A student of mine asked me why we are studying bundles in theoretical physics, and why it isn't enough to look at product spaces. I gave the example of the tangent bundle of the sphere, and of the magnetic monopole as a nontrivial circle bundle over the sphere. But as far as we know experimentally, in this one universe we don't live in a sphere and there are no magnetic monopoles. That's why she asked:
What is an example of a nontrivial bundle modeling an experimentally confirmed physical phenomenon?
I had to admit I don't know. Can anyone help me? The example can come from anywhere in physics, not just gauge theory.
(If this is better asked on the physics SE, feel free to move the question.)
Line bundles above the 2-torus are classified by the group $\mathbb{Z}$, i.e., there are non-trivial line-bundles over the 2-torus. Such line bundles (or more generally vector bundles) are used to model the topological aspects of the integer quantum Hall effect. For details see the TKNN paper.