## 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 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 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 (G̃_{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 10 + 1.

Original language | English (US) |
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Article number | 2350028 |

Journal | Reviews in Mathematical Physics |

Volume | 35 |

Issue number | 10 |

DOIs | |

State | Published - Nov 1 2023 |

## Keywords

- 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