M/F-theory as Mf-theory

Abstract

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 [Formula: see text] : — stems of homotopy groups correspond to charges of probe [Formula: see text]-branes near black [Formula: see text]-branes; — stabilization within a stem is the boundary-bulk transition; — the Adams d-invariant measures [Formula: see text]-flux; — trivialization of the d-invariant corresponds to [Formula: see text]-flux; — refined Toda brackets measure [Formula: see text]-flux; — the refined Adams e-invariant sees the [Formula: see text]-charge lattice; — vanishing Adams e-invariant implies consistent global [Formula: see text]-fields; — Conner–Floyd’s e-invariant is the [Formula: see text]-flux seen in the Green–Schwarz mechanism; — the Hopf invariant is the M2-brane Page charge ([Formula: see text]-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 [Formula: see text]-fluxes near M2-branes; — complex orientations lift these unit [Formula: see text]-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 [Formula: see text].