## Integrable Systems

As part of my project to not let orals actually degrade my sanity beyond recovery, today we’re going to do some physics! Well, kind of. We’ll be citing a lot of physics, at the least, and using it to motivate things. After all, a lot of cool stuff is coming into algebraic geometry via physics these days, and the best way to understand how these things fit is to understand at least the basics of the physics.

We’ll start out with classical mechanics. Back when Joe blogged here, we had a couple of posts on this topic. We’re going to have a bit more of a down-to-Earth view on the physics, though we’ll head right back into the stratosphere when the math starts up in earnest. Let’s take some system of particles moving around in some space. These particles will all obey Newton’s equation, which says that $F=ma$, and in fact, if $U$ is a potential energy function, we have $F=-\frac{dU}{dx}$ where $x$ is the variable for where in the space we are. Now, define the number of degrees of freedom as the smallest number $n$ such that there are functions $q_1,\ldots,q_n$ which determine the positions of all the particles. That is, each particle’s position is a function of $q_1,\ldots,q_n$ and might explicitly depend on time.

At this point, we can just assume that we have $n$ coordinate functions for our particles, and so we also need $n$ momentum functions $p_1,\ldots,p_n$. To incorporate the data of the potential properly, we don’t want to think of these $q_1,\ldots,q_n,p_1,\ldots,p_n$ as being QUITE coordinates on $\mathbb{R}^{2n}$. Rather, they’re coordinate functions on a $2n$-dimensional manifold. Now, locally they give us coordinates on affine space, but globally they may not. The point is that our problem has handed us a $2n$-dimensional manifold.

Ok, so we’ve got a system, and if we pick a point, Newton’s equation gives a unique solution, which means that our manifold, let’s call it $M$, is covered by curves, with each point lying on a unique one. These curves represent possible motions of the system for all time. We’ll call any function which is constant along each of these curves of motion a conserved quantity. Some of these we should all be familiar with. For instance, total energy, linear momentum, angular momentum, and the like are all conserved quantities.

As a quick aside, to relate this to the stuff that Matt’s talking about and getting at, we have an algebra of all smooth functions $M\to \mathbb{R}$, and we’ve got an action of the Lie group $\mathbb{R}$ on $M$ by translation through time. This induces an action on the algebra of smooth functions, which is denoted $C^\infty(M)$. The conserved quantities are then $C^\infty(M)^{\mathbb{R}}$, the invariant functions under this action.

So what we care about, for now, is how many independent conserved quantities there are, and what that even means. To get at this, we need to define the Poisson Bracket of two functions in $C^\infty(M)$. If $F,G:M\to\mathbb{R}$ are smooth, we define the Poisson bracket to be $\{F,G\}=\sum_{i=1}^n \frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i}-\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i}$. Now, the Poisson bracket of two conserved quantities is another conserved quantity. So, we define a noncommutative algebra generated by a collection of conserved quantities $F_1,\ldots,F_k$ by first taking all polynomials in these functions, and then taking all the Poisson brackets in them. We say that $F_1,\ldots,F_k$ are independent if none of them are in the algebra generated by the other $k-1$.

Now we require that our system have no EXPLICIT time dependence. That is, the $q_i,p_i$ are all dependent on $t$, but there are no free floating $t$‘s around, they’re all hidden in the coordinates and momenta. We call such systems autonomous. So now, we’re ready to really get started. A physical system is integrable in the sense of Liouville if it has the same number of degrees of freedom as independent conserved quantities. So we have $n$ functions $F_1,\ldots,F_n$ which generate the algebra of conserved quantities, but no subset of them does.

Now we pick a bunch of real numbers, $a_1,\ldots,a_n$. The conditions $F_i=a_i$ just give us values of the conserved quantities. They also make this look a bit more like the algebraic geometry we’ve gotten used to, because the collection of paths that have these values are the common roots of the system of equations $F_i-a_i$. So we get an $n$-dimensional subspace, lets call it $A$, for reasons that will become clear later. Now, there’s no reason to assume that the $a_i$ can just be any real numbers, but they do have to be in an $n$-dimensional manifold.

The upshot? An integrable system gives us a map $M\to B$, where $M$ is our $2n$-dimensional phase space and $B$ is a base space of some sort, with fibers $A$. Ahh, but we can actually do even better. Our phase space comes equipped naturally with a symplectic structure. Why does this help? A theorem of Liouville’s tells us that, in the physical systems case (yes, I know I’m being vague about hypotheses, but I’m doing physics for the moment and motivation, when we get to the math, I’ll be more careful) the fibers are all Lagrangian. Better, they’re actually tori.

So now let’s make it better for us mathematicians. The real numbers are hopeless to work over, so let’s make all things complex. We have complex coordinates and momenta, complex conserved quantities, etc. So we have a map of complex manifolds $M\to B$ with domain $2n$-dimensional (over $\mathbb{C}$!) and codomain $n$-dimensional. There’s a symplectic structure on $M$ such that the fibers are Lagrangian, but now they’re also complex tori. That is, compact, complex Lie groups.

Now we’re ready for the math. We define an algebraically integrable system to be a proper flat morphism of complex varieties $M\to B$ with Lagrangian fibers which, generically, are abelian varieties (thus explaining why I called the fibers $A$). So, in this case, the conserved quantities must be polynomials in the coordinates and momenta. Now, algebraically integrable systems are the objects we’re going to look at for a little bit longer.

We can, in fact, specialize even further (and many do) to integrable systems that come from families of curves. What’s going on these is that we take a flat family of curves over $B$. That’s the same as a map $B\to M_g$ where $\mathcal{M}_g$ is the moduli space of curves. Strictly, we’re pretending that this is a fine moduli space at the moment, but things happen to work out ok in this setting. The next thing we do is we map $\mathcal{M}_g$ into $\mathcal{A}_g$, the moduli space of dimension $g$ abelian varieties by taking each curve to its Jacobian. Then, we pull back the universal family of abelian varieties to $B$ to get a flat family of abelian varieties. We can then choose a symplectic structure that makes them all Lagrangian, and things are nice, modulo quite a few technical points that I’ve glossed over.

The biggest technical issue is solved by defining what’s called a Seiberg-Witten differential. First, though, we need the Gauss-Manin Connection. The Gauss-Manin Connection is the map $\nabla^{GM}=1\otimes d_B:R^1f_*\mathbb{Z}\otimes\mathscr{O}_B\to R^1f_*\mathbb{Z}\otimes \Omega^1_B$, where $R^1f_*$ is the first higher direct image of $f$, the map from the total space of the family of curves to $B$, which is defined by taking an injective resolution of the locally constant sheaf $\mathbb{Z}$, applying the pushforward $f_*$ to each thing, and then taking the first cohomology of this new complex (more about higher direct images later). So now, with the Gauss-Manin connection, we define a Seiberg-Witten differential to be a differential $\lambda$ on the total space of the family of curves such that all of its derivatives with respect to $\nabla^{GM}$ are holomorphic and the map $T_bB\to H^0(C_b,K_{C_b})$ given by taking $X\mapsto \nabla^{GM}_X\lambda|_{C_b}$. Yeah, the definition is a mouthful, with lots of jargon, but it solves things:

Theorem: A family of curves $\Sigma\to B$ gives rise to an algebraically integrable system of their Jacobians if and only if it admits a Seiberg-Witten differential.

Finally, we take a curve $C$ of genus $g\geq 2$, along with a complex reductive group $G$. We’ll define a Higgs bundle to be first a vector bundle with structure group $G$. That is, we require that the transition functions be in $G$ rather than all of $GL(n)$. The second part of the definition of a Higgs bundle is what is called a Higgs field, $\Phi\in H^0(C,\mathfrak{g}\otimes K_C)$, where $\mathfrak{g}$ is the Lie algebra of $G$.

The point of this is that it gives us a map $\Phi:P\to P\otimes K_C$. So next, we look at the ring of $G$ invariant polynomials on $\mathfrak{g}$. This is finitely generated (say, due to the Hilbert Basis Theorem) by $p_1,\ldots,p_n$, where $n$ is the rank of $\mathfrak{g}$. These give a map from the cotangent bundle to the moduli space of stable bundles with structure group $G$ (stable is a technical term, don’t worry about it at the moment) to $B=\oplus_{k=1}^n H^0(C,K_C^{d_k})$ where $d_k$ is the degree of $p_k$. The map, in fact, looks like $(P,\Phi)\mapsto \oplus_k p_k(\Phi)$. We call this the Hitchin map.

Hitchin himself proved that this map is proper and has Lagrangian fibers with respect to the canonical symplectic form on cotangent bundles. It’s also quite plainly algebraic in nature, and so is an algebraically integrable system. We call this the Hitchin system. A theorem of Markman says that its special in the cases $G=GL(n),SL(n)$.

Now, we’ve reached the end of my actual knowledge, but I do know that the Hitchins system is extremely important in the study of the Geometric Langlands Conjecture, which has a lot to do with moduli spaces of bundles and the like.