(u\) thermally and you get \(S^1\times S^2\) ── and \(S^2\) has no partner cycle to exchange with. The torus structure itself is absent.
Route A is structurally blocked. But that does not mean no Cardy-type formula has been built in higher dimensions.
| route | mechanism | required input |
|---|---|---|
| A: modular invariance | the \(SL(2,\mathbb{Z})\) of the torus | the central charge (unusable on \(S^1\times S^2\)) |
| B: thermal effective action | shrink the thermal circle, KK reduce, derivative expansion | anomaly coefficients, Casimir energy |
Route B is how it is actually done in higher dimensions. In the limit where the thermal circle is much smaller than the other scales, reduce dimensionally, and the coefficients of the derivative expansion encode the CFT data. Di Pietro–Komargodski derived Cardy formulae for supersymmetric theories in \(d=4,6\) along this route.
And taking route B changes what ① means again ── what is needed is not the central charge but the coefficients of the thermal effective action.
| step | what to do |
|---|---|
| 0 | Positive control. Reproduce the 3d BMS-Cardy in the original papers' convention. From \(c_M=3/G\), get the horizon entropy of flat cosmology. If this does not agree, the tool is broken → done, passed (§03). Relative deviation \(2.2\times10^{-16}\). But it failed once, and one misplaced coefficient turned up |
| 1 | The branch point. Does the supertranslation sector of BMS₄ have a \(c_M\) analogue? A pure algebra question, with a binary answer → done. The answer is NO. \(\dim H^2(\mathfrak{bms}_4)=0\) ── computed in Episode 13 |
| 2a | if yes → route A' (a 4d version of BMS-Cardy). But a substitute for the modular structure is needed → closed |
| 2b | if no → route B. Build a Carrollian thermal effective action on \(S^1_\beta\times S^2\) |
| 3 | Produce the density of states and check against \(A/4G\) |
Step 1 is a binary target. After Episode 2 (anomaly cancellation), Episode 6 (an algebra closing) and Episode 8 (lifting to twistor space), this is the fourth ── a target settled by whether it is satisfied or not. The form this series has repeatedly said "is the only kind that carries information."
No conclusion was reached. Even so, I think the state of things moved.
| up to last time | after this episode | |
|---|---|---|
| the central-charge situation | "\(c=0\), so it is a dead end" | \(c_L=0\) in 3d too. possibly not a dead end |
| the form of the question | "what is the central charge" | "is there a central term in the supertranslation sector" (binary) |
| the relation among ①②③ | we thought they were three independent items | ② determines ①. the order was reversed |
| the outlook for ② | "there is no modular analogue" | route A is blocked, but route B (thermal effective action) exists |
There were two pieces of bad news. The Virasoro central charge of extended BMS₄ is reported to be \(c=0\), and the central extension of BMS₄ is not a constant at all but a field-dependent Lie algebroid. Read plainly, the Cardy route is blocked.
But \(c_L=0\) in three dimensions too. Being a semidirect product, the BMS algebra admits two central charges, and in 3d Einstein gravity \(c_L=0\), \(c_M=3/G\). The first term of BMS-Cardy drops, and the entropy was carried entirely by \(c_M\) (the supertranslation sector).
So the question rewrites itself. Not "what is the celestial central charge" but ── "does the supertranslation sector of BMS₄ have a \(c_M\) analogue?" A question settled by a yes or no: the fourth "binary target."
② rewrote itself too. \(S^1\times S^2\) has no torus \(SL(2,\mathbb{Z})\), so route A (modular) is structurally blocked. In its place there is route B (thermal effective action), in which case the required input is anomaly coefficients rather than a central charge.
And ①②③ were not independent. Until ② is decided, ① is not even determined as "the coefficient of what." Listing them "one at a time" last time had the order backwards. No conclusion was reached, but the form of the question changed.
This document is Episode 11 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 BMS algebra is a semidirect product of superrotations and supertranslations; that BMS₃ admits two central charges \(c_L, c_M\); that in 3d Einstein gravity \(c_L=0\) and \(c_M=3/G\) (Barnich–Compère); that the BMS-Cardy formula consists of two terms; that in constructions of extended BMS₄ (with shadow symmetry) the Virasoro central charge is reported as \(c=0\); Barnich's centrally extended BMS₄ Lie algebroid; and Di Pietro–Komargodski's Cardy formula and thermal-effective-action method for \(d=4,6\) supersymmetric theories. The explicit forms of the commutators and of the Cardy formula differ by convention across the literature.
On the other hand, the reading that "the search leans too far towards the superrotation sector" is this series' author's inference and not an established point. Whether the supertranslation sector of BMS₄ has a \(c_M\) analogue is unsolved, including the possibility that it does not exist. This episode does not prove Episode 9's conjecture; it goes only as far as re-placing the question. The figure is a schematic of the relative contributions of the two terms of the BMS-Cardy formula, in arbitrary units.
Main series: Episode 1｜Episode 2｜Episode 3｜Episode 4｜Episode 5｜Episode 6｜Episode 7｜Episode 8｜Episode 9｜Episode 10 ｜ bonus: ①／②／③ ── to print, use your browser's "Print" → "Save as PDF."
Print / PDF: ⌘+P (Ctrl+P on Windows). The figure lets you check which central charge carries the entropy.