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 π, maximization over normalized torus ground states gives maxΟΞT2(Ο)=2logπ(π), attained in particular by the vacuum-flux state. The remaining problem is therefore to maximize total π at bounded categorical rank. For a simple WZW category π(g,k), write r(g,k) and π(g,k) for its categorical rank and total quantum dimension, and define FWZW(R)=supr(g,k)β€ R2logπ(g,k). We prove the sharp asymptotic law limRββFWZW(R)/(log2R)2 =7ΞΆ(3)/4Ο2. 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.
Related
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BibTeX
@misc{shen2026maximal,
title = {{Maximal Total Quantum Dimension at Bounded Rank in WZW Modular Tensor Categories}},
author = {Ce Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.20333},
eprint = {2608.20333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20333},
}
RIS
TY - UNPB
TI - Maximal Total Quantum Dimension at Bounded Rank in WZW Modular Tensor Categories
AU - Ce Shen
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2608.20333
UR - https://arxiv.org/abs/2608.20333
AB - 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 π, maximization over normalized torus ground states gives max_ΟΞ_T^2(Ο)=2logπ(π), attained in particular by the vacuum-flux state. The remaining problem is therefore to maximize total π at bounded categorical rank. For a simple WZW category π(g,k), write r(g,k) and π(g,k) for its categorical rank and total quantum dimension, and define F_WZW(R)=sup_r(g,k)β€ R2logπ(g,k). We prove the sharp asymptotic law lim_RββF_WZW(R)/(log_2R)^2 =7ΞΆ(3)/4Ο^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.
ER -