Understanding the index theorems with massive fermions

TL;DR

This paper explores how index theorems, traditionally dependent on massless fermions, can be understood within massive fermion systems, revealing new mathematical relations and simplifying boundary conditions, with implications for physics and condensed matter.

Contribution

It reformulates the chiral anomaly and index theorems using massive Dirac operators, connecting massless and massive fermions and simplifying boundary conditions in the Atiyah-Patodi-Singer index.

Findings

Revealed mathematical relations between massless and massive fermions.

Provided a boundary condition-free reformulation of the Atiyah-Patodi-Singer index.

Connected the massive formulation to anomaly inflow in physics.

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.