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

TL;DR

This paper explores a conjecture linking the Atiyah-Patodi-Singer index, domain wall Dirac operators, and Berry phase, providing explicit confirmation in a 2D case and highlighting their roles in topological insulators.

Contribution

It proposes a new conjecture connecting the APS index reformulation to Berry phase, with explicit confirmation in a specific 2D case.

Findings

Confirmed the conjecture in a 2D case with explicit calculations

Demonstrated Berry phase's division into bulk and boundary contributions

Linked Berry phase to the eta-invariant and Chern character

Abstract

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