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

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.

43pages, 1 figure, relation to anomaly descent equations added in the conclusions, references added, version published in PRD

References in corpus (4)

• Dai-Freed theorem and topological phases of matter
• The Schrödinger functional in lattice QCD with exact chiral symmetry
• Bulk-edge correspondence, spectral flow and Atiyah-Patodi-Singer theorem for the Z2-invariant in topological insulators
• Spectral functions of the Dirac operator under local boundary conditions

Cited by in corpus (5)

• Anomaly matching in QCD thermal phase transition
• Anomaly flow by an Aharonov-Bohm phase
• Mod-two APS index and domain-wall fermion
• Fractional skyrmion and absence of low-lying Andreev bound states in a micro fractional-flux quantum vortex
• The Anomaly Inflow of the domain-wall fermion in odd dimension