In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining techniques from homotopy theory and from supergeometry to what we call super-exceptional geometry within super-homotopy theory, we present an elegant joint solution to all three problems. This leads to a unified description of the Nambu-Goto, Perry-Schwarz, and topological Yang-Mills Lagrangians in the topologically nontrivial setting. After explaining how charge quantization of the C-field in Cohomotopy reveals D’Auria-Fré’s “hidden supergroup” of 11d supergravity as the super-exceptional target space, in the sense of Bandos, for M5-brane sigma-models, we prove, in exceptional generalization of the doubly-supersymmetric super-embedding formalism, that a Perry-Schwarz-type Lagrangian for single (abelian) N $$ \mathcal{N} $$ = (1 , 0) M5-branes emerges as the super-exceptional trivialization of the M5-brane cocycle along the super-exceptional embedding of the “half ” M5-brane locus, super-exceptionally compactified on the Hořava-Witten circle fiber. From inspection of the resulting 5d super Yang-Mills Lagrangian we find that the extra fermion field appearing in super-exceptional M-geometry, whose physical interpretation had remained open, is the M-theoretic avatar of the gaugino field.

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超特异几何：杂种M理论的起源和M5超特异嵌入构造

在探索 M 理论的数学公式时，我们考虑了三个主要的开放性问题：单（阿贝尔）M5 膜拉格朗日密度的第一性原理构造、异质 M 理论中规范场的起源，以及特殊 M 几何的超对称增强。通过将同伦理论和超几何技术结合到我们在超同伦理论中称为超异常几何的技术，我们为所有三个问题提供了一个优雅的联合解决方案。这导致了对拓扑非平凡环境中的 Nambu-Goto、Perry-Schwarz 和拓扑 Yang-Mills Lagrangian 的统一描述。在解释了同伦中 C 场的电荷量子化如何揭示 D'Auria-Fré 的 11d 超重力“隐藏超群”作为超异常目标空间后，在 Bandos 的意义上，对于 M5 膜 sigma 模型，我们证明，在双重超对称超嵌入形式主义的特殊推广中，对于单个（阿贝尔）N $$ \mathcal{N} $$ = (1 , 0) M5-膜随着“半”M5-膜轨迹的超异常嵌入，作为 M5-膜共环的超异常平凡化而出现，在 Hořava-Witten 圆纤维上超异常压缩。通过对由此产生的 5d 超级杨-米尔斯拉格朗日函数的检查，我们发现出现在物理解释仍然开放的超异常 M 几何中的额外费米子场是高吉诺场的 M 理论化身。对于单个（阿贝尔）N $$ \mathcal{N} $$ = (1 , 0) M5-膜的 Perry-Schwarz 型拉格朗日函数出现作为 M5-膜环沿超异常的超异常平凡化嵌入“半”M5 膜位点，在 Hořava-Witten 圆纤维上超异常压实。通过对由此产生的 5d 超级杨-米尔斯拉格朗日函数的检查，我们发现出现在物理解释仍然开放的超异常 M 几何中的额外费米子场是高吉诺场的 M 理论化身。对于单个（阿贝尔）N $$ \mathcal{N} $$ = (1 , 0) M5-膜的 Perry-Schwarz 型拉格朗日函数出现作为 M5-膜环沿超异常的超异常平凡化嵌入“半”M5 膜位点，在 Hořava-Witten 圆纤维上超异常压实。通过对由此产生的 5d 超级杨-米尔斯拉格朗日函数的检查，我们发现出现在物理解释仍然开放的超异常 M 几何中的额外费米子场是高吉诺场的 M 理论化身。