Consider classical electrodynamics on a spacetime of the type $\mathbb{R} \times S$ where $S$ is a 3-dimensional **closed** manifold. In the article 'K-theory in Quantum Field Theory' (see the statement after equation 3.6), Daniel Freed shows that in such circumstance, the total charge vanishes.

I understand the proof given there, but don't have a Physical intuition for it. Can somebody illuminate this better by providing some intuition ?

Also can this result be taken as an indication that the space slices of our Universe are compact, since we see as much positive charge around as negative charge ? I understand that the total charge *can* vanish even if $S$ is non-compact, but still if we somehow can show that space slices of our universe are compact, we would have an explanation of why the Universe is charge neutral.

Two more related questions :

Does this result hold in quantum field theory too ? And is there an analogous result for non-abelian gauge theories ?

Thanks !