Nice example of a topologically trivial bundle with nontrivial connection

Question

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?

Answer 1

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.

Answer 2

Let $M = {\mathbb R}^1$ be the time-line, with the trivialized bundle $M \times {\mathbb R}^1$ on it. There's a connection on it, called "inflation", such that parallel transport of $N$ at time $t$ to time $t'$ says how much $N$ dollars in year $t$ would be worth in year $t'$. This example comes from the frankly hilarious The Physics of Finance.