  Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for
understanding physics on the surface of materials in topological phases.
Although it is widely applied to physics, the mathematical set-up in the
original APS index theorem is too abstract and general (allowing non-trivial
metric and so on) and also the connection between the APS boundary condition
and the physical boundary condition on the surface of topological material is
unclear. For this reason, in contrast to the Atiyah-Singer index theorem,
derivation of the APS index theorem in physics language is still missing. In
this talk, we attempt to reformulate the APS index in a "physicist-friendly"
way, similar to the Fujikawa method on closed manifolds, for our familiar
domain-wall fermion Dirac operator in a flat Euclidean space. We find that the
APS index is naturally embedded in the determinant of domain-wall fermions,
representing the so-called anomaly descent equations.
