Abstract There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory. Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments, lacking a microscopic underpinning in M-brane charge quantization. We provide a framework for answering these questions, crucial not only for mathematical consistency but also for phenomenological accuracy of string theory, by formulating the M-theory C-field on flat M-orientifolds in the generalized cohomology theory called Equivariant Cohomotopy. This builds on our previous results for smooth but curved spacetimes, showing in that setting that charge quantization in twisted Cohomotopy rigorously implies a list of expected anomaly cancellation conditions. Here we further expand this list by proving that brane charge quantization in unstable equivariant Cohomotopy implies the anomaly cancellation conditions for M-branes and D-branes on flat orbi-orientifolds. For this we (a) use an unstable refinement of the equivariant Hopf-tom Dieck theorem to derive local/twisted tadpole cancellation, and in addition (b) the lift to super-differential cohomology to establish global/untwisted tadpole cancellation. Throughout, we use (c) the unstable Pontrjagin–Thom theorem to identify the brane/O-plane configurations encoded in equivariant Cohomotopy and (d) the Boardman homomorphism to equivariant K-theory to identify Chan–Paton representations of D-brane charge. We find that unstable equivariant Cohomotopy, but not its image in equivariant K-theory, distinguishes D-brane charge from the finite set of types of O-plane charges.

中文翻译：

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等变同伦意味着 orientifold 蝌蚪取消

摘要 弦理论中 RR 场蝌蚪对消条件的精确全局性质存在根本性的开放性问题。此外，M 理论中作为 M5/MO5 异常抵消的非微扰升力是基于间接合理性论证，缺乏 M 膜电荷量化的微观基础。我们提供了一个框架来回答这些问题，这不仅对数学一致性而且对弦理论的现象学准确性至关重要，方法是在称为等变同伦的广义上同调理论中在平坦的 M 方向折上制定 M 理论 C 场。这建立在我们之前关于平滑但弯曲的时空的结果的基础上，在该设置中，扭曲同伦中的电荷量化严格暗示了一系列预期的异常消除条件。在这里，我们通过证明不稳定等变同伦中的膜电荷量化意味着 M-膜和 D-膜在平坦轨道上的异常抵消条件来进一步扩展这个列表。为此，我们 (a) 使用等变 Hopf-tom Dieck 定理的不稳定改进来推导局部/扭曲蝌蚪对消，此外 (b) 提升​​到超微分上同调以建立全局/未扭曲蝌蚪对消。在整个过程中，我们使用 (c) 不稳定的 Pontrjagin-Thom 定理来识别在等变同伦中编码的膜/O 平面配置，以及 (d) 等变 K 理论的博德曼同态来识别 D-膜电荷的 Chan-Paton 表示。我们发现不稳定的等变同伦，但不是等变 K 理论中的图像，