M-理论、签名定理和几何不变量
M-theory, the signature theorem, and geometric invariants
摘要：
M-理论中C-场的运动方程和Bianchi恒等式都是用签名算符来编码的。然后，我们重新制定的拓扑部分的M-理论的行动使用的签名，这导致连接到几何的基础流形，包括正标量曲率。这导致了阿尔瓦雷斯-高美和威滕在12维空间中的神奇抵消公式的变化，并自然地导致了11维空间中的克勒克-斯托尔茨s-不变量。因此M-理论检测了11维（和7维）流形的同构类型，并在对可平行流形的限制中对拓扑11球进行了分类。此外，要求配分函数的相位是无异常的，这对s-不变量的允许值施加了限制。与10维的弦论相关，相当于把束缚理论看作一个盘丛，我们在这个公式中研究它的相应相。
英文摘要：
The equations of motion and the Bianchi identity of the C-field in M-theory are encoded in terms of the signature operator. We then reformulate the topological part of the action in M-theory using the signature, which leads to connections to the geometry of the underlying manifold, including positive scalar curvature. This results in a variation on the miraculous cancellation formula of Alvarez-Gaumé and Witten in twelve dimensions and leads naturally to the Kreck-Stolz s-invariant in eleven dimensions. Hence M-theory detects diffeomorphism type of eleven-dimensional (and seven-dimensional) manifolds, and in the restriction to parallelizable manifolds classifies topological eleven-spheres. Furthermore, requiring the phase of the partition function to be anomaly-free imposes restrictions on allowed values of the s-invariant. Relating to string theory in ten dimensions amounts to viewing the bounding theory as a disk bundle, for which we study the corresponding phase in this formulation.
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