M5 brane to D4 brane via cyclification of rational relative 3-cohomotopy

A bstract In this article, we start by reviewing the equations of motion and Bianchi identities for the abelian D4 brane worldvolume. Taking into account Hypothesis H that the 4-flux in M-Theory is flux quantized in a non-abelian cohomology theory called unstable 4-cohomotopy, and the three-flux on the M5 brane worldvolume in relative 3-cohomotopy or the quaternionic Hopf fibration, we immediately obtain the minimal model for the cyclification of the quaternionic Hopf fibration which encodes the Bianchi identities for the fluxes on the D4 brane worldvolume after double dimensional reduction. The two pictures can be mapped to each other, and thus, we derive from the implications of Hypothesis H, a non-abelian relative cohomology theory for the D4 brane, fibered over the 10d Type IIA spacetime fluxes. The point of this note is to highlight the power of non-abelian cohomology which easily captures the D4 worldvolume Bianchis as a relative cohomology fibered over the Type IIA spacetime fluxes, which is not known in the existing literature (to the best of our knowledge) if we take the Type IIA fluxes to be in twisted stable K theory.