Dijkgraaf-Witten 理论中有界秩下的最大总量子维度
Maximal Total Quantum Dimension at Bounded Rank in Dijkgraaf-Witten Theories
- Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究三维Dijkgraaf-Witten理论中有界秩下的最大总量子维度，证明上循环扭曲将初等阿贝尔群的增长从平方根提升至线性，并为多种群族建立上界，最后与WZW模型比较并讨论猜想。
AI中文摘要：
我们研究了三维 Dijkgraaf-Witten 理论中，在有界范畴秩截断 $R$ 下的最大总量子维度。该优化问题在物理上控制着在固定环面基态简并度下，态优化的环面拓扑纠缠熵的最大值，同时为分类这些拓扑有序相所需的搜索空间提供了定量约束。我们分析了拓扑扭曲（3-上循环）和规范群结构如何独立地控制这一权衡。对于固定素数 $p$ 的初等阿贝尔规范群，我们证明了引入上循环扭曲将最大总量子维度从平方根增长 $\Theta_p(R^{1/2})$ 提升到秩截断下的线性增长 $\Theta_p(R)$，这由循环和乐群的几何保护所驱动。对于无扭曲理论，我们建立了对称群和交错群的尖锐包络，并推导了跨越近单群、可解群和无根群族的群论界。特别地，近单群及其直积满足二次对数上包络：$\log\mathcal{D} = O((\log R)^2)$。最后，我们将这些尺度与 WZW 模型进行比较，并讨论了一般二次对数包络猜想的现状。
英文摘要：
We investigate the maximal total quantum dimension at a bounded categorical rank cutoff $R$ in three-dimensional Dijkgraaf--Witten theories. This optimization problem physically governs the maximal state-optimized torus topological entanglement entropy at a fixed torus ground-state degeneracy, while providing a quantitative constraint on the search space required to classify these topological order phases. We analyze how topological twists (3-cocycles) and gauge-group structures independently govern this trade-off. For elementary-Abelian gauge groups at a fixed prime $p$, we prove that introducing cocycle twists elevates the maximum total quantum dimension from square-root growth $Θ_p(R^{1/2})$ to linear growth $Θ_p(R)$ in the rank cutoff, driven by the geometric protection of cyclic holonomies. For untwisted theories, we establish sharp envelopes for symmetric and alternating gauge groups, and derive group-theoretic bounds across almost-simple, solvable, and radical-free families. In particular, almost-simple groups and their direct products obey a quadratic-logarithmic upper envelope: $\log\mathcal{D} = O((\log R)^2)$. Finally, we compare these scales against WZW models and discuss the status of the general quadratic-logarithmic envelope conjecture.
