Capturing the Atiyah-Patodi-Singer index from the lattice

TL;DR

This paper develops a lattice gauge theory formulation to compute the Atiyah-Patodi-Singer index, extending previous methods to more general boundary conditions and confirming its accuracy in the continuum limit.

Contribution

It introduces a new lattice formulation for the Atiyah-Patodi-Singer index that generalizes spectral flow methods to non-product boundary structures.

Findings

The lattice formulation accurately reproduces the continuum index at small lattice spacings.

The approach extends spectral flow techniques to more general boundary conditions.

The method is validated in the context of flat torus domains with compact boundaries.

Abstract

We construct a formulation of the Atiyah-Patodi-Singer index of Dirac operators in lattice gauge theory for domains with compact boundaries in a flat torus. The key idea is to exploit its equality to the spectral flow of the domain-wall fermion Dirac operators, which we generalize in this work to cases without product structure near the boundary. We prove that, for sufficiently small lattice spacings, this formulation correctly captures the continuum Atiyah-Patodi-Singer index.