TITLE: How to draw a complex line bundle?
QUESTION [7 upvotes]: 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.

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?

REPLY [4 votes]: 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.