[nilaykumar]

I'm not too familiar with the physics side of things, so maybe I'm misunderstanding the notion of gauge symmetry, but even in the absence of matter I think there is a U(1) gauge symmetry present. The homogeneous Maxwell's equations are expressed purely in terms of the curvature F of the connection, if I remember correctly. So you might add a flat U(1)-connection to A without changing the equations of motion.

As for the math: the key point is that the metric is globally defined on the spacetime manifold M. 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. They depend, in particular, on the choice of local gauge s:U \to P. So the question about covariance (or globalization or coordinate-independence or whatever you want to call it) of both sides of the equation g_{\mu\nu}=A_\mu A\nu very much involves the question of local gauge transformations.

Either the principal bundle is assumed to be trivial, in which case there exists a global gauge and the connection A can be identified with a global 1-form on M (connections form an affine space modeled on such 1-forms; a gauge effectively converts this affine space to a vector space by choosing an origin), or we need to check that the right-hand side A_\mu A_\nu is indeed a symmetric two-tensor on M. The latter is not clear to me.

I admit I haven't looked at the paper carefully, and physicists typically don't approach things in such an explicitly mathematical way. So perhaps there's some (physical?) justification for why that equation typechecks. I don't quite see it though.

[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… :)