Dirac Operators, APS Boundary Conditions, and Spectral Flow on a Finite Warped Cylinder

TL;DR

This paper investigates the spectral properties of the Dirac operator on a finite warped cylinder with APS boundary conditions, introducing a regularized family of boundary conditions to handle zero-mode crossings and analyze spectral flow.

Contribution

It identifies intrinsic endpoint operators for APS boundary conditions, derives a determinant characterization of the spectrum, and introduces a regularized boundary condition framework for continuous spectral flow analysis.

Findings

APS index vanishes in constant-gauge setting.

Discontinuous APS projector at boundary mode crossings.

Regularized boundary conditions enable continuous spectral flow analysis.

Abstract

We study the Dirac operator on a finite warped cylinder coupled to a background $U(1)$ gauge field. We identify the intrinsic endpoint operators defining the Atiyah-Patodi-Singer (APS) boundary condition and derive a determinant characterization of the modewise APS spectrum. In the constant-gauge, invertible setting, the endpoint reduced $\eta$ contributions cancel, so the APS index vanishes. For smooth gauge families, the APS projector becomes discontinuous when a boundary mode crosses zero. We therefore introduce a regularized APS-type family of self-adjoint endpoint conditions that remains continuous across such crossings. This regularized family admits a real-symplectic boundary formulation within the standard spectral-flow/Maslov framework: for nondegenerate regularization, the zero-mode set coincides with the boundary-zero set, and transverse boundary zeros give isolated regular crossings.