Flux Quantization on M-Strings

TL;DR

This paper explores flux quantization in M-theory involving M-strings and M5-branes, revealing a refined nonabelian cohomology framework that captures complex topological and geometric structures in supergravity.

Contribution

It introduces a doubly-relative twisted Cohomotopy framework for flux quantization with M-string probes, extending previous models to include nested brane hierarchies and geometric engineering.

Findings

Flux quantization described by twisted Cohomotopy in M-theory.

Hierarchical probe brane configurations admit refined flux quantization.

Geometric models relate to Chern-insulator phases via M-string configurations.

Abstract

The electric Gauss law in 11D SuGra is famously non-linear, whence its flux quantization must be in nonabelian cohomology. We have previously shown that the minimal admissible choice is 4-Cohomotopy, which in the presence of magnetized M5-probes takes its relative twistorial form. Here we discuss how this situation is further refined in the presence of M-string probes on the M5-worldvolume. Based on the superspace formulation of 11D SuGra, we find the nested Bianchi identities by iterating the superembedding construction for super p-branes. The resulting probe brane hierarchy (M1 on magnetized M5 in 11D bulk) turns out to admit flux quantization in a doubly-relative form of twisted Cohomotopy, classified by the factorization of the quaternionic Hopf fibration through the twistor fibration. The further equivariant refinement of this cohomology theory reduces on A-type singularities to a form of relative 2-Cohomotopy which geometrically engineers Chern-insulator phases on $\mathrm{M5}\cap \mathrm{A}_n$, with the M-string playing the role of gapped nodal lines.