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

TL;DR

This paper explores the mathematical relationship between M5 branes and D4 branes through the lens of rational cohomotopy theory, providing a new perspective on flux quantization and brane dynamics in M-theory and Type IIA string theory.

Contribution

It introduces a novel non-abelian relative cohomology framework linking M5 brane fluxes to D4 brane fluxes via cyclification of rational 3-cohomotopy, advancing the mathematical understanding of brane relations.

Findings

Derived equations of motion and Bianchi identities for D4 brane.

Computed minimal model for cyclification of quaternionic Hopf fibration.

Proposed a conjecture for a non-abelian relative cohomology theory for D4 branes.

Abstract

In this article, we start by re-deriving the equations of motion and Bianchi identities for the abelian D4 brane worldvolume. Noting that the 4-flux in M-Theory is rationally flux quantized in a non-abelian cohomology theory called 4-cohomotopy, and the three-flux on the M5 brane worldvolume in rational (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.