We give a very simple derivation of the Atiyah-Patodi-Singer (APS) index theorem and its small generalization by using the path integral of massless Dirac fermions. It is based on the Fujikawa’s argument for the relation between the axial anomaly and the Atiyah-Singer index theorem, and only a minor modification of that argument is sufficient to show the APS index theorem. The key ingredient is the identification of the APS boundary condition and its generalization as physical state vectors in the Hilbert space of the massless fermion theory. The APS η-invariant appears as the axial charge of the physical states.

中文翻译：

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来自轴向异常的 Atiyah-Patodi-Singer 指数定理

我们通过使用无质量狄拉克费米子的路径积分，给出了 Atiyah-Patodi-Singer (APS) 指数定理及其小推广的一个非常简单的推导。它基于 Fujikawa 关于轴向异常与 Atiyah-Singer 指数定理之间关系的论证，并且仅对该论证稍作修改就足以显示 APS 指数定理。关键要素是 APS 边界条件的识别及其在无质量费米子理论的希尔伯特空间中作为物理状态向量的推广。APS η-不变量表现为物理状态的轴向电荷。