标题： 物理学家友好的阿蒂亚-帕蒂加-辛格指标的重新表述及其数学证明 显示英文标题

标题： A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification

摘要： 阿蒂亚-帕蒂亚-辛格指标定理描述了对称保护拓扑绝缘体的体边对应关系。然而，该定理的数学设定与物理费米子系统没有直接关系，因为它人为地对费米场施加了一种非局部且不自然的边界条件，即所谓的“APS边界条件”。2017年，我们展示了可以通过域壁狄拉克算符的$\eta$不变量得到与APS指标相同的整数。最近，我们给出了一个数学证明，表明这种等价性不是偶然的，而是普遍成立的。在这篇为LATTICE 2019会议论文集撰写的稿件中，我们尝试以物理学家友好的方式解释整个故事。 显示英文摘要

摘要： The Atiyah-Patodi-Singer index theorem describes the bulk-edge correspondence of symmetry protected topological insulators. The mathematical setup for this theorem is, however, not directly related to the physical fermion system, as it imposes on the fermion fields a non-local and unnatural boundary condition known as the "APS boundary condition" by hand. In 2017, we showed that the same integer as the APS index can be obtained from the $\eta$ invariant of the domain-wall Dirac operator. Recently we gave a mathematical proof that the equivalence is not a coincidence but generally true. In this contribution to the proceedings of LATTICE 2019, we try to explain the whole story in a physicist-friendly way.