  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.
