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

TL;DR

This paper reformulates the Atiyah-Patodi-Singer index in a way that is more accessible for physicists, enabling straightforward application to lattice gauge theory and clarifying the bulk-edge correspondence of anomaly inflow.

Contribution

It introduces a local, physicist-friendly reformulation of the APS index using domain-wall fermions, simplifying its application in lattice gauge theory.

Findings

Index expressed without nonlocal boundary conditions

Connection established between domain-wall fermions and bulk-edge correspondence

Facilitates application of APS index in lattice gauge theories

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.