The n-Category Café

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.

So now we’re ready for:

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.