[codethief]

> but even in the absence of matter I think there is a U(1) gauge symmetry present

I don't think there is. The local U(1) gauge symmetry really comes from the (complex-valued) Dirac field and its coupling to the photon field (i.e. A). In classic electrodynamics you can add the 4-gradient of any function to the 4-potential A without changing the equations of motion, so the space of valid gauge transformations is infinite-dimensional. (Which is not that interesting – given that you can't measure the potential A –, so all those degrees of freedom are non-physical.)

> I agree that the A_\mu transform as the coefficients of a differential form (A is a connection after all), but the notation elides the fact that the A_\mu are defined only on U.

That's a very good point indeed! Though I think the authors simply interpreted A as a regular vector field on the manifold, meaning that it is defined everywhere. But following your train of thought for a moment: Could you solve this issue through a partition-of-unity argument? I.e. cover the manifold with neighborhoods where you have local gauges and then construct the metric locally as a finite sum of those A_{\mu,s} (s being the gauge).

The issue of a metric constructed this way not necessarily being a Lorentz metric (let alone non-degenerate) of course nonwithstanding. Then again, the authors didn't worry too much about this, either… :)