Local Systems: The connection perspective

Welcome to the next installation of my series on local systems. In this post I’ll be talking about connections. This post should require less sophistication than the last few — no schemes, no functors — I’ll almost be coming at the subject afresh. There will be another post later, explaining how you might get to connections if you started out thinking about the infinitesimal site.

To start out with, let’s talk about derivatives; ordinary, single variable calculus derivatives. We have a function $f$ of a variable $x$. Then the derivative of $f$ is the function $f'(x) := \lim_{h \to 0} (f(x+h) - f(x))/h$. There are two directions in which we might want to generalize this idea. The first is to work with functions on a manifold, on a space which has no inherent coordinate system. This is the subject of your standard Calculus on Manifolds course, and I am going to assume that my readers are at least vaguely familiar with it. The second is to work, not with functions, but with sections of vector bundles. That’s our subject in this post.

So, let’s think about a vector bundle $V$ on the line $\mathbb{R}$, and let $\sigma$ be a section of $V$. If we want to define $\sigma'$, we need to subtract $\sigma(x+h)$ and $\sigma(x)$, two vectors which live in different fibers. To think of it another way, we need to distinguish between $f(x)$, the point in the fiber over $x$, and $f(x)$, the constant function which assigns the same value at every point. Suppose that, for any $v \in V_x$, we had a local section $c_v$ of $V$ with $c_v(x)=v$; we think of $c_v$ as a constant function. Then we could define $\sigma'(x) = \lim_{h \to 0} \left( \sigma(x+h) - c_{\sigma(x)}(x+h) \right)/h$.

A local system gives us the constant functions $c_v$. (Indeed, in definitions A.2 and B.6, we took a local system to be the constant functions, along with the data of certain maps between them.) Today, we will take the fundamental object to be the operation of derivation, and see how to build everything else from it.

The infinitesimal site

This is a follow up to my blogpost Local Systems: The Infinitesimal Perspective. In it, I want to get into some very category-theoretic ways of looking at the ideas in that post. The level is going to be a bit higher here than in the rest of the series. Before, I’ve tried to make sure that people could follow the main picture if they only had an intuitive idea of schemes and sheaves; here I am going to need people to actually be fully comfortable with them.

I won’t refer to this material in the future local systems posts. (Well, hardly ever!) But I hope you’ll read it, because I find it really mind bending.

Before going in, let me explain why you should care about this, even if you don’t enjoy category theory for its own sake. In the previous post, we introduced $N^{\infty}(X)$ and described local systems in terms of vector bundles on $N^{\infty}(X)$. In that post, we gave an explicit description of $N^{\infty}(X)$. In this post, I will explain how working with $N^{\infty}(X)$ is equivalent to working in the category of nilpotent thickenings of $X$.

When we move to characteristic $p$, it is the latter notion which will generalize. There are some very strange nilpotent thickenings in characteristic $p$. For example, it is important to include $\mathrm{Spec} \ \mathbb{Z}/p^n$ as a thickening of $\mathrm{Spec} \ \mathbb{Z}/p$. If you don’t, you will get the wrong answers! get a cohomology theory with coefficients in a characteristic $p$ ring. So, for example, if you try to compute the number of fixed points of an automorphism using the Lefschetz fixed point theorem, you will only be able to get the number of points modulo $p$. (Revised due to Matthew Emerton’s comments below.)

When $X$ is defined over a field of characteristic $p$, the correct analogue of $N^{\infty}$ is a $p$-adic object. I believe that one can build this object explicitly; it is related to the ring of Witt vectors. The details are very hard though (I haven’t mastered them myself) and so the categorical approach descried below becomes important.

A warning: this is not the only difficulty in characteristic $p$. There are also problems coming from divided powers; some of which we will discuss in later posts.

By the way, my original plan was just to write a post on this stuff. Wow, would that have been incomprehensible!

Local systems: the infinitesimal perspective

This is the next in my series of posts on different ways to think about local systems. This time, we will consider an approach where we only build isomorphisms along infinitesimally short paths. If you pursued this line of thought long enough — and thought very hard about finite characteristic issues — you would come to the definition of a crystal. But I don’t plan to go nearly that far; I’ll just give you the intuitions in characteristic zero.

I think this is probably the hardest of the three perspectives I want to explore, but it logically comes second. Things will become a bit easier again when we get to our third perspective, connections.

To try to make this a bit easier, I’ll start with a nonstandard presentation; then reboot and give the actual definitions.

Local systems: the path groupoid approach

This is the first of the series of posts I promised, on different ways of getting local systems.

In this section, we’ll explain the approach which leads to étale sheaves. I’ll start out by describing the analogous ideas in the topological setting; and then sketch how to make them fully algebraic.

I’ve realized that I need a word for the data which I use to obtain a local system. Because I’m feeling uncreative, I’ll call it the input. Again, $X$ is a space of some sort on which we want to build a local system.

Geometry and triply graded knot homology

So, I thought I would actually get back to blogging my research (by which I mean, shameless self-promotion) a bit. Probably the problem that I’ve focused the most on in the past few years is how to understand knot homology geometrically. While this still has a lot of mysteries, Geordie Williamson and I are finishing up a pair of papers that I think are a big step forward in this area.

The one I’d like to talk about in this post is “A geometric construction of colored HOMFLYPT homology.” (It’s not on the arXiv yet, but we’re close. Comments would be helpful, hint, hint). I’m mostly just going to talk about the consequences of this paper for the triply graded homology of Khovanov-Rozansky, though I think one of its most exciting features is how easily it generalizes to the colored situation.

In my last post, I described a complex of sheaves $C_\beta$ on the group $\mathrm{GL}(n)$ for each braid $\beta$ of index n. I mentioned at the end of that post that there’s a certain functor which sends this sheaf to the complex of Soergel bimodules Khovanov used to define HOMFLY knot homology. This suggests that somehow, this knot homology can be extracted in a direct geometric way from $C_\beta$

And that’s exactly what we do; Continue reading

Woit on Geometric Representation Theory

Just wanted to point out to everyone that Peter Woit, of the blog Not Even Wrong is doing a great job blogging on the relations between representation theory of Lie groups, functions on Lie groups and differential operators. And he promises there will be physics before the end!

Hypertoric varieties and Koszul duality

So, on Wednesday, I gave a talk with the above title at IAS, about work in progress with Tom Braden, Tony Licata, and Nick Proudfoot.  I was hoping to get David Nadler to blog it for me, but he was *ahem* indisposed.  Failing that, I’ll direct you all to David Ben-Zvi’s notes (warning: freaking huge PDF).  Hopefully, that will whet your appetite for the forthcoming paper.

Tannakian construction of the fundamental group and Kapranov’s fundamental Lie algebra

This post is a report on a talk that Mikhail Kapranov gave in Berkeley a few weeks ago on the “fundamental Lie algebra” — it is a Lie algebra associated to a manifold and a point on the manifold much like the fundamental group. Before telling about what Kapranov said, I’d like to start with some background.

Let $M$ be a manifold and $x \in M$. Suppose we consider pairs $\mathcal{E} = (E, \nabla)$ where $E$ is a vector bundle and $\nabla$ is a flat connection on $E$. Two such pairs $\mathcal{E}, \mathcal{E'}$ may be tensored together to produce a new vector bundle with flat connection $\mathcal{E}\otimes \mathcal{E'}$.

Define a group $G$ by the following procedure. An element $g \in G$ is a collection $(g_\mathcal{E})$ of linear isomorphisms $g_{\mathcal{E}} : E_x \rightarrow E_x$ for each $\mathcal{E}$ which are natural with respect to maps $\mathcal{E} \rightarrow \mathcal{E'}$ and which obey the rule $g_{\mathcal{E} \otimes \mathcal{E'}} = g_{\mathcal{E}} \otimes g_{\mathcal{E'}}$.

In other words, an element of $G$ is a linear automorphism of the fibre at $x$ of any vector bundle with flat connection. So, what is $G$?

Pre-Talbot seminar

John Francis and I are organizing a seminar at MIT to prepare for the Talbot workshop at the end of March. The aim of the seminar is to help people understand Gaitsgory’s preprint about quantum geometric Satake and quantum Langlands, which should play a significant role in the workshop. As it happens, Joel posted about this very result last August.

I gave the first talk on Wednesday, giving an overview of the subject, and I just put some notes up on the seminar page. I tried to gloss over as many technical details as possible, because of the time constraints, but I’d like to hear about any actual errors, significant or not. The talk ended up bearing some structural similarity to Joel’s post (pure coincidence – although I said a bit more about tori). Speaker recruitment went perhaps a little too successfully, since I forgot that one of the weeks was spring break.

Addendum: If you’re a fan of Koszul duality, the notes from the Chicago talk (on the seminar page) have a sketch of a proof of the equivalence between factorizable sheaves and quantum group representations using a rather odd manifestation of Koszul duality.

Dispatches from the Conference on Gauge and Representation Theory

So, I got back from Scotland and my family Thanksgiving (mmm, turkey) just in time for an enormous conference here in Princeton on Gauge and Representation Theory. I don’t really have the energy at the moment for some full blown conference-blogging, but I thought I would mention what was going on for the benefit of those who couldn’t attend.

(EDIT: For those of you who find me too vague, David Ben-Zvi has posted some notes from the talks here.)

I’m afraid I can’t really comment on the physics talks. While my level of total lostness varied during them, the average was high enough that I have nothing intelligent to say. Peanut gallery?

On the subject of talks I did understand some of…. Continue reading