论文标题
Atiyah-Patodi-Singer索引的物理学家友好重新制定及其数学上的理由
A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification
论文作者
论文摘要
Atiyah-Patodi-Singer指数定理描述了对称性受保护的拓扑绝缘子的块状对应关系。但是，该定理的数学设置与物理费米子系统并不直接相关，因为它将其施加在费米亚字段上，即通过手动使用的非本地和不自然的边界条件称为“ APS边界条件”。在2017年，我们证明了与APS索引相同的整数可以从域壁式狄拉克运营商的$η$不变。最近，我们给出了数学证据，表明等效不是巧合，而是通常的。在2019年晶格会议记录的这一贡献中，我们试图以物理学家友好的方式来解释整个故事。
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 $η$ 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.
