  The M5-brane has been argued to potentially provide much-needed theoretical
underpinning for various non-perturbative phenomena in strongly-coupled quantum
systems and yet the primary non-perturbative effect already in its classical
on-shell formulation has received little attention: The flux-quantization of
the higher gauge field on its worldvolume.
  This problem appears subtle because of (1a.) the notorious self-duality of
the 3-form flux density in the small field limit, combined with (1b.) its
highly non-linear self-duality for strong fields, and (2a.) the twisting of its
Bianchi identity by the pullback of the 11d SuGra C-field flux density, which
(2b.) is subject to its own subtle flux quantization law on the ambient
spacetime.
  These subtleties call into question the tacit assumption that the M5 brane's
3-flux is quantized in ordinary cohomology and hence leaves open the rather
fundamental question of what the M5-brane's worldvolume higher gauge field
really is.
  Here we characterize the valid quantization laws of the M5-brane's 3-flux in
(non-abelian) generalized twisted cohomology. The key step is to pass to
super-spacetime and there to combine the character map with the
"super-embedding"-construction of the on-shell M5-brane fields, of which we
give a rigorous and streamlined re-rederivation.
  We show that one admissible quantization law of the 3-flux on M5-branes is by
4-Cohomotopy-twisted 3-Cohomotopy, as predicted by "Hypothesis H". Besides
quantizing the bulk fluxes (G4,G7) and the brane's H3-flux themselves, this law
also implies the (level-)quantization of the induced Page-charge/Hopf-WZ term
on the M5-brane, necessary for its action functional to be globally
well-defined.
  Finally, we demonstrate how with this flux quantization imposed, there
generically appear skyrmionic solitons on M5-branes and anyonic topological
quantum states on open M5-branes.
