It came out although nobody aimed at it ── by this series' criterion, the strongest kind of evidence
In Episode 1 we brute-forced 160,000-odd dimensionless quantities and got nothing. Through Episode 5 we kept writing that holography for flat spacetime is missing its centre. This series publishes little but negative results.
There is a reason for that. The discipline learned in Episode 1 ── a search that aims at coefficients comes out structurally empty. What you fitted is not evidence.
Which is exactly why, when the opposite happens, it should be handled with care. Something nobody was looking for that came out on its own. That is what this episode is about.
Episode 4 dealt with Weinberg's soft graviton theorem. Add one graviton to an amplitude and take \(q\to0\) and a \(1/q\) pole appears ── that was only the leading term.
Continue the expansion and it is a tower.
$$\mathcal{M}_{n+1}=\underbrace{\frac{S^{(0)}}{q}\,\mathcal{M}_n}_{\text{supertranslations}}+\underbrace{S^{(1)}\,\mathcal{M}_n}_{\text{superrotations}}+\underbrace{q\,S^{(2)}\,\mathcal{M}_n}_{?}+\underbrace{q^2 S^{(3)}\,\mathcal{M}_n}_{?}+\cdots$$Each term corresponds to the Ward identity of a different symmetry. The supertranslations (\(q^{-1}\)) and superrotations (\(q^{0}\)) seen in Episode 4 were only the first two rungs of an infinite tower.
The question is whether this tower forms a closed algebra. When you take the commutator of two rungs, do you land back inside the tower?
Strominger's (2021) result is this.
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 moreover the towers of soft photons and gluons transform irreducibly under it.
It closed. And not only gravity ── the soft sectors of gauge fields fit into representations of the same algebra.
The obvious question follows ── what is \(w_{1+\infty}\)?
A forbidding name, but the content is startlingly plain. It is the set of all operations that deform a two-dimensional surface without changing its area.
Here is how to build it. Pick a function \(f(x,y)\) on the plane. Think of it as a "Hamiltonian" and generate the flow.
$$\dot x=\frac{\partial f}{\partial y},\qquad \dot y=-\frac{\partial f}{\partial x}$$This is Hamiltonian mechanics itself, and by Liouville's theorem the area is conserved. Change \(f\) and you get a different deformation. There are as many deformations as there are functions \(f\) ── hence infinite-dimensional.
And the commutator of these deformations is the Poisson bracket.
$$\{f,g\}=\frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}$$Let us check that it closes. With monomials it is one line.
Put in \(f=x^{a}y^{b}\) and \(g=x^{c}y^{d}\).
$$\{x^{a}y^{b},\,x^{c}y^{d}\}=\bigl(ad-bc\bigr)\;x^{a+c-1}\,y^{b+d-1}$$the right-hand side is a monomial too
It closes. The structure constant is \(ad-bc\) ── a single 2×2 determinant. Relabel the indices and this takes the form of the standard \(w_\infty\) commutation relation \([w^p_m,w^q_n]=(m(q-1)-n(p-1))\,w^{p+q-2}_{m+n}\).
So \(w_\infty\) is nothing more than functions on the plane with a Poisson bracket. If you can differentiate, you can derive the structure constants yourself.
Move the figure to see what "preserving area" means.
Whichever you choose, the change in area stays within rounding error ── even though the shape is distorted past recognition.
Incidentally all three have exact solutions ── for \(f=xy\), \(x=x_0e^{t},\;y=y_0e^{-t}\); for \(f=x^2/2\), \(x=x_0,\;y=y_0-x_0t\); for \(f=x^3/3\), \(x=x_0,\;y=y_0-x_0^2t\). It is not numerical-integration error: the area really is preserved exactly.
This is the subject of the episode. \(w_{1+\infty}\) is not an algebra built for soft theorems. It had appeared earlier, in entirely different places.
| where it appeared earlier | context |
|---|---|
| area-preserving diffeomorphisms | pure geometry. what we built in §03 |
| self-dual gravity | the symmetry of the solvable sector of the Einstein equations (the Plebański equation) |
| twistor theory | Penrose's non-linear graviton construction: building self-dual gravity as a deformation of twistor space |
| 2d integrable systems | an algebra that recurs in systems with infinitely many conserved quantities |
| and now: soft theorems | the \(q\to0\) limit of 4d gravitational scattering amplitudes |
Only the fifth is an entirely different entrance from the other four. From a place that looks unrelated to geometry or twistors — the infrared limit of scattering amplitudes — the same algebra came out.
In fact the connection is understood. Self-dual gravity is integrable, and its symmetry was \(w_\infty\). And ──
The soft and collinear limits of gravity are effectively looking at the self-dual sector.
So \(w_{1+\infty}\), the symmetry of self-dual gravity, appears as the tower of soft theorems.
So infrared physics and twistor theory were pointing at the same thing through this algebra. Separate entrances led to the same room.
Let us hold this against the series' criterion.
In Episode 1 we brute-forced 160,000 dimensionless quantities and got nothing. In Bonus ③ we wrote that celestial CFT lacks content. The shared lesson was this.
A search that aims at coefficients comes out structurally empty.
The moment you say "hitting any one of \(p/q\) will do," hitting carries no information. Add targets and you only raise the coverage.
\(w_{1+\infty}\) is the exact opposite.
| Episode 1's brute force | \(w_{1+\infty}\) | |
|---|---|---|
| the target was | chosen afterwards (\(p/q\), \(\pi^k\)) | already there beforehand (area preservation, integrable systems) |
| freedom | add targets and you hit | the algebra closes or it does not: binary |
| look-elsewhere | applies (coverage = hit rate) | does not apply |
| kind | fitting | discovery |
In Episode 2 we treated anomaly cancellation as an example of "making a constraint the target." This is the same structure ── a target that is decided by a yes/no produces no look-elsewhere effect.
Whether the algebra closes leaves no room for choice. If it closed, there is structure there.
To avoid overvaluing it, let us state the limits plainly.
① It is tree level. Loop corrections deform the algebra. Work on "deformed soft algebras" is progressing, but what it becomes after deformation is unsettled.
② It is the self-dual sector. Restricted to positive helicity, and self-dual gravity is only a solvable special part of full gravity. The extension to the full theory with both helicities is unsolved.
③ It is a symmetry, not a construction of a theory. The distance between "we know the symmetry" and "we can build the dual theory" is still considerable. The independent definition of the boundary theory left empty in Episode 5 is not filled by this result.
In this series' language ── this is strong circumstantial evidence, not a proof. Keeping Episode 1's discipline #6, "never write 'solved'," that is the accurate way to put it.
Even so, for a series that has lined up nothing but negative results, having one thing that came out although we were not aiming at it is significant. It is a sign that the place we are looking is right.
The soft theorem has an infinite tower. The supertranslations (\(q^{-1}\)) and superrotations (\(q^0\)) treated in Episode 4 were only the first two rungs. And the tower forms a closed algebra ── the wedge subalgebra of \(w_{1+\infty}\) (Strominger 2021). The towers of soft photons and gluons transform irreducibly under it.
The content of \(w_\infty\) is plain. Just functions on the plane with a Poisson bracket ── the set of all area-preserving deformations. Computed on monomials, the structure constant is a single 2×2 determinant \(ad-bc\), and closure can be checked in one line.
And this algebra existed beforehand. Area-preserving diffeomorphisms, self-dual gravity (the Plebański equation), twistor theory, 2d integrable systems. The same algebra came out of an entirely different entrance, the infrared limit of scattering amplitudes. Not a coincidence: the soft and collinear limits of gravity are looking at the self-dual sector.
Why this is strong evidence for this series. Episode 1's brute force allowed the target to be chosen afterwards, so it was eaten by look-elsewhere. Whether an algebra closes is binary, and \(w_{1+\infty}\) had been defined earlier for another purpose ── discovery, not fitting. The same structure as anomaly cancellation in Episode 2.
But still not a proof. Tree level, the self-dual sector, and "we know the symmetry" rather than "we built the theory." The centre left empty in Episode 5 is still empty.
This document is Episode 6 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 existence of the infinite soft expansion; that the tower of positive-helicity soft gravitons organizes into the wedge subalgebra of \(w_{1+\infty}\) (Strominger, PRL 127, 221601, 2021); that \(w_\infty\) is the algebra of area-preserving diffeomorphisms; the monomial Poisson-bracket formula; and the integrability and twistor description of self-dual gravity (Penrose's non-linear graviton). This result is established mainly at tree level in the positive-helicity (self-dual) sector, and loop corrections deform the algebra. The extension to full gravity is unsolved. Also, identifying a symmetry is not constructing a dual theory, and an independent definition of the boundary theory still does not exist. The three flows in the figure use exact solutions, so area preservation is exact rather than a numerical approximation (the Jacobian is identically 1). The displayed area is computed by polygonal approximation, so a constant discretization difference remains.
Main series: Episode 1｜Episode 2｜Episode 3｜Episode 4｜Episode 5 ｜ 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 confirm that the area does not change even as the shape distorts.