Maximal Total Quantum Dimension at Bounded Rank in WZW Modular Tensor Categories

Abstract

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 $\max_\psi\Gamma_{T^2}(\psi)=2\log{\mathcal D(\mathcal C)}$, 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 $\lim_{R\to\infty}\frac{F_{\WZW}(R)}{(\log_2R)^2} =\frac{7\zeta(3)}{4\pi^2}.$ 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.