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.

arXiv comment43pages, 1 figure, relation to anomaly descent equations added in the conclusions, references added, version published in PRD

References in corpus (22)

• Exactly massless quarks on the lattice
• A Method for Simulating Chiral Fermions on the Lattice
• Chiral Fermions from Lattice Boundaries
• More about exactly massless quarks on the lattice
• Axial symmetries in lattice QCD with Kaplan fermions
• The index theorem in QCD with a finite cut-off
• Fermion Path Integrals And Topological Phases
• The "Parity" Anomaly On An Unorientable Manifold
• Chern-Simons Currents and Chiral Fermions on the Lattice
• Boundary effective action for quantum Hall states
• Edge-of-edge States
• Dai-Freed theorem and topological phases of matter
• Nonperturbative Regulator for Chiral Gauge Theories?
• Domain Wall Fermions and the Eta Invariant
• The Schrödinger functional in lattice QCD with exact chiral symmetry
• Chiral anomaly for local boundary conditions
• Bulk-edge correspondence, spectral flow and Atiyah-Patodi-Singer theorem for the Z2-invariant in topological insulators
• Higher (Odd) Dimensional Quantum Hall Effect and Extended Dimensional Hierarchy
• Fermion number anomaly with the fluffy mirror fermion
• Six-dimensional regularization of chiral gauge theories
• Axial U(1) current in Grabowska and Kaplan's formulation
• Spectral functions of the Dirac operator under local boundary conditions

Cited by in corpus (25)

• Why are fractional charges of orientifolds compatible with Dirac quantization?
• Gravitational parity anomaly with and without boundaries
• A generalized boundary condition applied to Lieb-Schultz-Mattis type ingappabilities and many-body Chern numbers
• Anomaly matching in QCD thermal phase transition
• The Atiyah-Patodi-Singer index and domain-wall fermion Dirac operators
• General anomaly matching by Goldstone bosons
• Gravitational instantons and anomalous chiral symmetry breaking
• Index Theorems and Domain Walls
• Higher Berry Phase of Fermions and Index Theorem
• Anomaly inflow and $p$-form gauge theories
• Atiyah-Patodi-Singer index on a lattice
• Curved domain-wall fermions
• Atiyah-Patodi-Singer index theorem from axial anomaly
• Anomaly flow by an Aharonov-Bohm phase
• Anomaly and Superconnection
• Atiyah-Patodi-Singer index theorem for domain-wall fermion Dirac operator
• Mod-two APS index and domain-wall fermion
• Topological physics in quantum critical systems
• Universality in Anomaly Flow
• Comments on the Atiyah-Patodi-Singer index theorem, domain wall, and Berry phase
• Fractional skyrmion and absence of low-lying Andreev bound states in a micro fractional-flux quantum vortex
• $K$-theoretic computation of the Atiyah(-Patodi)-Singer index of lattice Dirac operators
• Understanding the index theorems with massive fermions
• Non-Hermitian topological wall modes in rotating Rayleigh-Benard convection
• The Anomaly Inflow of the domain-wall fermion in odd dimension