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 $S_x(t)=e^{-t{A}_x}$ linked by transport operators $K_{\gamma }{S}_{\gamma (0)}(t)=S_{\gamma (1)}(\lambda (\gamma )t)K_{\gamma }$ along paths $\gamma$, we prove that the scaling field carries no monodromy: $\lambda (x,y)=\tau (x)/\tau (y)$ for a continuous positive function $\tau$, 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 ${\pi }_1(M)$ 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 ${\mathbb{Z}}_2$ orientation classes of Moebius eigenline bundles.

arXiv comment18 pages