AI 中文总结
本文研究希尔伯特丛上时间尺度交织余循环的规范刚性与和乐分类，证明标度场无单值性，并分类连续情形的和乐，包括绝热输运与平坦酉余循环的表示分类。
AI 中文摘要
我们将时间尺度交织余循环的离散规范刚性推广到连通流形M上的算子网络，并对连续情形所允许的和乐进行分类。对于由传输算子沿路径γ满足K_γ S_{γ(0)}(t)=S_{γ(1)}(λ(γ)t)K_γ关联的耗散半群S_x(t)=e^{-tA_x}，我们证明标度场无单值性：存在连续正函数τ，使得λ(x,y)=τ(x)/τ(y)，该函数在相差一个乘法常数意义下唯一，且其可乘性由交织关系强制导出而非预先假定。对于正则余循环，传输在M上的希尔伯特丛上定义一个有界算子联络；闭合回路的和乐被限制在生成子的交换子中，并分解为谱扇区，在每个扇区上，该和乐是绝热（Berry-Wilczek-Zee）输运，并乘以一个独立的交换子值扇区势；对于平坦酉余循环，和乐由π_1(M)在交换子中的酉表示分类。一个在S^1上的显式旋转网络展示出等于宇称算子的和乐，而每个能级的Berry联络一形式恒为零：单值性由Moebius本征线丛的Z_2定向类承载。
英文摘要
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 $\mathbb{Z}_2$ orientation classes of Moebius eigenline bundles.
Comments18 pages
