Index Theorems and Domain Walls

TL;DR

This paper extends the Atiyah-Patodi-Singer index theorem to configurations with domain walls, where gauge fields are discontinuous, linking bulk and boundary anomalies in such settings.

Contribution

It demonstrates that the APS index theorem remains valid for gauge field configurations with domain walls characterized by discontinuities.

Findings

APS index theorem applies to domain wall configurations

Boundary eta invariant captures parity anomaly

Bulk Pontryagin density relates to chiral anomaly

Abstract

The Atiyah-Patodi-Singer (APS) index theorem relates the index of a Dirac operator to an integral of the Pontryagin density in the bulk (which is equal to global chiral anomaly) and an $\eta$ invariant on the boundary (which defines the parity anomaly). We show that the APS index theorem holds for configurations with domain walls that are defined as surfaces where background gauge fields have discontinuities.