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arXiv · 2112.00261

A physicist-friendly reformulation of the Atiyah-Patodi-Singer index (on a lattice)

Hidenori Fukaya
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Abstract

The Atiyah-Singer index theorem on a closed manifold is well understood and appreciated in physics. On the other hand, the Atiyah-Patodi-Singer index, which is an extension to a manifold with boundary, is physicist-unfriendly, in that it is formulated with a nonlocal boundary condition. Recently we proved that the same index as APS is obtained from the domain-wall fermion Dirac operator. Our theorem indicates that the index can be expressed without any nonlocal conditions, in such a physicist-friendly way that application to the lattice gauge theory is straightforward. The domain-wall fermion provides a natural mathematical foundation for understanding the bulk-edge correspondence of the anomaly inflow.

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Hidenori Fukaya. 2021-12-01. A physicist-friendly reformulation of the Atiyah-Patodi-Singer index (on a lattice). https://arxiv.org/abs/2112.00261

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