Local Systems: The connection perspective

Local Systems: The connection perspective June 15, 2009

Posted by David Speyer in Algebraic Geometry, D-modules.
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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.

We start with a special case, and then build up to the whole. Let $V$ be a vector bundle on $\mathbb{R}$ . We’ll write $x$ for the coordinate on $\mathbb{R}$ . Of course, $V$ can be trivialized, but for the moment we don’t want to trivialize it. A connection on $V$ is a map $\nabla$ from sections of $V$ to sections of $V$ , such that

(1) For any two sections $\sigma$ and $\tau$ of $V$ , we have $\nabla(\sigma + \tau) = \nabla(\sigma) + \nabla(\tau)$ .

(2) For any section $\sigma$ of $V$ , and any scalar-valued function $f$ on $X$ , we have $\nabla(f \sigma) = (\partial f/\partial x) \sigma + f \nabla(\sigma)$ .

If we do trivialize $V$ , then the operation of taking the derivative with respect to $x$ obeys these axioms. The point is that these are axioms which hold without talking about any choice of a trivialization. If you’re an algebraist, the following might help you: a derivation is a map from an algebra (to something); a connection is a map from a module.

A section $\sigma$ of $V$ is locally constant if $\nabla(\sigma)=0$ . So this is how to go from connections to the more sheafy perspectives which are focused on locally constant sections.

I said early on that a local system is a vector bundle with isomorphisms between different fibers. How, in the setting of connections, do we build an isomorphism between one fiber of $V$ and another? Let $a$ and $b$ be two points of $\mathbb{R}$ and let $v_0$ be a point in the fiber $V_a$ . Solve the differential equation $\nabla(\sigma)=0$ , with initial condition $\sigma(x)=v_0$ . The isomorphism between $V_a$ and $V_b$ will send $v_0$ to $\sigma(v_0)$ . So, this is how to go from a connection to the path groupoid approach, when our as space is $\mathbb{R}$ . Later, when we work on more complicated spaces, we’ll have to keep track of the path along which we solve the differential equation, but everything else will look the same.

Let’s see this differential equation in coordinates. Choosing an arbitrary trivialization of $V$ in order to write things down, we have $\nabla(u) = \partial u/\partial x + A(x) \cdot u$ where $A(x)$ is an $n \times n$ matrix, varying with $x$ . (Exercise!) So we have to solve the differential equation $u' = - A u$ . If $n=1$ , this has the solution $u(t) = e^{-\int_a^{t} A(x) dx} \cdot v_0$ ; for larger $n$ , one usually cannot give a closed form solution.

I have now presented all the main ideas, in the case where our space is the real line. We will now move to the case of an arbitrary manifold $X$ . This will introduce two difficulties: we won’t have a natural coordinate system on $X$ , and there is a genuinely new phenomenon that happens when $X$ has dimension larger than one — the possibility of curvature.

Let’s start by addressing the lack of coordinates; this will just be a matter of careful notation. Let $D$ be a vector field on $X$ ; we will think of this as a derivation on the scalar-valued functions on $X$ . Then we want a way, $\nabla_D$ , of differentiating with respect to $D$ . This should obey

(1′) $\nabla_D(a\sigma + b\tau) = a\nabla_D(\sigma) + b\nabla_D(\tau)$ , where $a$ and $b$ are real constants and

(2′) $\nabla_D(f \sigma) = D(f) \sigma + f \nabla_D(\sigma)$ .

We also need a condition on how this depends on $D$ :

(0) $\nabla_{fD+gE}(\sigma) = f \nabla_D(\sigma) + g \nabla_E(\sigma)$ , where $f$ and $g$ are scalar valued functions.

You now have reached the definition of a connection. I always found it hard to remember the difference between (1′) and (0).
Why are the coefficients $a$ and $b$ just constants, while $f$ and $g$ get to be functions? Unfortunately, wordpress won’t let me do a hidden section inside a hidden section, so click on this asterisk to read the explanation*.

We won’t be interested in all connections, we will only be interested in the integrable ones. (Also known as flat, or zero-curvature.) This will be easiest to explain in coordinates. Let $x_1$ , $x_2$ , …, $x_d$ be coordinates on $X$ near $x$ . Then we expect $\nabla_{\partial/\partial x_i} \nabla_{\partial/\partial x_j} \sigma$ to equal $\nabla_{\partial/\partial x_j} \nabla_{\partial/\partial x_i} \sigma$ . After all, if we were doing honest differentiation of functions, we would have $\partial^2 f / \partial x_i \partial x_j = \partial^2 f / \partial x_j \partial x_i$ .
A connection is said to be integrable if we have $\nabla_{\partial/\partial x_j} \nabla_{\partial/\partial x_i} \sigma$ for some (equivalently any) set of coordinates. An equivalent condition is that $\nabla_D \nabla_E \sigma - \nabla_E \nabla_D \sigma = \nabla_{[D,E]} \sigma$ for any vector fields $D$ and $E$ .

It turns out that, if $\nabla$ is an integrable connection, then the differential equation $\nabla(\sigma)=0$ is uniquely solvable for any initial condition $\sigma(x) = v_0$ . These solutions give a local trivialization of our vector bundle, just like in the one dimensional case we discussed above.

If $\nabla$ is not integrable, then we can still solve $\latex \nabla (\sigma)=0$ along any path. But making even topologically trivial changes to the path may change the result. In other words, you get holonomy, not just monodromy.

Next up: some other ways to think about integrability, such as $D$ -modules, curvature and the deRham complex. Then, the relationship between connections and the infinitesimal perspective on local systems.

* Suppose we have a vector field $D$ , and a section $\sigma$ , and we want to compute the value of $\nabla_D(\sigma)$ at a point $x \in X$ . Then it is enough to know $D$ at $x$ . However it is not enough to know $\sigma$ at $x$ ; we have to know the first order variation of $\sigma$ .

This is part of a more general distinction that everyone should learn at some point. Suppose that $V$ and $W$ are two vector bundles on $X$ , and $\mathcal{V}$ and $\mathcal{W}$ are the sheaves of sections of $X$ . Suppose we have a map $\phi:\mathcal{V} \to \mathcal{W}$ . If $\phi(au+bv) = a \phi(u) + b \phi(v)$ for $a$ and $b$ constants, then $\phi$ is called a “map of sheaves” and the stalk of $\phi(v)$ will depend on the stalk of $v$ . But if $\phi(fu+gv) = f \phi(u) + g \phi(v)$ , then $\phi$ is called a “map of $\mathcal{O}_X$ -modules”, and the fiber of $\phi(v)$ will depend only on the fiber of $v$ . In other words, $\phi$ will be induced by a map of vector bundles $V \to W$ . So $\nabla$ has the first kind of linearity in $\sigma$ and the second (more local) kind in $D$ .

As another application of the above ideas, if $\nabla$ and $\nabla'$ are two connection on the same vector bundle $V$ , then $\nabla - \nabla'$ is a map of $\mathcal{O}_X$ -modules from $V$ to itself. Hey, I just solved on of your exercises for you! (New exercise: which one!)

Comments»

1. john mangual - June 19, 2009: Oh cool, so you can construct a bundle which is $S^1 \times \mathbb{C}$ but for which the constant functions are $\sigma_z(\theta) = e^{i k \theta} z$ for some $k \in \mathbb{R}$ . In fact, any connection of them form $A : S^1 \to SU(1)$ has to be homotopic to one of these, no? If we require there be no holonomy, then $k \in \mathbb{Z}$ . Are these classified by some kind of K-theory?
2. Scott Carnahan - June 19, 2009: John,
I think you meant to say that one can construct a trivial bundle on S^1 with fiber C, and can give a connection whose flat local sections have the exponential form (but the parameter k can be complex). SU(1) is a trivial group.

Since S^1 is one-dimensional, any connection on this bundle is flat, and the resulting monodromy representation is the same as one of the exponential form. The Riemann-Hilbert correspondence gives an equivalence between coherent sheaves with flat (aka integrable) connection and locally constant sheaves of vector spaces, so these connections are classified by representations of the fundamental group into automorphisms of the fiber. In the one-dimensional case, these are parametrized by homomorphisms from Z into $\mathbb{C}^\times$ , i.e., elements of $\mathbb{C}^\times$ .
3. john mangual - June 19, 2009: Yeah, I meant $U(1)$ . Since the base space, $S^1$ is 1-dimensional, we can’t have two independent vector fields so any connection is flat. The monodromy representation of $\pi_1(S^1) = \mathbb{Z}$ is determined by the image of the generator and can be any member $\mathbb{C}$ .
Actually, we can’t have 0 because $\int_0^{x_1} A(x) dx$ would have to be $-\infty$ .

My point was that by choosing $A(x) = A$ , the resulting constant sections are $u(x) = e^{ Ax } u_0$ , we can have constant sections that wind around the origin in $\mathbb{C}$ by choosing $A$ to be purely imaginary. Then we can look at the set of connections whose monodromy representation is trivial. These are classified by the number of times they wind around. Could I even say there is a map from the space of connections to the fundamental group of the orthogonal group of the fiber?

I would love to see a post on the Riemann-Hilbert correspondence.
4. john mangual - June 19, 2009: “formula does not parse” is just the integral get when solving for the constant sections. it’s the exponential of something finite so it can’t be zero.

I fixed it: you were missing curly braces on the exponent –NS
5. A semana nos arXivs… « Ars Physica - July 1, 2009: [...] Local Systems: The connection perspective [...]

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