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

Understand

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.