Symmetry-Fixed Holonomies and Spectral Isolation in Two-Cycle Photonic Geometries A Square Parent Manifold for a Qubit and a Hexagonal Qutrit Manifold
Abstract
A system with two periodic directions carries two commuting holonomies a=(u,v)∈R2/Z2. We determine their distinguished values while separating lattice, arithmetic, and observable effects. Maximizing the lowest twisted eigenvalue places a at a deep hole of the momentum lattice. For every rectangular torus the maximizer is antiperiodic, so complex multiplication is sufficient for torsion optima but not necessary. Let Gτ−1 be the dual metric and Dτ(a) the normalized zeta determinant of the twisted Laplacian. At the rotation-fixed deep holes of the square and hexagonal lattices, symmetry gives the exact determinant response −Hessalog Dτ=2π(ℑτ)Gτ−1. With spectral wavenumber κ=2π, the lowest manifolds are fourfold and threefold, with gaps 2κ2 and 4κ2/3; phase errors split them linearly while their centroids remain stationary. We then give a finite-device realization: an 8×8 microring lattice closed by two phase-controlled seams. At a reported coupling scale of 16 GHz, its exact square-lattice spectrum has a 17.32 GHz shell gap and a 1.92 GHz doublet separation for a 0.1 holonomy error; a triangular-link configuration gives a threefold qutrit manifold with an 18.11 GHz gap. This is a quantitative spectroscopy proposal, not a claim of topological protection or a completed device.
Versions
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BibTeX
@misc{planat2026symmetryfixed,
title = {{Symmetry-Fixed Holonomies and Spectral Isolation in Two-Cycle Photonic Geometries A Square Parent Manifold for a Qubit and a Hexagonal Qutrit Manifold}},
author = {Michel Planat},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.16501},
eprint = {2608.16501},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16501},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symmetry-Fixed Holonomies and Spectral Isolation in Two-Cycle Photonic Geometries A Square Parent Manifold for a Qubit and a Hexagonal Qutrit Manifold
AU - Michel Planat
PY - 2026
DA - 2026/08/17
PB - arXiv
DO - 10.48550/arxiv.2608.16501
UR - https://arxiv.org/abs/2608.16501
AB - A system with two periodic directions carries two commuting holonomies a=(u,v)∈R^2/Z^2. We determine their distinguished values while separating lattice, arithmetic, and observable effects. Maximizing the lowest twisted eigenvalue places a at a deep hole of the momentum lattice. For every rectangular torus the maximizer is antiperiodic, so complex multiplication is sufficient for torsion optima but not necessary. Let G_τ^−1 be the dual metric and D_τ(a) the normalized zeta determinant of the twisted Laplacian. At the rotation-fixed deep holes of the square and hexagonal lattices, symmetry gives the exact determinant response −Hess_alog D_τ=2π(ℑτ)G_τ^−1. With spectral wavenumber κ=2π, the lowest manifolds are fourfold and threefold, with gaps 2κ^2 and 4κ^2/3; phase errors split them linearly while their centroids remain stationary. We then give a finite-device realization: an 8×8 microring lattice closed by two phase-controlled seams. At a reported coupling scale of 16 GHz, its exact square-lattice spectrum has a 17.32 GHz shell gap and a 1.92 GHz doublet separation for a 0.1 holonomy error; a triangular-link configuration gives a threefold qutrit manifold with an 18.11 GHz gap. This is a quantitative spectroscopy proposal, not a claim of topological protection or a completed device.
ER -