I want to check whether my understanding about flux is correct.

So let me consider a $d$-dimensional spacetime manifold $M$ and a $(p+2)$-form flux (or field strength) $F$. Then there are two natural quantity we can define for flux: $$ \int_CF \qquad\text{and}\qquad \int_{C'}*F$$ where $*$ means Hodge dual. Here $C$ is $(p+2)$-cycle and $C'$ is $(d-p-2)$-cycle inside $M$. Also suppose I want to get a non-vanishing value of the above quantities.

Then does it correct to interpret as follows?

  • Flux with sources: Because of sources, we have defects in spacetime manifold $M$. So, in general, we can't deform a cycle $C'$ freely into a point, because of defects. That means we always have a non-vanishing value of the above integration.

  • Flux without sources: Even if we don't have defects in $M$, there may exist a non-trivial cycle (which belongs to $(d-p-2)$th homology class) $C'$ inside $M$. So if we integrate $*F$ over this non-trivial cycle $C'$, we can have non-vanishing value of the above integration.

The above is stated for an electric flux. For a magnetic flux we can apply the same logic.

One more somewhat related question: If we want to put a flux on a compact manifold supported by sources, does we always need to prepare a pair of sources? It seems like that without even number of defects, we can always deform a cycle into a point.

Many thanks for reading long question.


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