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

Abstract: In this article, we start by finding the equations of motion for the abelian D4 brane worldvolume. Assuming Hypothesis H that the 4-flux in M-Theory is flux quantized in a non-abelian cohomology theory called 4-cohomotopy, and the three-flux on the M5 brane worldvolume in (twisted) 3-cohomotopy, we compute 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 at the rational level, we conjecture a non-abelian relative cohomology theory for the D4 brane, fibered over the 10d Type IIA spacetime fluxes.