AI 中文总结
该研究针对双曲能带理论，区分两类全纯性，在三种K-理论情形下证明可观测量非分解，确定拓扑保留的量化配对与局域电荷，区分有限秩区与热力学体并分析算术同余塔中可观测量的收敛性。
AI 中文摘要
拓扑K-理论在对称性和稳定度固定时，为欧几里得晶体中的带隙自由费米子相提供了决定性的稳定分类原理。相比之下，在双曲能带理论中，它对全能带问题的决定性较弱：广义动量区通过模空间变化，但过渡到普通K-类会消除其几何和全纯变化。我们区分运动学全纯性（复几何组织区空间）与动力学全纯性（该几何为哈密顿量或投影子提供信息），并将由此产生的损失表述为可观测量分解问题。对于紧致双曲曲面X，我们在三种情形下证明了非分解：纤维K⁰(X)、占据态K⁰(B)以及算子代数K-理论。受影响的量包括谱与希格斯谱曲线、贝里和乐与量子度量、部分填充的霍尔响应，以及费米面和节点几何。我们还确定了拓扑保留的量化配对与局域电荷。在一个显式面积因子下，Kotani–Sunada的底带黑塞矩阵是H¹(X;ℝ)上的霍奇内积，结合整数相交形式，它可恢复同调标记的主极化雅可比，进而通过托雷利定理和单值化得到基础复与双曲曲面，但无法得到完整的泰希米勒标记。我们还将有限秩区与热力学体区分开，局部忠实覆盖精确再现多项式迹并控制连续谱可观测量。在算术同余塔中，解析可观测量以O(|Gₙ|⁻ᵟ)（δ>0）收敛，Cˢ类可观测量以O((log|Gₙ|)⁻ˢ)收敛，而已确立的大秩相干极限可恢复体态密度矩。
英文摘要
Topological $K$-theory supplies the decisive stable classification principle for gapped free-fermion phases in Euclidean crystals once symmetry and stabilization are fixed. In hyperbolic band theory, by contrast, it is less decisive for the full band problem: generalized momentum sectors vary through moduli spaces, but passing to ordinary $K$-classes collapses their geometric and holomorphic variation. We distinguish kinematical holomorphy, in which complex geometry organizes the sector spaces, from dynamical holomorphy, in which that geometry informs a Hamiltonian or projector, and formulate the resulting loss as an observable-factorization problem. For a compact hyperbolic surface $X$, we prove nonfactorization in three settings: fibrewise $K^0(X)$, occupied-state $K^0(B)$, and operator-algebraic $K$-theory. The affected quantities include spectra and Higgs spectral curves, the Berry holonomy and the quantum metric, the partially filled Hall response, and the Fermi surface and nodal geometries. We also identify the quantized pairings and local charges retained by topology. Up to an explicit area factor, the Kotani--Sunada bottom-band Hessian is the Hodge inner product on $H^1(X;\mathbb R)$. Together with the integral intersection form, it recovers the homology-marked principally polarized Jacobian and hence, by Torelli and uniformization, the underlying complex and hyperbolic surface, though not a full Teichmüller marking. We also separate finite-rank sectors from the thermodynamic bulk. Locally faithful covers reproduce polynomial traces exactly and control continuous spectral observables. In arithmetic congruence towers, analytic observables converge as $O(|G_n|^{-η})$ for some $η>0$, and $C^s$ observables as $O((\log |G_n|)^{-s})$, while established coherent large-rank limits recover bulk density-of-states moments.
Comments70 pages, 3 tables
