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 an 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.
Highlights
The APS index theorem states that the APS index is equal to the sum of the bulk integral of the Chern character and the eta-invariant, ind APSD =
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
We give a conjecture that the reformulated version of the Atiyah-Patodi-Singer index can be given from the Berry phase associated with domain wall Dirac operators when adiabatic approximation is valid
Summary
While the original APS index theorem (1.2) was formulated in terms of massless Dirac operators, it has recently been reformulated [22,23,24] in terms of domain wall Dirac operators. They replaced the cylinder with an infinite one but with a domain wall at xd = ±L2/2 and introduced a new index, the domain wall APS index, defined by ind DW. |x2|≤L2/2 d2x F12 = aL − aR holds Combining this with the result of the eta invariants, one finds that the APS index theorem holds in the case
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