Reflection Symmetry, APS Boundary Conditions, and Equivariant Spectral Flow on a Warped Cylinder
TLDR
This paper studies reflection symmetry and APS boundary conditions for twisted Dirac operators on a warped cylinder, detailing unitary symmetry conditions and spectral flow.
Key contributions
- Reflection lifts to unitary symmetry for twisted Dirac operators iff holonomy parameter $2A \in \mathbb{Z}$.
- In reflection-compatible cases, opposite angular modes are paired, and APS blocks are unitarily equivalent.
- For fixed gauge-trivial holonomy, spectral flow has an \(RO(O(2))\)-valued decomposition.
- For varying holonomy, \(O(2))\)-equivariance is lost, replaced by mod-two parity of APS crossing events.
Why it matters
This work provides fundamental insights into the interplay of symmetry and boundary conditions in quantum field theory, particularly for Dirac operators. It introduces new invariants for spectral flow, which are crucial for understanding topological phases and anomalies in condensed matter physics and high-energy theory.
Original 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.
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