Mod-two APS index and domain-wall fermion

TL;DR

This paper reformulates the mod-two Atiyah-Patodi-Singer index using domain-wall fermions, avoiding non-local boundary conditions and providing a clearer understanding of global anomaly inflow on closed manifolds.

Contribution

It introduces a new, fermion mass-based formulation of the mod-two APS index on closed manifolds, eliminating the need for non-local boundary conditions.

Findings

Mathematically proves equivalence of formulations

Separates edge and bulk mode contributions naturally

Provides a gauge-invariant description of anomaly inflow

Abstract

We reformulate the mod-two Atiyah-Patodi-Singer (APS) index in a physicist-friendly way using the domain-wall fermion. Our new formulation is given on a closed manifold, which is extended from the original manifold with boundary, where we instead give a fermion mass term changing its sign at the location of the original boundary. This new setup does not need the APS boundary condition, which is non-local. A mathematical proof of equivalence between the two different formulations is given by two different evaluations of the same index of a Dirac operator on a higher dimensional manifold. The domain-wall fermion allows us to separate the edge and bulk mode contributions in a natural but not in a gauge invariant way, which offers a straightforward description of the global anomaly inflow.