I would like to ask why the existence of a non-trivial p-cycle leads to a non-trivial flux. I would say that e.g. for a five-form $F_{(5)}$ field strength , the flux is: $$\int\limits_{\mathcal{C}^{5}}F_{(5)} $$ so in general: $$\int\limits_{\mathcal{C}^{p}}F_{(p)} $$
Is this correct? And why the geometry should be a p-cycle? Couldn't it be some other topology?
A p-cycle is a differential form that lives in $ker(\partial_p)$ for the differential $\partial_p$ (in grading $p$), and such a form is nontrivial if it is not in the image of $\partial_{p+1}$. Mathematically we can see this as a cycle that is not the boundary of anything, picture a circle around a torus that bounds no area on the torus. If one has a boundary we can have the Stokes' rule that $$ \int_{\partial M}\omega = \int_M d\omega. $$ This is seen in Gauss' law. For a cocycle we then have a form $\omega \ne d\xi$, it is not the result of a coboundary, but where it is closed with $d\omega = 0$. Physically this means the field content of fields is not due to another field. This has some bearing of gauge invariance with ${\bf A}\rightarrow {\bf A} + {\bf d}\xi$ is such that $d^2\xi = 0$ gives gauge invariance. This is a topological form of a similar thing, and is seen in BRST quantization.
The argument for p-cycles is that fields are not due to special conditions on a boundary, but are purely topological. This removes the need for auxiliary conditions in the theory.
