We consider the index of a Dirac operator on a compact even dimensional manifold with a domain wall. The latter is defined as a co-dimension one submanifold where the connection jumps. We formulate and prove an analog of the Atiyah–Patodi–Singer theorem that relates the index to the bulk integral of Pontryagin density and η -invariants of auxiliary Dirac operators on the domain wall. Thus the index is expressed through the global chiral anomaly in the volume and the parity anomaly on the wall.

中文翻译：

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域墙的Atiyah–Patodi–Singer指数定理

我们考虑带域壁的紧凑偶数流形上Dirac算子的索引。后者定义为连接跳转的一个维数子流形。我们公式化并证明了Atiyah-Patodi-Singer定理的一个类似物，该定理将指数与Pontryagin密度的体积分和畴壁上辅助Dirac算子的η-不变量相关。因此，该指数通过体积上的全局手性异常和壁上的奇偶异常表示。