作为 Mf-理论的 M/F-理论
M/F-Theory as Mf-Theory
AI总结：
本文基于假设 H 证明，M/F-理论的膜荷与通量可由稳定同伦、广义上同调和 Mf 配边理论的不变量、定向及配边结构系统刻画。
摘要：
在探寻 M-理论的数学基础时，“假设 H”——即通量在 Cohomotopy 理论中被量子化——意味着，在平坦但可能奇异的时空上，M-膜荷局域地组织为球的等变同伦群。这里我们展示这如何引出 M-理论中所猜想现象与稳定同伦、广义上同调和配边理论 Mf 中基础数学概念/结果之间的对应：同伦群的茎对应黑 b-膜附近探针 p-膜的荷；茎内的稳定化是边界-体转变；Adams d-不变量度量 G4-通量；d-不变量的平凡化对应 H3-通量；精细化 Toda 括号度量 H3-通量；精细化 Adams e-不变量辨识 H3-荷格；消失的 Adams e-不变量意味着一致的全局 C3-场；Conner-Floyd 的 e-不变量是 Green-Schwarz 机制中所见的 H3-通量；Hopf 不变量是 M2-膜 Page 荷（G7-通量）；Pontrjagin-Thom 定理将作为所有这些荷来源的极化膜世界体联系起来。第三稳定茎中的配边见证了 K3 曲面上的自发 KK-紧化；第三稳定茎的阶意味着 K3 上 M/F-理论中的 24 个 NS5/D7-膜。四元数定向对应 M2-膜附近的单位 H3-通量；复定向将这些单位 H3-通量提升为具有杂化线丛的杂化 M-理论。事实上，我们发现了维数有界的四元数/复 Ravenel 定向；并且我们发现该界为 10，正与时空维数 10+1 相符。
英文摘要：
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 H3-flux seen in the Green-Schwarz mechanism; the Hopf invariant is the M2-brane Page charge (G7-flux); the Pontrjagin-Thom theorem associates the polarized brane worldvolumes sourcing all these charges. Cobordism in the third stable stem witnesses spontaneous KK-compactification on K3-surfaces; the order of the third stable stem implies 24 NS5/D7-branes in M/F-theory on K3. 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.
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