Atiyah-Patodi-Singer index on a lattice

TL;DR

This paper introduces a non-perturbative lattice formulation of the Atiyah-Patodi-Singer index, connecting continuum topological invariants with lattice gauge theory through domain-wall fermions.

Contribution

It provides a novel lattice-based definition of the APS index using the $ ext{eta}$ invariant of the domain-wall Dirac operator, ensuring integer values at finite lattice spacing.

Findings

The lattice index matches the continuum APS index in the limit.

The curvature term arises from bulk modes, while boundary contributions come from edge modes.

The formulation is consistent with known topological invariants.

Abstract

We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS) index in lattice gauge theory, in which the index is given by the $\eta$ invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set of the free domain-wall fermion, we perturbatively show in the continuum limit that the curvature term in the APS theorem appears as the contribution from the massive bulk extended modes, while the boundary $\eta$ invariant comes entirely from the massless edge-localized modes.