A mathematical preprint reports that the optimized vacuum-flux torus topological entanglement entropy in a defined class of WZW theories grows quadratically with the binary logarithm of the allowed number of sectors. If R is the sector cutoff and q_R = log2 R, the reported asymptotic form is Γmax(R) ~ [7ζ(3)/(4π²)]q_R² as R becomes large.
A search across WZW families
The calculation ranges over modular tensor categories built from untwisted affine simple Lie algebras at positive integral levels. Their integrable sectors are counted exactly, and their total quantum dimensions are used to evaluate the entropy.
The study compares the classical A, B, C and D families with exceptional cases, while following balanced, proportional, boundary and stacked rank-level sequences. The optimization is carried out under the sector-count constraint r(g,k) ≤ R.
The benchmark is set by a squared logarithm
The leading coefficient is reported as 7ζ(3)/(4π²), or 0.2131391994 nats per squared topological qubit. The result is therefore an asymptotic capacity–entropy benchmark, rather than a formula describing every finite cutoff.
Balance is where the leading gain appears
An explicit type-C sequence, Sp(2n) at level n, reaches the global coefficient at leading quadratic order. In this family, integrable sectors can be represented as lattice paths in an n × k rectangle, or as n occupied sites among n + k ordered sites. The balanced choice n = k is half filling.
The paper’s entropy–spectral analysis proves that half filling uniquely maximizes proportional-limit efficiency: the relevant inequality is saturated only at s = 1/2. That conclusion concerns proportional asymptotics, not every finite rank-level pair.
Other families do not improve the quadratic benchmark
Type A reaches half the leading coefficient, while types B, C and D share the larger leading value. The paper attributes that factor-of-two difference to a second pair-root channel present in the latter three families. Their tie is only at leading proportional order, so the analysis does not settle their subleading finite-cutoff ordering.
Unbalanced sequences provide no competing normalized quadratic term: when the effective ratio tends to zero or infinity, log D divided by [log r]² tends to zero. Fixed-level cases either have bounded sector counts or grow too slowly to add a quadratic term, exceptional families have zero quadratic coefficient, and finite stacks cannot improve the leading bound.
A mathematical benchmark with clear boundaries
The endpoint being optimized is the vacuum-flux state for a torus cut into two cylinders, written in the paper as Γ_vac^T2 = 2 log D. The paper selects the vacuum representative because all definite Abelian-flux states maximize Γ_T2.
The theorem is restricted to untwisted WZW modular tensor categories at positive integral levels. The authors describe it as a benchmark for effective topological field theories and say it does not address the microscopic realizability of arbitrary Sp(2n) at level n phases.
Because the main statement is asymptotic, finite-cutoff corrections and subleading differences remain unresolved, including possible distinctions among the leading-order-tied B, C and D families. The document is arXiv:2608.20333v1, dated 20 August 2026, and prepared for submission to JHEP.
Paper data and sources
Original title: Maximal Torus Topological Entanglement Entropy in WZW Theories at Bounded Ground-State Degeneracy
Authors: Ce Shen
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text