Gauge Rigidity and Holonomy Classification of Time-Scaled Intertwining Cocycles on Hilbert Bundles
Abstract
We extend the discrete gauge rigidity of time-scaled intertwining cocycles to operator networks over a connected manifold M and classify the holonomy that the continuous setting admits. For dissipative semigroups Sx(t)=e−tAx linked by transport operators KγSγ(0)(t)=Sγ(1)(λ(γ)t)Kγ along paths γ, we prove that the scaling field carries no monodromy: λ(x,y)=τ(x)/τ(y) for a continuous positive function τ, unique up to a multiplicative constant, with multiplicativity forced by the intertwining relation rather than assumed. For regular cocycles the transport defines a bounded operator connection on a Hilbert bundle over M; the holonomy of a closed loop is confined to the commutant of the generator and decomposes into spectral sectors, on each of which it is adiabatic (Berry-Wilczek-Zee) transport twisted by an independent commutant-valued sector potential; for flat unitary cocycles the holonomy is classified by unitary representations of π1(M) in the commutant. An explicit rotating network over S1 exhibits holonomy equal to the parity operator while the Berry connection one-form of every level vanishes identically: the monodromy is carried by ℤ2 orientation classes of Moebius eigenline bundles.
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BibTeX
@misc{alexa2026gauge,
title = {{Gauge Rigidity and Holonomy Classification of Time-Scaled Intertwining Cocycles on Hilbert Bundles}},
author = {Anton Alexa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11965},
eprint = {2609.11965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11965},
}
RIS
TY - UNPB
TI - Gauge Rigidity and Holonomy Classification of Time-Scaled Intertwining Cocycles on Hilbert Bundles
AU - Anton Alexa
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.11965
UR - https://arxiv.org/abs/2609.11965
AB - We extend the discrete gauge rigidity of time-scaled intertwining cocycles to operator networks over a connected manifold M and classify the holonomy that the continuous setting admits. For dissipative semigroups S_x(t)=e^−tA_x linked by transport operators K_γS_γ(0)(t)=S_γ(1)(λ(γ)t)K_γ along paths γ, we prove that the scaling field carries no monodromy: λ(x,y)=τ(x)/τ(y) for a continuous positive function τ, unique up to a multiplicative constant, with multiplicativity forced by the intertwining relation rather than assumed. For regular cocycles the transport defines a bounded operator connection on a Hilbert bundle over M; the holonomy of a closed loop is confined to the commutant of the generator and decomposes into spectral sectors, on each of which it is adiabatic (Berry-Wilczek-Zee) transport twisted by an independent commutant-valued sector potential; for flat unitary cocycles the holonomy is classified by unitary representations of π_1(M) in the commutant. An explicit rotating network over S^1 exhibits holonomy equal to the parity operator while the Berry connection one-form of every level vanishes identically: the monodromy is carried by ℤ_2 orientation classes of Moebius eigenline bundles.
ER -