## May 12, 2016

### E8 as the Symmetries of a PDE

#### Posted by John Huerta

My friend Dennis The recently gave a new description of the Lie algebra of $\mathrm{E}_8$ (as well as all the other complex simple Lie algebras, except $\mathfrak{sl}(2,\mathbb{C})$) as the symmetries of a system of partial differential equations. Even better, when he writes down his PDE explicitly, the exceptional Jordan algebra makes an appearance, as we will see.

This is a story with deep roots: it goes back to two very different models for the Lie algebra of $\mathrm{G}_2$, one due to Cartan and one due to Engel, which were published back-to-back in 1893. Dennis figured out how these two results are connected, and then generalized the whole story to nearly every simple Lie algebra, including $\mathrm{E}_8$.

Let’s begin with that model of $\mathrm{G}_2$ due to Cartan: the Lie algebra $\mathfrak{g}_2$ is formed by the infinitesimal symmetries of the system of PDE $u_{x x} = \frac{1}{3} (u_{y y})^3, \quad u_{x y} = \frac{1}{2} (u_{y y})^2 .$ What does it mean to be an infintesimal symmetry of a PDE? To understand this, we need to see how PDE can be realized geometrically, using jet bundles.

A jet bundle over $\mathbb{C}^2$ is a bundle whose sections are given by holomorphic functions $u \colon \mathbb{C}^2 \to \mathbb{C}$ and their partials, up to some order. Since we have a 2nd order PDE, we need the 2nd jet bundle: $\begin{matrix} J^2(\mathbb{C}^2, \mathbb{C}) \\ \downarrow \\ \mathbb{C}^2 \end{matrix}$ This is actually the trivial bundle whose total space is $\mathbb{C}^8$, but we label the coordinates suggestively: $J^2(\mathbb{C}^2, \mathbb{C}) = \left\{ (x,y,u,u_x,u_y, u_{x x}, u_{x y}, u_{y y}) \in \mathbb{C}^8 \right\} .$ The bundle projection just picks out $(x,y)$.

For the moment, $u_x$, $u_y$ and so on are just the names of some extra coordinates and have nothing to do with derivatives. To relate them, we choose some distinguished 1-forms on $J^2$, called the contact 1-forms, spanned by holomorphic combinations of $\begin{array}{rcl} \theta_1 & = & d u - u_x d x - u_y d y, \\ \theta_2 & = & d u_x - u_{x x} d x - u_{x y} d y, \\ \theta_3 & = & d u_y - u_{x y} d x - u_{y y} d y . \end{array}$ These are chosen so that, if our suggestively named variables really were partials, these 1-forms would vanish.

For any holomorphic function $u \colon \mathbb{C}^2 \to \mathbb{C}$ we get a section $j^2 u$ of $J^2$, called the prolongation of $u$. It simply takes those variables that we named after the partial derivatives seriously, and gives us the actual partial derivatives of $u$ in those slots: $(j^2 u) (x,y) = (x, y, u(x,y), u_x(x,y), u_y(x,y), u_{x x}(x,y), u_{x y}(x,y), u_{y y}(x,y) ) .$ Conversely, an arbitrary section $s$ of $J^2$ is the prolongation of some $u$ if and only if it annihilates the contact 1-forms. Since contact 1-forms are spanned by $\theta_1$, $\theta_2$ and $\theta_3$, it suffices that: $s^\ast \theta_1 = 0, \quad s^\ast \theta_2 = 0, \quad s^\ast \theta_3 = 0 .$ Such sections are called holonomic. This correspondence between prolongations and holonomic sections is the key to thinking about jet bundles.

Our PDE $u_{x x} = \frac{1}{3} (u_{y y})^3, \quad u_{x y} = \frac{1}{2} (u_{y y})^2$ carves out a submanifold $S$ of $J^2$. Solutions correspond to local holonomic sections that land in $S$. In general, PDE give us submanifolds of jet spaces.

The external symmetries of our PDE are those diffeomorphisms of $J^2$ that send contact 1-forms to contact 1-forms and send $S$ to itself. The infinitesimal external symmetries are vector fields that preserve $S$ and the contact 1-forms. There are also things called internal symmetries, but I won’t need them here.

Amazing theorem 1. The infinitesimal external symmetries of our PDE is the Lie algebra $\mathfrak{g}_2$.

Like I said above, Dennis takes this amazing theorem of Cartan and connects it to an amazing theorem of Engel, and then generalizes the whole story to nearly all simple complex Lie algebras. Here’s Engel’s amazing theorem:

Amazing theorem 2. $\mathfrak{g}_2$ is the Lie algebra of infinitesimal contact transformations on a 5-dim contact manifold preserving a field of twisted cubic varieties.

This theorem lies at the heart of the story, so let me explain what it’s saying. First, it requires us to become acquainted with contact geometry, the odd-dimensional cousin of symplectic geometry. A contact manifold $M$ is a $(2n+1)$-dimensional manifold with a contact distribution $C$ on it. This is a smoothly-varying family of $2n$-dimensional subspaces $C_m$ of each tangent space $T_m M$, satisfying a certain nondegeneracy condition.

In Engel’s theorem, $M$ is 5-dimensional, so each $C_m$ is 4-dimensional. We can projectivize each $C_m$ to get a 3-dimensional projective space $\mathbb{P}(C_m)$ over each point. Our field of twisted cubic varieties is a curve in each of these projective spaces, the image of a cubic map: $\mathbb{C}\mathbb{P}^1 \to \mathbb{P}(C_m) .$ This gives us a curve $\mathcal{V}_m$ in each $\mathbb{P}(C_m)$, and taken together this is our field of twisted cubic varieties, $\mathcal{V}$. Engel gave explicit formulas for a contact structure on $\mathbb{C}^5$ with a twisted cubic field $\mathcal{V}$ whose symmetries are $\mathfrak{g}_2$, and you can find these formulas in Dennis’s paper.

How are these two theorems related? The secret is to go back to thinking about jet spaces, except this time, we’ll start with the 1st jet space: $J^1(\mathbb{C}^2, \mathbb{C}) = \left\{ (x, y, u, u_x, u_y) \in \mathbb{C}^5 \right\} .$ This comes equipped with a space of contact 1-forms, spanned by a single 1-form: $\theta = d u - u_x d x - u_y d y .$ And now we see where contact 1-forms get their name: this contact 1-form defines a contact structure on $J^1$, given by $C = \mathrm{ker}(\theta)$.

Many of you may know Darboux’s theorem in symplectic geometry, which says that any two symplectic manifolds of the same dimension look the same locally. In contact geometry, the analogue of Darboux’s theorem holds, and goes by the name of Pfaff’s theorem. By Pfaff’s theorem, there’s an open set in $J^1$ which is contactomorphic to an open set in $\mathbb{C}^5$ with Engel’s contact structure. And we can use this map to transfer our twisted cubic field $\mathcal{V}$ to $J^1$, or at least an open subset of it. This gives us a twisted cubic field on $J^1$, one that continues to have $\mathfrak{g}_2$ symmetry.

We are getting tantalizingly close to a PDE now. We have a jet space $J^1$, with some structure on it. We just lack a submanifold of that jet space. Our twisted cubic field $\mathcal{V}$ gives us a curve in each $\mathbb{P}(C_m)$, not in $J^1$ itself.

To these ingredients, add a bit of magic. Dennis found a natural construction that takes our twisted cubic field $\mathcal{V}$ and gives us a submanifold of a space that, at least locally, looks like $J^2(\mathbb{C}^2, \mathbb{C})$, and hence describes a PDE. This PDE is the $\mathrm{G}_2$ PDE.

It works like this. Our contact 1-form $\theta$ endows each $C_m$ with a symplectic structure, $d\theta_m$. Starting with our contact structure, $C$, this symplectic structure is only defined up to rescaling, because $C$ determines $\theta$ only up to rescaling. Nonetheless, it makes sense to look for subspaces of $C_m$ that are Lagrangian: subspaces of maximal dimension on which $d\theta_m$ vanishes. The space of all Lagrangian subspaces of $C_m$ is called the Lagrangian-Grassmannian, $\mathrm{LG}(C_m)$, and we can form a bundle $\begin{matrix} \mathrm{LG}(J^1) \\ \downarrow \\ J^1 \\ \end{matrix}$ whose fiber over each point is $LG(C_m)$. It turns out $LG(J^1)$ is locally the same as $J^2(\mathbb{C}^2, \mathbb{C})$, complete the with latter’s complement of contact 1-forms.

Dennis’s construction takes $\mathcal{V}$ and gives us a submanifold of $\mathrm{LG}(J^1)$, as follows. Remember, each $\mathcal{V}_m$ is a curve in $\mathbb{P}(C_m)$. The tangent space to a point $p \in \mathcal{V}_m$ is thus a line in the projective space $\mathbb{P}(C_m)$, and this corresponds to 2-dimensional subspace of the 4-dimensional contact space $C_m$. This subspace turns out to be Lagrangian! Thus, points $p$ of $\mathcal{V}_m$ give us points of $LG(C_m)$, and letting $m$ and $p$ vary, we get a submanifold of $LG(J^1)$. Locally, this is our PDE.

Dennis then generalizes this story to all simple Lie algebras besides $\mathfrak{sl}(2,\mathbb{C})$. For simple Lie groups other than those in the $A$ and $C$ series, there is a homogenous space with a natural contact structure that has a field of twisted varieties living on it, called the field of “sub-adjoint varieties”. The same construction that worked for $\mathrm{G}_2$ now gives PDE for these. The $A$ and $C$ cases take more care.

Better yet, Dennis builds on work of Landsberg and Manivel to get explicit descriptions of all these PDE in terms of cubic forms on Jordan algebras! Landsberg and Manivel describe the field of sub-adjoint varieties using these cubic forms. For $\mathrm{G}_2$, the Jordan algebra in question is the complex numbers $\mathbb{C}$ with the cubic form $\mathfrak{C}(t) = \frac{t^3}{3} .$

Given any Jordan algebra $W$ with a cubic form $\mathfrak{C}$ on it, first polarize $\mathfrak{C}$: $\mathfrak{C}(t) = \mathfrak{C}_{abc} t^a t^b t^c ,$ and then cook up a PDE for a function $u \colon \mathbb{C} \oplus W \to \mathbb{C} .$ as follows: $u_{00} = \mathfrak{C}_{abc} t^a t^b t^c, \quad u_{0a} = \frac{3}{2} \mathfrak{C}_{a b c} t^b t^c, \quad u_{a b} = 3 \mathfrak{C}_{a b c} t^c ,$ where $t \in W$, and I’ve used the indices $a$, $b$, and $c$ for coordiantes in $W$, 0 for the coordinate in $\mathbb{C}$. For $\mathrm{G}_2$, this gives us the PDE $u_{00} = \frac{t^3}{3}, \quad u_{01} = \frac{t^2}{2}, \quad u_{11} = t ,$ which is clearly equivalent to the PDE we wrote down earlier. Note that this PDE is determined entirely by the cubic form $\mathfrak{C}$ - the product on our Jordan algebra plays no role.

Now we’re ready for Dennis’s amazing theorem.

Amazing theorem 3. Let $W = \mathbb{C} \otimes \mathfrak{h}_3(\mathbb{O})$, the exceptional Jordan algebra, and $\mathfrak{C}$ be the cubic form on $W$ given by the determinant. Then the following PDE on $\mathbb{C} \oplus W$ $u_{00} = \mathfrak{C}_{abc} t^a t^b t^c, \quad u_{0a} = \frac{3}{2} \mathfrak{C}_{a b c} t^b t^c, \quad u_{a b} = 3 \mathfrak{C}_{a b c} t^c ,$ has external symmetry algebra $\mathfrak{e}_8$.

##### Acknowledgements

Thanks to Dennis The for explaining his work to me, and for his comments on drafts of this post.

Posted at May 12, 2016 6:30 PM UTC

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### Re: E8 as the Symmetries of a PDE

Cool!

In our work we studied the action of the split real form of $\mathrm{G}_2$ on a 5-dimensional manifold, namely the space of ways of getting a little movable ball to touch a fixed ball whose radius is three times as big. You wrote:

$\mathfrak{g}_2$ is the Lie algebra of infinitesimal contact transformations on a 5-dim contact manifold preserving a field of twisted cubic varieties.

Is this 5-dimensional manifold the same as ours, or almost the same?

I think it’s almost got to be. There could be some subtleties involving different real forms, covering spaces, etc. But there can’t be that many different 5-manifolds on which $\mathrm{G}_2$ acts that were studied by Engel!

I suspect that this contact structure would shed light on how our 5-dimensional manifold is a kind of ‘phase space’ for the physics of a ball rolling on a ball.

Posted by: John Baez on May 13, 2016 4:12 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

I will go a bit further and conjecture that Dennis The’s PDE for $\mathfrak{g}_2$ is related to the physics of the rolling ball problem.

Posted by: John Baez on May 13, 2016 4:14 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

There are actually two 5-dimensional manifolds closely tied to $\mathrm{G}_2$. One is the space of configurations of one ball rolling on another ball, and this comes to us with a rank 2 distribution that tells us how to roll without slipping or twisting. This distribution has infinitesimal $\mathrm{G}_2$ symmetry if and only if the ratio of radii is 1:3 or 3:1, as we discuss in our paper.

But that’s a rank 2 distribution, while a contact distribution is rank 4. This is not the contact manifold that Engel was talking about, but that contact manifold is lurking nearby. Consider a trajectory for our rolling ball system that rolls along a great circle at unit speed. The space of all such trajectories is also 5-dimensional, and it’s this space which carries the contact structure and the field of twisted cubic varieties.

I don’t instantly see what the contact structure or field of varieties correspond to in rolling ball language, but it would be fun to figure out.

Posted by: John Huerta on May 13, 2016 4:46 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

John H. wrote:

Consider a trajectory for our rolling ball system that rolls along a great circle at unit speed. The space of all such trajectories is also 5-dimensional, and it’s this space which carries the contact structure and the field of twisted cubic varieties.

Hmm! That’s the space of ‘lines’ in our rolling ball incidence geometry. I’d forgotten that was 5-dimensional, just like the space of points.

We were trying to geometrically quantize the space of points. All of a sudden it seems smarter to quantize the space of lines.

In the ‘covariant phase space’ approach to classical mechanics, a point is a particle’s trajectory, not its position and momentum at a fixed time. This sometimes gives a phase space isomorphic to the usual fixed-time approach, but it’s better because symmetries like time translation or Lorentz (or Galilean) transformations act in a geometrically obvious way.

It seems the space of trajectories you’re talking about is a bit like a covariant phase space. It could be a contact manifold rather than a symplectic manifold because we’re imposing the constraint that the ball moves at unit speed. Cutting one dimension off a symplectic manifold, or adding one extra dimension, often produces a contact manifold.

So, my new improved guess is that the contact structure on this 5-manifold is the natural contact structure on the covariant phase space of a ball rolling without slipping or twisting on a ball 3 times as big.

Here’s a small step toward checking this:

If you take any manifold $M$, its cotangent bundle $T^\ast M$ is a symplectic manifold with $\omega = d \theta$ where $\theta$ is the tautological 1-form on the cotangent bundle, also called the Liouville form. If $M$ is Riemannian, taking the union of the unit spheres in all its cotangent spaces gives the unit sphere bundle $S T^\ast M \subseteq T^* M$. And we restrict $\theta$ to the unit sphere bundle, it makes the unit sphere bundle into a contact manifold.

Thus, the phase space of positions and momenta for a particle moving at unit speed is always a contact manifold!

For more details, see the second and third examples here:

Of course there’s a lot more work to do, since we need to impose the ‘no slipping or twisting’ constraint, and also mod out by time translations, so that we just have a space of trajectories, not trajectories with a known position at $t = 0$.

Posted by: John Baez on May 13, 2016 6:40 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

The suggestion of interpreting the $G_2$ contact structure as the covariant phase space is a tantalizing one, and I think this should amount to interpreting suitably the so-double fibration relating the (cover of the) $3:1$ rolling distribution and the $G_2$ contact structure, suitably completed to be defined on a compact homogeneous space. This thus gives one partial answer to John H.’s question about interpreting the twisted cubic varieties in terms of the rolling distribution.

As both of you know, one way to build the $3:1$ rolling distribution (up to universal cover) is to observe that $G_2$ acts transitively on the cone of null lines in the imaginary split octonions $\mathbb{V}$. It thus acts transitively on the null quadric $\mathbb{Q}$ produced by projectivizing it, and the cross product on $\mathbb{V}$ determines algebraically a ($G_2$-invariant) $2$-plane distribution $D$ on the quadric $\mathbb{Q} \subset \mathbb{P}(\mathbb{V})$, which we may view as $G_2 / P_1$, where $P_1$ is the stabilizer in $G_2$ of a null line in $\mathbb{V}$. The subgroup $P_1$ is parabolic, which situates this geometry in the broader and well-studied class of “generalized flag varieties”.

Up to conjugacy (and topology), $G_2$ only has three such subgroups. The others are not hard to describe: The group $G_2$ acts on the space of totally isotropic $2$-planes in $\mathbb{V}$, but this actions has two orbits, according to whether the cross product restricts to zero on each $2$-plane. Bryant calls the planes for which the restriction is zero special, and the stabilizer $P_2$ of a special $2$-plane is also parabolic. Suggestively, the orbit of $\mathbb{N} = G_2 / P_2$ of special $2$-planes is $5$-dimensional, and the structure the $G_2$ algebra induces on $\mathbb{N}$ is precisely the $G_2$ contact structure! (We can view the variety $\mathbb{N}$ as a subvariety of $\mathbb{P}(\mathfrak{g}_2)$, by the way, but it turns out to be much topologically complicated than $\mathbb{Q}$ is.)

The third parabolic subgroup is just $P_1 \cap P_2$, which we can regard as the stabilizer of a pair $[L, E]$, where $L$ is a null line in $\mathbb{V}$ and $E \supset L$ is a special null plane in $\mathbb{V}$. The $6$-dimensional homogeneous space $\mathbb{I} = G_2 / (P_1 \cap P_2)$ of such pairs carries a somewhat more complicated structure, and it fibers naturally over each of the two homogeneous spaces, with $1$-dimensional fibers in each case: $G_2 / P_1 \leftarrow G_2 / (P_1 \cap P_2) \to G_2 / P_2 .$ In this setting we call $\mathbb{I}$ an correspondence space, and this point of view yields a sort of duality between the rolling distribution and G2 contact geometries:

For any point $n \in \mathbb{N}$, we can look at its $1$-dimensional preimage in $\mathbb{I}$ and push this down via the other projection to a line in $\mathbb{Q}$ that turns out to be tangent to the $2$-plane distribution (and hence corresponds to an admissible motion of the $3:1$ rolling ball system). The reverse construction takes just a little more explanation: One can read off essentially all of what I’ve said so far from the Dynkin diagram for $G_2$, and one can also divine this way that we can naturally identify the contact distribution with $S^3 H$ for some auxiliary, invariant rank-$2$ vector bundle $H \to \mathbb{N}$, and in particular, the twisted cubic variety at each point is just the variety of perfect cubes in $S^3 H$. Now, if we fix a point in $\mathbb{Q}$, pulling back to $\mathbb{I}$ and pushing down to $\mathbb{N}$ gives a line in $\mathbb{N}$ tangent everywhere to the cubic variety field.

We can conclude that if $C$ is a curve tangent everywhere to the $2$-plane distribution $D$ (i.e., an admissible motion of the rolling ball system), then we may interpret the projective tangent line to $C$ at each point as an element of $\mathbb{N}$ tangent to the cubic variety field, and as we move along $C$ this traces out a dual curve $C^*$ in $\mathbb{N}$. Likewise, a curve in $\mathbb{N}$ tangent to the cubic variety field determines a curve in $\mathbb{Q}$ tangent everywhere to the distribution, and these constructions are dual to one other. (This is the G2 analogue of the construction that maps a curve in the real projective plane to a curve in the dual real projective plane and also the other way around. In the language of correspondences determined by a semisimple group and two parabolic subgroups, this projective correspondence arises from the double fibration $SL(3, \mathbb{R}) / Q_1 \leftarrow SL(3, \mathbb{R}) / Q_{12} \to SL(3, \mathbb{R}) / Q_2$, where the $Q_{\bullet}$’s are appropriate parabolic subgroups.)

Most of what I’ve said and much more (including the projective correspondence just mentioned and an analogous correspondence for $SO(5, \mathbb{C}) \cong Sp(4, \mathbb{C})$) is explained in more detail in Bryant’s highly readable lecture notes Élie Cartan and Geometric Duality, and I’ve mostly aligned my notation here with his.

Posted by: Travis Willse on May 13, 2016 9:32 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

By the way: you mentioned jet bundles. I recently learned that these were discovered by Charles Ehresmann, the differential geometer and category theorist whose complete works have just been made available online.

Posted by: John Baez on May 13, 2016 4:17 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

The burning question to my mind is: why doesn’t the method work for $\mathfrak{sl}_2(\mathbb{C})$?

Posted by: David Roberts on May 14, 2016 2:36 AM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

$SL(2,\mathbb{C})$ sticks out like a sore thumb because its only quotient by a parabolic subgroup (the Borel subgroup, i.e. upper triangular matrices with determinant one) is $\mathbb{P}^1$, which is not a contact manifold. Every other (complex) simple Lie group $G$ admits a quotient $G/P$ (adjoint variety) that is a contact manifold, endowed with an additional ($G$-invariant) geometric structure on its ($G$-invariant) contact distribution. It’s this extra structure on a contact manifold that is the starting point for the constructions in the paper. So $SL(2,\mathbb{C})$ has to sit on the sidelines while the others have fun.

However (see Remark 3.6) in the paper. This says that if you look at the explicit formulas for the PDE symmetries that I obtain for all types except A and C (written in terms of generating functions), then these naturally specialise to the type A case by setting the cubic form to be trivial (and not distinguishing a 0-th coordinate anymore). Surprisingly, this all still works for $A_1 = \mathfrak{sl}_2$, where there are no $x^i$ or $u_i$ coordinates anymore. Namely, $\{ 1, 2u, u^2 \}$ is a standard $\mathfrak{sl}_2$-triple with respect to the Lagrange bracket (given in equation (3.6) in the paper).

Posted by: Dennis The on May 14, 2016 5:56 AM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

Thanks, Dennis! It occurs to me that at least at the level of maximal compacts, $SU(2)$ itself is a contact manifold :-) But that’s not a huge help…

Posted by: David Roberts on May 14, 2016 7:59 AM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

Some notes:

• The passage between the two 5-dimensional homogeneous spaces of $G_2$ appeared originally in Engel (1893) and in more detail in Cartan’s thesis (1894, p.149-151). And these days, the double fibration picture that Travis described is well-known and standard.

• Keizo Yamaguchi did not get a mention above, but his geometric theory of jet bundles was a crucial perspective. Moreover, in terms of homogeneous spaces, he generalised the $G_2$ case and identified the existence of such PDE. My approach is different to Yamaguchi’s, and the main point of my paper is how to make these PDE explicit by connecting to work of Landsberg and Manivel.

• If $B = P_1 \cap P_2$ (see Travis’ post), then $G_2 / B$ (6-manifold) is the $G_2$ PDE that John described above (Cartan’s, not mine) as a (local) homogeneous space. As John described above, there are three natural 1-forms on the tangent space that yield a geometric structure that is a rank 3 distribution $E$ with growth $(3,5,6)$. I use these numbers to indicate the growth of the weak derived flag, e.g. $E' = E + [E,E]$, $E'' = E + [E,E']$, etc. (i.e. always bracket with $E$ to get the next step). This PDE has $G_2$ external symmetry. However, its internal symmetry, i.e. symmetry of the distribution $E$ is infinite-dimensional: there is a symmetry lying inside $E$ (a “Cauchy characteristic”), so one can always reparametrize along its flow. (Quotienting by this gets down to the $(2,3,5)$-manifold.)

• However, there is a subtle point here: $G_2 / B$ carries another natural $G_2$-invariant geometric structure that is a $(2,3,4,5,6)$-distribution $D$. This is obtained via a tautological construction: realize $G_2 / B$ as a $P^1$-bundle over the $(2,3,5)$-manifold $G_2 / P_1$ by taking lines in the rank 2 distribution downstairs. (A point upstairs is a line downstairs, which I pullback by the differential of the projection map.) See more in Travis’ post.

• How are these two related? $E = D'$. For those familiar with root diagrams, I encourage you to stare at the coloured root diagrams in Figure 1 (pg.15) of my paper.

Posted by: Dennis The on May 14, 2016 6:28 AM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

Dennis, if you feel there are places in the post where I should be clearer about who did what, I’m open. Am I correct in understanding that the construction taking points in $\mathcal{V}$ to their “affine tangent spaces” was your idea?

Speaking of that construction, let’s adopt your notation and call the submanifold of $\mathrm{LG}(J^1)$ that you construct by taking affine tangent spaces $\hat{\mathcal{V}}$. Is $\hat{\mathcal{V}}$ the same as $\mathrm{G}_2/B$ only locally, or is there a more global sense in which this is true?

Posted by: John Huerta on May 16, 2016 7:31 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

Given a submanifold of a projective space, looking at affine tangent spaces is a natural thing to do. However, as far as I can tell, doing this for the twisted cubic to get the $G_2$ PDE does not appear in the Cartan / Engel 1893 papers, nor Cartan’s 1894 thesis, nor Yamaguchi’s $G_2$-paper from 1999. The subadjoint varieties (in particular, the twisted cubic) are known to be Legendrian, i.e. at any point, their affine tangent spaces are Lagrangian subspaces. I learned of this through work of Buczynski and Landsberg-Manivel. So for your first question, my contribution on this point was using the affine tangent space construction to write a PDE down explicitly (and arguing that the symmetry algebra isn’t any bigger than what one expects it to be).

For your second question, $\widehat{\mathcal{V}}$ in $LG(J^1)$ is indeed globally $G_2 / B$. Me referring to “local” in my post above refers to the fact that the PDE written as you did above is a local coordinate expression for the submanifold $\widehat{\mathcal{V}}$.

Relating to both of your questions: There is a passage in Sec.6.2 of Yamaguchi’s $G_2$-paper where he talks of lifting the $G_2$ action on the adjoint variety (i.e. $J^1 = G_2/P_2$) up to $LG(J^1)$ and obtaining an orbit decomposition, i.e. $LG(J^1) = O \cup R_1 \cup R_2$, where $O$ is the open orbit and $R_1$ and $R_2$ have codimensions 1 and 2 respectively. (Here, $R_2$ is the $G_2$ PDE you described above.) However, (i) thinking in terms of analogous orbit decompositions for the other simple Lie groups $G$ gets pretty hairy (I tried this before), (ii) even if you classify orbits abstractly, how do you describe them locally? (iii) for any given $G$-orbit, why can’t the symmetry be larger than $G$? Instead, thinking of the PDE as a naturally constructed object from additional data on $J^1$ led to the advance here.

Posted by: Dennis The on May 16, 2016 10:23 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

You write:

For your second question, $\widehat{\mathcal{V}}$ in $LG(J^1)$ is indeed globally $G_2 / B$.

I think you mean that, if we take the contact manifold $G_2/P_2$ with its field of subadjoint varieties $\mathcal{V}$, then $\hat{\mathcal{V}}$ is globally $G_2 / B$. What you wrote wasn’t quite correct because the manifold I call $J^1$ is $\mathbb{C}^5$, which is only the same as $G_2 / P_2$ locally, thanks to Pfaff’s theorem.

I still don’t quite understand this business with Cauchy characteristics. Why does it make the algebra of internal symmetries infinite dimensional? Is it because, given a Cauchy characteristic $X$, $f X$ is also a Cauchy characteristic for any smooth function $f$?

Is there a conceptual way to understand why the Cauchy characteristic appears that I’m missing here? I feel like if I put together your comment and Travis’s comment, I should see it, but I’m feeling lazy.

Posted by: John Huerta on May 18, 2016 3:45 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

Sorry, I abused some notation above. You are correct: $\widehat\mathcal{V}$ in the Lagrange-Grassmann bundle over $G_2 / P_2$ is globally the same as $G_2 / B$.

(Also, I noticed another typo I made above: $E$ has growth vector $(3,4,6)$ not $(3,5,6)$. Sorry, I wrote that too quickly.)

Regarding Cauchy characteristics:

• If $X$ is a CC for a distribution $E$, this means $X$ is in $E$ (i.e. is a section of $E$) and $[X,Y]$ is in $E$ for any $Y$ in $E$. Given this, $fX$ is also a CC.

• Conceptual explanation: Stare at the top-middle picture in Figure 1 of my paper. The vectors with green circles represent the rank 3 distribution. (What I labelled $E$ above, but $\mathcal{D}$ in the paper.) Amongst these, vector addition corresponds to taking a Lie bracket of corresponding vector fields in the distribution modulo the distribution itself. From this point of view, the vector that I’ve labelled as $Ch(\mathcal{D})$ corresponds to a CC.

• Note: The rank 2 distribution (described in my post above) on the 6-manifold $G_2 / B$ does not have any CC’s.

Posted by: Dennis The on May 18, 2016 5:07 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

With the talk of PDEs, prolongations and jets, I wonder if rendition of this material in the jet-comonad formalism would be illuminating (see, e.g., section 1.3.1.1, and also 1.1.1.2, 1.1.3.4, 5.3.8 of dcct).

Posted by: David Corfield on May 15, 2016 10:18 AM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

I just found that Hurwitz’s article proving that $e$ is transcendental is the next paper in the CR after Engel’s!

Posted by: David Roberts on May 16, 2016 11:55 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

For the more pedestrian of us, can somebody explicitly write down some of the (7?) vectors in $TJ^2$ (the tangent bundle to the 2nd jet bundle) in the basis $(\partial_x, \partial_y, \partial_u, \partial_{u_x}, \partial_{u_y}, \partial_{u_xx}, \partial_{u_xy}, \partial_{u_yy})$ which preserve both the contact structure and the solution submanifold $S$?

Posted by: Leo Stein on May 19, 2016 10:09 PM | Permalink | Reply to this

### Re: E8 as the Symmetries of a PDE

Hi Leo,

By Backlund’s theorem, all contact symmetries of $J^2$ are prolongations of contact symmetries of $J^1$. Any of the latter correspond to a function $f$ (called a “generating function”) on $J^1$ via a standard formula - see (3.5) in my paper. (The Lie bracket of vector fields transfers to the so-called “Lagrange bracket” on functions via (3.6).)

For the “equation submanifold” $\mathcal{E} \subset J^2$ corresponding to the pair of PDE $u_{xx} = \frac{u_{yy}^3}{3}, u_{xy} = \frac{u_{yy}^2}{2}$, the symmetry algebra is isomorphic to the Lie algebra of $G_2$ (14-dimensional). Some examples:

$1, x, y, u_1, u_2, 2 u - x u_1 - y u_2$

correspond to the contact symmetries:

$\partial_u, x\partial_u + \partial_{u_1}, y\partial_u + \partial_{u_2}, -\partial_x, -\partial_y, x\partial_x + y\partial_y + 2u\partial_u + u_1\partial_{u_1} + u_2\partial_{u_2}$.

These prolong to $J^2$ to have trivial action on the $u_{11}, u_{12}, u_{22}$ coordinates. (So these are obvious symmetries of $\mathcal{E}$.)

Another is $xu_2 + \frac{y^2}{2}$, with corresponding vector field on $J^2$:

$-x\partial_y+\frac{y^2}{2}\partial_u + u_2\partial_{u_1} + y\partial_{u_2} + 2 u_{12}\partial_{u_{11}} + u_{22} \partial_{u_{12}} + \partial_{u_{22}}$.

The most complicated symmetry is

$u(u - x u_1 - y u_2) - \frac{1}{6} y^3 u_1 + \frac{2}{9} (u_2)^3 x + \frac{1}{3} y^2 (u_2)^2$

Taking this together with the symmetries $x, y, u_1, u_2$ one can generate (using the Lagrange bracket) the entire 14-dim symmetry algebra of $\mathcal{E}$.

(In Maple’s DifferentialGeometry package, there’s a subpackage called JetCalculus. In it, the commands GeneratingFunctionToContactVector and Prolong allow you to get the corresponding vector fields explicitly. Then you can check the symmetry condition directly.)

These formulas are obtained by specializing those in my Table 6 to the $G_2$ case. Do this as follows:

• cubic form $\mathfrak{C}(t,t,t) = \frac{t^3}{3}$ and dual cubic $\mathfrak{C}^*(t,t,t) = \frac{4t^3}{9}$. (See Appendix B.)

• indices: $i = 0,1$, while $a=1$. (The “$0$“-th coordinate is distinguished for me, so I start there.)

One selling point for looking at geometric considerations is that to obtain these symmetries, one doesn’t actually have to work on $J^2$, i.e. one can avoid the cumbersome prolongation formula. Instead, one can do the entire symmetry computation on $J^1$ by looking at symmetries of the twisted cubic field.

Posted by: Dennis The on May 21, 2016 5:06 PM | Permalink | Reply to this

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