有界基态简并度下WZW理论中的极大环面拓扑纠缠熵
Maximal Total Quantum Dimension at Bounded Rank in WZW Modular Tensor Categories
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中文总结 AI 辅助
研究有界基态简并度下WZW理论的极大环面拓扑纠缠熵，通过分析单李代数、正整数级及相关序列，得出渐近系数及经典族最优性等结论。
中文摘要 AI 辅助
对于未扭结的Wess–Zumino–Witten模张量范畴$\boldsymbol{\frak C}(\frak g,k)$，设$r(\frak g,k)$为单对象数目，$\boldsymbol{\frak D}(\frak g,k)$为其总量子维度。被分成两个圆柱的环面上的真空通量态具有正的普适熵贡献$\boldsymbol{\frak \tau}_{T^2}=2\boldsymbol{\frak \tau}\boldsymbol{\frak D}$。在满足$r(\frak g,k)\boldsymbol{\frak \tau}\boldsymbol{\frak R}$的所有单李代数和正整数级中，我们最大化该量。若$q_R=\boldsymbol{\frak \tau}_2\boldsymbol{\frak R}$，则当$R\to\boldsymbol{\frak \tau}\boldsymbol{\frak \tau}$时，$\boldsymbol{\frak \tau}_{\boldsymbol{\frak \tau}\boldsymbol{\frak \tau}}(R)\boldsymbol{\frak \tau}\boldsymbol{\frak \tau}[7\boldsymbol{\frak \tau}(3)/(4\boldsymbol{\frak \tau}^2)]q_R^2$。序列$Sp(2n)_n$达到渐近系数。在秩与级相等时，C型根乘积的傅里叶展开丢失偶模式，留下$2\boldsymbol{\frak \tau}^{-2}\boldsymbol{\frak \tau}_{\boldsymbol{\frak m}\boldsymbol{\frak \tau}\boldsymbol{\frak \tau}}\boldsymbol{\frak m}^{-3}=7\boldsymbol{\frak \tau}(3)/(4\boldsymbol{\frak \tau}^2)$。熵-谱不等式证明其在经典族中的最优性，而秩-级对偶和固定秩估计控制不平衡序列和例外序列。
英文摘要
We determine the maximal torus topological entanglement entropy (TEE) at bounded torus ground-state degeneracy within simply connected untwisted Wess--Zumino--Witten theories. For a fixed modular tensor category $\mathcal C$, maximization over normalized torus ground states gives $\max_ψΓ_{T^2}(ψ)=2\log{\mathcal D(\mathcal C)}$, attained in particular by the vacuum-flux state. The remaining problem is therefore to maximize total $\mathcal D$ at bounded categorical rank. For a simple WZW category $\mathcal C(\mathfrak g,k)$, write $r(\mathfrak g,k)$ and $\mathcal D(\mathfrak g,k)$ for its categorical rank and total quantum dimension, and define $F_{\WZW}(R)=\sup_{r(\mathfrak g,k)\leq R}2\log\mathcal D(\mathfrak g,k)$. We prove the sharp asymptotic law $\lim_{R\to\infty}\frac{F_{\WZW}(R)}{(\log_2R)^2} =\frac{7ζ(3)}{4π^2}.$ The balanced symplectic sequence $Sp(2n)_n$ attains the coefficient. The constant arises from the odd Fourier modes of the type-$C$ root product at equal rank and level. A sharp entropy--spectral inequality, rank-level duality, and fixed-rank estimates give the global upper bound. A separate corollary extends the same leading law to semisimple WZW categories,equivalently finite Deligne products of simple factors. The unrestricted TQFT envelope remains open; the theorem is sharp within the stated WZW class and supplies a constructive lower bound for the general problem.
发表机构
- Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
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