The Dirac operator on a manifold with boundary requires boundary conditions to become self-adjoint. The Atiyah–Patodi–Singer (APS) conditions are the canonical choice: project onto the non-negative eigenspace of the boundary operator. The APS index theorem then counts the signed number of zero modes.
The paper on Dirac operators on a finite warped cylinder (arXiv: 2603.23275) works out the spectral theory explicitly for a cylinder with warped geometry coupled to a U(1) gauge field.
When the boundary operators are invertible (no zero modes on the boundary), the APS conditions are clean and the index vanishes — the positive and negative modes cancel. The interesting regime is when boundary modes cross through zero as the gauge field varies. At crossings, the APS projector becomes discontinuous: it jumps from projecting onto one subspace to projecting onto a different one.
The paper introduces a regularized family of boundary conditions that remains continuous through the crossings. The spectral flow — the net number of eigenvalues crossing zero as the gauge field varies — is tracked within the Maslov framework, connecting the analytic spectral flow to a topological intersection number.
The through-claim: boundary mode crossings are the singularities of the APS conditions, and regularization converts analytic discontinuity into topological invariance. The APS boundary conditions fail precisely when the boundary spectrum has a zero crossing. The regularized family smooths the failure, and the spectral flow counts the crossings — converting an analytic obstruction into a topological quantity.
2603.23275. Mathematical physics / Dirac operators / APS boundary conditions / spectral flow / Maslov index.