We introduce a differential refinement of Cohomotopy cohomology theory, defined on Penrose diagram spacetimes, whose cocycle spaces are unordered configuration spaces of points. First we prove that brane charge quantization in this differential $4$-Cohomotopy theory implies intersecting $p \perp (p+2)$-brane moduli given by ordered configurations of points in the transversal $3$-space. Then we show that the higher (co-)observables on these brane moduli, conceived as the (co-)homology of the Cohomotopy cocycle space, are given by weight systems on horizontal chord diagrams and reflect a multitude of effects expected in the microscopic quantum theory of $\mathrm{D}_p \perp \mathrm{D}(p+2)$-brane intersections: condensation to stacks of coincident branes and their Chan–Paton factors, fuzzy funnel states and M2-brane 3-algebras, $\mathrm{AdS}_3$-gravity observables and supersymmetric indices of Coulomb branches, M2/M5-brane bound states in the BMN matrix model and the Hanany-Witten rules, as well as gauge/gravity duality between all these. We discuss this in the context of the hypothesis that the M-theory C-field is chargequantized in Cohomotopy theory.

中文翻译：

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微分余伦意味着通过配置空间和弦图相交的膜可观测量

我们介绍了在彭罗斯图时空上定义的上同伦上同调理论的微分改进，其上循环空间是点的无序配置空间。首先，我们证明在这个差分 $4$-上次同伦理论中膜电荷量化意味着交叉 $p\perp (p+2)$-膜模量由横向 $3$ 空间中的点的有序配置给出。然后我们表明，这些膜模量的更高（共同）可观察量，被认为是上伦余循环空间的（共同）同源性，由水平弦图上的权重系统给出，并反映了微观量子中预期的多种效应$\mathrm{D}_p \perp \mathrm{D}(p+2)$-膜相交理论：凝聚成重叠膜及其 Chan-Paton 因子、模糊漏斗状态和 M2-膜 3-代数，$\mathrm{AdS}_3$-重力观测值和库仑分支的超对称指数、BMN 矩阵模型中的 M2/M5-膜束缚态和 Hanany-Witten 规则，以及所有这些之间的规范/重力对偶性。我们在 M 理论 C 场在上伦理论中被电荷量化的假设的背景下讨论了这一点。