Atiyah-Patodi-Singer index from the domain-wall fermion Dirac operator

TL;DR

This paper reformulates the Atiyah-Patodi-Singer index theorem in a way that is more compatible with physical fermion systems, using domain-wall fermions with local boundary conditions to simplify calculations.

Contribution

The authors introduce a new, physically intuitive formulation of the APS index using domain-wall fermions with local boundary conditions, avoiding non-local boundary conditions.

Findings

The index matches the APS index using a local boundary condition.

The new formulation simplifies index computation with the Fujikawa method.

The approach is applicable to gauge theories with $U(1)$ or $SU(N)$ groups.

Abstract

The Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. The mathematical set-up for this theorem is, however, not directly related to the physical fermion system, as it imposes on the fermion fields a non-local boundary condition known as the "APS boundary condition" by hand, which is unlikely to be realized in the materials. In this work, we attempt to reformulate the APS index in a "physicist-friendly" way for a simple set-up with $U(1)$ or $SU(N)$ gauge group on a flat four-dimensional Euclidean space. We find that the same index as APS is obtained from the domain-wall fermion Dirac operator with a local boundary condition, which is naturally given by the kink structure in the mass term. As the boundary condition does not depend on the gauge fields, our new definition of the index is easy to compute with the standard Fujikawa method.