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?
Best Answer
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.