A physicist-friendly reformulation of the mod-two Atiyah-Patodi-Singer index

TL;DR

This paper introduces a physicist-friendly reformulation of the mod-two Atiyah-Patodi-Singer index using domain-wall fermions, clarifying the physical roles of edge and bulk modes in gauge anomaly cancellation.

Contribution

It proposes a new reformulation of the mod-two APS index that is more accessible to physicists, avoiding non-local boundary conditions and elucidating anomaly cancellation mechanisms.

Findings

Reformulation using domain-wall fermions clarifies anomaly inflow.

Provides a gauge-invariant description of the mod-two index.

Enhances understanding of edge/bulk mode roles in anomalies.

Abstract

Gauge anomaly in 4-dimensions can be viewed as a current inflow into an extra-dimension, where the total phase of the fermion partition function is given in a gauge invariant way by the Atiyah- Patodi-Singer(APS) eta-invariant of a 5-dimensional Dirac operator. However, this formalism requires a non-local boundary condition, with which the physical roles of edge/bulk modes are unclear and how the causality of the theory is maintained is not obvious. In this work, we consider a special case where the Dirac operator is in a real representation and its eta invariant becomes the mod-two type APS index. We propose a physicist-friendly reformulation of the mod-two index using domain-wall fermion formalism, which naturally describes how the global anomaly is canceled between edge and bulk.