Comments on the Atiyah-Patodi-Singer index theorem, domain wall, and Berry phase

A bstract It is known that the Atiyah-Patodi-Singer index can be reformulated as eta invariant of Dirac operators with a domain wall mass which plays key role in anomaly inflow topological insulator boundary. In this paper, we give conjecture version given simply from Berry phase associated when adiabatic approximation valid. We explicitly confirm for special case two dimensions where an analytic calculation possible. The divided into bulk and boundary contributions, each gives integration Chern character eta-invariant.