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?



Nice example of a topologically trivial bundle with nontrivial connection
I think that what you say is not complitely correct: the local anomaly is the curvature of the connection, which can be non-trivial wether the bundle is trivial or not. The global anomaly is the holonomy of the connection, which can be non-vanishing even for flat connections. In fact, considering the case of a line bundle, in order not to have an anomaly, you must be able to write a section as a function on the base-space, up to an overall constant. Therefore, if you have a trivial holonomy, you can choose a global parallel section (up to a constant), and any other section, divided by the flat one, gives a function. If the curvature is zero but not the holonomy, you can do this only locally, therefore there is a global anomaly but not a local one.


For the example you require, let us consider the trivial line bundle $X \times \mathbb{C}$: any global 1-form $A: TX \rightarrow \mathbb{R}$ gives a connection defined by $\nabla\_{X}V = \partial\_{X}V + A(X) \cdot V$. The curvature is $F = dA$, which is trivial in cohomology, but not as a single form in general. Therefore, if $A$ is not closed, you have a local anomaly even on the trivial bundle. If $dA = 0$, you can project it to a cohomology class in $H^{1}(X,\mathbb{R}/\mathbb{Z})$, and this class is the flat holonomy. For example, if $X = S^{1}$, for any global 1-form $A$ which is not integral, i.e. such that $\int\_{S^{1}}A \notin \mathbb{Z}$ ($A = \alpha dt$ with $\alpha \notin \mathbb{Z}$), you have a global anomaly but not a local one. This can happen not only on the trivial line bundle, but also on a line bundle with torsion first Chern class.


