So, I've been trying to understand what exactly an anomaly is, and how they arise in physics. Apparently an anomalous theory is some theory whose action is given by a section of some bundle (rather than a function). Hence, only if the bundle is topologically trivial, thus allowing one to write the action as a function, can we then integrate the action over the moduli space; "giving" the quantum theory. Now, there is a paper by Freed "Determinants, Torsion and Strings" where he calls this a global anomaly (perhaps first coined by Witten, not sure), and goes on to say that there are also local anomalies due to the fact that a bundle can be topologically trivial without having a nontrivial connection. So, I have a question:

(1) What's a nice (nontrivial) example of a trivial bundle with nontrivial connection?