Atiyah-Patodi-Singer index and domain-wall eta invariants

TL;DR

This paper derives a new formula linking the Atiyah-Patodi-Singer index to domain-wall eta invariants, avoiding the need for invertible boundary Dirac operators and global spectral projections.

Contribution

It introduces a novel index formula that expresses the APS index via eta invariants of domain-wall operators without invertibility assumptions.

Findings

Established a formula relating APS index to domain-wall eta invariants.

Developed an asymptotic gluing formula for eta invariants in the adiabatic limit.

Showed the eta invariant decomposes into interior, boundary, and negligible error contributions.

Abstract

In this paper we establish a formula, expressing the generalized Atiyah-Patodi-Singer index in terms of eta invariants of domain-wall massive Dirac operators, without assuming that the Dirac operator on the boundary is invertible. Compared with the original Atiyah-Patodi-Singer index theorem, this formula has the advantage that no global spectral projection boundary conditions appear. Our main tool is an asymptotic gluing formula for eta invariants proved by using a splitting principle developed by Douglas and Wojciechowski in adiabatic limit. The eta invariant splits into a contribution from the interior, one from the boundary, and an error term vanishing in the adiabatic limit process.