M5 brane to D4 brane via cyclification of rational relative 3-cohomotopy
Pinak Banerjee
TL;DR
This work investigates how D4 brane worldvolume fluxes in Type IIA string theory can be encoded within a non-perturbative, topological framework built from M-theory cohomotopy. By deriving the D4 worldvolume equations of motion from the DBI and CS actions and performing circle reduction of the M5 brane, the authors connect these equations to the cyclification (loop-space quotients) of the quaternionic Hopf fibration, recovering the Type IIA flux Bianchi identities in a rational, relative cohomotopy setting. The key result is a consistent mapping between the D4 worldvolume Bianchi identities, including non-linear terms in the worldvolume flux, and the minimal model of the cyclified relative 3-cohomotopy, supporting a rational relative non-abelian cohomology for the D4 brane fibered over IIA fluxes. This framework clarifies the topological underpinnings of D4 flux quantization and suggests a path toward an integral lift, with the rational model anchored in LS^4//S^1 rather than KU_0//BU(1).
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.