Understanding the index theorems with massive fermions

Abstract

The index theorems relate the gauge field and metric on a manifold to the solution of the Dirac equation on it. In the standard approach, the Dirac operator must be massless in order to make the chirality operator well-defined. In physics, however, the index theorem appears as a consequence of chiral anomaly, which is an explicit breaking of the symmetry. It is then natural to ask if we can understand the index theorems in a massive fermion system which does not have chiral symmetry. In this review, we discuss how to reformulate the chiral anomaly and index theorems with massive Dirac operators, where we find nontrivial mathematical relations between massless and massive fermions. A special focus is placed on the Atiyah-Patodi-Singer index, whose original formulation requires a physicist-unfriendly boundary condition, while the corresponding massive domain-wall fermion reformulation does not. The massive formulation provides a natural understanding of the anomaly inflow between the bulk and edge in particle and condensed matter physics.

arXiv comment31 pages, 3 figures, invited review in IJMPA, references added

References in corpus (29)

• Higher-Order Topological Insulators
• Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation
• FLAG Review 2019
• The index theorem in QCD with a finite cut-off
• Fermion Path Integrals And Topological Phases
• The "Parity" Anomaly On An Unorientable Manifold
• Eta-Invariants and Determinant Lines
• Dai-Freed anomalies in particle physics
• Topology and the axial anomaly in abelian lattice gauge theories
• A New SU(2) Anomaly
• Boundary effective action for quantum Hall states
• Why are fractional charges of orientifolds compatible with Dirac quantization?
• Edge-of-edge States
• Dai-Freed theorem and topological phases of matter
• Gravitational parity anomaly with and without boundaries
• Atiyah-Patodi-Singer index from the domain-wall fermion Dirac operator
• A generalized boundary condition applied to Lieb-Schultz-Mattis type ingappabilities and many-body Chern numbers
• The Schrödinger functional in lattice QCD with exact chiral symmetry
• The Atiyah-Patodi-Singer index and domain-wall fermion Dirac operators
• Bulk-edge correspondence, spectral flow and Atiyah-Patodi-Singer theorem for the Z2-invariant in topological insulators
• Gravitational instantons and anomalous chiral symmetry breaking
• Higher (Odd) Dimensional Quantum Hall Effect and Extended Dimensional Hierarchy
• Index Theorems and Domain Walls
• Anomaly inflow and $p$-form gauge theories
• The KO-valued spectral flow for skew-adjoint Fredholm operators
• Anomaly and Superconnection
• Gauge interactions and topological phases of matter
• Atiyah-Patodi-Singer index theorem for domain-wall fermion Dirac operator
• Mod-two APS index and domain-wall fermion