M/F-theory as Mf-theory

Hisham Sati, Urs Schreiber

Research output: Contribution to journalArticlepeer-review


In the quest for mathematical foundations of M-theory, the Hypothesis H that fluxes are quantized in Cohomotopy theory, implies, on flat but possibly singular spacetimes, that M-brane charges locally organize into equivariant homotopy groups of spheres. Here, we show how this leads to a correspondence between phenomena conjectured in M-theory and fundamental mathematical concepts/results in stable homotopy, generalized cohomology and Cobordism theory Mf: — stems of homotopy groups correspond to charges of probe p-branes near black b-branes; — stabilization within a stem is the boundary-bulk transition; — the Adams d-invariant measures G4-flux; — trivialization of the d-invariant corresponds to H3-flux; — refined Toda brackets measure H3-flux; — the refined Adams e-invariant sees the H3-charge lattice; — vanishing Adams e-invariant implies consistent global C3-fields; — Conner–Floyd’s e-invariant is the H3-flux seen in the Green–Schwarz mechanism; — the Hopf invariant is the M2-brane Page charge (G̃7-flux); — the Pontrjagin–Thom theorem associates the polarized brane worldvolumes sourcing all these charges. In particular, spontaneous K3-reductions with 24 branes are singled out from first principles: — Cobordism in the third stable stem witnesses spontaneous KK-compactification on K3-surfaces; — the order of the third stable stem implies the 24 NS5/D7-branes in M/F-theory on K3. Finally, complex-oriented cohomology emerges from Hypothesis H, connecting it to all previous proposals for brane charge quantization in the chromatic tower: K-theory, elliptic cohomology, etc.: — quaternionic orientations correspond to unit H3-fluxes near M2-branes; — complex orientations lift these unit H3-fluxes to heterotic M-theory with heterotic line bundles. In fact, we find quaternionic/complex Ravenel-orientations bounded in dimension; and we find the bound to be 10, as befits spacetime dimension 10 + 1.

Original languageEnglish (US)
Article number2350028
JournalReviews in Mathematical Physics
Issue number10
StatePublished - Nov 1 2023


  • Adams invariants
  • Conner–Floyd invariant
  • Hopf invariant
  • Hypothesis H
  • M-theory
  • Pontrjagin–Thom theorem
  • Toda brackets
  • algebraic topology
  • cobordism theory
  • complex orientation
  • generalized cohomology

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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