We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS) index in lattice gauge theory, in which the index is given by the $\eta$ invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set of the free domain-wall fermion, we perturbatively show in the continuum limit that the curvature term in the APS theorem appears as the contribution from the massive bulk extended modes, while the boundary $\eta$ invariant comes entirely from the massless edge-localized modes.

中文翻译：

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格子上的 Atiyah-Patodi-Singer 指数

我们在格子规范理论中提出了 Atiyah-Patodi-Singer(APS) 指数的非微扰公式，其中指数由畴壁狄拉克算子的 $\eta$ 不变量给出。我们对索引的定义始终是具有有限晶格间距的整数。为了验证这个提议，使用自由畴壁费米子的本征模集，我们在连续统极限中微扰地证明了 APS 定理中的曲率项表现为来自大量扩展模式的贡献，而边界 $\eta$不变量完全来自无质量边缘局部模式。