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The most basic example of a topologically non-trivial real line bundle is the well-known Möbius strip. Everyone who learns about vector bundles will be confronted by it, if only because it has the distinguished advantage that we can draw a picture of it.

enter image description here

I would like to draw pictures of other line bundles, too. In particular, I have a complex line bundle which I would like to visualize somehow. How do I do that?

To be more specific:

  • The base manifold is the torus $M = S^1\times S^1$. It should be fine to visualize it as a rectangle, though.
  • The complex line bundle has structure group $U(1)$.
  • It is given as a direct summand of the trivial bundle $M \times L^2(\mathbb R^3)$. In other words, it is embedded in an infinite dimensional Hilbert space bundle. In particular, there is an induced connection coming from the hermitian form (scalar product).
  • (The bundle arises from an analysis of the Quantum Hall Effect.)

My questions:

1) Are there any example drawings of complex line bundles?

I imagine that one attaches a plane to every point of the base manifold, but it is not clear how to me how to arrange them such that one obtains a qualitative picture of the fact that they represent complex numbers.

2) Is there a minimal dimension $N$ such that every complex line bundle can be embedded into $\mathbb R^N$ in a suitable fashion?

It is probably the case that $N \geq 4$, so this won't be of much use, but it might still shed some insight on the problem, in particular because we are also given a connection.

3a) Any ideas of how one might go about drawing a complex line bundle?

3b) Any ideas on how to best visualize the connection coming from a hermitian form?

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+1 for relevant to my interests. –  AnonymousCoward Feb 18 '11 at 17:51
I have no idea of what «a qualitative picture of the fact that they represent complex numbers» is! –  Mariano Suárez-Alvarez Feb 18 '11 at 18:35
As far as I know, the standard way of visualizing this sort of thing is via sparse pointwise illustrations of the structure - for instance, the Wikipedia picture of surface normals ( en.wikipedia.org/wiki/File:Surface_normal.png ) shows the simplest example of this sort of visualization. –  Steven Stadnicki Feb 18 '11 at 18:53
@Steven: How does that apply to non-trivial vector bundles? I mean, the normal bundle of a sphere is trivial, hence easy to draw. In contrast, trying to draw a Moebius strip on a 2D piece of paper via sparse pointwise illustration seems like a hopeless task? –  Greg Graviton Feb 18 '11 at 20:11
@Greg: The version of the Moebius strip I've seen basically draws a local bundle (in the Mobius case, a line) and then parallel transports it around the strip to show the non-trivial nature. Of course, this mostly works because there's an almost-trivial holonomy involved; you can show the global nature at one point. In your case, if the connection isn't essentially flat that could be a severe complication... :/ –  Steven Stadnicki Feb 19 '11 at 0:05
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2 Answers

In any case... try making pictures of the bundle of unit circles associated to your complex line bundle, which has one dimension less. If the base space is the torus $S^1\times S^1$, then the total space of the circle bundle can be constructed by identifying faces of a cube, and that is something one can make pictures of.

If you described the actual bundle of which you want a picture, then someone could come up with something sensible. The general problem of «drawing complex line bundles» is pretty hopeless.

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The line bundle arises as a collection of 1D eigenspaces $X_k$ of the Hamilton operator $H(k) = \frac1{2m}(-i\hbar\partial_x+\hbar k_x+By)^2 + (-i\hbar\partial_y+\hbar k_y)^2$ in case that helps. :-P I'm happy to do the calculations myself, it's just that seeing some (literal) prior art would greatly simplify the task of figuring out what to draw. The thing is that I want to somehow "draw" the connection, too, but I'm not even sure that I can choose a global gauge (first Chern number is non-zero). What does a bundle of circles look like that doesn't have any globally non-zero sections?? –  Greg Graviton Feb 18 '11 at 20:03
@Greg: that does not tell me anything. Unless you can provide some concrete imnformation about how the bundle is put together, you are not going to be able to make any picture at all... –  Mariano Suárez-Alvarez Feb 18 '11 at 20:31
You mean a system of local trivializations with corresponding transition functions? I haven't made the effort of calculating one yet. (Which could be a good idea, thanks.) A priori, my line bundle is given by a family of projections, very much like the tangent bundle to the sphere $TS^2 \subseteq S^2\times\mathbb R^3$ is given by the family of projections $P_x v = v - \langle x,v\rangle x$ where $x \in S^2, v \in \mathbb{R}^3$. –  Greg Graviton Feb 18 '11 at 21:56
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Apparently, Mario Serna has produced pictures of $U(1)$-bundles on his webpage and in his paper "Riemannian Gauge Theory and Charge Quantization". Here an example

The image represents a trivial $\mathbb R^3$ over some rectangular base manifold. The $U(1)$ bundle which we want to visualize is shown as an $\mathbb R^2$-sub-bundle: the disks indicate the 2-dimensional fibers at each point, to be understood as subspaces of small 3-dimensional boxes at each point (not shown).

It seems that the disks are also meant to give an impression of the connection, but I don't fully understand how parallel transport is supposed to work here.

He cites a result by Narasimhan and Ramanan which says that every $U(1)$ bundle can be embedded into a trivial $(2d+1)$-dimensional complex vector bundle where $d = \text{dim} M$ is the dimension of the base manifold. Fortunately, the dimension is lower in the cases drawn.

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