Flux quantization in string and M-theory constrains the allowed field configurations on branes. The constraint is topological: the flux through a cycle must be an integer (or more generally, must take values in a specific cohomology group). Which cohomology theory applies depends on the brane and the background geometry.
Banerjee, Sati, and Schreiber show that flux quantization on M-theory strings — M2-branes ending on M5-branes — lives in doubly-relative twisted Cohomotopy. This is not ordinary cohomology. Cohomotopy classifies maps to spheres rather than cocycles, and the “doubly-relative” and “twisted” modifiers encode the boundary conditions of the M2-brane ending on the M5-brane.
The classification is topological: the allowed flux configurations form the doubly-relative twisted Cohomotopy group, and this group is computed by factoring the quaternionic Hopf fibration S^7 -> S^4. The quaternionic Hopf map — a fundamental object in homotopy theory connecting the 7-sphere to the 4-sphere — governs the flux quantization through its fiber structure.
The factorization determines which flux configurations are topologically distinct: two brane configurations that differ by a factor of the quaternionic Hopf map carry the same flux quantum. The Hopf fiber is S^3, so the flux takes values in a group controlled by the homotopy of S^3 — the quaternionic structure of the underlying geometry.
Flux quantization as homotopy theory. The integers that count flux quanta are not arbitrary — they come from the quaternionic Hopf fibration, a specific topological map that encodes the geometry of M-theory branes.