M/F-theory as Mf-theory

• Published in: Reviews in mathematical physics Vol. 35; no. 10
• Main Authors: Sati, Hisham, Schreiber, Urs
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.11.2023; World Scientific Publishing Co. Pte., Ltd
• Subjects: Branes; First principles; Fluxes; Homology; Hypotheses; Invariants; M theory; algebraic topology; Toda brackets; Conner–Floyd invariant; generalized cohomology; Hypothesis H; cobordism theory; M-theory; Hopf invariant; Pontrjagin–Thom theorem; complex orientation; Adams invariants
• ISSN: 0129-055X; 1793-6659
• DOI: 10.1142/S0129055X23500289

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 M f : — 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 G 4 -flux; — trivialization of the d-invariant corresponds to H 3 -flux; — refined Toda brackets measure H 3 -flux; — the refined Adams e-invariant sees the H 3 -charge lattice; — vanishing Adams e-invariant implies consistent global C 3 -fields; — Conner–Floyd’s e-invariant is the H 3 -flux seen in the Green–Schwarz mechanism; — the Hopf invariant is the M2-brane Page charge ( G ̃ 7 -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 H 3 -fluxes near M2-branes; — complex orientations lift these unit H 3 -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 1 0 + 1 .