Atiyah-Patodi-Singer index from the domain-wall fermion Dirac operator
Hidenori Fukaya, Tetsuya Onogi, Satoshi Yamaguchi
TL;DR
The paper tackles the mismatch between the mathematical APS index, defined with a nonlocal boundary condition, and physical fermion systems in topological phases. It replaces the APS setup with a domain-wall Dirac operator D_DW = D + M ε(x4), defining an index through a regulated eta invariant that pairs bulk and edge contributions via Pauli-Villars regularization. The authors show that this domain-wall construction reproduces the APS index, with the edge modes' η-invariant accounting for the boundary term and the bulk term giving the usual instanton density; this furnishes a physically natural, gauge-invariant route to compute the APS index in flat space and provides robustness to mass asymmetry. The approach clarifies the origin of the boundary contribution in terms of edge states and suggests lattice realizations and extensions to anomaly descent in higher dimensions, offering practical insights for topological materials and related field-theoretic structures.
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.