[Intuition] Why does the total charge vanish in classical electrodynamics on compact spaces ?

Question

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 !

Comments

All this activity is meaningless. First, CED in our world is still ill-defined. CED on compact "spaces", is it well defined? Does it have physical solutions? What about experimental input? It is experiment who "dictates" the equations, not our "generalizations" or "extrapolations".

Even though the total charge is zero, it maybe distributed non uniformly due to other forces, so there may be strong fields within the "space".

So many open questions..., no intuition is possible.

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To partially address the first part of the post, for a closed manifold intuition suggests the total charge would be zero because continuity of the field implies any nonzero field is divergenceless. The article linked asserts that the total charge vanishes for a compact manifold, for which it is not clear to me how this can be easily seen.