Reflection Symmetry, APS Boundary Conditions, and Equivariant Spectral Flow on a Warped Cylinder

TL;DR

This paper investigates reflection symmetry, APS boundary conditions, and spectral flow for twisted Dirac operators on a warped cylinder, revealing conditions for symmetry lifting, spectral flow decomposition, and invariants related to holonomy variations.

Contribution

It provides new criteria for symmetry lifting, analyzes spectral flow decomposition under symmetry, and introduces a residual invariant for varying holonomy in twisted Dirac operators.

Findings

Reflection lifts to a unitary symmetry iff 2A is integer.

Reflection pairs opposite angular modes and APS blocks are unitarily equivalent.

Spectral flow decomposes into an RO(O(2))-valued invariant or a mod-two parity invariant.

Abstract

We study reflection symmetry and Atiyah-Patodi-Singer (APS) boundary conditions for twisted Dirac operators on a finite warped cylinder. For a complex line twist with holonomy parameter $A$, we show that the reflection lifts to a unitary symmetry of the twisted Dirac setting if and only if $2A\in\mathbb Z$. In the resulting reflection-compatible fixed-holonomy case, reflection pairs opposite shifted angular modes, and the paired APS blocks are unitarily equivalent. The reflection trace on the APS harmonic space localizes to the unique self-paired zero-mode sector. We then turn to parameter-dependent versions of the model. For fixed gauge-trivial holonomy, the family remains pointwise \(O(2)\)-equivariant, and its spectral flow admits an \(RO(O(2))\)-valued decomposition. For genuinely varying holonomy, pointwise \(O(2)\)-equivariance is lost along the path. The representation-ring-valued invariant is then replaced by a residual sign-level invariant: the mod-two parity of the APS crossing events.