发表机构
Osaka Dental University（大阪牙科大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究平面开书的Calabi-Yau填充判据及其在Ising表示中的量子操作，证明CY因子分解长度最小化拓扑量，并揭示接触拓扑与量子纠缠的关联。
AI 中文摘要
我们研究通过圆盘D^2的分支覆盖提升辫子所获得的平面开书，以及Ising表示中的量子操作。我们给出了相关的Stein填充为Calabi-Yau（CY）的一个判据。在固定单演的正因子分解中，每个CY因子分解具有最小长度，且其填充最小化χ和b_2。应用于Baykur最近的例子，得到一个平面接触3流形，具有无限多个非同胚的CY填充。那里的覆盖度为≥6。在度为4时，单个CY填充迫使所有填充都是CY，并且在某种情况下填充是唯一的；在度≤3时，填充是唯一的，并且在温和条件下是CY。覆盖圆盘的容许切割将开书分解为子开书。每个正因子分解随后局部化到各片，并且CY条件恰好在其局部成立时成立。这使得状态空间成为由兼容奇偶选择索引的张量积的直和，当片的度≤4时每个因子中至多有两个量子比特，这也是CY条件仅依赖于单演的范围。对于特定的度4覆盖，两量子比特doily的点、线和旗由它的子开书系统实现。那里的十五个可提升辫子共享相同的量子操作和相同的Stein填充，并且仅通过它们所允许的子开书系统来区分。因此，张量积结构的选择由提升携带，而非由辫子群表示携带，这暗示了接触拓扑与量子纠缠之间的联系。
英文摘要
We study planar openbooks obtained by lifting braids through branched covers of D^2, together with the quantum operations in the Ising representation. We give a criterion for the Stein fillings to be Calabi-Yau (CY). Among positive factorizations of a fixed monodromy, a CY one has minimal length and its filling minimizes χand b_2. Applied to Baykur's recent examples, this gives a planar contact 3-manifold with infinitely many non-homeomorphic CY fillings, the cover there having degree \ge 6. At degree 4 a single CY filling forces every filling to be CY, and in one case the filling is unique; at degree \le 3 it is unique, and CY under a mild condition. Admissible cuts of D^2 decompose the openbook into subopenbooks. Every positive factorization then localizes, and the CY condition holds exactly when it holds locally. The state space is then a direct sum of tensor products indexed by the compatible parity choices, with at most two qubits per factor when the pieces have degree $\le 4$, the same range in which the CY condition depends only on the monodromy. For one degree 4 cover, the points, lines and flags of the two-qubit doily are realized by its subopenbooks. Fifteen liftable braids there share the same quantum operation and Stein filling, and are separated only by the subopenbook systems they admit. A choice of tensor-product structure is thus carried by the lift, not by the braid group representation, suggesting a link between contact topology and quantum entanglement.
CommentsCompanion paper: arXiv:2609.34200. 18 pages. Generative AI (ChatGPT, Claude) used for literature search and verification, checking elementary computations, and mathematical discussion; see the declaration before the references
