In this question, I'd like to go over the physics - math dictionary occurring in the geometric structure (Principal bundle/spin bundles etc.) of Maxwell electrodynamics and the Dirac field. Consider the action for classical electrodynamics on Minkowski spacetime $(M,\eta)$: $$S_{QED} = \int_{M} \big[ -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \psi^{\dagger} \gamma^{0}(i\gamma^{\mu} D_{\mu} - m) \psi \big] dV_{\eta}$$ where the $\gamma$ symbols are the Dirac matrices, $D_\mu = \partial_\mu - ieA_\mu$ is a so-called gauge covariant derivative in which $A_\mu$ is the electromagnetic four potential. (I will italicize physics terminology that we will translate).
The above theory is a $U(1)$-gauge theory, i.e. a geometrical physical theory formulated using a principal bundle - thus one which must "respect" the mathematical redundancy of describing a principal bundle (its isomorphism class). This is analogous to the fact that isometric Lorentzian manifolds (if there are fields, we pull these back naturally too) describe the same spacetime (and fields). Namely that $$S_{EH}[(M,g)] = S_{EH}[(N,h)]$$ when $(M,g)$ is isometric to $(N,h)$. Anyway here is the structure which I so far see regarding QED before quantization:
We have a principal $U(1)$ bundle over $\mathbf{R}^4$ (so necessarily trivial $\equiv$ there exists a global smooth section) on which there is an principal $U(1)$ connection $\omega$. Now a trivialization of this bundle $(E,\pi,M)$ over a neighboorhood $\mathcal{U} \subset M$ is called a local gauge. A local gauge transformation is a change in this trivialization and hence equivalently the related section (the local smooth section equivalent to the trivialization, which can also be called a local gauge or mathematically, a trivializing section). Anyway, it is a standard fact that such local gauge transformations are in one to one correspondence with bundle automorphisms of $\pi^{-1}(\mathcal{U})$.
The $A_\mu$ above is the pullback of the principal $U(1)$ connection to the trivializing neighborhood by some trivializing section, so $s^{*}\omega$. As in the above discussion, if one wants to do a local gauge transformation, one needs only change the local trivializing section. So we replace $s$ by the section $s^g$ where pointwise it is defined by $s^g = s \cdot g$ where $g: \mathcal{U} \to E$ is a smooth function (all local sections are given this way). From here we get the famous formula $$(s^g)^{*}\omega = Ad(g)^{-1} s^{*} \omega + g^{-1} dg$$. This is the transformation formula for the connection one forms.
Next, recall the Faraday tensor is simply the pullback of the curvature of $\omega$.
Question one Given this set up, how do we show geometrically that the Faraday tensor term is "gauge invariant".
Question two Considering the Dirac term is geometrically formulated in terms of a (associate) spin bundle and all that machinery, how does the gauge transformation act on that end? Also what is the gauge covariant derivative geometrically?