A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification

TL;DR

This paper offers a physicist-friendly reformulation of the Atiyah-Patodi-Singer index theorem, connecting it to physical fermion systems and domain-wall Dirac operators, with mathematical justification for the equivalence.

Contribution

It provides a new, physically intuitive reformulation of the APS index theorem and proves its mathematical equivalence to the domain-wall Dirac operator approach.

Findings

Reformulation makes the APS index more accessible to physicists.

Mathematical proof confirms the equivalence between the APS index and domain-wall Dirac operator invariant.

Clarifies the physical interpretation of the index theorem in topological insulators.

Abstract

The Atiyah-Patodi-Singer index theorem describes the bulk-edge correspondence of symmetry protected topological insulators. The mathematical setup for this theorem is, however, not directly related to the physical fermion system, as it imposes on the fermion fields a non-local and unnatural boundary condition known as the "APS boundary condition" by hand. In 2017, we showed that the same integer as the APS index can be obtained from the $\eta$ invariant of the domain-wall Dirac operator. Recently we gave a mathematical proof that the equivalence is not a coincidence but generally true. In this contribution to the proceedings of LATTICE 2019, we try to explain the whole story in a physicist-friendly way.