まあ「U(1)-gauge theory」という事で。Abelian gauge theory. The phase
factor of the vector potential is the case. Therefore, if you are
curious, you can take a look at the action integral which involves
all those F_{\mu ,\nu}'s. It is straightforward to see that those
physical quantities are gauge invariant. The curvature 2-form as a form
could depend on the choice of the local coordinate. However,
the cohomology class which represents the differential 2-form doesn't
depend on the choice of the non-intrinsic informations like local
coordinates and the differential structures. Its a topological invariant
which is nothing but the 2nd Chern class.

--neko--